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. 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.

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