The mathematicians’ view of Galileo

What did 17th-century mathematicians such as Newton and Huygens think of Galileo? Not very highly, it turns out. I summarise my case against Galileo using their perspectives and a mathematical lens more generally.

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I’m going to conclude my case against Galileo with this final episode on this subject. Here’s a little anecdote I found that can be used to frame the overall point that I have made. Galileo was sentenced by the church in 1633. And to go along with this there was a bit of a crackdown on Galileo sympathisers. Somebody in Florence was going to publish a book that made reference to the “most distinguished Galileo.” But the Inquisition intervened and demanded that this phrase should be changed. Instead of “most distinguished Galileo,” the phrase should be changed to: “Galileo, man of noted name.” I am not generally on the side of the Inquisition, but I have come to the conclusion that this particular decree is sound. Instead of “Galileo, father of modern science,” we would be better off saying “Galileo, man of noted name.”

That’s what I have argued before. Today I will offer a bit of a roundup with some new perspectives on these issues. And that will be the end of my 18-episode rant against Galileo.

My main claim has been that Galileo was a poor mathematician. Historians are still blind to this fact. People still speak of “Galileo’s mathematical genius.” That persistent myth must certainly die. John Heilbron, the UC Berkeley historian of science, published an authoritative biography of Galileo in 2010. There Galileo is called “the greatest mathematician in Italy, and perhaps the world” in his time.

Galileo was no such thing. In reality, tell-tale signs of mathematical mediocrity permeate all his works. Many pages of Galileo would not be out of place somewhere in the middle of the piles of slipshod student homework that some of us grade for a living.

A number of Galileo’s numerous mathematical errors even concern some of his core achievements. I have discussed all of his notable scientific contributions and found much to object in every single case. For instance, Galileo uses “his” law of fall erroneously on a number of occasions: when he tries to explain the orbital speeds of the planets, when he tries to calculate how long it would take for the moon to fall to the earth, when erroneous claims to have proved that centrifugal whirling could never throw objects off the earth regardless of speed, and when he erroneously describes the path of a falling object in a reference frame not rotating with the earth. Galileo is praised for having discovered the law of fall, but the fact is that he derived as many false conclusions from it as correct ones.

He also not infrequently presents arguments that are demonstrably inconsistent with his core beliefs, such as his tidal theory contradicting his own principle of relativity, his Joshua argument contradicting his own principle of inertia, and his objection to the geocentric explanation of sunspots being inconsistent with his own heliocentrism.

I should say that, of course, other people made mistakes too. It was the early days of science after all. Suppose I concede that everyone has an equal comedy of errors to their name. Even so, this would still prove my point that Galileo was a dime a dozen scientist and not at all a singular “father of modern science.” But I do not in fact need to concede this much. Galileo’s sum of errors are not just par for the course. They are exceptionally poor, and in matters of mathematics especially they are astonishing.

We have seen time and time again that virtually all of “Galileo’s” achievements were either anticipated or at least made independently by others. To name just the most striking case:

“Let us hypothetically assume that a scholar contemporary to Galileo pursued experiments with falling bodies and discovered the law of fall as well as the parabolic shape of the projectile trajectory, that he found the law of the inclined plane, directed the newly invented telescope to the heavens and discovered the mountains on the moon, observed the moons of the planet Jupiter and the sunspots, that he calculated the orbits of heavenly bodies using methods and data of Kepler with whom he corresponded, and that he composed extensive notes dealing with all these issues. In short, let us assume that this man made essentially the same discoveries as Galileo and did his research in precisely the same way with only one qualification: he never in his life published a single line of it. As a matter of fact, the above description refers to a real person, Thomas Harriot.”

Actually these discoveries are not identical with those of Galileo but rather go beyond them, because Galileo never “calculated the orbits of heavenly bodies using methods and data of Kepler,” as Harriot did, who was a better mathematician.

So the history of science would have been much the same without Galileo, because people like Harriot and others were doing all of that stuff independently anyway.

It’s instructive to compare Galileo to Kepler in these kinds of terms. We can find independent contemporary discoveries for almost everything Galileo did, but not so for Kepler’s achievements, even though many of them are still central in modern science. Harriot was a “second Galileo” and you could go on to a third or a fourth stand-in without much loss. It would be much harder to find a “second Kepler.”

In my view it is not hard to see why: Kepler was an excellent mathematician who worked on difficult things, while Galileo didn’t know much mathematics and therefore focussed on much easier tasks. The standard story has it that Galileo’s insights were more “conceptual,” yet at least as deep as technical mathematics. On this account it is imagined that basic conceptions of science that we consider commonsensical today were once far from obvious: we greatly underestimate the magnitude of the conceptual breakthroughs required for these developments because we are biased our modern education and anachronistic perspective.

But if this is true, how come that Galileo’s ideas—for all their alleged “conceptual” avant-gardism—spontaneously sprung up like mushrooms all over Europe? And how come all of those ideas can easily be explained to any high school student today, if they are supposedly so profound and advanced? The same cannot be said for Kepler’s ideas. They were neither simultaneously developed by dozens of scientists, nor can they be taught to a modern student without years of specialised training. Perhaps this contrast between Galileo and Kepler says something about what genuine depth in the mathematical sciences looks like.

In my opinion, mathematicians at the time realised this perfectly well. I have already spoken before about the very harsh words that Descartes and sometimes Kepler had for Galileo. “He is eloquent to refute Aristotle but that is not hard,” as Descartes said. There are a number of quotes like that from mathematicians. And of course they spotted numerous mathematical blunders in Galileo, which they condemned.

Let’s look at what some other competent mathematicians thought of Galileo.

Christiaan Huygens was perhaps the greatest physicist and mathematician of the generation between Galileo and Newton. He is often portrayed as continuing the scientific program of Galileo. Huygens’s collected works is 22 thick volumes. Go ahead and try to find any strong praise of Galileo in there, let alone anything remotely like calling him a “father of science.” Somehow Huygens never got around to saying any such thing, in these tens of thousands of pages on physics and mathematics and astronomy that we wrote. Hmm, what a mystery.

The closest Huygens ever gets to mentioning Galileo favourably is in the context of a critique of Cartesianism. In the late 17th century, the teachings of Descartes had attracted a strong following. In the eyes of many mathematicians, the way Cartesianism had become an entrenched belief system was uncomfortably similar to how Aristotelianism had been an all too dominant dogma a century before. Huygens makes this parallel explicit:

“Descartes had a great desire to be regarded as the author of a new philosophy [and] it appears that he wished to have it taught in the academies in place of Aristotle. [Descartes] should have proposed his system of physics as an essay on what can be said with probability. That would have been admirable. But in wishing to be thought to have found the truth, he has done something which is a great detriment to the progress of philosophy. For those who believe him and who have become his disciples imagine themselves to possess an understanding of the causes of everything that it is possible to know; in this way, they often lose time in supporting the teaching of their master and not studying enough to fathom the true reasons of this great number of phenomena of which Descartes has only spread idle fancies.”

It is in direct contrast with this that Huygens slips in a few kind words for Galileo: “[Galileo] had neither the audacity nor the vanity to wish to be the head of a sect. He was modest and loved the truth too much.” Historians have observed that Huygens in all likelihood quite consciously intended this passage to apply to himself as much as to Galileo. Perhaps this is why Huygens is surely too generous in praising Galileo’s alleged “modesty.” Galileo was anything but modest, of course.

In any case, it is very interesting to see what Huygens says about Galileo’s actual science in this passage. Let us read it, and keep in mind that this is as close as Huygens ever gets to praising Galileo, and that the context of the passage—a scathing condemnation of Cartesianism—gives Huygens a notable incentive to put Galileo’s scientific achievements in the most positive terms for the sake of contrast.

In light of this, Huygens’s ostensible praise for Galileo is most remarkable, I think, for how qualified and restrained it is. Huygens’s praise begins like this:

“Galileo had, in spirit and awareness of mathematics, all that is needed to make progress in physics …”

Interesting phrasing. Huygens seems to be saying: Galileo said all the right things about about mathematics and scientific method, but he didn’t actually carry through on it. Given Galileo’s rhetoric, he ought to have been able to do it, but be didn’t.

Interestingly, Huygens does not say that Galileo had great mathematical ability or demonstrable achievements, only that he was “aware” that mathematics is necessary for physics. In this respect, Galileo “had all that is needed to make progress in physics,” Huygens says. Why not simply say that Galileo *made* great progress in physics, instead of this convoluted and qualified “he had what was needed to do so”? So really Huygens’s ostensible praise for Galileo is actually quite backhanded. At least that’s how it seems to me.

Let’s continue reading because the Huygens quote goes on. Here is the rest of the sentence:

“… and one has to admit that he was the first to make very beautiful discoveries concerning the nature of motion …”

Galileo wasn’t the first, as we now know. Huygens didn’t know about the unpublished work of Harriot etc., so he is overly generous in that regard. But never mind that. Huygens’s formulation is still very restrained in an interesting way: did you notice that strange phrase “one has to admit”? “One has to admit” that Galileo was the first to make certain discoveries. Who speaks of their greatest hero in such terms? One “has to admit” that he made some discoveries? That seems more like the kind of phrasing you use to describe the work of someone who is overrated, not someone you esteem as the founder of science.

Huygens wrote in French. The phrase is “il faut avouer.” I’m not a linguist but I think the translation I gave is the most natural one. “Il faut avouer”: “one has to admit”; it suggests a reluctance to concede the point. I’m not sure if it’s possible to argue that taken in context it could also be construed as “even a Cartesian would have to admit” or something like that. If you’re an expert of 17th-century French I would like to hear your opinion about this.

Let’s see, the Huygens quote continues even further and here’s how it concludes. After this remark about Galileo having made discoveries concerning motion, Huygens adds:

“ … although he left very considerable things to be done.”

Well, yes. That’s my point exactly. What is most striking and remarkable about the work of Galileo is not the few discoveries he “admittedly” made, but how very little he actually accomplished despite all his posturing about mathematics and scientific method. It seems to me that Huygens and I agree on this. Even in his most pro-Galilean sentence in all his works, Huygens is undermining Galileo as much as he is praising him.

What about Isaac Newton? What did he think of Galileo?

Newton famously said that “if I have seen further it is by standing on the shoulders of giants.” Many have erroneously assumed that Galileo was one of these “giants.” One scholar even proposes to explain that “when Newton credits Galileo with being one of the giants on whose shoulders he stood, he means …” blah blah blah. We do not need to listen to what this philosopher thinks Newton meant, because the first part of the sentence is false already. The assumption that Galileo was one of the scientific giants in question has no basis in fact.

The closest Newton gets to praising Galileo is in the Principia, his most important work. After introducing his laws of motion, Newton adds some notes on their history.

“The principles I have set forth are accepted by mathematicians and confirmed by experiments of many kinds. By means of the first two laws and the first two corollaries Galileo found that the descent of heavy bodies is in the squared ratio of the time and that the motion of projectiles occurs in a parabola.”

The laws and corollaries in question are: the law of inertia, which Galileo did not know, as we have seen; then Newton’s second law, the force law F=ma, which Galileo also did not know; and the composition of forces and motions, which was established in antiquity.

Note that Newton doesn’t say that Galileo was the discoverer of these laws. All Newton says is that Galileo used these laws to find the path of projectiles. Indeed, as one historian has pointed out, “Newton’s Latin contains some ambiguity” for it “can have two very different meanings: that the two laws were completely accepted by Galileo before he found that projectiles follow a parabolic path, or that these two laws were already generally accepted by scientists at the time that Galileo made his discovery of the parabolic path.”

Either way, Newton is wrong. Of course, once you are looking at the world though Newtonian mechanics it is natural to think that surely Galileo must have had these laws, because that is so obviously the right way to think about parabolic motion. Therefore, as Dijksterhuis says, “[according to] the myth in which he appears as the founder of classical dynamics, [Galileo] must surely have known the proportionality of force and acceleration. But to those who have become acquainted with Galileo through his own works, not at second hand, there can be no doubt that he never possessed this insight.”

Quite so, and indeed Newton was not acquainted with Galileo’s work directly. As I.B. Cohen says, “Newton almost certainly did not read [Galileo’s] Discorsi until some considerable time after he had published the Principia,” if ever. On the other hand “early in his scientific career, [Newton] had read [Galileo’s] Dialogo”—but that is of course his work on Copernicanism, not his work on mechanics and the laws of motion, which is what Newton is referencing in the Principia.

“Hence Newton (rather too generously, for once!) allowed to Galileo the discovery of the first two laws of motion.” And the reason for Newton’s excessive charity is not hard to divine. To quote I.B. Cohen again, Newton’s Principia is marked by an obvious and vehement “anti-Cartesian bias.” “Because of his strongly anti-Cartesian position, Newton might have preferred to think of Galileo rather than Descartes as the originator of the First Law.” Whereas, “in point of fact, the Prima Lex [that is, the law of inertia] of Newton’s Principia was derived directly from the Prima Lex of Descartes’s Principia”—that is, the correct law of inertia. Descartes stated the correct law of inertia with crystal clarity in this published key work, while Galileo never stated it anywhere, nor believed it.

Clearly, then, Newton’s attribution of these laws to Galileo means next to nothing. Galileo demonstrably did not know these laws; Newton hadn’t read Galileo anyway; and Newton had an obvious bias and incentive to overstate Galileo’s importance in order to belittle the influence of Descartes which he did not want to admit.

Newton’s words aren’t high praise in any case. In fact, that becomes ever clearer if we read on in Newton’s text. For when Newton continues his historical discussion he says on the very same page:

“Sir Christopher Wren, Dr. John Wallis, and Mr. Christiaan Huygens, easily the foremost geometers of the previous generation, independently found the rules of the collisions and reflections of hard bodies.”

So evidently Newton was in the mood when writing this to point out who “the foremost geometers” of the past were. Yet on the very same page he had no such words for Galileo. A telling omission. Altogether there is no evidence that Newton regarded Galileo particularly highly, let alone considered him anywhere near a “father of modern science.”

The time has come to wrap up my Galileo story. Perhaps I can sum it up like this.

Say you go to the library and find the shelves with philosophical texts ordered chronologically. You pick the books up one by one and see what they have to say about science. Century after century you find the same thing. Aristotle, Aristotle, Aristotle. Then commentaries on Aristotle. Then commentaries on commentaries on Aristotle. Then people who ostensibly try to think more independently, yet cling desperately to Aristotelian concepts and terminology as if their life depended on it, even when they try to challenge isolated claims of Aristotelian dogma. Then, suddenly: Galileo. What a breath of fresh air this is. The Aristotelian shackles are emphatically discarded, and all the nowadays familiar principles of modern science are articulated in lucid and entertaining prose. At once after him everyone is a scientist. The Aristotelianism that ran rampant for centuries had suddenly stopped dead in its tracks. How can one not admire this singular father of the scientific worldview, this pivotal hero who divides the entire history of thought into two disjoint worlds separated by such an abyss?

Alas, you made one mistake. You went to the philosophy shelves. You should have gone to subbasement 3, where the mathematics books are kept. This may not have been an obvious choice. Perhaps you were educated in the humanities and therefore naturally drawn to the sprawling and well-attended shelves in your part of the library. The out-of-the-way mathematics section never caught anyone’s eye. Isn’t it just for nerds in training who need to double-check their formulas? Apparently there are a few books there from Greek times, but blink and you miss them between thick modern textbooks on algebraic topology and partial differential equations. And if you do open one of those old math books, it’s full of technical diagrams and equations anyway. Who would ever think to look there for man’s view of the world? Surely that’s what philosophy is for?

In reality, we sent Galileo to your shelves because he wasn’t good enough for ours. Galileo wasn’t the first to do anything except explain what mathematicians had always known in such basic terms that even philosophers could understand. Galileo once wrote to a fellow philosopher:

“If philosophy is that which is contained in Aristotle’s books, you would be the best philosopher in the world. But the book of [natural] philosophy [or science] is that which is perpetually open to our eyes. But being written in characters different from those of our alphabet, it cannot be read by everyone; the characters of this book are triangles, squares, circles, spheres, cones, pyramids and other mathematical figures, the most suited for this sort of reading.”

That is Galileo’s advice to the philosophers of his day. I say much the same thing to modern scholars. If the history of science is that which is contained in philosophical books, you would be the best historians in the world. But the real truth is perpetually open to our eyes, if only we take the trouble to read mathematics.