**Transcript**

Pick up a rock and throw it in front of you. It makes a parabola. The path of its motion is parabolic. That’s Galileo’s great discovery, right? Well, not really. Galileo does claim this but he doesn’t prove it. Even Galileo’s own follower Torricelli acknowledged this. The result is “more desired than proven,” as he says, very diplomatically.

And the reason why Galileo doesn’t prove it is a revealing one. It is due to a basic physical misunderstanding.

The right way to understand the parabolic motion of projectiles like this is to analyse it in terms of two independent components: the inertial motion and the gravitational motion. If we disregard gravity, the rock would keep going along a straight line forever at exactly the same speed. That’s the law of inertia. But gravity pulls it down in accordance with the law of fall. The rock therefore drops below the inertial line by the same distance it would have fallen below its starting point in that amount of time if you had simply let it fall straight down instead of throwing it. A staple fact of elementary physics is that the resulting path composed of these two motions has the shape of a parabola.

Galileo does not understand the law of inertia, and that is why he fails on this point. If the projectile is fired horizontally, like for instance a ball rolling off a table, then Galileo does prove that it makes a parabola. He proves it the right way, they way I just outlined, by composition of inertial and gravitational motion.

But if you throw the rock at some other angle, not horizontally, then Galileo doesn’t dare to give such an analysis. This is because he thinks the law of inertia is maybe not true for such motions. He thinks, if you throw a rock at an upward angle, then maybe the rock won’t have such an inertial disposition to keep going in that direction with that speed. Instead, we thinks maybe the motion is going to slow down gradually, like a ball struggling to roll up a hill or an inclined plane.

Galileo asserts neither this wrong form of inertia nor the right one. He equivocates and never takes a stand, because he isn’t sure. And this is why he cannot give a correct proof of the theorem of parabolic motion. Even though such a proof was very much within his reach. In fact, Cavalieri, who was a better mathematician, had already published this proof, the correct analysis of parabolic motion, before Galileo wrote his book.

So it’s not that this stuff was beyond the reach of the mathematical and scientific methods of the time by any means. On the contrary, it was already explicitly spelled out completely and correctly in a published book that Galileo was aware of. And still Galileo gets it wrong in his famous work. He’s just not a very good physicist.

Ok, so that’s the big picture on parabolic motion. Now I want to go into more detail on these things. First let’s take a step back and look at inertia generally.

Here is Newton’s law of inertia: “Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.” That’s from Newton’s great Principia of 1687. It’s Law 1 of that work. A cornerstone of the whole thing.

In Galileo there’s nothing like that. Even the most ardent Galileo admirers admit this. Here’s Stillman Drake, Galileo’s great defender. Even he, and I quote, “freely grant that Galileo formulated only a restricted law of inertia” and that “he neglected to state explicitly the general inertial principle” that everyone knows today, which was instead correctly “formulated two years after his death by Pierre Gassendi and René Descartes.”

That’s the charitable interpretation. That’s the view of Galileo’s most committed supporters. And it is rather too kind, in my opinion. Trying to attribute to Galileo some kind of “restricted law of inertia” is a dubious business. Stillman Drake tries to do so, and here is what he says: “in my opinion the essential core of the inertial concept lies in the ideas of a body’s indifference to motion or to rest and its continuance in the state it is once given. This idea is, to the best of my knowledge, original with Galileo.”

You could very well argue that that’s not really inertia at all because it doesn’t involve the straightness of the direction of the motion, nor does it explicitly say that the motion keeps going at a perpetual uniform speed. It only focusses on indifference of motion versus rest and preservation of the state of motion.

So that’s “the essential core of the inertial concept” according to Galileo’s defenders. That’s very convenient. Galileo got half the properties of inertia right and half wrong, so his supporters try to spin it and say that the parts he did get right are “the essential core”, you see, and the other stuff is just secondary anyway so it doesn’t really matter that Galileo was wrong about all of that.

Sure enough, if you’re allowed to pick and choose like this which half of inertia you think is important then you can find some evidence for that part is Galileo. For example, Galileo says quite correctly: “No one could say why a thing once set in motion should stop anywhere; for why should it stop here rather than there? So that a thing will either be at rest or must be moved ad infinitum, unless something more powerful get in its way.”

Sure enough, that’s indifference of motion versus rest and preservation of the state of motion, the alleged “core” of the inertial concept. How much credit do you think Galileo deserves for this? For getting half of inertia right? Maybe you think that was the difficult step, the conceptual revolution, and then it was easy for Newton and others to fill in the details by just continuing what Galileo started.

Actually I tricked you. The quote I just read is not from Galileo at all. I lied. The quote is from Aristotle. It’s from Aristotle’s physics, written two thousand years before Galileo. So if you think that’s “the essential core of the inertial concept,” then Aristotle was the pioneering near-Newtonian who conceived it, not Galileo.

This claim is rather isolated in Aristotle and didn’t really form part of a sustained and coherent physical treatment of motion comparable to how we use inertia today. Aristotle as usual is focussed on much more philosophical purposes. So you might say: that’s a one-off quote taken out of context which sounds much more modern than it really is.

Indeed. But then again the same could be said for Aristotle’s so-called law of fall that Galileo refuted with so much fanfare. This too is only mentioned in passing very briefly and plays no systematic role in Aristotle’s thought. Yet Galileo takes great pride in defeating this incidental remark, and his modern fans applaud him greatly for it. So if we want to dismiss Aristotle’s inertia-like statement as insignificant, then, by the same logic, we ought to likewise dismiss all of Galileo’s exertions to refute his law of fall as completely inconsequential as well. If we argue that statements such as those of Aristotle don’t count as scientific principles unless they are systematically applied to explain various natural phenomena, then we would have to conclude that there was no Aristotelian science of mechanics at all. This, of course, would be a disastrous concessions to make for advocates of Galileo’s greatness, since so much of Galileo’s claim to fame is based on contrasting his view with so-called “Aristotelian” science.

So take your pick. Here are the three options:

Option 1. Galileo’s understanding of inertia was very poor.

Option 2. Galileo’s understanding of inertia was pretty good, but so was Aristotle’s.

Option 3. Galileo’s understanding of inertia was pretty good, but not Aristotle’s, because Aristotle’s statements, even though they say pretty much what Galileo says, should be disqualified because they are philosophy rather than science.

I, of course, advocate the first solution: throw Galileo under the bus. He and Aristotle were both stupid. Problem solved.

If you want to preserve Galileo’s reputation you’re in a trickier position. Are you going to admit that Aristotle understood inertia? But then what was Galileo’s contribution, and how could it be revolutionary, if that kind of stuff was already well understood two thousand years before? Or do you want to say: No, Aristotle didn’t really understand this, because his text wasn’t meant as science anyway. Well, then what is the value in Galileo spending hundreds of pages of his most important works arguing against Aristotle?

You tell me how you’re gonna solve these puzzles. Trying to maintain Galileo’s alleged greatness, it just doesn’t add up. You’re left having to bend over backwards with these inconsistent rationalisations.

What about the *rectilinear* character of inertia? The thing keeps going *straight*. Is that in Galileo? The following passage may appear to suggest as much. Quote: “A projectile, rapidly rotated by someone who throws it [like a rock in a sling], upon being separated from him retains an impetus to continue its motion along the straight line touching the circle described by the motion of the projectile at the point of separation. The projectile would continue to move along that line if it were not inclined downward by its own weight. The impressed impetus, I say, is undoubtedly in a straight line.”

That’s Galileo, and it’s straight up rectilinear inertia, right? Done and dusted. No, not so. It’s not inertia, it’s impetus. That’s what Galileo calls it. The projectile has “impetus” to go straight. But what does that mean? What is “impetus”? Is it the same thing as inertia? Will “impetus” run out, for example? Is the motion caused by the “impetus” perpetual and uniform? Galileo doesn’t say, and most likely he didn’t believe so.

In many other sources at the time, “loss of impetus by projectiles was likened to the diminution of sound in a bell after it is struck, or heat in a kettle after it is removed from the fire,” as Drake remarks. This conception is perfectly compatible, to say the least, with what Galileo writes. In fact, Galileo nowhere asserts the eternal conservation of rectilinear motion. On the contrary, he explicitly rejects it: “Straight motion cannot be naturally perpetual.” That’s an exact quote form his major work. “It is impossible that anything should have by nature the principle of moving in a straight line.” Again, a literal quotation right out of Galileo’s main work. It is easy to understand, then, why Galileo’s defenders are so eager to insist on characterising “the essential core of the inertial concept” in a way that does not involve its rectilinear character, since Galileo clearly and explicitly *rejected* rectilinear inertia.

If there’s any inertia in Galileo it is horizontal rather than rectilinear inertia. Here are some quotes from Galileo.

“To some movements [bodies] are indifferent, as are heavy bodies to horizontal motion, to which they have neither inclination or repugnance. And therefore, all external impediments being removed, a heavy body on a spherical surface concentric with the earth will be indifferent to rest or to movement toward any part of the horizon. And it will remain in that state in which it has once been placed; that is, if placed in a state of rest, it will conserve that; and if placed in movement toward the west, for example, it will maintain itself in that movement. Thus a ship, for instance, having once received some impetus through the tranquil sea, would move continually around our globe without ever stopping.”

Another one:

“Motion in a horizontal line which is tilted neither up nor down is circular motion about the center; once acquired, it will continue perpetually with uniform velocity.”

Again, as with the sling and the projectile, one can debate whether this is inertia per se. In Newtonian mechanics too a hockey puck on a spherical ice earth would glide forever in a great circle, even though this is not inertial motion. But this agreement with Newtonian mechanics only holds if the object is prevented from moving downward, as the puck is by the ice, or the ship by the water. Galileo seems to have believed horizontal inertia to hold also for objects travelling freely through the air, which is of course not compatible with Newtonian mechanics. For example, Galileo says:

“I think it very probable that a stone dropped from the top of the tower will move, with a motion composed of the general circular movement and its own straight one.”

Once again it is not entirely clear that this is supposed to represent inertia at all. It is conceivable that, in Galileo’s conception, the circular motion itself is not a force-free, default motion, but rather a motion somehow caused or contaminated by gravity-type forces. Who knows? Galileo just isn’t clear about these kinds of things. Newton and Descartes, like the good mathematicians that they are, state concisely and explicitly what the exact fundamental assumption of their theory of mechanics are. Their laws of inertia are crystal clear and specifically announced to be basic principles upon which the rest of the theory is built. Galileo never comes close to anything of this sort. He uses the casual dialogue format of his books to hide behind ambiguities. One moment he seems to be saying one thing, then soon thereafter something else, like an opportunist who doesn’t have a systematically worked out theory but rather adopts whatever assumptions are most conducive to his goals in any given situation.

Let’s get back to parabolic motion. Some have tried to argue that “if Galileo never stated the law [of inertia] in its general form, it was implicit in his derivation of the parabolic trajectory of a projectile.” That’s a quote from Stillman Drake. It would have been a very good argument if Galileo had treated parabolic trajectories correctly. But he didn’t, so the evidence goes the other way: Galileo’s bungled treatment of parabolic motion is actually yet more proof that he did not understand inertia.

His restriction of inertia to horizontal motion only is clear in his treatment of projectiles. He speaks unequivocally of “the horizontal line which the projectile would continue to follow with uniform motion if its weight did not bend it downward.” But he does *not* make the same claim for projectiles fired in non-horizontal directions. Rather he studiously avoided committing himself on that point because he was afraid it wasn’t true, like we said.

Since he only trusted the horizontal case, Galileo tried to analyse other trajectories in terms of this case. To this end he assumed, without justification, that a parabola traced by an object rolling off a table would also be the parabola of an object fired back up again in the same direction. In other words, “he takes the converse of his proposition without proving or explaining it.” That judgement is in fact a quote from Descartes, a mathematically competent reader who immediately spotted this blatant flaw in Galileo’s book.

Here’s another interesting point that Descartes makes: Galileo “seems to have written [this theory] only to explain the force of cannon shots fired at different elevations.” That is to say, Galileo made no theoretical use of his theory of projectile motion whatsoever. For example, he makes no connection to the motion of planets, the moon, comets; nothing like that. That’s a huge missed opportunity.

Instead Galileo erroneously claimed that his theory would be practically useful for people who were firing cannons. That’s quite naive, as Descartes pointed out. Here’s a quote on this by the historian A. Rupert Hall: “In many passages Galileo remarks that the theory of projectiles is of great importance to gunners. He made little or no distinction between his theory and useful ballistics; he believed—though without experiment—that he had discovered methods sufficiently accurate within the limitations of military weapons to be capable of direct application in the handling of artillery.”

This belief, however, was completely wrong. A contemporary put the matter to experimental test, and reported as follow: “I was astonished that such a well-founded theory responded so poorly in practice. If the authority of Galileo, to which I must be partial, did not support me, I should not fail to have some doubts about the motion of projectiles, and whether it is parabolical or not.” That’s a follower of Galileo writing shortly after his work was published.

Galileo foolishly thought his theory would work without testing it. This is evident for example from the extensive tables that he printed as an appendix to his big book: ballistic range tables based on his theory. These long tables make no sense at all other than as a practical guide for firing cannons. So clearly Galileo thought his theory was practically viable, which it is absolutely not.

Here’s a more theoretical issue related to inertia: the relativity of motion.

When teaching basic astronomy at Padua, Galileo explained to his students that Copernicus was undoubtedly wrong about the earth’s motion. The earth doesn’t move, Galileo explained. Because, if the earth moved, a rock dropped from a tower would strike the ground not at its foot but some distance away, since the earth would have moved during the fall. In support of this claim, “Galileo observed that a rock let go from the top of a mast of a moving ship hits the deck in the stern.” This had indeed been reported as an experimental fact by people who actually carried it out.

Of course this is completely backwards and the opposite of Galileo’s later views that he is famous for. To be sure, these lectures do not necessarily say anything about Galileo’s personal beliefs. In all likelihood he simply taught the party line because it was the easiest way to pay the bills. But at least the episode does show that the simplistic narrative that “the experimental method” forced the transition from ancient to modern physics is certainly wrong. On the contrary, experimental evidence was among the standard arguments for the conservative view well before Galileo got into the game.

In his later works Galileo of course affirms the opposite of what he said in those lectures: the rock will fall the same way relative to the ship regardless of whether the ship is standing still or travelling with a constant velocity. He gives a very vivid and elaborate description of this principle. I’ll quote in it full, it’s a long quote but it’s quite fun:

“Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other.”

Ok, so that’s that famous passage. Galileo’s prose is as embellished with fineries as this little curiosity-cabinet of a laboratory that he envisions. But is it any good of an argument? Insofar as it is, the credit is perhaps due to Copernicus himself, who had already made much the same point a hundred years before. Here are his words:

“When a ship floats over a tranquil sea, all the things outside seem to the voyagers to be moving in a movement which is an image of their own, and they think they themselves and all the things with them are at rest. So it can easily happen in the case of the movement of the Earth that the whole world should be believed to be moving in a circle. Then what would we say about the clouds and the other things floating in the air or falling or rising up, except that not only the Earth is moved in this way but also no small part of the air [is moved along with it]?”

So Galileo’s relativity argument, like so much else he says, is old news. The primary contribution of his version is literary ornamentation. Adding some butterflies and whatnot, while saying nothing new in substance.

Perhaps one could argue that Galileo goes beyond Copernicus’s passage by asserting more definitively that no mechanical experiment of any kind could prove that the ship is moving. Today the so-called “Galilean” principle of relativity says that the phenomena in the cabin cannot be used to distinguish between the ship being at rest or moving with constant velocity in a straight line. But Galileo clearly has another scenario in mind: he sees the ship as travelling along a great circle around the globe. This is the kind of motion he believes cannot be distinguished from rest, in keeping with his misconceived idea of horizontal inertia. This principle of relativity—the actually “Galilean” one—is of course false. In fact it’s even worse than that. Galileo’s purpose with this passage about the ship is to argue, erroneously, that the rotation of the earth cannot be detected by physical experiments, which in fact it can. The Foucault pendulum is a device that can detect this.

So the attribution of the principle of relativity of motion to Galileo in modern textbooks is doubly mistaken. First of all, relativity of motion and the idea of an inertial frame had been noted long before and was invoked by Copernicus to much the same end as Galileo. Moreover, Galileo’s principle is wrong in itself (because it’s about motion in a great circle, not in a straight line), and furthermore his purpose in introducing it is to draw another false conclusion from it (namely that the earth’s motion is undetectable). So errors at every turn as usual with Galileo. And there’s plenty more where that came from.

]]>**Transcript**

In 1971, Apollo 15 astronauts conducted a famous experiment on the moon. Here’s a bit of the original recording:

“In my left hand I have a feather. In my right hand a hammer. And I’ll drop the two of them here and hopefully they will hit the ground at the same time. How about that? Mr Galileo was correct.”

Actually, no. Mr Galileo was not correct. What the astronauts should have said is: Mr Galileo was wrong. According to Galileo, the moon has an atmosphere like the earth. So the feather should fall more slowly then, just like on earth. Galileo even claimed that this is “obvious” that the moon has an atmosphere. Obvious! That’s his word. This is in the Sidereus Nuncius, one of his famous published works. It is “obvious” that “not only the Earth but also the Moon is surrounded by a vaporous sphere.” Those are Galileo’s own words.

So if the astronauts wanted to test Galileo’s theory they should not have dropped a hammer and a feather. They should have taken off their helmets and suits and tried to breathe. That would have showed you how “right” Galileo really was.

But ok, let’s put the issue of the moon’s atmosphere aside. Heavy objects fall as fast as light ones, if we ignore air resistance. We are often told that this is one of Galileo’s most fundamental discoveries, and also that he supposedly destroyed the Aristotelian theory of physics on this point by simply dropping some objects of different weight from the Leaning Tower of Pisa. That was allegedly an eye-opening moment in which the world realised that empirical science is more reliable than philosophy and the words of ancient authorities. So goes the story-book version. Let’s see how much truth there is to these things. If any.

Galileo indeed often portrays himself as defeating obstinate philosophers who would rather cling to the words of Aristotle than believe empirical evidence and experimental demonstration. He imagines his enemies to say things like: “You have made me see this matter so plainly and palpably that if Aristotle’s text were not contrary to it, I should be forced to admit it to be true.” That’s a quote from one of Galileo’s dialogues. Galileo likes to pretend that his enemies are like that. It makes life easy for him. But in reality Galileo’s “anti-Aristotelian polemics were directed only at straw men.” Galileo concocted these caricature Aristotelians in order to “let them play the buffoon in his dialogues, and thus enhance his own image in the eyes of his readers,” as one historian has aptly put it.

Galileo’s ploy was well calculated. It tricks many of his readers to this day into believing the fairytale of Galileo the valiant knight singlehandedly fighting for truth in world beset by dogmatism. But “excessive claims for Galileo the dragon slayer have to be muted,” as one overly diplomatic historian puts it. Such ridiculous characterisations of Galileo should not only be “muted” but actively reversed.

This is clearly seen in the case of that famous question: do heavy objects fall faster than lighter ones? Aristotle had answered: yes. Twice as heavy, twice as fast. According to Aristotle.

Legend has it that Galileo shocked the world when he dropped some balls of different weight from the tower of Pisa and revealed them to fall at the same speed. But the notion that it required some kind of radical conceptual innovation by a scientific genius like Galileo to realise that one could test the matter by experiment is ludicrous and idiotic. Of course one can drop some rocks and see if it works: this much has been obvious to any fool since time immemorial. In fact, Philoponus—an unoriginal commentator—had clearly and explicitly rejected Aristotle’s law of fall by precisely such an experiment more than a thousand years before Galileo. This is just a plain fact. We have the source.

So if you want to believe that experimental science and empirical verification was a radical new insight then this puts you in quite a pickle. If that was a revolution, then why is it found for the first time in this extremely mediocre commentator from the 6th century? If these people had the key to science, then why did sit around and write commentary upon commentary on Aristotle? Their contributions to mathematics and science is otherwise zero. Do you really expect anyone to believe that the principles of scientific method escaped the many first-rate minds of the age of Archimedes, only to be discovered by utter nobodies in an age of vastly, incomparably lower intellectual quality? How likely is it that such elementary scientific principles eluded generations of the best mathematical minds the world has ever seen, only to be then discovered by derivative and subservient thinkers in an age where the pinnacle of mathematical expertise extended little further than the ability to multiply 3-digit numbers?

It doesn’t make any sense. So that’s a proof by contradiction that testing things empirically was never revolutionary. It is a trivial idea that any fool has always considered obvious.

Indeed Galileo was far from original in his own age either. Have you ever heard anyone calling Benedetto Varchi “the father of modern science”? Yet here is his statement of “Galileo’s” great insight, expressed two decades before Galileo was even born. Quote:

“The custom of modern philosophers is always to believe and never to test that which they find written in good authors, especially Aristotle. [But it would be] both safer and more delightful to descend to experience in some cases, as for example in the motion of heavy bodies, in which both Aristotle and all other philosophers without ever doubting the fact, have believed and affirmed that according to how much a body is more heavy, by so much more [speed] will it descend, the test of which shows it not to be true.”

All of that is a quote from a run-of-the-mill humanist writing in 1544, long before Galileo. That just goes to show what an obvious idea it was.

It is not clear whether Galileo did in fact carry out such an experiment from the tower of Pisa when he was teaching at the university there, as legend would have it. Personally I find the story plausible since a field day throwing rocks surely had great appeal to a professor who wasn’t very good at thinking. But be that as it may. It is in any case perfectly clear that many people carried out such experiments around that time, independently of Galileo. Simon Stevin, for example, in the Netherlands, certainly did, and published his results years before Galileo made his experiment, if indeed he ever did. Stevin used lead balls of different weights. He dropped them from a height of 30 feet. And on the ground he had placed some metal sheet or something that would make a lot of noise. So you could hear whether they banged down at the same time or not. So again a way of avoiding the need for stopwatches or high-speed cameras by relying on our natural hearing which is pretty good at this. And Stevin was not the only one. In Paris, Mersenne was doing the same thing. He was dropping weights out out Parisian chamber windows before he ever heard of Galileo.

Here’s a quote from Butterfield: “To crown the comedy, it was an Aristotelian who in 1612 claimed that previous experiments had been carried out from too low an altitude. In a work published in that year he described how he had improved on all previous attempts—he had not merely dropped the bodies from a high window, he had gone to the very top of the tower of Pisa. The larger body had fallen more quickly than the smaller one, and the experiment, he claimed, had proved Aristotle to have been right all along.”

Galileo himself once cited another Aristotelian philosopher who conducted similar experiments. In order to investigate whether lead falls faster than wood, “we took refuge in experience, the teacher of all things,” says this author, and hence “threw these two pieces of equal weight from a rather high window of our house at the same time. The lead descended more slowly. Not only once but many times we tried it with the same results.” That’s a work from 1575.

So much for Aristotelians hating experiments. On the contrary, appeal to experiments were commonplace long before Galileo. But these guys got the result wrong, you say. Maybe they didn’t experiment at all, or if they did they messed it up somehow.

Well, so did Galileo. You remember how he got the wrong value for the area of the cycloid? So also for falling bodies. He messed that up too at first. In his earlier notes on this he writes: “If an observation is made, the lighter body will, at the beginning of the motion, move ahead of the heavier and will be swifter,” but if the fall is long enough the heavier body will eventually overtake it. Galileo devotes a full chapter to following this up, in which, in his words, “the cause is given why, at the beginning of their natural motion, bodies that are less heavy move more swiftly than heavier ones.”

So when Galileo started experimenting on this he got the wrong result, and he also believed himself to a have a good theory “explaining” those false results. We now know that the true cause for the erroneous results was not a theoretical one like Galileo imagined, but more likely a more pedestrian circumstance. Namely, that we are not good at dropping one object from each hand at the same time. Modern experiments show that people tend to drop the lighter object sooner, even though we feel that we have dropped them at the same time. This is why Galileo and others ended up thinking that lead balls started out slower than lighter bodies and only then picked up speed.

With experiments being so inconclusive, then, it is no wonder that Galileo relied more on a theoretical argument in his published account. His supporters would have us believe that “Galileo showed that Aristotle’s rule could be refuted by logic alone.” His argument is supposedly a “splendid and incontrovertible” model example of “cast-iron reductio ad absurdum reasoning.” Those are quotes from Galileo scholars. And they are all wrong.

Here is the quote from Galileo: “By a short and conclusive demonstration, we can prove clearly that it is not true that a heavier moveable is moved more swiftly than another, less heavy. If we had two moveables whose natural speeds were unequal, it is evident that were we to connect the slower to the faster, the latter would be partially retarded by the slower. But the two stones joined together make a larger stone; therefore this greater stone is moved less swiftly than the lesser one. But this is contrary to your assumption. So you see how, from the supposition that the heavier body is moved more swiftly than the less heavy, I conclude that the heavier moves less swiftly.”

“From this we conclude that both great and small bodies, of the same specific gravity, are moved with like speeds.” Furthermore, “if one were to remove entirely the resistance of the medium, all materials would descend with equal speed.”

All of that is Galileo, his celebrated argument. With this argument Galileo allegedly exposes a fundamental logical inconsistency in the Aristotelian theory of fall. Except he doesn’t. Aristotle is perfectly clear: heavier objects fall faster. So when you put the heavy and the light together they will fall faster. The inconsistency arises only when one inserts the additional assumption that when you put two bodies together the lighter will retard the heavier and slow it down. But there is no basis whatsoever for this latter assumption in Aristotle. It is a fiction that Galileo has made up. Only by dishonestly misrepresenting the view he is trying to refute in this way is he able to draw his triumphant conclusion.

A more honest form of the argument, which doesn’t depend on misrepresenting Aristotle, is the following.

Consider two identical bricks. If you drop them at the same time they would fall side by side. Of course. If a loose string connected them that wouldn’t make any difference. What if you glued the bricks together? According to Galilean logic, “no reason appears why this double brick of double weight should fall faster than two bricks tied together—or either one alone.”

Galileo was aware of this form of the argument, it’s in some notes of his. Although he decided to use the more dishonest version in his published work.

Anyway, what this shows, then, is that the real crux of the argument is the claim that two bricks held side by side should behave the same way in terms of fall whether they are glued together or not. This is not a bad argument, but it is not a matter of “logic alone” as the Galileo fans would have us believe. For instance, imagine you are taking a basketball free throw. You can choose between trying to hit the hoop with either two bricks glued together or two bricks merely held side by side as you throw them. Would you really say that “no reason appears” why nature should treat the two cases differently, so you might as well go with the loose bricks? I don’t think so. Then why accept this assumption in Galileo’s case?

So, if we’re being honest, we are back to having to rely on experiment after all. Galileo indeed discusses the experimental side of the matter too in his treatise. He admits that actual experiments do not come out in accordance with his law because of air resistance. But, he says, the fit is much better than for Aristotle’s law. He gives specific numbers for this. Exact measurements of how much the slower ball lags behind the heavier one. But this is fake data. He cooks the numbers to sound much more convincing in favour of his theory. The actual lag or difference between the two bodies is more than 20 time greater than the fake data Galileo reports in his published so-called masterpiece. “In no case could Galileo have consistently achieved the results he reported,” as one scholar says.

Nevertheless it remains true that Galileo’s law doesn’t fare as poorly as that of Aristotle in this experiment. So wow, what a hero, the great Galileo. He managed to improve on a two-thousand-year-old claim, made by a non-mathematician, which not a single mathematician ever believed. And which Aristotle himself obviously did not intend as quantitative science. Aristotle only introduced his so-called law very passingly and parenthetically as a stepping-stone toward making the philosophical point that there can be no such thing as an object of infinite weight. That’s one paragraph buried somewhere in the middle of his voluminous metaphysics, never to be used again.

Aristotle did not claim or intend his law as a definitive or quantitative treatment of falling bodies. Only later fools who clung to his words like gospel because they could not think for themselves made a big thing of this so-called law of fall. Kind of like some people seized upon some obscure remark in the Bible and gave it all kinds of significance beyond the apparent intent of the text.

Nobody was so foolish in antiquity as far as we know. As we saw, even people like Philoponus, who were so obsessed with Aristotle that they wrote long commentaries on his every word, even people like that rejected the law. In fact, “Galileo’s” discovery, so-called, that in the absence of air resistance, all objects fall at the same speed regardless of weight—that law is in fact not first stated by Galileo, as so many people believe. Rather it is explicitly stated by Lucretius, well over a thousand years before Galileo. “Lucretius was correct”; that’s what the astronauts should have said. As ever, Galileo gets credit for elementary ideas that are thousands of years old. In his own time too, a number of people discovered “Galileo’s” law of fall independently of him: certainly Thomas Harriott and Isaac Beeckman; arguably Descartes as well.

So we see how dependent Galileo is on the framing of him versus Aristotle. No wonder he clings to this and uses it as the trope of his dialogues. Galileo’s entire case rests on his readers considering Aristotle to be a great authority. If we admit the truth—that Aristotle’s law had been refuted more than a thousand years before, and that the ridiculous idea of relying on Aristotle for quantitative science would never have entered the mind of any mathematically competent person in Galileo’s time or in antiquity—then what does Galileo have to show for himself? An unproven claim that doesn’t even fit the fake data Galileo has specifically concocted for the purpose, let alone the many experiments that proved the opposite, including his own before he knew which way he was supposed to fudge the data. Galileo likes to portray himself as doing the world a great service by defeating the rampant Aristotelianism all around him. The truth is that he is rather doing himself a great service by pretending that these Aristotelian opinions are ever so ubiquitous, so that he can inflate the importance of his own contributions, the feebleness of which would be all too evident if he addressed actual scientists instead of straw men Aristotelians.

So, with all these arguments we have done quite a bit of damage to Galileo’s claim to credit for his law of fall, I believe. But we’re only getting started. There’s plenty more to come, including several howlers Galileo made when he tried to apply his law. But that’s for another day.

]]>**Transcript**

Quiz! Who said the following:

“The study of mechanics is eagerly pursued by all those interested in mathematics.”

What is the source of that quote? And here is another one:

“We attack mathematically everything in nature.”

Who said that? Surely it is after Galileo anyway, right? Since he is the one who invented the mathematisation of nature?

No. The quotes are from Pappus and Iamblichus. Greek authors, ancient authors. Well over a thousand years before Galileo.

Much modern scholarship would have you believe that such attitudes never existed. This is what you get if you read too much Plato and Aristotle and not not enough mathematicians. In Plato, mathematics is purer than snow. To apply it to the physical world is to defile it. So that’s quite an obstacle to science. If you glorify pure and abstract thought as the only worthwhile pursuit of rational beings, and deride empiricism as fit only for unphilosophical beasts, then you’re not going to get a whole lot of science done.

Aristotle too has many teachings antithetical to proper science. There’s his big book on “Physics”, so called, which is really metaphysical philosophy. Aristotle doesn’t care about the phenomena or laws of motion. Instead he cares about pseudo-profound philosophy puzzles like: Does any motion have a cause? Or: Does motion presuppose the existence of a cause of that motion? And by the way, there are four different kinds of causes, you see: the efficient cause, the material cause, and blah blah blah, whatever the other ones are, who cares. And you can sit around and split hairs between them all day long. Typical philosophy boilerplate stuff. Obviously a completely wrong turn on the road to science.

So that’s Plato and Aristotle in a nutshell. If that’s all you know about Greek scientific thought then Galileo is a breath of fresh air, sure enough. But Plato and Aristotle do not speak for “the Greeks”. You don’t get a very good idea of 20th-century science by reading Sartre and Heidegger. Likewise, if you want to understand Greek scientific thought, forget about Plato and Aristotle. On mathematical subjects they are derivative commentators at best.

There is zero evidence that those Platonic and Aristotelian ideologies that I outlined had any influence whatsoever on any mathematically competent person in the classical and Hellenistic eras of antiquity. People often refuse to accept this. Leading scholars. Their entire worldview is centered around the assumption that Plato and Aristotle are the alpha and the omega of Greek thought. Any suggestion to the contrary they will fight with tooth and nail as if their life depended on it. Which it actually kind of does, as I explained before.

But to take Aristotle’s ultra-philosophical physics as the state of the art of Greek science is ridiculous. It’s like saying that Wittgenstein was a leading 20th-century quantum physicist. Read the mathematicians, if you want to know what was really going on.

We know for a fact that even Aristotle’s own successors as heads of his own Lyceum departed right away from his teachings on a number of scientific questions. And that’s philosophers from his own school. Just imagine how little the mathematicians cared about Aristotle’s ideas. Only much later, in the Middle Ages, which was a time of vastly lower intellectual quality, did people come up with the imbecile notion of taking Aristotle to be an authority on physics.

But ok, enough ranting about that. Now let’s look at some actual Greek science instead. Let’s think about some of Galileo’s main results and investigate whether there was anything like that in Greek times.

Falling bodies, for example. The laws of motion of falling bodies, that’s a big Galilean triumph, right? Well, not so fast.

Maybe the Greeks already knew those things. We don’t know because a huge part of the Greek scientific corpus is lost. Burnt down in various library fires and what not. Or just disintegrated and discarded. It was a fragile thing. These works had to be hand-copied over and over to survive. Just think about books from the 19th century and what state they are in today. They’re already falling apart. Ancient Greek works had to beat long odds to make it.

Take Euclid’s Elements, for example. The oldest manuscript we have is closer to us in time than to Euclid. From ancient times we have only the tiniest scraps. A quarter of a page here and three lines there. Basically nothing.

It was a long time ago. How many of the books we print today are going to be around in more than 2000 years? Not many, I bet.

But we know something about what was lost through references in other works. And as far as the science of falling bodies is concerned we have some very intriguing indications.

We know for a fact that Strato wrote a treatise on falling bodies. It did not survive. But here’s what we know. Strato was an avid experimenter. To prove that falling objects speed up, he said: Pour water slowly from a vessel. At first it flows in a continuous stream, but then further down its fall it breaks up into drops and trickles. This is because the water is speeding up. So the water spreads out, like cars let loose on a highway after a congested area.

And here’s another experiment Strato used. Stones dropped into a sand bed from various heights. The stone makes craters of different depths depending on height fallen. Again showing that the thing is picking up speed as it goes. Did Strato make actual numerical measurements of this? Of the sand craters and other ways of quantifying fall? Maybe. Who knows?

And here’s another guy who wrote on falling bodies: Hipparchus. That’s an author who commands respect. Probably the greatest mathematical astronomer of antiquity, though his works are lost. Certainly a way better mathematician than Galileo, that’s for sure.

So we know that he wrote a treatise on falling bodies, which is lost. Of course he didn’t follow Aristotle’s ridiculous theory of falling bodies. In fact, a commentator on Aristotle explicitly says: “Hipparchus contradicts Aristotle regarding weight.”

But what exactly did he say? The indications are that he argued that weight depend on distance from center of the earth. He seems to have been engaged in questions like: if there was a tunnel through the center of the earth and an object fell down it, what would happen as it approached the middle? Superficial commentators have picked up on these striking aspects of his work that can be presented as a kind of gimmick in isolation. In the original they would surely have been incorporated in a mathematical treatment.

These things are very much in line with 17th-century physics. Dropping stones into sand, thinking about how gravity varies on a super-terrestrial scale and inside the earth. That is literally exactly what scientists spent a lot of serious effort on in the 17th century.

So here’s what we know. Greek scientists, who were excellent mathematicians and keen experimenters, wrote several lost treatises on falling bodies. What are the chances that this included good chunks of so-called Galilean science? Maybe including for example “Galileo’s” law of fall? They are appreciable. It is squarely within the realm of possibility and then some.

Here’s what a modern scholar says about Galileo’s law of fall: “The ease of stumbling upon this discovery renders it highly improbable that natural philosophers had ever searched for the law of fall [before the 17th century].” Yes, either that, or: they did search and they did find it. If it’s easy to stumble upon, what are the chances that these very sophisticated first-rate minds, who wrote entire treatises on this exact subject, somehow missed it? It’s food for thought.

The law I’m talking about here is the one that say that the acceleration of a freely falling object is constant. Equivalently, its velocity is proportional to time, and the distance fallen is proportional to time squared. Motion on an inclined plane is closely related to this. A ball rolling down a slope is basically a slow-motion version of falling. The ball will acquire the same speed as it would have in free fall through the same vertical distance. As we would say today, in anachronistic terms, since all balls covering the same vertical distance trade in the same amount of potential energy, they get the same amount of kinetic energy out of it.

This slow-motion of falling is easier to deal with experimentally. Here’s Stillman Drake the Galileo scholar again; he says: “In a way it is surprising that the law for the spontaneous descent of heavy bodies had not been recognized long before the 17th century. Measurements sufficient to put the law within someone’s grasp are quite simple. Equipment for making them had not been lacking---a gently sloping plane, a heavy ball, and the sense of rhythm with which everyone is born.”

The sense of rhythm is like a natural clock. You can tell when a musician is off beat. We can use this innate skill to verify the law of fall. No stopwatch required.

Just get your inclined plane, and put markers at interval that the ball ought to cover in equal times according to your hypothesised law, and then put little bells there at those points, and roll the ball down the slope. Do the bells ring at equal intervals or not? That’s quite easy to tell.

I did this experiment for your benefit. I’m going to play it for you. I used a very simple and primitive setup, using just some stuff I had around the house. For my inclined plane or ramp I found a suitable piece of wood in my basement; it used to be part some furniture or other. So I used a kitchen knife and cut some markings into it at places a rolling ball should cover in equal times. That just means putting a measuring tape down and marking at all the squares. I made marks at 4, 9, 16, and 25 units from the starting point. You can use whatever unit you like. There is no need to know anything about a gravitational constant or anything like that. Just put your ruler down and make these marks at the square numbers and that’s it.

So then I put the ramp in position on my living room coffee table. Then I got four wine glasses that I placed along the ramp at the markings I had made. I just propped them up with some books as needed. Then I had a glass marble lying around that happed to be suitable to the purpose. I positioned everything so that the marble would just about touch the wine glasses as it rolled down the ramp.

So, here we go, I’m going to play it for you. The idea is that you will hear four equally spaced clinking sounds.

—

It worked pretty well, I’d say. The thump at the end is the marble crashing into a bundled-up t-shirt that I put at the end of the ramp. I also made some recordings with the glasses in the wrong positions. I put them about equally spaced instead of spaced like square numbers. So I will play that recording as well and see if you can tell the difference.

—

I think the difference is pretty clear. Considering that I threw this experiment together very sloppily in about ten minutes I’d say it is viable and easy to build a quantitative theory of falling bodies this way. Obviously it would be easy to improve the accuracy of the experiment a lot from my extremely primitive setup. The way I did it the marble whacked into the wine glasses pretty good. That probably slowed it down by perhaps a non-trivial amount. If I was doing this for real I would of course use a heavier ball, a longer ramp, and, for the markers, tiny bells or something that isn’t heavy enough to impact the ball.

But I think even my setup kind of worked. I made a number of takes with each configuration in fact. I’ll play a couple of them and then as a blind test we will see if you can hear which are the correct configuration with the equal intervals and which are the bad ones with unequal intervals. Here we go.

—

We don’t know if the Greeks did this something like this. There is no evidence that they did. They certainly could have. But let’s put that question aside.

There’s a more general point to be made here. This stuff about the inclined plane, it’s very simple. The law is simple; the mathematics is simple; the experiment is simple; the idea is simple. You can explain everything to a child. If Galileo was the first to discover stuff like this, it must be considered a revolution or breakthrough of a conceptual nature. Certainly not a mathematical one, because mathematically it’s trivial. This goes for all or almost all of Galileo’s so-called achievements.

The underlying question is: Are there conceptual revolutions like that? Is the history of science a story of conceptual revolutions that made previously unimaginable things suddenly obvious? Some people are very willing to accept this. Many scholars are. It’s quite fashionable even, I would say, to insist on this.

Historians make a big mistake here in my opinion. They start from a reasonable premiss. They observe that those who have little or no expertise in history often interpret historical episodes in a naive way. These historical beginners take current ways of thinking for granted, and small-mindedly interpret past thought in terms of those narrow categories. That’s a massive mistake of course. And it is very stupid because it defeats the whole purpose of studying history in the first place, namely that history shows us other perspectives, other ways of thinking.

So that’s all fine and well. Modern historians are very much correct in condemning this kind of thing. However, they then go too far. They think to themselves: since it is bad to involve too much current ways of thinking when looking at the past, the best historian must be the one who does the exact opposite of that. That is to say, the best historian must be the historian who always emphasises conceptual differences whenever possible, and who always blows the tiniest indications of differences way out of proportion as if it was a very big deal. Since this is the opposite of the naive approach, it must be the pinnacle of sophistication, right? That’s what many historians seem to think.

This is of course a bad idea, since it is just as naive and dogmatic as the erroneous mentality it is supposed to counter. But it’s not my purpose to criticise it now, only to consider what it means in the case of Galileo.

I said I could explain the rolling ball and the law of fall to a child. Any child; modern or ancient. This is what the modern historians are eager to deny. They do not believe in such universalism. They say: No, the Greeks lived in a different conceptual universe; their way of thinking is just fundamentally, qualitatively different from ours. Even the most obvious or evident thing to us may very plausibly have been completely outside of their conceptual sphere. They couldn’t even think it, because the way they approached the world was just inherently and profoundly different from ours.

Galileo’s status stands and falls with our willingness to accept this radical relativism. His discoveries are so basic and obvious that the only way to consider them profound is to maintain that they were once *not* basic and obvious. In other words, that they are fundamentally different in character from anything the Greeks were doing, for example.

If we think there is only one common sense, and that mathematical truth and thought is the same for everyone, then we are strongly inclined to see Galileo’s achievements as quite basic, and we are strongly inclined to think that the works of for example Hipparchus on falling bodies was probably very similar to what 17th-century authors said on the same subject.

On the other hand, if we reject the very notion of a universal scientific common sense, then we are primed to think that Galileo opened up an entirely new world with his style of science.

So studying Galileo is a mirror to much larger questions. Either you are a cultural relativist and you think Galileo was a revolutionary, or you think mathematical thought is the same for you, me and everybody who ever lived, and then you think Galileo was just doing common-sense stuff. Those are the two possibilities. You have to pick sides. You can’t mix and match. You can’t have both mathematical universalism *and* Galileo being a revolutionary. These two basically contradict one another.

So once again Galileo is at the heart of fundamental questions about history. All the more reason to study him in detail. Which is exactly what we are going to do. Next episode it’s into the weeds on Galileo’s work on falling bodies.

]]>**Transcript**

Those who can’t do, teach. I’m sure you have heard this saying. It sums up Galileo’s role in the history of scientific thought, in my opinion. Galileo’s books are “Science for Dummies”. He drones on and on about elementary principles of scientific method because he lacks the mathematical ability to do anything more advanced.

It is precisely *because* he is so bad at mathematics that he is forced to waste so many words doing something so trivial.

Galileo was not much of a mathematician but he knew a thing or two about rhetoric. He saw a way to make a virtue out of necessity. He realised very well that he could not hold a candle to Archimedes. So he chose to play a different sport. He went after Aristotelian philosophers instead. Sure enough he scores some points against these fools, but that’s fish in a barrel.

Here is what Descartes said about Galileo: “He is eloquent to refute Aristotle but that is not hard.” Yes, exactly. Descartes hits the nail on the head right there. If Descartes had a podcast I wouldn’t have to make mine. Relative to the philosophers Galileo is a big step forward, yes, that’s true. But relative to the mathematicians he is just saying obvious things that everyone had already known for thousands of years.

Descartes continued in the same vein: “I see nothing in his books to make me envious, and hardly anything I should wish to avow mine.” Galileo’s mathematical demonstrations in particular did not impress Descartes: “he did not need to be a great geometer to discover those,” Descartes says.

That’s right on the money. Do you think my take on Galileo is crazy and unbalanced? Well, then you think Descartes is crazy too. Because he agreed with me. And so did other mathematically competent people at that time. People who knew Archimedes, unlike many who write on Galileo today.

Galileo’s claim to fame rests on the assumption that Archimedes does not exist and that everyone but Galileo was a raving Aristotelian. Galileo himself went out of his way to ensure this framing. His two big books are both dialogues. One character is a mouthpiece for Galileo and another an Aristotelian simpleton. This is the contrast class Galileo wants us to use when evaluating his achievements. And no wonder. Refuting Aristotle is not hard, as Descartes said, so of course Galileo can score some zingers against this feeble opposition.

Galileo tries to pass himself off as a rebel truth-teller taking on the supposedly all-pervasive Aristotelian establishment. If one buys into this deceptive framing one may very well come away with the impression that Galileo has done something of value. But no. Galileo has rigged the game. He has pitted himself against a convenient punching bag.

Here’s a quote: “The philosophers of our times philosophise as men of no intellect and little better than absolute fools.” That’s Galileo. And he’s right. Fools, the lot of them, those philosophers. But Galileo was not the only one to see this.

Descartes, for example, said of the Aristotelians that they were “less knowledgable than if they had abstained from study.” Galileo’s claim to fame is that he refuted people who were “less knowledgable than if they had abstained from study.” Hardly the pinnacle of intellectual achievement.

Another example: Tycho Brahe. A leading astronomer some decades before Galileo, and a rather conservative guy. He too complained about “the oppressive authority of Aristotle” as he called it. “Aristotle’s individual words are worshipped as though they were those of the Delphic Oracle,” he says.

This kind of attitude was universal among mathematicians. No use writing several thick books hammering home this point and little else, which is what Galileo did. That’s just beating a dead horse as far as the mathematicians are concerned.

Mathematicians had a clear sense of “us versus them” in this respect. The mathematicians versus the philosophers. It is remarkable how they have complete faith in the judgement of other mathematicians, and utter contempt for everyone else. This attitude is everywhere in Galileo’s time and even before. Probably already in Greek times, for all we know.

Consider Copernicus, for instance. That’s a hundred years before Galileo. Here’s what he says when he introduces his theory that the sun is in the center of the solar system: “I have no doubt that talented and learned mathematicians will agree with me.” For I will, he says, make everything “clearer than day—at least for those who are not ignorant of the art of mathematics.”

And those who are ignorant of mathematics? He also addresses them. “If perchance there are certain idle talkers wholly ignorant of mathematics dare to attack my work; they worry me so little that I shall scorn their judgements.” These are the kinds of people who “on account of their natural stupidity hold the position among philosophers that drones hold among bees.” “The studious need not be surprised if people like that laugh at us. Mathematics is written for mathematicians.” That’s all from the introduction to Copernicus’s great masterpiece. He doesn’t mince words, does he?

It’s the same in Kepler, Galileo’s contemporary. “Let all the skilled mathematicians of Europe come forward,” he implores. Evidently he has complete confidence that mathematical reason compels them to speak with one voice.

Like Copernicus, Kepler also addresses the non-mathematicians. In his great masterpiece, the Astronomia Nova, he puts a section in the introduction with the heading “advice for idiots.” There he says things like: “whoever is too stupid to understand astronomical science, I advise him that he mind his own business and scratch in his own dirt patch.”

This is all before Galileo has published anything. Mathematically competent people were united and had nothing but complete contempt for Aristotelian philosophy and the like. By the time Galileo comes along and belabours this point it has been old news for hundreds of years.

You gotta admire the guts of these people. These quotes from Copernicus, Kepler, they are from their scientific masterpieces. Not a passing remark in confidential personal correspondence to blow off steam, not a reply to a particular provocation, not something said in the heat of the moment. No, the decided to put this “advice for idiots” right at the heart of their scientific masterpieces; their crowning accomplishments that were written for the ages.

Galileo joked about his “lack of tact” as he called it. And he was not alone in this. Mathematicians of this age were not much for tact. “The presence of good tone means the absence of good sense,” says Schopenhauer. The mathematicians of Galileo’s time had a lot of good sense.

In fact, this has always been the way of the mathematician. Just imagine today a philosopher walking into a mathematics department, and starting telling them what to do and how to think. Of course no mathematicians listen to that. Not today, not in Galileo’s time, not ever.

In antiquity there’s Ptolemy, for example, the astronomer. Here’s what he has to say: “only mathematics can provide sure and unshakeable knowledge”; other “divisions of theoretical philosophy should rather be called guesswork than knowledge.”

Mathematicians have always taken this for granted. And this is why, from a mathematical point of view, Galileo is nobody, because he did little else than provide redundant proofs of this self-evident truth.

Now: a conspiracy theory of sorts. As we have seen, Galileo needs us to assume that Aristotelianism and philosophy was the state of scientific knowledge in his day, and that no one had ever heard of Archimedes. Only then does his so-called accomplishments come off looking any good.

Ask yourself: Who is inclined to go along with such an assumption? I’ll tell you who: Someone who doesn’t know any Archimedes but is very comfortable with Aristotle and other philosophers. People from the humanities, in other words.

So Galileo is in luck. He needs an audience with certain blind spots and predispositions, and he gets exactly that. Modern academia is set up in his favour. History of science is nowadays a humanities field. The default training of historians of science is not higher mathematics and physics; it is reading seminars based on non-mathematical authors such as Aristotle.

So the people tasked with being Galileo experts are by design the people most inclined to accept Galileo’s deceit. Pretending that Archimedes does’t exist serves both their purposes and Galileo’s. They share Galileo’s aversion to proper mathematics, so they are more than happy to write off Archimedes as a genius to be sure but a very specialised one, who is just doing some esoteric math stuff that doesn’t really matter to the history and philosophy of science.

I tried to quantify this a bit. The History of Science Society publishes an annual bibliography of works in the history of science. I thought I would use it to compare Aristotle and Archimedes. I compiled the number of entries for the past 15 years. I found the following. Archimedes: 42 entries. 42 books and articles written about him in the past 15 years. Aristotle: 482 entries. Well over ten times as many as Archimedes. This concerns Aristotle’s role in the history of science only, mind you. It’s not counting works on Aristotle altogether, of which there are many thousands more.

Actually the ratio in favour of Aristotle should be doubled because the bibliography also has another 339 for Aristotelianism. Which means dogmatic followers of Aristotle, basically. There were many of those in the middle ages and still in Galileo’s time. They get a lot of attention.

There has never been any entry on Archimedeanism in the bibliography. But if you’re writing one then count me in as a reader.

So, anyway, that’s 20 times as many works on Aristotelian thought as on Archimedes. And what would you expect? If you put the history of science in the hands of humanities people, that’s what they’re gonna do.

In fact, how could they do otherwise? Suppose I’m right and that the 20 to 1 ratio in favour of Aristotle over Archimedes is foolish and distorts the true nature of the development of scientific thought. Suppose for the sake of argument that I am right about that. Even so, the humanities people could not very well say so, even if they believed it. That would basically amount to saying: Please don’t give us any more money. You trusted us to study the development of scientific thought but we have come to the conclusion that it is best understood from a scientific and mathematical point of view than the philosophical training that we have. Please give our research money to them instead. Please fire half of the people in my department because there is too much work on Aristotle being done already. They are not going to say that, are they? Even if they believed it, they would be fools to say it.

So the way modern historical scholarship is set up plays right into Galileo’s hands. Actually that’s true in more ways than one. A heavy bias toward philosophy and away from mathematics is one thing. But here’s another one. Contextualism versus universalism.

The issue comes down to this: Do great minds think alike? I say they do. I say there is a spiritual unity of scientific thought from ancient to modern times. I say that what is obvious to us was obvious to the Greeks. I say it is ludicrous to think that generations of Greek mathematical geniuses of the first order, with their extensively documented interest science, all somehow failed to conceive basic principles of scientific method. I say these things because I can feel it in my bones. I say these things because I have spent my life in mathematics departments and experienced so many times the profound sense of thinking exactly alike with another person. Young or old, student or professor, when we talk about mathematics our minds are one. Mathematics has this power, to make brethren of us all. This is why Copernicus and Kepler had unshakeable confidence that mathematicians would ultimately agree with them. For the same reason Galileo says with conviction that “if Aristotle were now alive, I have no doubt he would change his opinion.” Such is the historiographical outlook of mathematicians.

The idea that modern science was born in a Galilean revolution, on the other hand, is based on seeing history as soaked in cultural relativism. Following this school of thought, you must approach ancient texts as if they were mysterious communiqués from an alien life form on the other side of the universe. You must banish any notion of unity in human thought and instead view old and new as worlds apart, separated by a conceptual abyss that no intuition can bridge. This is the worldview and historiographical approach of many who are far removed from mathematics. From this point of view it is completely natural to think that basic principles of scientific method that seem so obvious today were in fact once completely outside the conceptual universe of even extremely sophisticated mathematical scientists like Archimedes.

This is why Galileo is the idol of the humanists and the bane of the mathematicians. The philosophers say he invented modern science; the mathematicians that he’s a poor man’s Archimedes. The issue cuts much deeper than merely allotting credit to one century rather than another. Much more than a question of the detailed chronology of obscure scientific facts, it is a question of worldview and how one should approach and understand history.

I say that the traditional view of the “Galilean” scientific revolution is not only historically wrong but fundamentally inconsistent with the nature of mathematical thought. When I say Galileo boo, Archimedes yay, my point is not who was the “first” or who should “get credit” for this or that. That’s not so interesting. But Galileo is a window into more important things.

What is the relation between mathematics and science? Was mathematics before Galileo a technocratic enterprise? Compartmentalised, limited to certain computational tasks, blind to its own potential? Was mathematics stuck in that ditch until it was liberated by a “conceptual” breakthrough from without, so to speak; from philosophy? Or was mathematics always an expansive, empirically informed, interconnected study of all quantifiable aspects of the world?

The latter, of course, if you ask me. In any case, Galileo is ground zero for grappling with these questions. And that is why we study him.

]]>**Transcript**

Galileo is the most overrated figure in the history of science. That’s the thesis of Season 1 of this podcast. It’s going to be proper revisionist history fun. If you enjoy scholarly polemics and received wisdom turned on its head, then this is the story for you.

But it’s also more than that. Galileo is at the heart of fundamental questions. What is the relation between science, mathematics, and philosophy? Between ancient and modern thought? What’s the history of our scientific worldview, of scientific method? All this big-picture stuff. Galileo is right in the thick of the action is on all of these issues. So that’s all the more reason to study him.

But let’s start small. Here’s a simple snapshot of Galileo at work.

The cycloid. It’s a famous geometrical curve. The cycloid is the path traced by a point on a rolling circle. So, in other words, say you have a bicycle wheel. And you attach a piece of chalk to the rim of it. Then you roll the wheel along a wall, so that the chalk is drawing on the wall as it’s rolling. That makes a cycloid. It’s a kind of arch shape.

What’s the area of the cycloid? That was a natural question in Galileo’s time. Finding areas of shapes like that is what geometers had been doing for thousands of years. Archimedes for instance found the area of any section of a parabola, and the area of a spiral, and so on. The cycloid was a natural next step. It fit right into this tradition.

Of course nobody cared about the area of the cycloid as such. That’s not the point. Think for example of portraits painted by great artists. Of course the value in such a painting is not that it accurately depicts some prince and whatever pompous costume he was wearing. Obviously the value of this as art is not the subject but the method. How does the artist manipulate composition and subtle detail to achieve a particular impact? How does the artist build on tradition, yet innovate beyond it? Those are the things we admire when we view the works of great artists.

And it’s the same in mathematics. Archimedes, when he found his areas, gave clever geometrical arguments; gorgeous proofs. The point is not that he gave you a “formula” to compute various areas. How often have you needed to know the area of a spiral anyway? Never, of course. The point is not the result. The point is that Archimedes took human thought to a new level. His proofs are beautiful; they are logically flawless. They give you a sense that you are at the pinnacle of what the human mind can achieve. Everybody wanted to see more that kind of thing.

So solving problems like the area of the cycloid then, in this sense, in the Archimedean sense, was the way to prove yourself a worthy geometer. So Galileo tried. And failed. All those brilliant feats of ingenuity that Archimedes and his friends had blessed us with; it just wasn’t happening for Galileo. He just wasn’t any good at it.

In fact he said so himself. Here is a quote from Galileo on Archimedes: “Those who read his works realise only too clearly how inferior are all other minds compared with Archimedes’, and what small hope is left of ever discovering things similar to those he discovered.”

That’s Galileo. And he’s quite right. Except maybe it’snot that *all* other minds are inferior to Archimedes. Although certainly Galileo’s is.

So picture Galileo sitting in his study, racking his brain, staring at the cycloid. The books of Archimedes lie open on his desk. Agh, all that math, it’s giving him a headache. Thinking isn’t really working out for Galileo. It’s not his strong suit.

Desperate, frustrated, he turns to the failed mathematician’s last resort: trial and error. He cuts the cycloid out of thick paper and starts weighing it. It’s much like in modern mathematics classrooms where certain students prefer pushing buttons on a calculator and have some device do the thinking for them instead of trying to figure stuff out for themselves. So also Galileo starts fiddling around with scales and cardboard cut-outs as a substitute for thinking.

It is striking to compare Galileo’s take on this problem with that of his contemporaries. They in fact did solve it. Roberval in France, Torricelli, Galileo’s countryman in Italy, and also Descartes. They all solved it. And not with some middle school cut-and-paste project. They solved it as mathematicians, using reasoning and proof. These are contemporaries of Galileo. If they could do it, why couldn’t he?

Galileo himself says the cycloid area is a “very difficult” problem. “I worked on it fruitlessly,” he says. Compare that with Descartes’s reaction. Descartes, a mathematically competent person. When Descartes receives the problem—it was being passed around in correspondence at this time, as a challenge to various mathematicians—when Descartes receives it, he at once replies: “I do not see why you attribute such importance to something so simple, that anyone who knows even a little geometry could not fail to observe, were he simply to look.” And he back this up by immediately sending his solution, which he composed on the spot.

Now, Descartes, he is not famous for his humility, to put it mildly. So we should not necessarily read too much into those words there. But even so, the contrast with Galileo is very striking. The fact is that a number of mathematicians solved the cycloid problem with relative ease, while Galileo was fumbling with scissors and glue.

And Galileo got it wrong, too. Not only did he fail to find one of those beautiful Archimedean proofs, which was the whole purpose of the exercise in the first place. Not only that, but he also got the result itself wrong. The area of the cycloid is in fact 3 times the area of the generating circle, as these mathematicians showed. Galileo through specifically concluded based on his bumbling experiments that it was not exactly 3 but a bit off from 3. By relying on experiments unchecked by proper mathematics, Galileo got the answer wrong, and not for the first time nor the last.

That just goes to show why mathematicians have little respect for experiment. Christiaan Huygens, another very competent mathematician, once said: “Do not think that I am relying on experiments, because I know they are deceitful.” That was in a different context, nothing to do with the cycloid. But all the same. This is a universal attitude among mathematicians. And for good reason, as we see in the Galileo case. Haphazard trial and error has to be superseded by rigorous demonstration, as able mathematicians have always known.

So the moral of the story is this. In the case of the cycloid, it is undeniable that Galileo turned to empiricism precisely because he lacked the mathematical ability to tackle the problem any other way. It is beyond any shadow of a doubt that if he had had the ability to compose a mathematical proof, like some of his contemporaries did, he would have loved nothing more than to do so. In the case of the cycloid, these are facts.

This leads us to my revisionist thesis. I suggest that the cycloid episode is typical of Galileo’s science altogether. As with the cycloid, so with science. Galileo was bad at mathematics. And it is precisely because he was so bad at mathematics that he was so keen on experiments. He was not a pioneer of scientific method. He was not the father of modern science. He was not a heroic knight defeating dogmas and superstitions with the light of empirical truth. No, he was none of those things. Galileo was, first and foremost, a failed mathematician. This is the key to understanding his role in the history of science, in my opinion.

Galileo’s contribution to the history of thought is to cut off mathematical reasoning at the training-wheels stage; to air in public what true mathematicians considered unworthy scratch work at best. He experiments because he cannot think. He cannot reach insights by reason, so he turns to more simplistic methods, hands-on methods. In physics this blatant shortcoming has been mistaken for methodological innovation. But in the case of the cycloid we see its true colours. We see that it is a sign of failure rather than genius. Galileo’s empiricism is the last resort of a failed mathematician. It is not science being born; it is science being dumbed down.

I will argue for this conclusion in considerable detail in many future episodes. In any case, the cycloid example obviously fits this thesis like a glove. So that’s certainly food for thought at the very least.

It is instructive to compare Galileo with Archimedes, who was a proper mathematician. Consider Archimedes’s work on floating bodies. This is science done right, almost two thousand years before Galileo. Archimedes’s treatise is an outstanding masterpiece of science by the standards of any age. Only the mathematically illiterate could fail to grasp its immense significance. As indeed they have.

Archimedes gives a thorough theory of the floatation behaviour of paraboloids. That’s the shape generated when a parabola is rotated about its axis. Think of a wine glass or a champagne glass for example. The cups of those glasses are kind of paraboloids. Suppose you put a shape like that in water. Sometimes it will float upright, sometimes it will tip over, and so on. Under certain conditions the thing is in equilibrium, in other conditions not so it will wobble one way or the other.

Three parameters determine the state of the system. One is the inclination or tilt; so the angle the axis of the paraboloid makes with the water. Another is density. These are solid paraboloids. The material they are made of is either heavier or lighter than water in some certain ratio. Then there’s the steepness of the paraboloid. So whether it is more round like a wine glass or pointy like a champagne glass.

The equilibrium conditions of floating paraboloids depend on these parameters in highly nontrivial ways. It is a complicated and complex matter to know what’s going to happen if you vary any of these values. Perhaps you are familiar with dynamical systems catastrophe theory stuff from the 20th century? It’s like that. So you have a phase space and the stable states make some complicated surface and so on.

Here’s an example of an application. Consider an iceberg. It’s floating upright, let’s say. But it’s melting. It’s becoming thinner and thinner, it goes from a wine glass to a champagne glass kind of shape. There will be a critical point when upright floating is no longer an equilibrium state. The iceberg topples over. Even though the parameter changed only very slowly, the effect on the floating position was at first nothing for a long time and then a dramatic collapse all of a sudden as some critical threshold was met.

Archimedes does all of this. Well, he doesn’t mention applications like icebergs but he has all the mathematics of the thing. He gives detailed, exact, quantitative predictions of the floatation behaviour of paraboloids. When will it float, when will it tip over. Archimedes tells you all of that.

And he’s right. All of these very specific empirical predictions that he derives, they are spot on correct.

Now, poor Archimedes, he is often misunderstood. So many people who don’t care for mathematics, they hardly even know this work exists. But if they do look at it they say: What’s this? It’s just a bunch of technical geometry about parabolas and stuff. Archimedes says not a word about any experiment, not a word about any empirical data, nothing about testing his theory, nothing science-y like that at all. It’s just a bunch of intricate geometry. You might as well have opened any work on geometry, maybe Euclid or Apollonius or whoever. It looks just like that, like more geometry. It seems to have very little to do with the real world.

Archimedes has a few very basic postulates in the beginning: some common-sense assumptions about how fluids push on submerged objects. That’s the only link to the physical world. After stating these two or three assumptions at the beginning, it’s all-out geometry from there on.

So it has been easy for people to say: Well, let some specialised historian of mathematics read that technical gibberish, I’m a historian of science, I don’t care about that because it says nothing about empirical science. The Greeks may have been excellent geometers but they didn’t really do science, you see. They were speculative thinkers, philosophers. They were great with abstract stuff but they didn’t have the sense to ground their fanciful theories in reality.

That attitude is completely wrong, in my opinion. Ask yourself: What are the odds that Archimedes got his detailed, quantitative theory of floating bodies absolutely spot-on right if he was ignorant of empiricism and experiment and scientific method? Was he just sitting around doings speculative armchair geometry and, whoops, it just happened to come out exactly equal to empirical facts in a range of far from obvious ways? Are we supposed to believe that was just dumb luck? It doesn’t make any sense.

Of course Archimedes knew about the scientific method. Of course Archimedes tested his theory by experiment. That’s obvious. His text doesn’t say that because he was too good of a mathematician to think that kind of kid’s stuff counted for much of anything. He only published the actual theory, not the obvious tests that any fool with half a brain could do for themselves.

Galileo, though, was precisely that fool with that half brain. He spent his whole life spelling out those parts that Archimedes thought were too trivial to mention. People ignorant of Archimedes are readily tricked into thinking that this was somehow profound. But mathematicians know better.

This is why we need the mathematician’s point of view represented in historical scholarship. And that’s what you will get in this podcast. And that will be the basis for my revisionist interpretation of Galileo’s contribution to the history of science.

]]>The geometrical algebra issue, at bottom, stems from this conundrum: How different are the Greeks from us? Can we use our “mathematician’s empathy” to make sense of their work, and view differences in style and expression as superficial? Or is this the cardinal error of historical scholarship? In order to understand past thinkers, does sound historiography require us to put our own conceptions aside and start from the assumption that the Greeks inhabit a conceptual universe completely different than our own? When the Greeks do things that feel like algebra to us, are we feeling the underlying gist of their thinking, or are we merely seeing a reflection of our own ways of thinking that were not in the sources until we projected them there?

In my paper I argued for a fundamental unity rather than a fundamental chasm between ancient and modern thought. Specifically I defended these two theses:

GA1. The Greeks possessed a mode of reasoning analogous to our algebra, in the sense of a standardised and abstract way of dealing with the kinds of relations we would express using high school algebra. By and large, whenever we find it natural to interpret Greek mathematics in algebraic terms, the Greeks were capable of a functionally equivalent line of reasoning. If an algebraic interpretation of a Greek mathematical work suggests to us certain connections, strategies of proof, etc., then the Greeks could reach the same insights in a similarly routine fashion. This was an abstract, quantitative-relational mode of thought that was not confined to concrete geometrical configurations and not dependent on geometrical visualisation or formulation; in particular, it was obvious to the Greeks that the exact same kind of reasoning could just as well be applied to numerical relations as geometrical ones.

GA2. The Greeks were well aware of methods for solving quadratic problems (such as those exhibited in the Babylonian tradition). Books II and VI of the Elements contain propositions intended as a formalisation of the theoretical foundations of such methods.

In perfect agreement with these theses, Sidoli observes that many mathematicians from late antiquity onwards indeed switched back and forth between numerical and geometric perspectives, treating them as trivially equivalent:

“At least by the classical Islamicate period, and probably from much earlier, this blending of the geometrical and arithmetical readings of geometrical books of Euclidean works was commonplace.” Heron and Ptolemy did things of this nature and “there is no indication that they thought of their use of metrical analysis as in any way new, or innovative.” (§5.3)

Nevertheless Sidoli suspects that this was a later appropriation of the Euclidean material and not Euclid’s original intention:

“On balance I think [the geometrical algebra reading of Euclid] is less likely than the claim the Data was originally composed to address the needs of geometrical problem-solving and was then later repurposed as a means to justify and generalize metrical arguments.” (§6)

Sidoli’s arguments for this are as follows:

1. The propositions in Euclid that can be read in a GA way have a geometrical character that does not readily match up step-by-step with the arithmetical-algorithmic procedures the are, according to the GA interpretation, supposed to formalise. E.g.: “The only way this theorem could have any use to us in metrical, or arithmetic, problem-solving, is if we already knew, through independent considerations, what sorts of arithmetic operations to carry out, but were interested in an unrelated geometric approach as a purely theoretical justification of these operations.” (§3.4)

My reply: Indeed, which is why, in GA2, I spoke of “a formalisation of the theoretical foundations of such methods,” as opposed to a practical recipe version of them. Similarly, no one would use Elements Definition V.5 to compare magnitudes or Proposition I.2 to transfer lengths in everyday situations. Of course Euclid’s treatment is a highly refined formal system, far removed from giving practical recipes and very much concerned with investigating subtleties involved in founding all of mathematics on a minimalistic set of principles. It is indeed a matter of “a purely theoretical justification” just as Sidoli says. This is not in conflict with the GA theses.

2. The Data is not a systematic exposition of GA-type material. “At the very least, we must accept that if, in fact, Euclid had devised [the Data] in order to justify arithmetic procedures, he did so in a rather disorganized way.” (§6)

My reply: The Data is not a systematic exposition of anything. It’s a weird hodgepodge of theorems by anyone’s reckoning. No one, as far as I am aware, has ever made sense of the Data as a cogent treatise leading in a coherent way to any kind of natural goals, or given a good account of why Euclid selected the particular propositions he did. Even Sidoli himself admits that “the Data is more of a compilation than the Elements and it is really only the first half to three quarters of the text that can be read as articulating a single program” (source, p. 404). So the fact that the Data is not a perfect fit for the GA hypothesis is neither here nor there.

3. The correspondence between the arithmetical operations and their geometric equivalents is not systematically expounded by Euclid. Thus: “While Data 3 and 4 could be taken to justify adding and subtracting for general quantities, for multiplying and squaring, taking square roots and dividing, we must turn to Data 52, 55 and 57—but, as we saw, the proof of these theorems rely on the geometric construction of a square and a similar triangle.” (§6) “Proponents of [GA2] must contend with the fact that this highly geometrical, and frankly rather peculiar, proposition [Data 52] is the only candidate in the text for propositions demonstrating that the product of two given numbers is given.” (§3.4)

My reply: Euclid does after all establish the result, even if Sidoli thinks his manner is “peculiar,” so there is perhaps not much to discuss here. Especially not since the Data is an enigma on any reading whatsoever. But let us put those points aside for the sake of argument. Even so I do not think this proves much.

GA2 has to do with the fact that the very same extremely basic ruler-and-compass methods that suffice to build up all the geometry of the Elements are also sufficient to solve any quadratic equation and carry out various other manipulations of quadratic expressions. This is a sophisticated, nontrivial point regarding the scope of ruler-and-compass constructions. I do not see why this would need to be bundled, as Sidoli’s argument assumes it should be, with a pedantic foundational account of spelling out the elementary idea that products of numbers corresponds to areas of rectangles and so on. As Sidoli himself says: “there is no clear evidence that Greek mathematicians thought that the basic arithmetical operations needed to be justified, so there is no reason for us to believe that Euclid felt the need to engage in such a project” (§6). Indeed, so why could Euclid not simply have taken this for granted? I do not see how this speaks against the claim that Euclid was also interested in proving the nontrivial result that having a ruler and a compass means being able to solve any quadratic equation.

This does not contradict my point that Euclid’s concerns are foundational, because his theory remains internally foundational through and through. The intuitive association between arithmetical operations and geometrical constructs is not needed internally but only externally to identify the formal theory with other things. Such associations take place in a different arena of reasoning, not subject to the internal stringency of the theory. Just as in modern real analysis, say, mathematicians reason with the utmost care when developing the theory internally, but then switch to much more intuitive and informal notions when they apply it to differential equations arising from a physics problem. While the theory of real analysis is built up from set theory in a stringent, formal way, its association with its physical applications is taken for granted in an intuitive, common-sense way that is not incorporated in the formal theory itself. So also Euclid can sensibly do his formalised version of the theory of quadratic equations without having to specify internally within his system how this maps to “real world” ways of doing these things.

For all of these reasons I am not convinced that any conclusions regarding GA can be drawn from Euclid’s Data.

Beyond his main focus on the Data, Sidoli makes a few very brief general points on GA that are very interesting. Basically he argues “against” GA by saying that GA1 is so great that GA2 is not needed:

“GA1 is simply another way talking about the theory of the application of areas, which is not in any dispute.” (§3.4) “Almost no one would argue that it is not possible to make a reading of Elements [II] and VI as motivated by and justifying computational problem-solving. The question is rather whether such a reading [=GA2], or that through the theory of the application of areas [=GA1], is more broadly successful in explicating the ancient sources.” (§6)

This novel take is news to me. Traditionally, GA1 and GA2 have been considered naturally and closely associated. People have either accepted them both or rejected them both. Sidoli, instead, seems to construe them as mutually exclusive hypotheses in direct competition with one another.

I’m not sure what Sidoli means when he says “GA1 is not in any dispute.” Does he mean that he does not dispute it? Or that no one disputes it? The latter is certainly false. People absolutely do dispute GA1. They give arguments of the form “if the Greeks had truly been able to think algebraically [=GA1] then they would have done so-and-so differently than they did.” Such arguments are clearly a direct denial of GA1. The bulk of my GA paper is devoted to rebutting arguments (by Unguru, Mueller, Saito, Grattan-Guinness) of exactly this type.

So it is strange for me to see Sidoli claiming to disagree with me when he happily accepts GA1. In my eyes, if everyone accepts GA1 this is basically a win for GA, regardless of whether they also agree on GA2.

]]>I came up with this taxonomy when trying to explain certain conflicts and research choices that shaped the early history of the calculus. Below are a number of episodes that fit these categories very well. (For more details on the earlier systems, see this useful essay.)

Maestro philosophy: “Few but ripe.” It is a mark of class to focus only on elegant, simple, important, emblematic masterpieces. Write enough to give a definitive, impeccable treatment of the subject, but not more. It will be evident that this is merely the tip of an iceberg, resting on a solid body of technical expertise. But the tedious explication of the latter—the scaffolding, the tricks of the trade—is left to lesser Technocrat mathematicians.

E.g. Archimedes:

17th century Maestro examples:

Huygens: tautochrone = cycloid = evolute of cycloid.

Leibniz:

Non-examples:

Johann Bernoulli:

Not Maestro because not classically motivated and self-contained; presupposes “nerd” Technocrat interest in evaluating everything that can be symbolically formulated. Same with Euler’s so-called beautiful (actually only Technocrat) formula

A typical Maestro versus Technocrat conflict/misunderstanding: Leibniz versus the English on power series in the 1670s.

Leibniz typical Maestro, cares about singular, beautiful results: “I possess certain analytical methods, extremely general and far-reaching,” but “exquisite” series “especially is most wonderful.”

English typical Technocrat, care about plug-and-chug-ready formulas, criticise Leibniz for merely giving special cases. Collins: “infinite Series to be generally fitted to any equation proposed, so that an Algebraist being furnished with his Stock, will quickly fitt a Series.” Newton: I gave “a general Method of doing in all Figures,” whereas “Leibnitz never produced any other Series than numerical Series deduced from them in particular Cases.”

But Leibniz has no interest in Technocrat that doesn’t lead to Maestro: “I too used this method [of series inversion] at one time, but after nothing elegant had resulted in the example which I had by chance taken up, I neglected it forthwith with my usual impatience.”

Later Newton turns from Technocrat to Maestro, because more classical and elegant (and perhaps associated with a certain snobbery and sense of superiority): “He thought Huygens’s stile and manner the most elegant of any mathematical writer of modern times, and the most just imitator of the antients. Of their taste, and form of demonstration, Sir Isaac always professed himself a great admirer: I have heard him even censure himself for not following them yet more closely than he did; and speak with regret of his mistake at the beginning of his mathematical studies, in applying himself to the work of Des Cartes and other algebraic writers.”

Euler disapproves, goes back to Technocrat, values toolbox adaptability more than beauty: “I always have the same trouble, when I might chance to glance through Newton’s Principia: Whenever the solutions of problems seem to be sufficiently well understood by me, yet by making only a small change, I might not be able to solve the new problem using this method.”

Leibniz is by nature a Visionary. The Maestro tendencies in the series episode are coloured by the influence of Huygens, who, in typical Maestro manner, praised the series as “a discovery always to be remembered among mathematicians.”

Later Leibniz resisted Maestro and saw it as a distraction from his main task of Visionary. This is why, for example, he fights not to get drawn into the brachistochrone problem (a true Maestro problem): “The problem draws me reluctantly and resistingly to it by its beauty, like the apple did Eve. For it is a grave and harmful temptation to me.”

Visionary need Technocrat to spell out the details of their systems. E.g. Descartes: Van Schooten; Leibniz: l’Hôpital, Johann Bernoulli.

Leibniz: “I wish there were young people who would apply themselves to these calculations. With me it’s like the tiger who lets run whatever he does not catch in one or two or three attempts.”

Leibniz is no more than 5% Technocrat: “Had I 20 heads, or better yet 20 good friends, I would put one of them toward working out the theory of conics.”

Huygens: I will learn calculus but only for Maestro, not for Technocrat. “I still do not understand anything about ddx, and I would like to know if you have encountered any important problems where they should be used, so that this gives me desire to study them.” “[Natural] curves merit, in my opinion, that one selects them for study, but not those [curves] newly made up solely for using the geometrical calculus upon them.”

Leibniz: Agree, Technocrat calculus worth little. “You are right, Sir, to not approve if one amuses oneself researching curves invented for pleasure.” But the difference between Visionary and Maestro is that Visionary is more focussed on general methodological insights, which is why Leibniz adds: “I would however add a restriction: Except if it can serve to perfect the art of discovery.”

L’Hôpital’s Rule: typical Technocrat result of the sort condemned here.

Systematic theory of integration by partial fractions: a Technocrat topic needed for Visionary, namely “a question of the greatest importance: whether all rational quadratures can be reduced to the quadrature of the hyperbola and the circle” (Leibniz). This forces Leibniz, reluctantly and contrary to his nature, to do some Technocrat work, with poor results (Leibniz erroneously believes that “ can be reduced to neither the circle nor the hyperbola by [partial fractions], but establishes a new kind of its own”). A typical Visionary, Leibniz clearly has very little interest in actually evaluating integrals, and only cares about giving a big-picture methodological-foundational account of integration in general.

Myth: Early Leibnizian calculus driven by applications; lacks attention to rigour. Hence typically Pragmatist.

Reality: The exact opposite: Early Leibnizian calculus primarily Visionary. It was actually indifferent to applications and was consumed by rigour, only rigour meant something completely different at that time. This is one of the main points of my book.

]]>But times change. French politics soured and foreigners were chased out of the country. Leibniz had to return to Germany. Huygens withdrew to his family mansion in the Netherlands. The Academy descended into reactionary mediocrity.

But Huygens did not retire to feed the ducks in his estate gardens. Though old and frail at this point, he kept up with the latest mathematics. This meant learning the new calculus developed by his former protégé Leibniz. The student had become the master, as the saying goes. But perhaps more interestingly, the master had become the student.

What a treat of history this is. Reading the correspondence between Huygens and Leibniz during these years, we get to see learning in action. We get to see how the calculus is taught by its inventor, and how a sage mathematician of the highest credentials goes about learning it. We get to see the former director of scientific research at the Academy of Sciences take a seat in the front row of Calculus I, pencils sharpened and notebook in hand. It’s a naked view of calculus genesis, unique in history.

Huygens proved a feisty pupil. He was not the kind of student who copies down the formulas and asks questions about the homework problems he got stuck on. Proofs do not impress him much either. What he demands most of all is motivation. He wants new mathematics to be thoroughly justified, not in the narrow sense of being logically correct, but in the broader sense of being a worthy human enterprise.

Thus, after having mastered derivatives, he wonders if second derivatives are just a formalistic indulgence or if they’re actually good for something. He writes to Leibniz:

“I still do not understand anything about ddx, and I would like to know if you have encountered any important problems where they should be used, so that this gives me desire to study them.”

Tell me why I would want to study second derivatives, Huygens demands. Not the formal rules for working with them, and the proofs thereof, and artificial problems specifically invented for them. No, not that. Any mathematician can make up such mathematics ad infinitum. A new mathematical theory must prove itself not by solving its own internal problems, but by proving itself on a worthy, honest-to-god problem recognised in advance.

Leibniz understands well, and replies:

“As for the ddx, I have often needed them; they are to the dx, as the conatus to heaviness or the centrifugal solicitations are to the speed. Bernoulli employed them for the curves of sails. And I have used them for the movement of the stars.”

We don’t care about second derivatives because the symbolism suggested we could do derivatives once over. We care about them because they are the right way to tackle mathematically a rich range of fascinating and important phenomena. Do you want to understand the shape of a sail bowed by the wind? Do you want to describe how planets move around the sun? Then you want to understand second derivatives.

This is not about pure versus applied mathematics. For example, in his big book on the pendulum clock, Huygens took inspiration from this concrete situation to develop a thoroughly mathematical theory treating evolutes and involutes abstractly and exhaustively. He gives a general proof, for example, that the evolute of any algebraic curve is algebraic. This is the kind of theorem that would make even the most doggedly purist mathematician proud.

The point is not that mathematics needs to be applied. It is that it needs to be motivated. We don’t study nature because we refuse to admit value in abstract mathematics. We study nature because she has repeatedly proven herself to have excellent mathematical taste, which is more than can be said for the run-of-the-mill mathematicians who have to invent technical pseudo-problems because they can’t solve any real ones. Says Huygens:

“I have often considered that the curves which nature frequently presents to our view, and which she herself describes, so to speak, all possess very remarkable properties. Such as the circle which one encounters everywhere. The parabola, which is described by streams of water. The ellipse and the hyperbola, which the shadow of the tip of a gnomon traverses and which one also encounters elsewhere. The cycloid which a nail on the circumference of a wheel describes. And finally our catenary, which one has noticed for so many centuries without examining it. Such curves merit, in my opinion, that one selects them for study, but not those [curves] newly made up solely for using the geometrical calculus upon them.”

Leibniz agrees: “You are right, Sir, to not approve if one amuses oneself researching curves invented for pleasure.”

If only modern calculus books lived by the same rule! Flip to the problem section at the end of any chapter in any standard calculus textbook and you will find a thousand problems “made up solely for using the calculus upon them”—exactly what Huygens condemns. Perhaps it should give us pause for thought when both the inventor of the calculus and its most able student ever are in complete agreement that our way of writing textbooks is stupid.

Modern students may well sympathise with Huygens again when he makes a similar point regarding exponential expressions:

“I must confess that the nature of that sort of supertranscendental lines, in which the unknowns enter the exponent, seems to me so obscure that I would not think about introducing them into geometry unless you could indicate some notable usefulness of them.”

Leibniz shows him how such expressions can solve certain problems, but Huygens is still not impressed: “I do not see that this expression is a great help for that. I knew the curve already for a long time.” Again: first show me what your technical thing can do, or else I have no reason to study it. And if I can do the same thing by other means then you have still failed.

I for one wish we had more little Huygenses in our calculus classrooms today. And I am worried we alienate the few we have by requiring them to swallow a style of exposition that a master like Huygens unequivocally rejected as bad mathematics unworthy of his time.

The study concerns the evaluations of precisely two teachers: one male, one female. Obviously no sane human being would draw general conclusions about gender from two individuals. But that is apparently what passes for peer-reviewed research in this field.

In the reported data, the female instructor had 1,169 students and the male instructor 357, in one semester. Could it be that having a more than three times higher teaching load has a negative impact on teaching quality? And that this rather than gender was the key difference between these two instructors? This possibility is not considered by the authors.

Actually the researchers threw away at least half of the actual data. We have no idea what it said. Here is their justification for this:

> Students had a tendency to enroll in the sections with the lowest number initially (merely because those sections appeared first in the registration list). This means that section 1 tended to fill up earlier than section 3 or 4. It may also be likely that students who enroll in courses early are systematically different than those who enroll later in the registration period; for example, they may be seniors, athletes, or simply motivated students. For this reason, we examined sections in the mid- to high- numerical order: sections 6, 7, 8, 9, and 10. (Supplementary materials, page 4)

This is crazy. The authors are openly admitting that they purposefully selected a non-representative sample, which by their own admission is likely to exclude certain types of students. Why on earth would you do this? Why not sample for instance all odd-numbered sections and hence get a sample that includes early- and late-registering students in representative proportions?

I can think of one reason. The authors of course knew that if they found no gender bias their study would go unpublished and would have been a waste of time, whereas if they found gender bias they could get it published in a Cambridge University Press journal and featured on Slate. So they had every incentive in the world to ensure that the data came out the way they wanted. And if you are allowed to study only two instructors, and arbitrarily discard half the data on nonsensical grounds, then it is not difficult to prove anything you want.

Also extremely problematic is that the teachers in question were the researchers themselves. This is obviously a terrible idea methodologically speaking. It is not far-fetched to think that their obvious incentive, as researchers, to find gender bias influenced their behaviour as instructors. The standard practice of keeping studies double-blind exists precisely to prevent the risk for such contamination. This study is about as far from double-blind as you can get.

For instance, the authors make a big fuss about how the evaluations more often referred to the male instructor as “professor” and the female instructor as “teacher” despite equal credentials. This is obviously the kind of thing that can be easily manipulated by the instructor. Throughout the semester they would have had every opportunity to plant the language they want the students to use.

If the authors wanted to ensure the desired outcome of the study, they could, for example, distribute the evaluations with different prompts. The male instructor can tell the students: “These evaluations are important. The university uses them to decide which professors get their positions renewed.” Now you have planted the terminology of “professor”, and also made students apprehensive about being critical since you have made them think about the possibility of you being fired. The female instructor, meanwhile, might say while handing out the evaluations: “Honest feedback is important to me as a teacher.” Now you have primed the students to refer to you as a “teacher”, and encouraged them to speak freely without holding back, since you have implied that the evaluations are for your own use and that you value honest feedback. The danger of such contamination of the data is vastly greater when the instructors in questions have a blatant vested interest in ensuring a particular outcome of the study, as in the case of this study.

]]>In my opinion, the field would be better off if it was open for debate and diverging viewpoints, instead of simplistically insisting that there is only one Right Way that no rational person could possibly disagree with. But the world of educational research and policy prefers the latter framing.

Edu-people are an ideologically homogenous group. They have very definite opinions on what is right and wrong in education, and virtually no one in the field ever disagrees on these core beliefs. 99 times out of a hundred their research confirms these opinions that they already held. This is either because they are brilliant and objectively right, or because they are biased and shield themselves from alternative viewpoints and critical thought because of the echo-chamber uniformity in the field. To tell which, we should look at the quality of their research, as I have done in many cases. From such investigations I have concluded that, in my view, we would do well to regard educational research with suspicion to say the least. But anyone with such opinions cannot get into the world of edu-people so the consensus stands.

A number of STEM faculty share my sceptical view of educational research. The way edu-people deal with such opposition rubs me the wrong way and only gives me all the more reason to be apprehensive about their claims.

For one thing, edu-people constantly refer to their own opinions as fact. They don’t say “we believe” but rather “research shows”. One phrase they have devised to this end is “evidence-based”. This phrase is repeated manically hundreds of times in the report. Instead of saying “that person has a different view of teaching than me”, edu-people say “that person’s views are not evidence-based”. It is hardly the hallmark of objectivity and open-mindedness to systematically use such a blatantly value-laden yet ostensibly factual term to refer to one’s own opinions.

It makes one wonder whether a statement such as the following is not basically tautological: “In the committee’s view, improving the quality of undergraduate STEM education will require wider use of evidence-based STEM educational practices and programs.” (1-7) Since “evidence-based practices” is effectively code for “our opinions”, the statement basically reads: in the committee’s view, more people need to agree with the committee.

Or put it this way: among the many critics of the edu-people consensus, has anyone ever said: “I don’t agree with you because I don’t think evidence should be taken into account when making instructional and policy decisions.” Of course not, that would be ridiculous. Yet edu-people insist on such a framing, hammered home with hundreds of repetitions of such phraseology. We may want to ask ourselves why edu-people are so attached to this rhetoric, by which they imply that anyone opposed to them must be ignorant of evidence.

Let’s keep this meaning of “evidence-based” in mind when we read the following quote from the report:

“A growing body of research indicates that many dimensions of current departmental and institutional cultures in higher education pose barriers to educators’ adoption of evidence-based educational practices.” (3-12)

Translation: Our opinions have still not achieved complete hegemony. To fix this we have spent a lot of time analysing why.

“A well-established norm in some STEM departments [is that of] allowing each individual instructor full control over his or her course. … One recent analysis found that the University of Colorado-Boulder science education initiative made little progress in shifting ownership of course content from individual instructors to the departmental level because of this dimension of departmental culture.” (3-12) Thankfully, Michigan State University offers an encouraging model where “leadership … from the provost” led to “increased coordination of instructional and assessment practices.” (3-13)

Translation: There is too much democracy and decentralised power in academia. Our sect is in control of administrative positions, but our power is not yet great enough to force everyone to agree with us, though soon we hope to achieve this goal.

Professors being in control of course content is apparently an evil that cannot be eradicated soon enough. Apparently it is better if a provost—a career bureaucrat—is in charge and bosses around the professors who are the actual experts in the field. That’s apparently what “evidence-based” “best practices” demand. I do not see the rationale for this insistence on uniformity, other than the one I have inferred in my translation. Or do these people also wish to replace all local small businesses with a McDonald’s and a Walmart, by the same logic?

The curious emphasis on instructional homogeneity is perhaps all the more jarring when juxtaposed with another perennial edu-slogan: “instructor diversity provides educational benefits to all students” (4-10); indeed, “the benefits of instructor diversity are clearly demonstrated by available research” (4-11). It serves edu-people well that the term “diversity” has been emptied of any meaning but the modern politicised one, for else this would square poorly with their explicitly announced intent to eradicate diversity of pedagogical approaches among instructors.

]]>I think Carman is wrong. I think his argument is very implausible for an obvious reason that he does not acknowledge. Namely: Why would Aristarchus have affirmed and written a treatise on heliocentrism if it had nothing but disadvantages? What possible reason could he have had done for doing so? None, in fact. Yet this is exactly what Carman proposes.

Let’s go though it from the beginning. The basic facts are as follows. Copernicus’s heliocentric system has a number of advantages, including the determination of planetary distances. It also has a number of disadvantages, notably the absence of annual parallax, which means that the stars must be very far away to explain why they don’t look sometimes close and sometimes more distant as the earth changes position in the course of a year. In other words, there is a lot of “wasted space” in the universe. This was commonly considered implausible, and hence an argument against heliocentrism.

Aristarchus wrote a treatise arguing that the earth revolves around the sun. Archimedes mentions it when discussing the size of the universe, in a way that shows that Aristarchus was well aware of the parallax issue.

The great bulk of first-rate Greek astronomical works are no longer extant, including Aristarchus’s treatise on heliocentrism and virtually everything from the very active century following it (which could very well have included a dozen skilled heliocentrists for all we know).

According to Carman, Aristarchus’s treatise most likely concerned only the sun and the earth and said nothing about the planets. Or, if it did consider the planets, it most likely made the planets go around the earth rather than around the sun. Either way, the treatise would amount to nothing but a trivial point about relativity of motion, namely that A moving in a circle about B is equivalent in terms of relative positions to B moving in a circle about A. Thus either hypothesis could account for the same phenomena.

I say: there is no reason for Aristarchus to write such a treatise, and plenty of reasons for him not to. For one thing, the result in question is rather trivial and has nothing to do with the sun and the earth specifically—it’s a result about circular motion generally. Indeed, as Carman himself notes, it is found as such in Euclid’s Optics. Carman somehow tries to construe this as support for his reading: “Aristarchus’s treatise on Heliocentrism could be understood as an application of these propositions of Euclid’s.” This does not make sense to me. Of course one possibility in Euclid’s theorem is to take A=sun and B=earth. How could you possibly fill an entire treatise making this elementary point that I just expressed in a single sentence? Furthermore, why apply it to the sun and the earth, rather than, say, the earth and the moon?

Which brings us to the core problem of Carman’s account. Let’s say for the sake of argument that, as Carman supposes, Aristarchus’s treatise only talked about the earth-sun system and proved that either orbiting the other would give rise to the same phenomena as far as their relative positions are concerned. He would then have faced the inevitable and obvious follow-up question: Which of the two hypotheses is the right one?

How would Aristarchus have answered this question? As far as one can tell from Carman’s account, only one relevant consideration was known to Aristarchus: the parallax problem, which he explicitly recognised, as Carman himself admits. This strongly suggests that it is the earth that is stationary. Thus Aristarchus should clearly have concluded against heliocentrism.

Yet we know for a fact that Aristarchus not only discussed the hypothesis of the earth’s motion about the sun, but also asserted it as physical reality, as Carman also admits. Why? Why would Aristarchus write a treatise proposing this bold hypothesis, discuss a major argument against it (the parallax argument) and no arguments for it, and then conclude that the hypothesis is true? And why, furthermore, would Archimedes, who was perhaps the greatest mathematician of all time, cite this treatise with tacit approval as a viable description of physical reality? It makes no sense.

The only reasonable explanation is that Aristarchus recognised an advantage of placing the sun in center. And the obvious guess for what this was is that he saw the same advantages as Copernicus did, including the argument from planetary distances. Indeed, we even have Aristarchus’s only surviving treatise on the relative distances of the sun, earth, and moon, which proves that he was a highly competent mathematician very much concerned with celestial distances. What are the odds that, despite this, he somehow failed to put two and two together and make the straightforward connection between his heliocentrism in one treatise and his preoccupation with celestial distances in the other? It seems to me extremely unlikely that this connection could somehow have escaped the attention of Aristarchus, not to mention the century of highly competent mathematical astronomers who followed him.

Furthermore, note that Aristarchus’s surviving treatise also treats the sizes of the sun, earth, and moon. Combined with his heliocentric hypothesis, this means that smaller bodies orbit bigger ones, rather than conversely as in the geocentric system. This is arguably a physical plausibility argument in favour of the heliocentric theory. Again it is very difficult to imagine that this could somehow have escaped Aristarchus’s attention even though it was right under his nose. Much more likely is that Aristarchus explicitly made this connection too, which would immediately have suggested to him that the planets revolve around the sun as well.

Alas, the authors do not say a word about another very obvious alternative explanation: research productivity. The authors found that women constitute 31% of the speakers, which is supposedly significant underrepresentation. But what proportion of publications are authored by women? Well, surprise surprise, it’s about 31%.

It is not difficult to see why the authors conveniently neglected to mention this well-known fact that is obviously highly relevant. The authors of course want to suggest that “colloquium committees … unwittingly favor men” because of “bias” and “stereotypes.” It would thus be inconvenient to admit that speaking frequency simply mirrors publication productivity, since manuscripts submitted for publication are anonymised. But of course that is only relevant if you are actually trying to investigate facts in an honest way, not if you are trying to concoct evidence for a predetermined conclusion that is ideologically agreeable to you.

The authors’ literature review is of course full of references to many papers I have discussed before. This was a new one for me though:

“Participants who read a lecture, which was posited as having been written and delivered by a male or female professor, rated the lecture by the male (versus the female) professor significantly more positively.”

Sounds like damning evidence! But let’s see what happens if we actually look up the paper cited. Then we find right in the abstract:

“Students … evaluated an identical written lecture by a male and female professor on pay disparities between men and women in the workforce suggesting sex discrimination.”

Are you kidding me?! What the authors deceitfully refer to simply as “a lecture” was in fact a politicised opinion piece that specifically argued that women are discriminated against. Obviously this is an absolutely idiotic way of testing whether students are biased by gender when evaluating “a lecture.” Obviously the students had every reason in the world to be a little more apprehensive when this case came from a female lecturer. It is one of the most elementary principles of critical thinking to be less trusting of information coming from a self-serving source, as everyone knows except gender bias researchers apparently.

Another recent study (featured at Nature’s news blog) makes a similarly dubious case for gender bias. The researchers looked at success rates of grant applications and found that women and men have about equal chances when only the research proposal itself is being evaluated, but that men are favoured when the calibre of the researcher is part of the evaluation criteria.

This of course proves absolutely nothing about gender bias since the higher ratings of male applicants is very plausibly due to better publication records (a possibility the authors themselves admit; pp. 9-10), which is exactly the sort of thing that the evaluation system is designed to take into account, and very reasonably so.

But, as usual, nothing is more biased than gender bias researchers. The title the authors cooked up for their paper is blatantly dishonest and fanatically ideological: “Female grant applicants are equally successful when peer reviewers assess the science, but not when they assess the scientist.”

With this deceitful phrasing the authors manage to insinuate that taking into account people’s scientific track record actually means judging people on factors other than “the science.” It is pathetic that Nature eagerly picks up this propaganda nonsense in their headline. This kind of “research”, and the cheerleading reception it receives, is a disgrace to academia and an insult to critical thought.

]]>I have been trying to track down the source of this quote I once heard attributed to Leonardo da Vinci, and I believe this is it:

It’s a bit less colourful than the translation I heard, but close enough.

Words like these have a bad reputation these days, but I believe they express a healthy rejection of authority that was key in the Renaissance and scientific revolution. But who should get credit for it? Not Leonardo, evidently, for he attributes the saying to an ancient source, as we see. According to one editor, no such passage can be located in extant works. This suggests that: Leonardo had access to ancient works we don’t; he copied from them liberally; the things he copied are now erroneously attributed to him; Leonardo and his contemporaries were happy to raid ancient sources for cool ideas but took little care to preserve them whole.

Interestingly, this is exactly Russo’s hypothesis regarding Leonardo’s scientific works generally. Thus he writes for example: “Leonardo’s ‘futuristic’ technical drawings … was not a science-fiction voyage into the future so much as a plunge into a distant past. Leonardo’s drawings often show objects that could not have been built in his time because the relevant technology did not exist. This is not due to a special genius for divining the future, but to the mundane fact that behind those drawings … there were older drawings from a time when technology was far more advanced.”

I was always inclined to believe Russo, and all the more so now that my investigation of the fart quotation independently vindicated his view.

]]>The author emphasises the Gutiérrez case and is perhaps right as far as that goes, but as a general dismissal of critiques of mathematics education I think it is very problematic. There are many legitimate critiques of mathematics education research, and dismissing them in this manner is in my view counterproductive and not consistent with the author’s call for humility.

Consider an analogy. A hundred years ago there was much scholarly literature justifying oppression of women, homosexuals, certain ethnic groups, etc. Today everyone agrees that that entire body of scholarly literature was just plain stupid. The most admirable people were the small minority who dared to say so.

The logic of the AMS blog post would not be on the right side of history. It mirrors the voice of the oppressors, who defended the bigoted research. At that time the same rhetoric would have gone: Stop with your “inflammatory” “knee-jerk” critique of fine researchers who explain why blacks and homosexuals are inferior human beings. These are “experts in their fields” who “deserve respect” dammit!

Just as idiotic, bigoted “research” a hundred years ago was rotten to its core, so we must be open to the possibility that the same is true for mathematics education today. Therefore we must allow criticisms of the field, and not ban critical thinking in the name of “respect.”

It is not healthy to ban outsider critiques of scholarly fields. Saying that it’s “arrogant” for people who are not “experts in the field” to criticise it is a recipe for intellectual stagnation.

This framing is basically tantamount to banning critical thinking altogether. For to become an “expert in the field” you must pass through a PhD program ran by people in the field, publish papers peer-reviewed by people in the field, get hired by people in the field, etc. By construction, therefore, an “expert in the field” is one who thrives in the status quo. The current fashions loves them, and they love the current fashions. That’s what their success means, virtually by definition.

Someone critical of the field, on the other hand, cannot pass through these screening stages, and hence will never, by definition, be an “expert in the field.” I, for example, spent two years in PhD program in mathematics education. But I left the field because I was critical of its methods. The field maintains its cozy consensus by weeding out people like me long before they become “experts.” No wonder, then, that all “experts” agree with each other.

Hence outsider critiques is the only genuine critique there is. To dismiss it as “arrogant” “knee-jerk reactions” of people who should show more deferential “respect” toward “experts” amounts to banning critical thinking from the field of mathematics education.

]]>The study’s data set is impressive: 19,952 student evaluations of university faculty in courses where students were randomly allocated to instructors. Female faculty were rated lower, despite producing the same outcomes in terms of grades.

A major problem, however, is this: The evaluation forms completed by students never actually asked them to judge whether the teacher was good or bad. Here is what the students were actually asked (39):

T1: “The teacher sufficiently mastered the course content”

T2: “The teacher stimulated the transfer of what I learned in this course to other contexts”

T3: “The teacher encouraged all students to participate in the (section) group discussions”

T4: “The teacher was enthusiastic in guiding our group”

T5: “The teacher initiated evaluation of the group functioning”

When the authors say female faculty received lower evaluations, they mean lower average score on these five items. But these five items are very poorly conceived as a way of capturing teaching quality, for the following obvious reasons.

T1 is a bad measure of teaching quality since you can master the content and still be a lousy teacher.

T4 is a bad measure of teaching quality since a teacher can be enthusiastic but ineffectual, or dry but effective.

T3 is very dubious since the pedagogical strategy of calling on reluctant students is not necessarily positive.

T5 is a bad measure of teaching quality since it’s pointless if the group worked fine already. The data suggests that groups on the whole worked fine (39). If the instructor saw this and hence for this reason did not “initiate evaluation of the group functioning,” then it obviously makes no sense to punish this teacher in the course evaluations for not wasting class time on a needless group evaluation.

The instructor’s performance on T2 can by definition not be checked by controlling for course grade. It could be that female faculty were simply worse at this. The conclusions of the study follow only if we agree that the equality of grade outcomes prove that female faculty performed equally well. But T2 specifically asks for things that go beyond the course, i.e., things that do not count toward the course grade. Hence we have no way of telling whether the students’ assessment of T2 were biased or accurate.

In sum, the supposed evaluative measure of teaching quality is not a measure of teaching quality at all. The assumption—essential for the study’s conclusions—that equality of grade outcomes means equality of instructor performance on T1-T5 is unwarranted.

There are some grounds to nevertheless maintain the authors’ interpretation. One is that the bias seems to cut somewhat uniformly across T1-T5, suggesting that the students harbour blanket or generic depreciation of female faculty rather than giving thoughtful and reliable answers to each item separately. At least this is indicated by the only data we have showing a breakdown of the items T1-T5 one by one (Table B3). Unfortunately, we have such data only for graduate student instructors. There is reason to think that this is the instructor group that most confirm the authors’ thesis of gender bias. For the bias against female faculty “is larger for mathematical courses and particularly pronounced for junior women” (abstract). This could be due to stereotype bias. Alternatively, it could be due to gender bias in favour of women in graduate student recruitment. The fact that evaluations are lowest among junior female instructors and in mathematical fields would then be a reflection of the fact that these fields have lately been very aggressive in recruiting women at all costs.

Another argument for the authors’ interpretation is the fact that the gender bias is “driven by male students’ evaluations” (abstract). If female faculty were genuinely worse, wouldn’t female students too recognise this? Maybe. But an alternative explanation could be that female faculty are especially supportive of female students, so that the differing evaluations by student gender reflect a genuine difference in the quality of instruction received. The authors themselves note that this is by no means an outlandish hypothesis: “Female students receive 6% of a standard deviation higher grades in non-math courses if they were taught by a female instructor compared to when they were taught by a male instructor. … This might be evidence for gender-biased teaching styles.” (30) Note also that it is easy to imagine how T3 in particular could reflect such bias.

One reason to think that the students are not entirely off the mark in their evaluations is how their judgement develops over time. “The bias for male students is smallest when they enter university in the first year of their bachelors and approximately twice as large for the consecutive years. For female students, we find that only students in master programs give lower evaluations when their instructor is female, but not otherwise.” (30) You would think that students would get better rather than worse at judging teaching quality in the course of their education.

Here’s another point:

“Strikingly, despite the fact that learning materials are identical for all students within a course and are independent of the gender of the section instructor, male students evaluate these worse when their instructor is female.” (3)

Two possible explanations suggest themselves:

(a) The students are blinded by bias and cannot evaluate the course materials objectively. They let their predjudice against the female instructor cloud their judgement even on this question which had nothing to do with her.

(b) Female instructors were less good and hence unable to highlight and bring out positives and insights in the course materials, thereby making the course material seem less good. Hence lower evaluations of instructors and course materials go hand in hand.

Of course the authors suggest (a). But the supposed logic behind this is somewhat dubious. If male students hate women, shouldn’t their evaluation of the textbook be based on the gender of the textbook author? If they are driven by and seek to express their dislike of the female instructor, and the textbook was written by a male author, shouldn’t they rate the textbook higher rather than lower, so as to convey that it was the particular instructor rather than the course materials that were at fault? In fact, if the students had done precisely this, then that too could have been used as evidence of their blatant gender bias. Thus two completely different outcomes could both be spun as clear evidence of gender bias. This suggests that we should be careful before jumping to the conclusion that the data confirms our favoured hypothesis.

]]>> Students rated the male identity significantly higher than the female identity, … demonstrating gender bias. (291)

How can this be? Simple: the authors were so determined to prove gender bias that they decided to cheat and move the goalposts, as they admit in a footnote:

> While we acknowledge that a significance level of .05 is conventional in social science and higher education research, … we have used a significance level of .10 for some tests where: 1) the results support the hypothesis and we are consequently more willing to reject the null hypothesis of no difference; 2) our hypothesis is strongly supported theoretically and by empirical results in other studies that use lower significance levels; 3) our small n may be obscuring large differences; and 4) the gravity of an increased risk of Type I error is diminished in light of the benefit of decreasing the risk of a Type II error. (288)

In other words: We decided to call things significant even when they’re not if: 1) it agrees with what we already decided in advance that the results of the study should be; 2) everyone already knows we’re right anyway so we don’t need any of that pesky “scientific evidence” stuff (even though finding such evidence is ostensibly the whole purpose of our paper); 3) our study is so ridiculously small that the results could mean anything; and 4) we may be dead wrong but on the other hand maybe we’re not.

Only with this sham, unprecedented definition of significance did the authors manage to find a so-called “significant” pro-male bias.

In any honest universe, the title of the paper would be “Failure to Expose Gender Bias in Student Ratings of Teaching” and the abstract would say “Students did not rate the male identity significantly higher than the female identity, … demonstrating that no gender bias can be inferred.”

The study does, however, demonstrate one clear and undeniable bias, namely that of the authors in favour of their preconceived hypothesis.

Actually, the authors were not even content with this amount of cheating. They lie even more when they say:

> These findings support the argument that male instructors are often afforded an automatic credibility in terms of their … expertise. (300)

In reality, they specifically asked the students to rate how “knowledgable” their instructor was, and the results (299) showed no significant gender effect even with the authors’ sham definition of significance. Since this didn’t stop them from concluding the exact opposite, one wonders why they bothered gathering any data at all.

]]>One of the authors I challenged was Eberhard Knobloch (who, incidentally, was just awarded the most prestigious prize in the field). He replied with indignation in a letter to the editors, calling my paper “completely unacceptable.” I have submitted a reply which will hopefully appear soon (update: here it is).

In the meantime, a review of my paper just appeared in the Zentralblatt. The review is written by Paolo Bussotti, who spent three months as a guest researcher hosted by Knobloch in 2014. This so-called review is not really a review of my paper at all, but rather a regurgitation of Knobloch’s letter to the editors, which Bussotti follows slavishly.

Bussotti cites Knobloch’s letter parenthetically, but in no way indicates that “his” critique is in fact nothing but a point-by-point regurgitation of everything Knobloch said in his letter. Bussotti’s phraseology will lead readers to think that he is offering an independent judgement, when in reality he is parroting Knobloch’s letter. For instance, Bussotti writes:

> [Blåsjö’s] main theses can be summarized in two items: 1) … 2) … let us start from what I have indicated as item 2)

This summary of my view in terms of these two theses is due to Knobloch, who even explicitly labelled them (1) and (2). But those who do not have Knobloch’s letter in front of them will surely be mislead by this kind of phrasing into believing that Bussotti has carried out his own independent analysis, rather than simply transcribed almost literally the exact view of his friend, who is one of the parties in the conflict.

Although Bussotti obediently follows Knobloch on every single point of substance, he does manage to introduce some absurd misunderstandings of his own. For instance, he writes:

> The whole question turns around the interpretation of the sentence translated by Knobloch as “It serves, however, to lay the foundations of the whole method of indivisibles in the soundest way possible” and by Blåsjö as “Whence it will be permissible to use the method of indivisibles proceeding by spaces formed by steps or by sums of ordinates as strictly demonstrated”. The two translations are not significantly different and the whole question concerns the interpretation of that “it”.

It is difficult to fathom how Bussotti could have gotten it into his head that these two quotes are “two translations” of the same passage and even that they are “not significantly different.” They are of course completely different quotes and obviously do not refer to the same passage in Leibniz. The relevant quote in my paper is on a different page altogether (137), with a translation that follows Knobloch virtually verbatim.

In any case, the notion that “the whole question” comes down to this one sentence (as Bussotti claims twice) is absurd. It does, however, square well with Knobloch’s letter, which opens with a critique of my reading of this passage.

As for the substantive point at stake, it concerns whether the “it” in question refers to Proposition 6 (as the standard view has it) or to the idea of its proof (as I claim). Bussotti regurgitates (without saying so) Knobloch’s argument that it must be the former, for reasons of Latin grammar. I do not deny that the “it” is Proposition 6 grammatically speaking. But this proves nothing. As seen in my paper, in the very same passage Leibniz uses the very same “it” as follows: “In it, it is demonstrated in fastidious detail that …” Thus Leibniz is obviously using “it” (i.e., “Proposition 6”) quite loosely as a way of referring to the whole passage of text (somewhat like a chapter heading, say), rather than to the propositional statement per se (which is what Knobloch’s interpretation needs). Thus my interpretation is not at all inconsistent with the text.

Knobloch also raised a quibble about whether Leibniz’s proposition should be called a foundation of infinitesimal geometry or of infinitesimal calculus. Knobloch tried to allege that he spoke only of the former and that the latter is an anachronistic misnomer introduced by me. Bussotti duly parrots the same point:

> infinitesimal geometry [is] partially different from infinitesimal calculus, a difference which [Blåsjö] seems, at best, to underestimate, as he uses indifferently both expressions.

But the insinuation that I somehow introduced this false equivocation is absurd. The notion that Leibniz’s proposition provides a foundation for the calculus is clearly and explicitly present in the works I criticise. In fact, later, when it suits his purposes, Bussotti himself goes on to reaffirm exactly this:

> Leibniz’s proposition 6 offers a general foundation to integral calculus …, no doubt about this.

Why, then, is he bitching that I spoke of calculus instead of infinitesimal geometry, if he himself uses the same terminology and thinks there is “no doubt” that it is accurate? Bussotti’s critique is not even coherent, let alone sound.

Bussotti’s review ends with an accurate and revealing observation:

> The approach of [Blåsjö] does not seem favourable to edify new and collaborative researches in the line traced by Knobloch and by the other scholars who have studied Leibniz [for] many years. … My conviction is that new insights as to the concept of rigour in Leibniz can be achieved taking into account that the general picture traced by these authors is basically correct.

This seems to me an accurate description of a kind of implicit axiom of modern historiography, namely that scholarship should be collaborative rather than critical. You should pat your friends on the back, not question them. This may be a sound policy if we want academia to be a feel-good social club. But as a recipe for intellectual progress I think it is fundamentally misconceived.

]]>It would be sad indeed if the leading scientists in history committed such elementary blunders and couldn’t avoid even the crudest forms of social bias. If so, we should have to accept the postmodern historiography that takes science to be a social construct determined by the powers that be rather than by objective truth.

Let’s see if this is the right conclusion on Darwin. Saini bases her point on this quote:

“The chief distinction in the intellectual powers of the two sexes is shewn by man’s attaining to a higher eminence, in whatever he takes up, than can woman.” (Descent of Man, 361)

It is indeed hard to dispute, as Darwin observes, that “lists … of the most eminent men and women in poetry, painting, sculpture, music …, history, science, and philosophy” have mostly men in them.

But is this due to biology? Saini evidently thinks it would be a naive fallacy to assume as much. So what is the alternative? That these inequalities are arbitrary social constructs with no biological basis? Then how to explain that they have persisted across cultures and millennia? Did men just happen to obtain the upper hand once upon a time thousands of years ago and then doggedly managed to maintain their arbitrarily constructed advantage without interruptions across countless revolutions, bloody wars, religious upheavals, and the rise and fall of empires? And also the same chance occurrence took place many times over in one geographically isolated civilisation after another?

I don’t think you have to be “blinded by bias” to infer that there are biological factors at play here. Of course this does not mean that men are more intelligent than women. Maybe they just have more muscles and maintain their advantage by force, for example. So is Darwin’s mistake that he assumed “intellectual powers” to be the explanation? His view is rather more nuanced. He in fact explicitly denies that there is an innate difference in this regard:

“It is, indeed, fortunate that the law of the equal transmission of characters to both sexes prevails with mammals; *otherwise*, it is probable that man would have become as superior in mental endowment to woman, as the peacock is in ornamental plumage to the peahen.”

Instead, he attributes the advantage of men to “higher energy, perseverance, and courage.” If there is any innate gender difference, says Darwin, it is this: “Man is the rival of other men; he delights in competition, and this leads to ambition which passes too easily into selfishness. These latter qualities seem to be his natural and unfortunate birthright.”

Today, “although men do not now fight for their wives, and this form of selection has passed away, yet during manhood, they generally undergo a severe struggle in order to maintain themselves and their families; and this will tend to keep up or even increase their mental powers, and, as a consequence, the present inequality between the sexes.” Therefore, “in order that woman should reach the same standard as man, she ought, when nearly adult, to be trained to energy and perseverance.”

In sum, Darwin denies that men are inherently more intelligent than women. Instead he attributes their higher prominence in intellectual pursuits to differences in attitude, and notes that present societal circumstances play a large part in this. Altogether, I do not think this warrants the conclusion that Darwin’s reflections are self-serving “Victorian male” make-believe rather than science.

]]>Some day I will write a follow-up paper, but for now let us consider Ragep’s most overarching argument. He thinks independent discovery by Copernicus is implausible because:

> Perhaps most importantly, why would someone seek to start from scratch when it was certainly known in the fifteenth and sixteenth centuries that Islamic astronomers still had much to teach their European counterparts? (194)

Is this true? Did Europeans at the time consider Islamic astronomers way more advanced than themselves? Is there even a shred of evidence that Copernicus ever held such an opinion? No.

Ragep opts to back up his claim with one single, peculiar reference: the chapter by Feingold in Ragep (ed.), Tradition, Transmission, Transformation. Here’s what Feingold has to say:

> Most of those who sought access to Arabic science were animated by … “reductionist” motives: They viewed “the achievement of Islamic scientists … merely [as] a reflection, sometimes faded, sometimes bright, or more or less altered, of earlier (mostly Greek) examples.” Certainly they recognized the existence of a considerable body of scientific knowledge available in Arabic, but it was usually adjudged either as derivative of the Greeks or, at best, the fruit of sheer drudgery. (445)

A few had “great hopes” to find “most precious stones for the adornment and enriching of my syntaxis mathematike” “in that happy Arabia” (447), and set out to learn Arabic for the purpose. But this was soon followed by a “rapid decline of such studies” (448).

> Some were simply disillusioned by what they viewed as the small return on their investment. John Greaves, for example, griped … that the drudgery he had put himself through editing Abulfeda’s Geography was simply not worth it: “to speak the truth, those maps, which shall be made out of Abulfeda, will not be so exact, as I did expect; as I have found by comparing some of them with our modern and best charts. In his description of the Red sea, which was not far from him, he is most grossely mistaken; what may we think of places remoter?” (448)

Others too lamented “how greate the losse of time was to study much the Eastern languages” and no longer “much care for to trouble myself about the keys [to oriental learning] when there was no treasure of things to be come at.” (449) Francis Bacon agreed:

> “The sciences which we possess come for the most part from the Greeks. … Neither the Arabians nor the schoolmen need be mentioned; who in the intermediate time rather crushed the sciences with a multitude of treatises, than increased their weight.” (443-444)

> Thomas Sprat, the official historian of the [Royal] Society, was willing to admit that the Arabs were “men of deep, and subtile Wit,” but he also felt it unnecessary to discuss them in surveying the progress of knowledge because their studies “were principally bent, upon expounding Aristotle, and the Greek Physitians.” Besides, “they injoy’d not the light long enough. … It mainly consisted, in understanding the Antients; and what they would have done, when they had been weary of them, we cannot tell.” (454)

> More disparaging was Joseph Glanvill who faulted the Arabs principally for their blind devotion to Aristotle. … “These Successors of the Greeks did not advance their Learning beyond the imperfect Stature in which it was delievered to them.” (454)

> William Wotton [held that the Arabs] “translated the Grecian Learning into their own Language [but] had very little of their own, which was not taken from those Fountains.” … “There is little to be found amongst them, which any Body might not have understood as well as they, if he had carefully studied the Writings of their Grecian Masters. … There are vast Quantities of their Astronomical Observations in the Bodleian Library, and yet Mr. Greaves and Dr. Edward Bernard, two very able Jugges, have given the World no Account of any Thing in them, which those Arabian Astronomers did not, or might have not learnt from Ptolemee’s Almagest, if we set aside their Observations which their Grecian Masters taught them to make.” (455)

> Theophilus Gale … [argued that] it is not Aristotle … who should be blamed for breeding that “Sophistic kind of Disputation, which now reigns in the Scholes.” This was the doing of his Arab commentators, Averroes and Avicenna in particular, “who, being wholly unacquainted with the Greek Tongue, were fain to depend upon the versions of Aristotle, which being very imperfect, left them under great darknesse and ignorance touching Aristotle’s mind and sense; whence there sprang a world of unintelligible Termes and Distinctions, with as many Sophistic Disputes and Controversies. These the Scholemen (more barbarous than the Arabians) greedily picked up … and incorporated with their Theologie.” (456)

All of this is quoted from the one article Ragep himself singled out as support for his claim that it made no sense for people like Copernicus to think for themselves since they had so much to learn from the much wiser Arabic sources. If this is the evidence in favour of his claim, you can imagine for yourself what evidence against it would look like.

]]>If the truth is simple, and the right interpretation is half a page long, you can’t publish it. But if you “problematise” the question and bring in an assortment of irrelevant material, chances are that you can put together twenty pages of subtleties and footnotes. And you can certainly publish that, because everybody knows that’s what scholarship is supposed to look like.

Repeat this for a few generations and the papers with the erroneous view have now become forty pages apiece since they have to include baroque analyses of each other in addition to the misconceived primary evidence the mistaken view was based on in the first place.

The further this goes on, the more naive you will look if you speak the simple truth. “But there’s an enormous literature on that!” people will exclaim with indignation. Experts upon experts have piled on the footnotes and devoted entire careers to the issue. Surely so many eminent scholars cannot be wrong. Meanwhile, the simplistic view you espouse has not been expressed by anyone with the proper titles and credentials since practically the age of the dinosaurs (i.e., more than half a century ago).

But the fact that a certain view dominates the latest papers in the latest journals doesn’t mean it has won the day by merit, only that academic evolution is bound to produce organisms that thrive on the excrement of another. The law of the academic jungle is not survival of the fittest; it’s survival of the most publishable.

It is only natural that bottom feeders become more hostile to outsiders with every passing generation. The more established they become, the greater their stake in insisting that quantity of footnotes is a proxy for expertise. Then those pesky people who speak the simple truth are simpletons by definition, and no one needs to face the unpleasant prospect that they’ve been living in the wrong ditch for generations.

These forces make it natural and predictable that historical research will take us further and further from the truth. In time, as academics invest more and more in their erroneous interpretations and build entire schools upon them, they even develop an instinctive hostility toward the truth, since, at that point, accepting the truth is tantamount to challenging the territorial hegemony and survival of their entire tribe.

]]>This definition is clearly meaningless drivel. How can such a masterful work, which is clearly written by a top-quality mathematician, open with such junk?

Russo proposed a compelling answer to this conundrum. It goes as follows.

Euclid didn’t define “straight line” at all. His focus was on the overall deductive structure of geometry, and for this purpose the definition of “straight line” is essentially irrelevant, as indeed shown by the fact that the utterly useless Definition 4 is never actually used anywhere in the Elements.

Archimedes agreed with Euclid as far as the Elements were concerned, but in the course of his further researches he found himself needing the assumption that among all lines or curves with the same endpoints the straight line has the minimum length. He therefore stated this property of a straight line as a postulate in the context where it was needed.

The Hellenistic era, which included Euclid and Archimedes, was one of superb intellectual quality. Unfortunately it did not last forever.

Heron of Alexandria lived about 300 years later. This was a much dumber time. Fewer people were capable of appreciating the great accomplishments of the Hellenistic era. But Heron was one of the best of his generation. He could glimpse some of the greatness of the past and tried his best to revive it.

To this end Heron tried to make Euclid’s Elements more accessible to a less sophisticated audience who didn’t have the background knowledge and understanding that Euclid’s original readers would have had. He therefore wrote commentaries on Euclid, trying to explain the meaning of the text.

To these new, dumber readers of geometry, it was necessary to explain for example what a straight line is. Heron realised that Archimedes’s postulate captured well the essence of a straight line. However, it is not itself suitable as a definition, because a line should have the property of being the shortest distance between any two of its points, not be phrased in terms of only two fixed points, as in Archimedes’s postulate.

Heron therefore explained that “a straight line is [a line] which, uniformly in respect to [all] its points, lies upright and stretched to the utmost towards the ends, such that, given two points, it is the shortest of the lines having them as ends.” The phrase “uniformly …” obviously refers to the universality of the shortest-distance property applying to any two points on the line.

Now fast-forward another 300 years. Intellectual quality has now plunged deeper still. Geometry is in the hands of rank fools. Euclid’s Elements, which was once written for connoisseurs of mathematical subtlety, is now used by schoolboys who rarely get past Book I and “learn” that only by mindless rote and memorisation. A time-tested way (still in widespread use today) to “teach” advanced material to students who do not have the capacity to actually understand anything is to have them blindly memorise a bunch of definitions of terms.

In this context, therefore, there is a need for an edition of the Elements which includes many definitions of basic terms, which must be short and memorisable, and which don’t need to make mathematical sense.

In this era of third-rate minds, some compiler set out to put together an edition of the Elements that would satisfy these conditions. Heron’s commentary on the Elements is appealing in this context since it affords opportunities to focus on trivial verbiage instead of hard mathematics. But Heron’s description of a straight line is still too complicated. It’s too long to memorise as a “soundbite” and the mathematical point it makes is moderately sophisticated.

The compiler therefore makes the decision to simply cut Heron’s description off after the bit about “uniformly in respect to [all] its points.” This solves all his problems in one fell swoop. The only drawback is that the “definition” becomes utter and complete nonsense, but since the whole purpose of it is nothing but blind memorisation anyway this doesn’t matter anymore.

This is how the ridiculous Definition 4 — a mutilated vestige of what was once a very good definition — ended up in “Euclid’s” Elements.

I have included excerpts from Russo’s paper in my History of Mathematics Reader. There you can find the above argument in his own words.

]]>I wrote these notes to supplement a conventional book, because in my opinion typical intro to proofs courses are fundamentally questionable in their very nature. The premise of such a course is rather like that of My Fair Lady: uncivilised students must be taught to “talk the talk.” It’s the dress code and table etiquette of mathematics. In their fanatical devotion to the clinical, sterilised, Bourbaki way of doing mathematics, these courses have forgotten that precision elocution is pointless unless you have something to say. These courses deem actual mathematical content “too messy” and instead feed the student only fake theorems specifically concocted for the sake of being amenable to the desired mould of what a mathematical proof should look like.

Consider these notes, then, my pauper’s rebellion. Down with haut bourgeoisie snobbery for snobbery’s sake, down with fake perfume and powdered wigs. Let us have the courage to tackle real mathematics as it occurs in nature. Let us put meaning and purpose and exciting ideas first, and let us accept the airs and graces of the mathematical aristocracy only after they have proved their worth in this enterprise.

]]>I maintain that it is crucial to keep the mathematician’s point of view alive in historical scholarship. Technical acumen and an empathic sense of how a mathematician thinks are essential tools for understanding past mathematics. And this is best done by one who lives and breathes mathematics on a daily basis in the classroom, at the colloquium, and by the coffee machine — in short, from within a department of mathematics.

Fields-medalist David Mumford put it well when relating his “personal experience reading Archimedes for the first time”: “after getting past his specific words and the idiosyncrasies of the mathematical culture he worked in, I felt an amazing certainty that I could follow his thought process. I knew how my mathematical contemporaries reasoned and his whole way of doing math fit hand-in-glove with my own experience. I was reconstructing a rich picture of Archimedes based on my prior. Here he was working out a Riemann sum for an integral, here he was making the irritating estimates needed to establish convergence. I am aware that historians would say I am not reading him for what he says but am distorting his words using my modern understanding of math. I cannot disprove this but I disagree. I take math to be a fixed set of problems and results, independent of culture just as metallurgy is a fixed set of facts that can be used to analyze ancient swords. When, in the same situation, I read in his manuscript things that people would write today (adjusting for notation), I feel justified in believing I can hear him ‘speak’.”

This way of doing history has widespread resonance in the global mathematical community. All mathematicians know the feeling of struggling to understand a mathematical work until it “clicks” and one feels certain that one has experienced the same idea as the author, regardless of whether he be centuries or millennia removed from us. Those of us who approach mathematical texts in this way know not to pay too much attention to superficial aspects of the presentation: scribbles of various kinds are merely imperfect representations of the author’s thought, whereas the digested “aha” insights we reach when we understand it are its true content. Mumford is right that modern historians, by contrast, are trained to categorically reject such a “gut feeling” approach and stick slavishly to the exact written word as if it were a veritable alien communiqué for which no concordance with our own ways of thinking may be assumed.

The new historiography has greatly advanced the field by offering more specialised perspectives than mathematicians alone ever could, such as histories deeply informed by broader social context and meticulous work on sources and editions according to the highest standards of textual critical apparatus. But amidst the zeal to exploit these new frontiers the field has been left with a leadership vacuum in its traditional core dominion. The time is ripe for a resurgence of the mathematician’s perspective, whose cross-fertilisation with modern developments will bring great fruits.

Here in Utrecht we keep alive this mathematical tradition of historical scholarship. We have a legacy of generations of quality history of mathematics being done in a Mathematical Institute that commands the highest international respect. We are widely recognised as the natural heirs of this way of doing history. It is not for nothing that Jan Hogendijk was awarded the European Mathematical Society’s inaugural Otto Neugebauer Prize, epitomising our continuity with the Göttingen mathematical tradition, while Henk Bos was awarded the Kenneth O. May Prize, the highest honour of the International Commission for the History of Mathematics.

I have taken up the role of torch-bearer of this movement in both words and deeds. I offered a big-picture vision for its enduring relevance and importance in a programmatic paper on the historiography of mathematics, and my more specialised works instantiate these ideals. A notable example is my paper reviving and defending the geometrical algebra interpretation of the history of Greek geometry: an issue where the battle lines have traditionally run along departmental divisions, and the older interpretations being advanced by mathematicians like van der Waerden and Weil have been singled out for criticism as emblematic of the dangers of the mathematically-oriented approach to history. With no mathematicians forthcoming anymore to challenge them, the humanistic historians who dominate the field today had been lulled into a consensus, to the detriment of the vitality of our field. The same dynamic is at play in many other cases as well, which is why the mathematician’s point of view has much to offer, not only in terms of subject-matter insights, but also for stimulating diverse and critical thought in the field.

My first publication, on the isoperimetric problem, symbolises how my point of origin is mathematics itself. Although it is a work of history in that it gives an exhaustive survey of historical solutions of the problem, it is clearly driven by a mathematician’s delight at beautiful proofs drawn from wide-ranging fields of mathematics, such as complex analysis and integral geometry.

My conception of the history of mathematics as being first and foremost about the development of mathematical ideas gives my work educational and expository appeal for a broad mathematical audience. My work on the history of the calculus, for instance, is fertile soil in this regard, and I have made the most of this in my free calculus textbook and several papers making classroom-relevant aspects of my research accessible to a wide readership.

This grounding in the mathematician’s point of view serves me well in my more historical work, where an intuitive sense of what makes sense mathematically often leads me to different interpretations than those who are guided more by contextual considerations external to the mathematical argument itself. I have taken on historians and philosophers along such lines for instance in my Copernicus paper and my paper on Leibniz’s early work on the foundations of the calculus, to name but two prominent examples.

Thinking like a mathematician also gives me a perspective on broader issues missed by historians and philosophers who keep technical mathematics at arm’s length. For example, in my dissertation I studied Leibniz and argued that from the corpus of his technical mathematical works there emerges a clear and unmistakeable picture of his conception of the purpose and method of geometry. This has wide-ranging implications for understanding the scientific and philosophical thought of that era generally, yet this perspective has been neglected since it is expressed “only” implicitly in the mathematical works. But as Albert Einstein said: “If you wish to learn from the theoretical physicist anything about the methods he uses, I would give you the following advice: Don’t listen to his words, examine his achievements.”

Mathematicians conversant with the history of science are also much needed to analyse technical issues. My paper on Copernicus is a case in point. In this article I refute an argument due to Swerdlow that has been considered crucial for over forty years and has been widely cited as decisive by historians who had not themselves worked through its technical mathematical basis.

On the other hand, mathematical understanding is not all it takes to do history of mathematics. Mathematicians who turn to history without background and training in this field often make grave errors of their own. I expose and refute many such errors of anachronism in my dissertation and elsewhere. One example is my paper on what is often called Leibniz’s proof of the fundamental theorem of calculus but which is actually nothing of the sort. The notion that this is Leibniz’s proof of this theorem is widely repeated in numerous sources. It is a notion that seems very plausible to anachronistic eyes looking only at a short piece of Leibniz in isolation, but in reality it is simply false, as becomes clear when the work is studied in its proper context. Another example is my paper rehabilitating Jakob Steiner’s geometry from anachronistic misjudgements. To clear up these kinds of things the field desperately needs proper professional expertise in both history and mathematics.

A key theme emerging from my dissertation, which I intend to build on in future work, is the influence of classical mathematics on general scientific and philosophical thought in the early modern period, which was much more comprehensive than recognised today. It was a widespread conviction at the time that if you seek truth, you must do what the geometers did; you must replicate their method and extend it to other branches of learning and philosophy. Descartes’s Discours de la méthode (1637) is explicitly written for this very purpose; indeed this famous manifesto on the method of doubting everything clearly proclaims that “I did not doubt” that “only mathematicians” had struck upon the right way of reasoning. Likewise Hobbes writes in his Leviathan (1651) that “geometry is the only science that it hath pleased God hitherto to bestow on mankind,” and proceeds to expressly fashion his philosophy in its image. Spinoza’s Ethica (1677) declares in its very title that its is “ordine geometrico demonstrata.” Newton opens his Principia (1687) with a preface outlining what “the glory of Geometry” consists in, in order to use its example to justify his innovative scientific methodology.

But what exactly did these authors mean when they spoke of “the geometrical method”? The complexities of this question are poorly understood by scholars and historians today. The 17th-century thinkers who invoked geometry were not referring to some superficial idea of geometry as conveyed by Plato or Aristotle. They were referring to the rich picture of the geometrical method that emerges from a thorough study of technical corpus of Greek geometry, as conveyed by advanced technical writers such as Pappus. Indeed they frequently refer to this technical tradition even in works that go well beyond geometry itself: Descartes cites Pappus in his Discours; Hobbes does the same in his Elements of Philosophy; Newton cites Pappus even in the very first sentence of his Principia. These authors were thoroughly versed in the technical Greek tradition, as their mathematical works show. By citing Pappus and other technical Greek material they are signalling very clearly that when they call upon geometry they mean not some simplistic idea of axiomatic-deductive method but a rich methodology only conveyed implicitly in the finest ancient works of advanced geometry.

Unfortunately modern scholars do not share these 17th-century thinkers’ excellent technical mastery of advanced classical geometry. Consequently, current scholarship has failed to appreciate the extent to which conceptions of the geometrical method permeates 17th-century thought. One indication of this is that the crucial Book 4 of Pappus’s Collection was translated into English for the first time only in 2010. Even more deplorably, Leibniz’s published mathematical works have never been translated into English at all, even though their crucial importance is universally acknowledged. Meanwhile, any philosophical treatise of even a fraction of the importance of these works has invariably been translated multiple times, betraying the skewed and anti-mathematical emphasis of modern scholarship.

Much of the richness and impact of the mathematical perspective has therefore been missed by modern scholars since it is not spelled out in philosophical prose, neither in Greek nor in early modern times. But mathematics speak loud and clear to anyone who cares to listen, and anyone who was serious about philosophy in those eras was obviously expected to know their geometry — much in the spirit of the famous inscription above the entrance to Plato’s academy. To understand 17th-century thinkers it is time for us to start taking their appeals to geometry seriously and recognise the full scope of the rich methodological picture they drew from advanced Greek geometry.

]]>This proves that there is gender bias in academic science hiring, or so we are supposed to conclude.

My concern is this: Many faculty members want studies to prove that there is gender bias. It fits their own political and ideological beliefs. They are happy when they see studies prove this. They like to refer to such studies. I know because I follow them on Twitter.

This raises the question: Did the faculty members in the study answer truthfully, or did they “second guess” the purpose of the study and submit whatever answer would produce their own preferred outcome? They may indeed have thought to themselves: “Although I’m not biased, I am convinced that a lot of my colleagues are, so I better answer as if I was too, so that attention is drawn to this important problem and progressive measures can be taken.”

Of course the faculty members knew they were being studied and of course they had no actual stake in their replies, unlike when they’re doing actual hiring. And if we look at the prompt the faculty evaluators received, the purpose of the study is quite transparent. So they had no incentive to be truthful, but some incentive to ensure the study produced the results they favoured.

Which hypothesis is right, mine or the authors’? We could try to test it by looking at gender differences among the evaluators. If the authors are right, and there is unfair discrimination against women due to bias and prejudice, one might expect this bias to be stronger among male evaluators, since women who are themselves established scientists might be expected to be open to promising female students. If my hypothesis is the operative one, on the other hand, one might expect the opposite; that is, that female evaluators would be even more biased than men, since they arguably have a greater stake in “gaming the study” to make sure it shows gender bias. The latter is in fact what happened, though the difference is not great.

Meanwhile, if one looks at real data instead of contrived experiments, “actual hiring shows female Ph.D.s are disproportionately … more likely to be hired” (source, page 5365). We see the same thing by looking at the official data from the American Mathematical Society regarding hiring and PhDs in the mathematical sciences in the United States. In the latest data, women constitute 31% of PhDs awarded and 32% of positions filled. However, women constitute only 28% among PhD recipients who are U.S. citizens. This is perhaps the more relevant ratio since, among those who do their doctorate in the U.S., those who are U.S. citizens are surely significantly more inclined to aim for a job in U.S. academia. It therefore seems that hiring institutions have a preference for women, as they indeed often state openly.

]]>That is what I and others suspect. An article in the latest issue of Isis claims to disprove us:

> This essay seeks to explain the most glaring error in Ptolemy’s geography: the greatly exaggerated longitudinal extent of the known world as shown on his map. The main focus is on a recent hypothesis that attributes all responsibility for this error to Ptolemy’s adoption of the wrong value for the circumference of the Earth. This explanation has challenging implications for our understanding of ancient geography: it presupposes that before Ptolemy there had been a tradition of high-accuracy geodesy and cartography based on Eratosthenes’ measurement of the Earth. The essay argues that this hypothesis does not stand up to scrutiny. The story proves to be much more complex than can be accounted for by a single-factor explanation. A more careful analysis of the evidence allows us to assess the individual contribution to Ptolemy’s error made by each character in this story: Eratosthenes, Ptolemy, ancient surveyors, and others. As a result, a more balanced and well-founded assessment is offered: Ptolemy’s reputation is rehabilitated in part, and the delusion of high-accuracy ancient cartography is dispelled. (687)

As an aside, here’s a pro tip for academic novices: If you have little of substance to offer, make sure to lay it on thick with self-congratulatory posturing about how your work is supposedly based on “more careful analysis” and “more balanced and well-founded assessment,” showing everything to be “much more complex” than others think. After all, who would dare disagree with someone who is so careful and balanced and ever so sensitive to complexities?

Contrary to his smug proclamations, the author’s case is flimsy. For one thing he immediately admits that, indeed, Ptolemy’s error can be solved in a single stroke by recalculating his map with Eratosthenes’s excellent value for the circumference of the earth, yielding an “uncannily accurate” map (692). But he alleges that this is a mere “coincidence” (691). What is his evidence for this?

His main argument concerns the value of the length unit “stade.” A “‘short’ stade … is implied by the high accuracy of Eratosthenes’ value for the circumference” (694), for which there is no explicit evidence in the record, unlike a “long” or “common” value for a stade that is mentioned in some sources. Of course this is not strange since our hypothesis is based on precisely the claim that many excellent sources are lost. But there is in fact implicit evidence for the short stade, as the author himself in effect admits:

> The main argument for the “short” stade is based on comparison between ancient and modern distances: those measured on a modern map are divided by their ancient counterparts in stades, giving the length of an average stade. This comparison has been undertaken repeatedly, … and invariably the average stade comes out to be much closer to the “short” value of 157.5 m than to the “common” one of 185 m. On this basis, many researchers suggest that for practical purposes ancient surveyors used a special short stade, one never directly attested in extant sources. … This stade is often termed the “itinerary stade.” This result might have been regarded as a brilliant confirmation of the “short stade hypothesis” were it not for one “but”: strangely enough, in comparing ancient and modern distances, a crucial factor has been lost sight of——namely, “measurement error.” The proponents of the “itinerary stade” proceed from a tacit assumption that ancient distances were measured almost as accurately as modern ones. However, this cannot be true, for two main reasons. First, with rare exceptions, there is no indication that distances given by ancient sources were actual measurements on the ground, rather than rough estimates deduced, for example, from the duration and the average speed of travel. Second, and most important, even when ancient distances do derive from actual and quite accurate measurements, they were certainly measured not as a crow flies but including all the twists and turns of the route. (701-702)

The author’s counterargument is very weak. It merely asserts what he is trying to prove, namely that distance measurements in the time of Eratosthenes would have been poor and naive. The obvious reply, which the author does not consider, is that it is very possible that Eratosthenes and others used much more sophisticated mathematical methods such as triangulation. Instead the author expects us to believe that the generation that gave us the geometry of Archimedes was too stupid to account for “twists and turns of the route” when estimating distances for geographical purposes. There is also some reason to think that Eratosthenes defined a new stade based on his earth measurement——another possibility ignored by the author.

> We can apply a simple test: the same comparative approach may be used to determine the length of the Roman mile. Since it has been firmly established as equal to 1,480 m, such comparison will yield the average accuracy of Roman distance measurements. … For this purpose I have examined more than 160 distances given in Pliny’s Natural History. … If these distances are believed to be accurate, then, by the same logic that has led us to the “itinerary stade,” we have either to conclude that Pliny’s mile was equal to circa 1,190 m, which is impossible, or to assume that Roman measurements of distances were much less accurate than Greek ones, which is hard to believe. Otherwise, we have to conclude that Pliny’s distances were overestimated by 25 percent on average——that is, by approximately the same amount that Eratosthenes’ and Strabo’s distances must have been if they were expressed in the “common” stades of 185 m. (702-703)

Apparently it is “hard to believe” that the greatest generation of geometers who ever lived were better at measuring distances than a second-rate encyclopedist from a civilisation that never contributed an iota to mathematics during its entire lifespan.

The author also maintains that “the hypothesis … contains numerous logical fallacies” (693), namely:

> The match between the recalculated Ptolemy coordinates and the modern ones, however close, does not in itself mean that the longitudinal distortion in Ptolemy’s map was due entirely to a single cause—namely, the wrong value for the Earth’s circumference. Nor does it warrant discarding two other possible causes: the exaggeration of distances and the lengthening of the stade. Another crucial point to stress is that Ptolemy’s recalculated map turns out to be accurate only in terms of spherical coordinates. This does not mean that the actual distance measurements underlying it and Eratosthenes’ value for the Earth’s circumference were equally accurate. (693)

Calling these things “logical fallacies” is just ridiculous. Obviously no one ever claimed that these things were logical implications, only that they were the likeliest explanations. The author does however commit a logical fallacy himself when he draws the non sequitur that our hypothesis is a “delusion” from the fact that Ptolemy’s map errors can be explained by other means.

Altogether, the author’s preposterously overblown claims exaggerate his case by a mile and then some. I am surprised that such wording was allowed to stand in a respectable journal. The author has not taken our hypothesis seriously, let alone given a “careful” and “well-founded” demonstration that it “proves” to be a “delusion.”

]]>Our coach wanted to make us master passers so that we could play tiki-taka. So we spent hours and hours every week on one specific drill: playing football without goals. Without the “distraction” of trying to score goals, we could focus purely on passing and possession play. That was the idea.

In reality, what happened is that we got sick and tired of this boring drill that took all the fun and excitement out of the game and turned it into a pointless drudge. Instead of making us master passers it made us disinterested slackers who didn’t see any reason to put in our best effort.

Unfortunately much curriculum planning in mathematics is based on the same hare-brained logic. Again and again we see the same pattern: in order to do B you need A, but teaching A and B together in one course would be too much, so A is detached and drilled at length in a prerequisite course.

The problem is that B is the only reason anyone is interested in A in the first place, so now you are teaching an entire course on a topic A which serves no purpose whatsoever in and of itself. By severing A from B you are guaranteeing that your course has zero intrinsic motivation. You are not helping students by giving them “a good foundation in A” before moving on to B. Rather you are obliterating the meaning and purpose of mathematics and forcing your students to approach it as an empty chore.

A notable example is the plague of “intro to proofs” courses where the pedantry and mechanics of proof writing is detached from any context where these skills serve an actual purpose. In the same way we detach integration rules from their purpose, which is solving differential equations, and we detach rings and ideals from their purpose, which is number theory, and so on.

The fallacy is one of short-sighted, non-organic thinking. “I’m a teacher of B. I find that my students are lacking in A. Let’s solve my problem by having them do tons of A in a prerequisite course.” This is B-centered thinking that zooms in on one particular issue and wreaks havoc with the cohesion and integrity of the curriculum as a whole in order to “fix” it.

The A-course is rarely a success, because it’s hard to learn something well without knowing what it’s for, and it’s hard to stay motivated and excited when there is no purpose to what one is doing. But, unfortunately, the same blinders that led to this course in the first place also means that its proponents are blind to its failures. The disastrous outcomes only leads them to double-down on their short-sighted scheme. “Now they have a whole course on A and they still don’t get it! Obviously this proves how essential it is to drill and drill and drill A before going on to B.” Thus, as with medieval bloodletting, the failure of the treatment is taken as evidence that more of it is needed.

The naiveté is the same one that led to the myth that vitamin C cures and prevents the common cold. Compare: “Vitamin C is essential to the immune system. Therefore, taking tons of vitamin C pills will keep us from ever catching a cold.” “In higher mathematics it is essential to understand concepts and techniques relating to writing proofs. Therefore, making all our students cram these skills in a dedicated course detached from any content will have them flying through later courses without impediment.”

Just as a well-rounded diet gives us all the vitamins we need in a natural way, so also a well-rounded mathematics curriculum automatically incorporates any necessary material in an organic and natural manner. Unfortunately mathematical curriculum planners do not believe in teaching mathematics they way it grows naturally and organically. Instead they would rather play the role of a hubristic doctor in a sci-fi dystopia who thinks he can “improve” on nature by replacing all organic foods with artificial capsules.

]]>At the level of lecture discussion it makes students eager to attend class and attentively follow your reasoning since this will give them “free answers.” For instance, you are introducing multivariable functions and want to convey the idea that a great way of analysing them are by their cross-sections with horizontal and vertical planes. So you pose this problem and work out at least part of it on the board. Students are primed to appreciate your point since it answers a direct need of theirs. And by stopping short of giving away the final answer you force students to pay attention to the underlying method since they will need it to complete the problem.

You also want to create substantive students discussions in pairs or small groups. To this end it is nice to have conceptual questions that allow for multiple reasonable standpoints. An example is this “paradox” on how one integral can have several “different” answers. You can ask students to work in pairs, one checking one method, the other the other, and then try to convince each other that they are right. Heated discussion ensues, ultimately leading to some reflection on the meaning of the answer––a lesson that cannot be taught often enough.

Full-class discussion or group work is especially stimulating for problems that involve more open-ended conceptual thinking, interpretation, and reflection, rather than single-track computation. This and this are examples that work very well.

Exam-oriented thinking is a plague that prevents students from learning and teachers from teaching. Many a traditional course shoots itself in the foot already before it starts by being structured around the idea of a final exam consisting of a fixed number of highly standardised, computational problems. This corrupts the teacher, who in this mindset asks questions that are “good practice for the exam” instead of asking what lines of inquiry are best for actually learning mathematics in a meaningful way. It also corrupts the students, who quickly conclude that rote computations is all they “really need to know” and hence zone out at any attempts by the teacher to explain underlying reasoning.

The worksheet model frees us from this tyranny. Teachers are no longer crippled by the straightjacket of having to ask only “exam-type” questions, and students find that a large part of their grade comes from a variety of questions involving genuine thought rather than a restrictive set of archetype calculations. We are free to pursue interesting “one-off” problems that make you think, instead of having to discard them as “unexaminable” just because they are not replicable ad infinitum with different formulas and numbers. Since a large part of the graded work takes part in a formative, discussion-oriented setting, we are not constrained to ask self-contained, unambiguous questions of a fixed level of difficulty, as a traditional high-stakes exam requires. Instead of designing our course with the exam in mind, we can design it with mathematical thinking and learning in mind.

Here are a number of examples of problems in this vein. These questions make you think about the material from various vantage points: the “why” behind certain formulas; visual, intuitive, qualitative interpretation of what you are doing; and at the end even some “cultural interest” connections.

These types of problems can be incorporated in a class in various ways depending on the format of the class. In a larger lecture setting they can be used as the basis for the lecture, in which case the students have an extra incentive to follow along since they need the answers. They can also be used to break up the lecture for a few minutes of reflection and discussion among students. In some settings the boundary between class discussion material and exercise assignments need not be sharp: one can assign a number of these kinds of problems and let student requests determine which get discussed in class and which are left as homework. A small class could even be entirely student-driven thanks to the structure that a well-thought-out sequence of questions affords.

The worksheet format is also suited for longer “story” problems such as these, which allow us to work out substantial problems from first principles, such as setting up a differential equation before solving it. These problems too can be incorporated in various ways, from making homework more interesting to making extended discussions of applications viable in a lecture (since it is now truly part of the [graded] course content rather than “enrichment” material as in a traditional course). In classes of moderate size one can also assign such problems to individual students to present to the class. Since everyone needs to enter the answers in the online system, it’s everyone’s points on the line and the class will listen attentively and try to catch any errors. Assigning such problems based on individual student interests is also a way of drawing on existing expertise and connecting the course with other parts of their study program.

The worksheet format also allows us to break down the traditional division between “theory” and “practice.” Again, this very unfortunate and harmful aspect of conventional teaching is in large part a product of examination needs: having students run through computation problems that can be multiplied at will is very convenient for examination purposes, whereas asking for explanations and conceptual reasoning is very messy. But with the WeBWorK worksheet model we can make the latter realistically implementable.

One useful way of getting students engaged with proof-oriented thinking is asking them to evaluate purported proofs, like this. In a traditional course, the teacher and textbook may model many examples of good proofs, but students are seldom confronted with erroneous reasoning. Therefore they often come to associate proofs more with superficial aspects such as phraseology than with actual content. Reasoning-evaluation problems like this forces them to look deeper and cultivates a healthy critical mindset for reading proofs in general.

Here’s another theory example: I introduce the fundamental theorem of calculus, give an intuitive proof, and then ask some follow-up questions that should be easy if you followed the proof, but often prove not so easy since students have so little experience with this type of mathematical reasoning––which is exactly why we need these kinds of questions. In a classroom I might go through the given proof on half a board and then ask the students to complete the proof of the follow-up case in parallel on the other half of the board, mimicking the steps of the first proof with minor adjustments as needed. I might ask for a volunteer student to come to the board to carry this out with the help of suggestions from the class and maybe some leading questions from me if needed. If I refuse to do it any other way (i.e., explain it myself) the students will be pushed to go along with it: after all, if they don’t they will have to do this as a homework problem, which will be much harder and more work.

The second follow-up problem asks for much greater conceptual insight. I marked it with a dagger , signalling that it is a challenge problem. I like to include some problems like this for ambitious students to puzzle about, while others do not need to worry about them since the grading scale will reflect that these kinds of problems are for those aspiring to the very highest grades.

Much other theoretical material will be of this “ type”: it is not at all required for average students in introductory courses, but on the other hand you want to encourage students with substantial mathematical aspirations to start reflecting on more theoretical aspects as early as possible. One way of doing this is to include some of the more theoretical material as -marked readings with various interspersed comprehension questions. Here are a number of examples of how this can be implemented in WeBWorK. With such an “interactive textbook” type of presentation, ambitious students are rewarded for reading the theory and given a “training wheels” guide to reflective reading of mathematical texts. These are some examples.

All of the above problems I have written myself. WeBWorK comes with a large library of standard practice problems (which I also use), but to reach all the goals I highlighted above we must go beyond this restricted notion of what an online homework system is for.

]]>I always knew I wanted to study mathematics and I followed a full-time mathematical course load from my first day to my last back home at Stockholm University. But I also craved more. I had a broad intellectual appetite. So in addition to my mathematics courses I took art history, philosophy, and so on. My first semester I took an evening course on the history of classical music. The lecturer would weave a grand narrative and ever so often sit down at the piano and play some select bars to illustrate a point. This remained with me as an image of how I longed to experience mathematics: the technical masterpieces and their broader context each heightening the appreciation for the other.

My mathematical courses, meanwhile, were plain-vanilla technical courses. I could do the exercises and pass the exams easily enough, but I was frustrated that bigger why-questions were not being addressed. I could play the formal game of definitions and proofs, but I wanted to know why anyone would want to define, say, principal and maximal ideals in the first place, and what the purpose of these theorems about them was supposed to be. So I went to the library to find out. And that is what brought me to the history of mathematics, for it was only there that I could find the answers to my why-questions. It was only by studying the history of the subject that I realised, for instance, that all the abstract gobbledygook about rings and ideals that I was being fed in my courses was really just number theory with all the interesting applications left out. Thus my interest in the history of mathematics was always a means to an end; it was always subordinated to teaching and mathematical understanding. And, fundamentally, it still is. In this way the problem of teaching has remained the root of all my scholarly work ever since.

It was no accident, of course, that I found the answers I was looking for in the history books. Mathematicians do not make up arbitrary definitions and start proving theorems about them aimlessly. They work on interesting and natural problems, and introduce new concepts only when they serve a credible purpose in this pursuit. That is how history works, and I think a good argument can be made that it is how teaching should work also. This is why, to my mind, mathematics and its history—and thus in my case teaching and research—form an organic whole.

It was in this mindset that I finished up my masters degree, so naturally I was more excited about teaching than going on to do a Ph.D. right away. I got a two-year position teaching at Marlboro College in Vermont, which was a lovely experience. I had long perceived my affinity with the liberal arts ideal, and I at once felt at home in this environment. With these people I debated Kant in the dining hall, traced conics in the snow, cast horoscopes according to ancient principles, and, most of all, taught mathematics in a thinking rather than robotic fashion. The small class sizes and absence of centrally fixed curricula gave me ample opportunity to translate my long-held ideals into teaching practice. From this moment, if not sooner, I knew that this was a passion I wanted to make a career of.

But first I had to get my Ph.D., of course, and I was still more excited about understanding mathematics historically and contextually than about joining the frontiers of research. So I went to the LSE in London to do a second masters degree in philosophy and history of science. This afforded me the opportunity to study in their own right the kinds of big-picture questions that had kept cropping up in my mind as a reflective mathematics student. My view of mathematics was greatly enriched by these perspectives, but doing such a Ph.D. would have taken me too far from home. I was now a full-fledged humanist who could speak to philosophers and historians like a native, but in my heart I was still a mathematician.

So I went back to the root, the problem of teaching that had started it all. I enrolled in a Ph.D. program in educational mathematics at the University of Northern Colorado. My two years spent there made me a native in the world of mathematics education research as well. I studied what research has shown about learning and cognition, I was inducted into social science research methodology, and I saw reform teaching of all kinds in action at the hands of its most enthusiastic advocates. This was all very valuable, and I was always prepared to devote my life to mathematics education. But ultimately I felt that the current paradigm of mathematics education research was too intellectually restrictive for my tastes. So I left.

From there I went to my current position at Utrecht University, where I completed my Ph.D. in the history of mathematics. This has been the perfect Ph.D. for me. It has brought together all the themes that drove me during my searching years. It has allowed me to be an intellectual and a philosopher, a humanist and a reader of books, while remaining with both feet on solid mathematical ground. My path may not always have seemed the most direct but in retrospect its logic is clear: in writing my thesis on the history of the calculus I have gone full circle back to my early university days and made it my daily bread to explore precisely that elusive richness that I always felt was hiding behind the austere façade of the modern mathematical curriculum.

]]>If, like me, you find the face-value implications of this evidence rather depressing, you may want to look for some way of explaining them away. That’s what I tried to do, and here is what I found.

First we may ask ourselves: Do the test scores really measure quality of learning? Maybe direct instruction is a good way of “teaching to the test,” leading to good scores on artificial standardised tests that do not really measure what we really aim for in education. It seems to me that this is not a convincing answer in this case. Judging by the published samples of actual test items used, the questions seem sound and certainly like the kind of thing one would want students to know. They don’t seem at all focussed on “cram”-type knowledge, like standardised vocabulary and rote calculations. If anything, they seem like exactly the kinds of questions advocates of enquiry-based learning would prefer. So we got beaten at our own game, as it were.

Next we can ask ourselves: How does PISA define and measure teacher-directed and enquiry-based learning anyway? Again, this seems to have been done in a very reasonable way that does not leave any room for invalidating the findings on methodological grounds.

To measure how teacher-directed a class was, “PISA asked students how frequently … the following events happen in their science lessons: The teacher explains scientific ideas; A whole class discussion takes place with the teacher; The teacher discusses our questions; and The teacher demonstrates an idea.” (63)

Enquiry-based was measured instead by the following statements: “Students are given opportunities to explain their ideas”; “Students spend time in the laboratory doing practical experiments”; “Students are required to argue about science questions”; “Students are asked to draw conclusions from an experiment they have conducted”; “The teacher explains how a science idea can be applied to a number of different phenomena”; “Students are allowed to design their own experiments”; “There is a class debate about investigations”; “The teacher clearly explains the relevance of science concepts to our lives”; and “Students are asked to do an investigation to test ideas.” (69)

This seems quite in order. In fact, once the analysis is split into these sub-statements, the case for direct instruction is even much stronger than the above table suggests. For it is the most direct-instruction-y part of each group that works best, and the most equiry-y part that is least successful, as we see in the tables on pages 65 and 73.

We may also ask: what is “adaptive instruction”? This sounds like something reformers would approve of, and it is highly correlated with success. However, once we look into the details it is not so uplifting: in a nutshell, it seems “adaptive instruction” may in practice be more like the traditionalist tricks of “teaching to the test” and “dumbing it down,” for the statements PISA used for this measure were: “The teacher adapts the lesson to my class’s needs and knowledge”; “The teacher provides individual help when a student has difficulties understanding a topic or task”; and “The teacher changes the structure of the lesson on a topic that most students find difficult to understand”. (66)

Finally, we must ask ourselves whether the results highlighted in the table are false correlations somehow, or at least not causations. After all, we see in the table that “after-school study time” is strongly associated with negative scores. Surely this is a matter of correlation rather than causation: students who spend a lot of time studying outside of class are weaker on average, but this is not the reason they are weaker, one would hope. Could something like this hold for enquiry-based learning too? This would have to mean that weaker students are exposed to enquiry-based learning to a greater extent. Is there any evidence for this? I’m not sure. The study includes tables on which countries do most direct instruction (64) and which do most enquiry-based learning (72). It turns out that leaders in enquiry-based learning are not the self-proclaimed avant-garde in the rich West but rather the Dominican Republic, Peru, Jordan, Lebanon, Algeria, etc. The correlations control for other variables so this does not appear to be the whole explanation, unless these controls operate only on a within-country level, which seems a possibility (e.g., socio-economic profile seems to be defined relative to the country, not relative to the world). So it seems possible, but I can’t tell how likely, that the disastrous results for enquiry-based learning are due to between-country differences rather than within-country differences. If so, that would undermine the face-value implications of the data, since within-country differences would be much more relevant for pedagogical decision making (as it would better measure what happens when the two teaching methods are applied to comparable students).

More confusingly, these tables actually seem to show that the dichotomy of direct instruction versus enquiry-based learning is a false one according to this data. Because many countries do either lots of both or very little of each, which makes no sense if we are picturing it as and either-or situation. Korea, for instance, are dead last by a wide margin in the use of teacher-directed instruction, yet they are somehow also second to last on enquiry-based learning. What on earth are these Koreans doing in their classes then, if it’s neither one nor the other? Many other countries exhibit similarly paradoxical results. This suggests that this data is poorly suited for making judgements about one teaching style versus the other. For this purpose, it would have been better to have asked the students questions that forced them to pick a point on this continuum instead of asking them about each separately, in a non-exclusive way.

So there are some grounds for casting doubt on the data, but by and large I think we have to admit that this PISA report is very damning evidence for the fashion of the day in educational reform ideology.

]]>The data is as follows (Table 1, page 18):

The final exams were double-blind graded by a third party (48), so they can be considered objective.

In a nutshell: Male students like male teachers much better, even though they do not do any better under them. Female students like male teachers a bit better, even though they do a bit better under female teachers.

Thus: Evaluations are biased in favour of male instructors. “The first main result is that gender biases exist.” (5)

Maybe so, but let’s look at the more fine-grained evaluation components in the table. To which specific aspects of teaching do students attribute their higher ratings of male teachers? Note that the answer is quite unequivocal: men are significantly better on “animation & leadership,” “current issues,” and “intellectual development,” and pretty much equal to women on the rest. Both genders of students largely agree on this. This clear pattern suggests that there are underlying differences in teaching approaches between male and female teachers, not just crude, across-the-board gender bias.

In fact, note that “current issues” and “intellectual development” are things that may be very desirable in an education generally but not directly relevant to a specific course examination. So it could be that the male teachers were better at imparting those more general skills, but no better at course-specific aspects like “organization,” “instructional materials,” etc. This hypothesis would give a rational explanation for virtually the entire table without the need to assume that the students were biased. According to this hypothesis: male and female teachers were equally good at the course-specific aspects; therefore they produced equal outcomes at the final exams; but male teachers were better at the bigger-picture aspects “current issues” and “intellectual development,” which, although not directly relevant to the exam, were nevertheless valued by students.

Even differences by student gender can explained by this hypothesis: male teachers and male students alike have a particular interest in those bigger-picture aspects, while female teachers and students alike are more focussed on the particular course at hand. If so, that would explain why male students gave the male teachers a greater “boost” in evaluations than the female students did.

It seems to me that this hypothesis is a simple and unified way of explaining many nuances of the data that are completely unexplained on the crude hypothesis of across-the-board gender bias.

I do not particularly believe this hypothesis. My point is not that it is true and that there is no gender bias. I think there is a fairly decent case to be made for some gender bias here. My point is only that if we are to work objectively with data then we must have an open mind toward alternative hypotheses such as this, instead of jumping to the gender-bias conclusion with blinders on. Gender bias may be a problem, but so is confirmation bias in favour of predetermined conclusions.

The only main point of the data that my hypothesis does not explain is the evaluation item called “quality of animation & ability to lead.” This seems to me a very stupid item, and I am inclined to dismiss it altogether. First of all, animation and leadership are surely very different things, so they should not be combined in one item. Furthermore, in the appendix where the author supposedly gives the actual questionnaire given to the students, the corresponding questions simply reads: “How do you evaluate your teacher’s class leadership skills?” (61) Is this what students were asked? Then why has the author inserted “quality of animation” in the data table at this item heading?

What is “class leadership” anyway? Does it mean leading discussions? If so it would seem to be related to many of the other items, like “communication skills,” “organization of classes,” “usefulness of feedback,” etc. But men score high on “leadership” but not on these other things, so it seems leadership must mean something else. Could it mean something like being strict and firm with deadlines for example? But then shouldn’t it be related to “clarity of assessment,” which it is not in the data? In sum, the question of “class leadership” seems to me much too vague and poorly formulated to be taken seriously. With such a vague question, it would not be surprising if students, lacking any way to answer it meaningfully, fell back on some stereotype about men being “leaders.” But this would say nothing at all about the course.

The author in fact has her own way of explaining the more fine-grained scores on the student evaluations, which goes as follows:

“The second main result I find is that students rate teachers in different dimensions of teaching according to gender stereotypes of female and male characteristics. … Students give more favorable ratings to women for teaching skills that require a lot of work outside of the classroom, such as the preparation and the organization of the course content, the quality of instructional materials, and the clarity of the assessment criteria. … Male teachers, however, tend to obtain more favorable ratings by both male and female students in less time-consuming dimensions of teaching, such as quality of animation and class leadership skills.” (5)

In my opinion, this explanation doesn’t fit the data at all (and certainly not as well as my hypothesis above). The author is trying to force gender stereotypes into the picture, but her attempt is quite absurd. Why would “clarity of assessment criteria” “require a lot of work outside of the classroom”? On the contrary, a lazy teacher will typically make up simple assessment criteria that are very clear indeed, since this makes life easy for the teacher. And I suppose no one ever works on their “leadership” and “animation” skils, they are just innate? What a strange assumption. And in any case it is very dishonest and deceptive to focus on them and claim they are representative, when, as we clearly see from the data, male teachers also scored much higher on “ability to relate to current issues,” which is clearly something that could cost the teacher a lot of preparation time: it is much easier to just “teach from the book” the same way year after year.

In conclusion, I believe the research literature is biased in favour of the gender bias interpretation. Although, as we have seen, other interpretations are perfectly plausible, the author pushes only the gender bias interpretation. And she pushes it too far when she writes that “male students give much higher scores to male teachers … in all dimensions of teaching” (5), and when she claims that gender stereotypes explain item-by-item variation in the evaluations. Yet these unwarranted conclusions are the takeaways from the study picked up and cited by others.

]]>“Teachers consistently underrate girls’ math skills.” – New York University

“Teachers consistently underrated girls’ math skills, even when boys and girls behaved and performed in similar ways academically.” – PBS

“Perhaps most unsettling is the study’s finding that teachers perceive girls with nearly identical mathematical abilities—and identical behavioral profiles—to be significantly less able than their male counterparts.” – Quarts (retweeted by the MAA)

Judging by these quotes you might think that the study:

1. Measured the mathematical ability of children.

2. Measured their teachers’ perceptions of their mathematical ability.

3. Compared the above two results.

4. Found that teachers rate the ability of girls lower than their actual test performance.

But not so. The researchers did steps 1-3 alright, but they did not find 4. They found the opposite. What the data actually says is that teachers judge ability accurately, by and large. If anything, teachers tend to overestimate the ability of girls. For example, “at the very top of the distribution, teachers rate the math proficiency of girls higher than that of boys — a pattern that sharply contradicts the direct cognitive assessment pattern. That is, whereas the direct assessment finds that only about 33% of students at or above the 99th percentile are female, teachers rate girls to be over 60% of the top students.” (p. 10 of the study)

So an alternative soundbite version of the study could have been: “mathematical ability of girls overrated by teachers.” But good luck getting featured on PBS and retweeted by the MAA saying that. It may be what the data says, but that’s irrelevant. People don’t want to hear that conclusion. Everybody knows that anyone who is “progressive” in education is fighting against rampant gender bias. And if you want to get anywhere in academia (and certainly if you want a sweet faculty gig at New York University like the lead author of the study) then you better toe the “progressive” party line.

The purpose of educational “research” is to reach ideologically desirable conclusions. The purpose of educational “research” is to tell people what they have already decided they want the answer to be. Antiquated notions like critical thought and questioning preconceptions have no place in “progressive” educational “research.”

This being so, the authors of the study of course found themselves in a pickle when the data clearly contradicted The Only Acceptable Truth. How annoying that teachers didn’t play ball and show that clear gender bias that the researchers needed to find in order to conform to the ideologically predetermined outcome of the study!

But the researchers found a clever way of getting around this problem. They also asked the teachers about the behaviour of the students, and naturally the teachers rated girls as better behaved than boys. So, by doing some statistical trickery to shift the baseline accordingly, the researchers managed to make the desired gender bias appear out of nowhere by comparing teachers’ assessment of ability not with actual test scores but with test scores after behaviour scores had been subtracted or factored out.

This puts the sensationalist quotes above in a whole new light. What they seemed to be saying was that girls were consistently underestimated even when they performed at the same level, and furthermore even when they displayed the same behaviour. We are supposed to react: Unbelievable! What more do these girls need to do to be respected and taken seriously?!

But the reality if very different. It’s not: gender bias is rampant, EVEN when girls behave the same way and everything. Rather it’s: there is NO bias against girls at all, UNLESS a dubious “behaviour” score is used to shift the baseline. The behaviour condition does not make the claim of gender bias stronger; it makes it much, much weaker, not to say altogether unsustainable.

Indeed, the problem with this baseline shift is blatantly obvious: How do we know that teacher assessments of behaviour themselves are not massively biased? After all, isn’t “good behaviour” a more nebulous concept than mathematical ability as measured by standard tests? Indeed the researchers had no other measure of “behaviour”: they only had the teachers’ opinions and no way whatsoever to check it objectively. And since the teachers were, if anything, biased in favour of girls on ability, could they not also be biased in favour of girls on behaviour? This could in fact explain the entire effect detected, so that, ironically, the alleged research finding of bias against girls could just as well be due to a bias in favour of girls.

The authors themselves in fact say exactly this in a parenthetical “caveat”: “One caveat to consider is that teachers’ ratings of student behavior might be biased by student gender. For example, if teachers rate girls’ behavior as better than that of equally behaving boys, then this bias would contribute to the gender gap we see in teacher ratings of girls and boys as well as to our findings regarding the underrating of ‘equally’ behaving and equally performing girls and boys. ... The possibility of biased ratings of behavior suggests that caution is warranted in interpreting results.” (pp. 13-14)

But this little “caveat” is buried on page 13 of the actual, peer-reviewed article, and the called-for “caution” is of course nowhere to be found in the sensationalist drivel being peddled by journalists and on Twitter because it caters to the reigning ideology. Which is why when people say “research shows...” one must understand that this means “because of my preconceived beliefs I refuse to consider any other possibility than...”

]]>Outsiders often criticise philosophy, but they shouldn’t because philosophy is hard and outsiders don’t understand it. Just look at science: everyone knows it’s hard and no outsiders are foolish enough to think they can criticise it.

To which I reply:

Should politicians, Wall Street traders, and predatory corporate CEOs be immune from criticism from outsiders on the grounds that their work involves some technicalities that outsiders do not fully comprehend? According to the logic of Humphreys’ terrible argument the answer must be: yes.

A more constructive way of looking at it is that critical thought from a diversity of viewpoints is a good thing. From the fact that no outsiders criticise the hard sciences Humphreys tries to infer that no outsiders should criticise philosophy either. But why not instead make the inference that it would be better if more people criticised the hard sciences too? That would make sense, unless one thinks that scientists and philosophers are infallible and have nothing to learn from outside viewpoints.

In fact, the hidden assumption in Humphreys’ argument speaks volumes about the doctrinal assumptions of analytic philosophy. Analytic philosophers are so infatuated with the sciences and so obsessed with the idea that they are doing the equivalent of hard science that they confuse “X is how it’s done in science” with “X is the right and proper state of affairs.”

A self-rationalised refusal to listen to critiques from anyone one deems an outsider is a recipe for insular and dogmatic thought. Humphreys is openly calling for exactly this. Yet at the same time he is baffled that everyone “from Nobel prize winners to Amazon reviewers” is critical of his field. But instead of putting two and two together he doubles down on insularity and urges philosophers to bury their heads even deeper in the sand.

In the Middle Ages, scholastic Aristotelianism completely dominated the entire field of philosophy. It too was ever so technical, so these philosophers too could run Humphreys’ argument. If the scholastic philosophers had had their way, and no one had been allowed to criticise them who did not know the immense and pedantic scholastic literature inside and out, then we would still be in the Middle Ages today. The right way forward was to overthrow scholastic philosophy from without. Those who finally made watershed progress came at it from another point of view and very often had nothing but contempt for the established philosophies–––in short, they were the equivalent of those pesky “Nobel prize winners and Amazon reviewers” whom Humphreys want to drive out of philosophy.

Or take a contemporary example: religion. Analytic philosophers are generally strongly committed atheists. But of course they generally lack the technical expertise of a professional cleric, so the latter could use Humphreys’ argument against the philosophers: “stop dabbling in fields outside of your expertise.” Evidently in this case analytic philosophers consider it perfectly plausible that an intelligent outsider can have a more correct view of the matter than a well-versed expert. So why could it not be the same with philosophy? Why is it not conceivable that a Nobel prize winner could be able to spot misguided assumptions and systemic problems with the field more clearly than those entrenched in its belief system?

In sum, Humphreys’ argument ultimately rests on the unwarranted assumption that current academic philosophy has reached the end of history, i.e., that it will never again need the kinds of overhauls from without that have characterised virtually all of its greatest advances in the past. Philosophy would be better off if it was open to the possibility of its own fallibility. And this means being open to listening to outsider critiques of the field instead of trying to write them off as a pathology due to some “peculiar psychological attitude” among non-philosophers, as Humphreys does.

]]>> Blåsjö’s argument that Copernicus as a skilled geometer could easily have discovered the Tusi couple leaves unanswered the question why it was not discovered much earlier in Europe by astronomers working with the Ptolemaic model, or by other astronomers in the Arabic tradition before al-Tusi. (72)

It is not strange that earlier astronomers did not discover the Tusi couple if they had no reason to do so and if it served no purpose to them. Copernicus and al-Tusi were trying to eliminate the equant, and in this context the Tusi couple was valuable to them. But if earlier astronomers were not trying to eliminate the equant they had no reason to make this discovery. So if Bala’s argument has any weight it comes only from the following point:

> In fact al-Tusi himself was motivated to make this discovery because he accepted Ibn al-Haytham’s demand nearly two centuries earlier that astronomers should do away with the equant. But the fact that in the Arabic world it required two centuries to accomplish this discovery makes Blåsjö’s claim that any skilled geometer could easily have discovered the theorem extremely implausible. (72)

This would be a good argument if in fact two centuries worth of skilled astronomers worked hard trying to eliminate the equant. But nothing of the sort has been proved. The fact that Ibn al-Haytham said that someone ought to do so doesn’t mean that everyone spent two hundred years trying. “How influential Ibn al-Haytham’s treatise was is not certain” [Sw, 45]. And in any case the conflict between equant and uniform circular motion had already been obvious to everyone since antiquity.

> Blåsjö seems to presume that Copernicus would have independently discovered the Urdi lemma. Again this is extremely implausible since if he had done so he would have mentioned it as a discovery of a new mathematical theorem. (73)

Would he? Why should we believe this completely unsubstantiated assertion? In reality, the Urdi Lemma is not an interesting “new mathematical theorem.” It is a piece of applied mathematics interesting only in its specific context. In applied mathematics one constantly derives various relations relevant to the specific matter at hand without calling them “new mathematical theorems” in the abstract.

> It is particularly striking that although Copernicus deployed the Tusi couple and the Urdi lemma in his astronomical model, he did not make any mention of the fact that he independently discovered these theorems quite unknown to the Greeks. (73)

There is nothing “striking” about this. Copernicus does state and prove the Tusi couple theorem. The notion that he should have stopped and said “oh, by the way, I discovered this” is absurd. How many mathematical treatises have you read where, in the middle of the mathematical exposition, the author chimes in and says “hey, I discovered this, you know”? If this is a required mark of originality we would have to infer that Archimedes and Gauss, for example, never made any mathematical discoveries either, because such remarks are lacking in their works too.

Furthermore, Copernicus may well have thought that the Greeks did know the theorem. A passage in Proclus suggests as much, and in fact Copernicus himself explicitly cites this passage in this connection (albeit only many years after first using the Tusi couple). [Ve] [diB]

> Even after Ibn al-Haytham pointed to the problem [of variation in distance in Ptolemy’s lunar model], it took centuries of work by a chain of astronomers for a solution to be found. … Despite his undoubted genius al-Haytham was unable to solve the problem which was taken up by a string of mathematical astronomers who followed him, including al-Tusi and Urdi, until it was finally solved by Ibn al-Shatir more than two centuries later. (74)

I believe this is false. I believe Ibn al-Haytham did not in fact point this out at all. Although Ibn al-Haytham criticised Ptolemy on other points, “Curiously, he says nothing about the great variation of distance in the lunar theory, the one case in which the model … is obviously incorrect” [Sw, 45]. Nor am I aware of any long “string” of works devoted to trying to resolve this issue before Ibn al-Shatir.

> Since Ptolemy himself would have been aware of the problem, it makes one wonder why a solution that appears so obvious to Blåsjö would not have been adopted by Ptolemy himself. (74)

I agree that Ptolemy would most likely have been aware of this, but it is less clear to what extent he would have been interested in addressing it. Ptolemy’s model is basically just as good as that of Copernicus and Ibn al-Shatir as long as one cares only about the angular position of the moon, which is the essential thing for virtually all astronomical purposes. Their improvement is only needed if one is concerned not only with meeting the traditional needs of computational astronomy but also with arriving at a physically “true” model. Since the latter concern is completely separable from traditional technical astronomy, Ptolemy could quite possibly have considered it more or less irrelevant.

> Blåsjö’s conclusion is questionable because there are many ways of constructing non-Ptolemaic astronomical models that do away with the equant by deploying the Tusi-couple and the Urdi lemma. The al-Shatir model is merely one possible approach. In the sixteenth century Shams al-Din al-Khafri (died 1550) produced four such different models for Mercury’s motion. None of them were similar to the others in mathematical construction but all of them were able to account for the same set of observations. This raises the question: Why did Copernicus propose a model so similar to the Ibn al-Shatir model and not any of the other possible models that al-Khafri had devised? (75)

In fact “Copernicus considered no less than four models for Mercury” [Sw, 405], not just the one which can be interpreted as equivalent to Ibn al-Shatir. So Copernicus had a bunch of models and Islamic astronomers had a bunch of models–––what does that prove? Nothing.

Incidentally, it is perhaps not surprising that Copernicus’s models would have more in common with Ibn al-Shatir’s than with al-Khafri’s. Copernicus and Ibn al-Shatir both correct Ptolemy’s lunar model in the same way. This involves getting rid of Ptolemy’s “crank” arrangement (i.e., the small circle near the center of the orbit in [Bl, figure 4]) for moving the body back and forth in terms of distance from the earth. In this way they eliminate the distance problem in Ptolemy’s theory. Copernicus’s Mercury model does something very much analogous to Ptolemy’s Mercury model: here too a “crank” construction has been eliminated [Bl, figure 5]. Is it so surprising, then, that Copernicus and Ibn al-Shatir have much in common when it comes to both the Moon and Mercury, when these cases are in some respects analogous? Meanwhile, al-Khafri differs from them in that his lunar model retains the absurd distance implications of Ptolemy’s model [Sa, 24]; and this is perhaps not the only respect in which his work is a step backwards from Ibn al-Shatir.

REFERENCES

[Ba] Arun Bala, The Scientific Revolution and the Transmission Problem, Confluence: Online Journal of World Philosophies, Issue 4, 2016. [link]

[Bl] Viktor Blåsjö, A Critique of the Arguments for Maragha Influence on Copernicus. Journal for the History of Astronomy, 45(2), 2014, pp. 183–195. [link]

[diB] Mario di Bono, Copernicus, Amico, Fracastoro and Tusi’s Device: Observations on the Use and Transmission of a Model, Journal for the History of Astronomy, XXVI, 1995, 133-154. [link]

[Sa] George Saliba, A Sixteenth-Century Arabic Critique of Ptolemaic Astronomy: The Work of Shams al-Din al-Khafri, Journal for the History of Astronomy, XXV, 1994, 15-38. [link]

[Sw] Noel Swerdlow and Otto Neugebauer, Mathematical Astronomy in Copernicus’ De Revolutionibus, Springer, 1984.

[Ve] I. N. Veselovsky, Copernicus and Nasir al-Din al-Tusi, Journal for the History of Astronomy, IV, 1973, 128-130. [link]

]]>Place yourself in the boiling desert heat of the Orient. What branch of science is best suited for such a climate? Astronomy, of course. The nighttime science. Who can think during the day, when splashing your face with water from your courtyard fountain is all you can do to keep from perishing altogether? The night sky, on the other hand, is so considerate as to bring a moderate temperature to go with its planetary food for thought.

One Thousand and One Nights is the literary masterpiece of the region: nights, not days, nota bene. A Scheherazade who had tried the same feats of creativity in the exhausting midday heat would have ran out of steam a lot sooner and paid with her head.

Only Eratosthenes, in Egypt, managed to do a little daytime science, but only by making the scorching noon sun an integral part of his method for measuring the earth.

Cross the Mediterranean and the climate is no longer so oppressive. In toga and sandals, a summer at a Greek beach is altogether pleasant. So much so that you might become restless with so much comfort and feel the urge to pick up a stick and start drawing geometrical figures in the sand to entertain yourself. Thus geometry is born.

Northern Italy is more temperate yet, allowing Renaissance scientists to climb the Leaning Tower of Pisa with a rock under each arm without dying of heatstroke. But they’re still doing “outdoor physics”: focus the rays of the sun with a lens, study the path of a canon ball, climb mountains to measure air pressure, and so on.

Keep going north and outdoor physics becomes less and less viable. No wonder the powdered gentlemen scientists of Paris turned to an indoor science–––a “physique de salon.” Academies, collections, fine-tuned instruments, amphitheatrical experiment and dissection halls: these are the resorts of the scientist on rainy or cold day.

Go further north still and scientists feel compelled to shut themselves up in a laboratory wearing lab “coats,” thus bringing with them to science the strategies they developed to live with six-month winters in their everyday lives.

If we go as far north as my home country Sweden we see these tendencies taken to an unhealthy extreme. It is as if the science here is so afraid of the cold that it becomes unhealthily shut up in its labs and loses sight of the greater horizons. Witness for instance how the Swedes stupidly gave Einstein his Nobel Prize not for his obviously most important work on big-picture theories of the geometry of the universe but instead for the indoor-experimental photoelectric effect–––a kind of science very congenial to those in a cold, dark country looking for sparks of light deep in the catacomb sublevels of their concrete laboratories. (I should know: I spent my youth as a mathematics and science student in the underground bowels of Stockholm University, as seen for instance in this selfie.)

]]>In reality this doesn’t prove much of anything. As always with statistics you can “prove” whatever you like by cherry-picking whatever supports your preferred conclusion. Let’s say for example that I wanted to prove the opposite: student evaluations are biased against men. Easy. Take the very same data set used to establish the conclusion above and search for other evaluative words instead. Sure enough, it turns out that for instance “idiot,” “fool,” “crap,” “junk,” and “insensitive” are more often used in evaluations of male instructors more than female ones. Meanwhile female instructors are much more often called “terrific,” “splendid,” “lovely,” “loved,” “wonderful,” which doesn’t exactly seem to square so well with the narrative of patriarchal oppression. Perhaps the strongest bias at work here is that determining which conclusions are fashionable enough to be featured in the New York Times.

]]>First of all they assume that one can teach form without substance–––or in other words that it is a good idea to force students to slavishly follow the pedantic rituals of proof-writing without showing any of the actual mathematics that motivated the techniques in question. Thus we spend most of the course proving that if is even then is even and other extremely obvious and trivial results. The idea is to teach the craft of proof-writing without being distracted by difficult mathematics, but the reality is that we are forcing students to follow pedantic rituals without giving them any rational reason to do so. The obvious message is that mathematics is about pedantry for pedantry’s sake.

By teaching the ritualistic pedantry of proof-writing without content we are driving a wedge between normal, common-sense, intuitive reasoning and mathematical proofs. In this way we are crushing any inclination on the part of our students to think in a creative and independent way. We crush their attempts to try to grasp mathematics in a way that makes sense to them. We crush creativity and curiosity, and we impose unquestionable uniformity. By asking them to prove trivial things in a pedantic way we are telling them that their ways of reasoning are not mathematics, and that to become a mathematician they must transport themselves to a parallel universe where nothing that is natural or intuitive to them counts for anything, and where it is a capital offence to call an even number without specifying that is a whole number even though every normal person understand exactly what they mean.

Since we do not convey any credible purpose of what we teach, the students must accept the pedantic rules we insist upon even though they are completely unjustified. Making people follow arbitrary rules is a terrific way of crushing independent thought and instilling a crippling sense of fear in your subjects, as many dictators have been aware. Like an oppressed population fearing that the secret police will pounce upon them at any moment with arbitrary accusations, so our students are taught to live in fear of our pedantic mathematics. They soon learn to mimic the officially sanctioned examples and constantly ask the teacher’s permission before they attempt the slightest deviation: “Can you do that?” “Is that allowed?” But it is not careful analytical reasoning that makes them ask, it is fear of arbitrary rules and a complete alienation from any sense that they themselves can figure out what makes makes sense and what doesn’t.

We also learn to “prove” that using a ton of pretentious epsilonic machinery, without, of course, ever reaching anything remotely like a result for which such a machinery is appropriate. The technique of -proofs was developed in the 19th century to answer questions such as: Is the limit function of a convergent series of continuous functions continuous? Is every continuous function differentiable almost everywhere? Is every continuous function integrable? It is for these kinds of questions that an -approach makes sense. To use this technique instead (and only) to prove that is continuous at is a parody of mathematics.

It is madness to imply to our students that such pompous proofs are somehow superior to common-sense, intuitive reasoning as far as the results we study are concerned. Using extremely pretentious language to say obvious things is the hallmark of the worst kind of drivel philosophy produced by poseurs in Paris. It’s a sad day when mathematicians of all people are beating these self-important faux-thikers at their own game.

Intro to Proof courses also have much in common with the idiotic “New Math” movement of the 60’s that everyone now agrees was disastrous. Just as “New Math” consisted in little more than expressing trivial ideas in pretentious Bourbaki-inspired terminology, so Intro to Proof courses fetishise the superficial trappings of mathematics without backing it up with any meaning or purpose. In fact, our book even has a whole bunch of “exercises” quibbling about the number of elements of and suchlike, exactly in the manner of “New Math” madness.

And did you know that, when proving a theorem of the form , a proof is called “trivial” if it shows that is always true regardless of , and “vacuous” if it shows that is always false? Who on earth has ever used such ridiculous terminology for any credible purpose? It would be better to say that in either such case the theorem is called “stupid” and leave it at that. If is always true you would be better off just making that your theorem instead of giving a pointless name to , now wouldn’t you? But of course our textbook gives a hundred “exercises” on this ridiculous nonsense that has nothing to do with real mathematics, because it’s easier to teach stupid terminology than meaning and purpose.

Another problem with these kinds of books is that they select exercises the popular form of pseudo-learning in which students unthinkingly mimic an example template, changing only superficial details. Thus when studying induction we set out to prove a thousand pointless and artificially concocted arithmetic formulas. The only thing that is different from proof to proof is some basic algebra, which the students already know how to do, so by having them blindly go through the motions of writing proofs in this way one can create the illusion that learning is taking place, even though the exact opposite is achieved: the only part the students need to alter from proof to proof is the only part they didn’t need to practice, while the actual substance supposed to be taught (the structure of an induction proof) is the same from proof to proof and can therefore be ritualistically copied without thinking.

Another example of this is -proofs: we spend virtually all of our efforts doing algebra to find the specific expression for needed to prove the (obvious) continuity of some specific function at some specific point. In other words, we completely miss the point of what -proofs are all about, and instead focus on exactly the one part that is the most insignificant, because that’s the approach that lends itself best to students mindlessly learning to go through the motions of drill exercises instead of actually doing any serious thinking.

Convergence or divergence of power series is done in the same mindless-algebra-crunching way. In fact, astonishingly, the book never even mentions why anyone would care whether a series is convergent or divergent, despite posing a hundred exercises on this exact topic. You do not “learn to write proofs” this way. What you learn is that meaning and purpose has no place in mathematics; that asking why anything is interesting or worth proving in the first place is a sure sign that you do not have a mathematical mind. This is the worst possible lesson you could teach young students, yet it is as if these kinds of books were determined to drive home this exact point.

By the time the course finally gets around to some actual substance toward the end, it has successfully eradicated any misconceptions in the students’ minds that mathematics might involve human communication and trying to figure things out using natural reasoning. Instead they will have learned that a mathematical proof is a sacred ritual and that no one is allowed into the sect of they do not learn to chant the spells in exactly the manner of the elders.

Consider for example this theorem and proof (the essential ingredient of the proof of the Schröder-Bernstein Theorem) from our textbook:

Someone who wanted to understand what is going on and feel why it’s true might put it instead like this:

Theorem. Let and suppose there is a bijection . Then there is also a bijection .

Proof. Think of these sets as the populations of Sweden (), Europe (), and the world (). Imagine that all of these populations are infinite, and that every person lives alone in one apartment, and that every apartment in the world is occupied.

What does the bijection mean in this context? It means everyone in the world can be matched up with a unique Swedish person. We might be tempted to think of it as “everyone marries a Swedish person.” But this would be misleading, because marriage is mutual: if I marry you, you marry me. But that is not what the bijection is saying. The bijection is not only pairing up non-Swedish with Swedish people. Swedish people are themselves part of the world population , so they too get assigned another Swedish person in the course of the bijection. Therefore it is better to think of as matching members of the world population with Swedish apartments. Since there was precisely one Swedish person per Swedish apartment to begin with this just another way of looking at the same bijection. This is a better analogy than marriage because it is not mutual: if I move into your apartment, you do not have to move into mine.

So the bijection says that the whole world can move into precisely one apartment each in Sweden. This will involve Swedish people moving around within Sweden to “make room” even though there was already one person per apartment (in the manner of “Hilbert’s Hotel”).

Given such a bijection, we want to find a bijection . That is to say, we want a scheme for everyone in the world to move to Europe in such a way that every European apartment has precisely one person in it.

Maybe some natural disaster made much of the world uninhabitable, and the United Nations hired some mathematicians to figure out how to move everyone to Sweden, which they thought would be the only inhabitable country. But then it turned out the scale of the disaster was not as bad as thought, so actually all of Europe would still be inhabitable.

Therefore we are now facing the problem of how to adapt the original move-everyone-to-Sweden scheme and turn it into a move-everyone-to-Europe scheme. Intuitively we feel it must be doable since it should be “easier” than moving everyone to Sweden, but how exactly should we do it? We might be tempted to say: Everyone outside of Sweden but in Europe just stay put, and everyone in the rest of the world and Sweden move according to the original scheme. Sure enough this will get everyone to Europe. But it won’t fulfil the condition that no apartment be left empty. For in the original scheme Swedish people moved about to make room for other Europeans, and since those Europeans are now staying home these apartments will become empty.

Instead we must issue the following instructions. First everyone outside Europe move to the Swedish apartments assigned to them under the original scheme. This means a number of Swedish people will be “bumped” out of their apartments. Let them also move to the apartment assigned to them under the original scheme, and the same for those they bump out in turn, and so on. Let such repercussion moves play out in as many steps as needed, and let everyone not affected simply stay put where they are.

In this way we are left with precisely on person per apartment in Europe. For it is clear that no apartment is left empty since people only move after someone has taken their place. And it is clear that no two moving people are assigned the same apartment, since all moves take place according to the original moving scheme, which by assumption assigned everyone a unique apartment.

We have thus constructed a bijection between the world population and European apartments. By associating each European apartment with its original occupant (when there was precisely one European per European apartment before the moves), this gives a bijection between world population and Europeans, as required. QED.

This is the same proof in different words. It shows in plain and intuitive terms why the theorem holds. It also shows how to arrive at the solution, unlike the textbook proof, which pulls it out of a hat like magic. But God forbid, of course, that you should ever utter any reasoning of this form in a proofs course! That would be sacrilege of the first degree! No clearer proof could exist that you were not cut out to be a mathematician!

Is this the message we want to send our students? That it is more important to slavishly follow the sanctioned “house style” than to try to think and reason in an open-minded fashion that is convincing to yourself and to others? Or that using pretentious terminology and notation somehow makes an argument “more rigorous”?

]]>Paradox 1. By definition, educational research is completely ignorant of content. By design, it cannot say anything about whether one explanation a teacher may give is better than another, or whether a given topic should be taught at all, because that’s not based on “data.”

Any normal person would find this absurd. Aren’t these the very core questions of education? How can you call yourself a “department of education” if you explicitly forbid any of your Ph.D. students to address such questions? Suppose we are in fact teaching a certain topic (say for instance l’Hôpital’s rule in calculus) to hundreds of thousands of people every year, but that this topic is actually worthless and serves no meaningful purpose and should not be taught at all. Is it not preposterous and absurd that, in such a case, educational research by definition and design could never speak of this fact? But in the idiotic world of education “research” that’s how it goes.

You want to teach l’Hôpital’s rule to one class by lecture and another by group work, and see who scores better on a test? Excellent research question!

You want to analyse different ways of proving l’Hôpital’s rule from a pedagogical standpoint? You want to discuss the purpose of teaching l’Hôpital’s rule in the first place? You want to investigate how and to what end l’Hôpital’s rule was developed and applied historically, in order to inform teaching? Impossible! None of those things are “research,” you fool!

Paradox 2. The notion of “data” on which educational research is based is incoherent and irrational.

If I spend a lot of time thinking about how to teach l’Hôpital’s rule and write down insightful reflections, then that’s of course just “opinion” and not “data” and therefore “not research” and it would be impossible for me to get anywhere as an education researcher. But if you interview me about how I teach l’Hôpital’s rule and write a paper about “collegiate mathematics instructor’s attitudes and strategies for teaching l’Hôpital’s rule” then you are a good data-based researcher sure to rise in the ranks. As I observed to no avail in a graduate seminar in mathematics education: if we each publish our own opinions there is no data and no research, but if we go in a circle and publish descriptions of one another’s opinions then suddenly all our opinions have become “data” and now it’s all “research,” even though we have accomplished nothing except to filter and garble our thoughts through imperfect interpreters.

Likewise, if I study 17th-century texts on problems using l’Hôpital’s rule and draw pedagogical lessons from this, then I am also “not doing research” since there is no “data.” But if I put a student in front of a camera and have him think aloud while trying to solve the same problems, then this is “data”; and if I draw from this pedagogical lessons in the exact same way I did for the historical texts then that is now “research” all of a sudden. Is it conceivable that it could be more insightful to study the historical development of an idea, reflecting the same insights we want our students to reach, than to overanalyse a video of an unprepared student randomly rambling his way through a problem? We better hope not, for if so then educational research has shot itself in the foot before it even begun.

None of this makes any sense. The dogmatic obsession with this naive notion of “data” makes educational research rotten at its very core.

The rationale for the empiricist dogma is supposedly that research should be based on “facts” rather than “opinions,” which may sound like a reasonable idea. But, as these examples show, the main consequence in practice of this dogmatic empiricist interpretation of this principle is that it makes it impossible to address any questions that actually matter, while incentivising the investigation of stupid questions that can be address by this narrow-minded conception of “research” methodology.

Meanwhile, if the goal of the empiricist dogma was to eliminate empty opinions it is not working: educational research is in general blatantly and massively biased, as I have demonstrated in a hundred cases. Worse yet, the empiricist dogma has made it very difficult to expose these biases, or indeed to have any rational and critical discussion, since researchers hide behind the idea that their research is all based on “facts” and “data” and is hence by definition objective and beyond criticism.

The most widely touted conclusion of educational research is “lecture doesn’t work.” Indeed, lecture is a rather irrational way to teach. But the right way to prove this is the rationalistic way, using argument and thought. The wrong way to prove it is to naively proclaim it as an empirical “fact” proved by “data” and “research.” But the latter, of course, is the ruling dogma. See for instance the recent report on Active Learning in Post-Secondary Mathematics Education by the Conference Board of the Mathematical Sciences (an umbrella organisation that includes basically anyone who’s anyone in U.S. higher education). “A wealth of research has provided clear evidence that active learning results in better student performance and retention than more traditional, passive forms of instruction alone,” the report reads, referring in particular to the “landmark” meta-analysis of 225 research studies which I have criticised before.

I therefore call upon everyone concerned with education to resist dogmatic empiricism and to admit a sensible role for rationalism, that is, allowing that one can learn some things through thought and reflection. In other words:

Stop wasting massive resources on proving the obvious. Don’t spend several lifetime’s worth of work piling up 225 studies and a mountain of data to prove that we shouldn’t lecture when reason proves it in a minute.

Stop hiding behind “data.” Stop preempting debate by pretending that your opinion is a “fact” proved by “research.” Instead, have the courage to allow educational matters into the arena of rational discussion.

Stop letting methodological dogmas dictate the direction of your research. Have the courage to face and tackle the questions that truly matter. Start there, not with a preconceived idea of what “research” ought to look like.

]]>(a) I write it down and give it to you in print.

or:

(b) I tell you to show up at a given place on a given time for me to blurt it out as a long monologue and hope you catch it all.

Option (b) has number of disadvantages built into its very format. The transmission of information is artificially paced at the rate at which I talk, so if your natural rate of absorbing this material is either higher or lower than this then you will be bored or overwhelmed accordingly. The fact that I go on talking continuously also means you cannot reflect on the information I give you until later, forcing you into passivity of thought during the time we are together. The presentation is also uncompromisingly linear: if a point depends on a previous one in a way you did not anticipate, you cannot go back an reconsider the earlier point in this new light, potentially making it impossible for you to understand the latter point. These obstacles to learning are logical consequences of (b), but they are completely needless since they can all be avoided by opting for (a).

Add to this the logistic absurdity of (b) applied to an entire class. Every single student has to structure their life around this one time slot. They set their alarm clock (even though sleeping would have made them much better mentally prepared to take in the information), commute during the worst hours, and rearrange work and family obligations in complicated ways. And for what? So that they can all sit in the same room at the same time to listen to someone basically dictate a textbook to them while they take “notes.” The university, meanwhile, incurs great costs for the large lecture hall and the expensive textbook-parroter, especially since the charade has to be repeated again year after year. Why not save a ton of money and human costs on all this logistic madness by using a video lecture instead, or better yet skip the parroting-step altogether and just have the lecturer write the notes straight away? That would be option (a), which is, by contrast, incredibly efficient, since a text is amazingly portable across time and space: it can be shared with anyone anywhere, as well as preserved for future generations, at virtually no cost.

But there is a way to make sense of this very costly enterprise of herding students together in one room. Namely: use the possibilities afforded by people learning something together; use the human resources and potential that we have paid so dearly to bring together. Discuss, think aloud, question, collaborate. Indeed, this goes very well with option (a), for reading in advance and having the text concretely in front of us is the natural starting point for such discussion and active engagement. Only of we take advantage of this precious opportunity for students to engage with their co-learners is it rational to have a scheduled class in the first place. Otherwise we could all just as well sit at home and read a book and save ourselves a massive amount of trouble and money.

]]>If we grant this as an axiom of teaching, it follows that traditional Calculus I courses must be reformed as follows.

**Cut limits.** Limits are a hotbed of pseudo-problems that do not serve any meaningful purpose. Monstrous fractions involving , double substitutions, triple l’Hôpital’s rule problems and god knows what else: these problems are fake, fake, fake. They are not taught because they are genuinely needed to address genuinely interesting questions. They are made up for the sake of making up drill problems. They create unreflective and slavishly rule-following students, because any student who thinks for himself will immediately come to the conclusion that the class is a meaningless drill with no purpose.

**Cut proto-“real analysis.”** To the above it will be objected that limits caters to thinking students in that it addresses why the calculus works. So it is alleged, but it is not true. First of all the mass of drill problems assigned have absolutely nothing to do with such why-questions, so this false motivation is a dishonest bait-and-switch. Furthermore the why-questions at hand can be understood perfectly well without the pretentious machinery of limits. This is proved by history, where all leading mathematicians for over a hundred years understood the calculus perfectly without ever bothering with limits.

If you are serious about addressing why-questions you start with why-questions and develop the theory needed to answer them. It then soon becomes apparent that a few basic and intuitive notions are enough to deal with the matter to everyone’s satisfaction, which is indeed precisely what happened historically.

The charade continues with supposedly “conceptual” questions about a barrage of artificially contrived, piecewise defined functions pock-marked with a plethora of discontinuities and medley of different types of non-differentiability. A strange form of unconscious communism seems to be at the bottom of this approach: if “all functions are equal” then indeed it makes sense to give these fake examples as much screen time as the sine and the logarithm. But anyone who starts with interesting and meaningful questions, instead of making up problems to fit their preconceptions of what they have to teach, it is evident enough that some functions are more interesting than others, and that wasting half a calculus course nitpicking about exceptional qualities of the most artificial ones is a pointless exercise in pedantry.

To pretend that these kinds of things are needed to understand “the foundations” of basic calculus is a lie. I suspect most students realise as much, in their guts if not fully consciously.

Limit theory, exceptional functions, and real analysis rose together in the 19th century, more than a hundred years after the calculus had already flourished and produced everything a calculus student has any reason to be interested in. To be sure, all these things served a meaningful purpose then. The 19th-century conception of the foundations of the calculus is profound and important and served to resolve important enigmas that had arisen in the meantime. Of course I do not belittle or deny the value of such investigations in any way––on the contrary I would love to teach them in a real analysis class––but I also observe that they have nothing to do with basic calculus.

Real analysis is a wonderful subject, and calculus is a wonderful subject, but they are two completely different subjects. Don’t try to mix them in some Frankensteinian fashion, and most of all don’t lie to your students and pretend that analysis is the thinking student’s calculus. In reality it is the opposite: only the gullible, subservient student buys this bogus myth.

**Include differential equations.** The usual battery of integration techniques are usually accompanied by “”-style problems, much like the limit sections attacked above. Should they therefore be committed to the flames also? Not at all. The situation could not be more different, although students (and perhaps not a few teachers) in a traditional calculus course wouldn’t know it. Unlike the nonsense real analysis material artificially shoehorned into Calculus I, integration techniques do serve a very credible purpose that is very easily made evident to students. It only takes one simple reshuffling of the order of the topics: teach differential equations as early as possible, as I do in my book. This simple recipe at once changes the entire nature of drill problems on integration techniques. Without differential equations the students will conclude, with good justification, that these problems are nothing but a cruel obstacle course with no purpose. But with differential equations the student cannot draw the same conclusion without denying the value of studying population dynamics, the motion of rockets and projectiles and planets, and a thousand other fascinating and useful things besides. Hence we must either teach differential equations in Calculus I, or accept widespread hatred of mathematics as a rational outcome of our own doing.

A textbook should strive not to explain, but to stimulate the reader to explain for himself:

“Even if occasionally we had been able very easily and conveniently to find in a book a truth or view which we very laboriously and slowly discovered through our own thinking and combining, it is nevertheless a hundred times more valuable if we have arrived at it through our own original thinking. Only then does it enter into the whole system of our ideas as an integral part and living member; only then is it completely and firmly connected therewith, is understood in all its grounds and consequents, bears the colour, tone, and stamp of our whole mode of thought, has come at the very time when the need for it was keen, is therefore firmly established and cannot again pass away. ... [Whereas] the truth that has been merely learnt sticks to us like an artificial limb, a false tooth, [or] a nose of wax.” (§260)

The student must resist the temptation to demand that things be explained to him, and to seek out an “answer key” as soon as he is stuck:

“The mind is deprived of all its elasticity by much reading as is a spring when a weight is continually applied to it; and the surest way not to have thoughts of our own is for us at once to take up a book when we have a moment to spare.” (§258) “We should, therefore, read only when the source of our own ideas dries up, which will be the case often enough even with the best minds. On the other hand, to scare away our own original and powerful ideas in order to take up a book, is a sin against the Holy Ghost. We then resemble the man who runs away from free nature in order to look at a herbarium, or to contemplate a beautiful landscape in a copper engraving.” (§260)

The student should be aware that nothing could be easier than spotting one who is speaking thoughts not genuinely his own:

“According to these observations, it will not surprise us to learn that the man who is capable of thinking for himself and the book-philosopher can easily be recognized even by their style of delivery; the former by the stamp of earnestness, directness, and originality, by all his ideas and expressions that spring from his own perception of things; the latter, on the other hand, by the fact that everything is second-hand, consists of traditional notions, trash and rubbish, and is flat and dull, like the impression of an impression.” (§263)

Those who dislike thinking for themselves are those who approach learning with dishonest and ignoble intent:

“Thus we can divide thinkers into those who think primarily for themselves and those who think at once for others. The former ... are the real philosophers. For they alone take the matter seriously; and the pleasure and happiness of their existence consists in just thinking. The others are the sophists; they wish to shine and seek their fortune in what they hope to obtain from others in this way; this is where they are in earnest. We can soon see from his whole style and method to which of the two classes a man belongs.” (§270)

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