Mathematics versus philosophy, then and now

Divergent interpretations of Galileo’s alleged greatness cut across disciplinary divides: mathematics versus philosophy, science versus humanities. Understanding Galileo means dealing with these fundamental tensions.

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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.