Galileo, Ignoramus: Mathematics versus Philosophy in the Scientific Revolution Viktor Blåsjö Abstract I offer a revisionist interpretation of Galileo’s role in the history of science. My overarching thesis is that Galileo lacked technical ability in mathematics, and that this can be seen as directly explaining numerous aspects of his life’s work. I suggest that it is precisely because he was bad at mathematics that Galileo was keen on experiment and empiricism, and eagerly adopted the telescope. His reliance on these hands-on modes of research was not a pioneering contribution to scientific method, but a last resort of a mind ill equipped to make a contribution on mathematical grounds. Likewise, it is precisely because he was bad at mathematics that Galileo expounded at length about basic principles of scientific method. “Those who can’t do, teach.” The vision of science articulated by Galileo was less original than is commonly assumed. It had long been taken for granted by mathematicians, who, however, did not stop to pontificate about such things in philosophical prose because they were too busy doing advanced scientific work. Contents 4 Astronomy 38 4.1 Adoption of Copernicanism . 38 1 Introduction 2 4.2 Pre-telescopic heliocentrism . 40 4.3 Tycho Brahe’s system . 42 2 Mathematics 2 4.4 Against Tycho . 45 2.1 Cycloid . .2 4.5 The telescope . 46 2.2 Mathematicians versus philosophers . .4 4.6 Optics . 48 2.3 Professor . .7 4.7 Mountains on the moon . 49 2.4 Sector . .8 4.8 Double-star parallax . 50 2.5 Book of nature . .8 4.9 Moons of Jupiter . 51 2.6 Calculus of speeds . .9 4.10 Rings of Saturn . 52 4.11 Sunspots . 53 3 Physics 13 4.12 Sunspots and heliocentrism . 54 3.1 Fall and weight . 13 4.13 Phases of Venus . 57 3.2 Law of fall . 16 4.14 Comets . 62 3.3 Gravitational constant . 17 3.4 Planetary speeds . 18 5 Salon science 63 3.5 Moon fall . 19 5.1 Galileo, populariser . 63 3.6 Resistance of the medium . 19 5.2 Three dimensions . 64 arXiv:2102.06595v1 [math.HO] 12 Feb 2021 5.3 Babylonian eggs . 65 3.7 Inertia . 20 5.4 The Bible . 67 3.8 Relativity of motion . 22 5.5 First Inquisition . 69 3.9 Centrifugal force . 23 5.6 Second Inquisition . 70 3.10 Circular path of fall . 24 3.11 Projectile motion . 27 6 Galileo evaluated 72 3.12 Practical ballistics . 29 6.1 Conclusion of the above . 72 3.13 Catenary . 29 6.2 Reception . 74 3.14 Pendulum . 31 6.3 Descartes . 74 3.15 Brachistochrone . 31 6.4 Kepler . 75 3.16 Floating bodies . 31 6.5 Huygens . 77 3.17 Tides . 34 6.6 Newton . 78 1 7 Before Galileo 79 8.1 Mathematics and nature . 89 7.1 Plato . 80 8.2 Empirical method . 94 7.2 Aristotle . 81 8.3 Experimental method . 95 7.3 Golden age . 81 8.4 Technology and craft knowledge . 96 7.4 Lost and ignored . 83 8.5 Instrumentalism versus realism . 101 7.5 Greek mechanics . 84 7.6 Greek astronomy . 85 8.6 Mechanical philosophy . 104 8.7 Continuity thesis . 105 8 “Father of Science” 89 8.8 Epilogue . 108 1 Introduction Galileo is overrated, I maintain. I shall systematically go through all his major achievements and all stan- dard arguments as to his alleged greatness, and offer a critical counter-assessment. I go after Galileo with everything but the kitchen sink, but I hasten to add that assembling the case against him is ultimately a means to an end. The stakes are much higher than Galileo’s name alone. Galileo is at the heart of fun- damental questions: What is the relation between science, mathematics, and philosophy? Between ancient and modern thought? What is the history of our scientific worldview, of scientific method? Galileo is right in the thick of the action on all of these issues. Consequently, I shall use my analysis of Galileo as a fulcrum to articulate a revisionist interpretation of the history of early modern science more broadly. But let’s start small, with a simple snapshot of Galileo at work. 2 Mathematics 2.1 Cycloid The cycloid is the curve traced by a point on a rolling circle, like a piece of chalk attached to a bicycle wheel (Figure 1). Many mathematicians were interested in the cycloid in the early 17th century, including Galileo. What is the area under one arch 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. Galileo wanted nothing more than to join their ranks. The cycloid was a suitable showcase. It was a natural next step following upon the Greek corpus, and hence a chance to prove oneself a “new Archimedes.” There was only one problem: Galileo wasn’t any good at mathematics. Try as he might, he could not for the life of him come up with one of those clever geometrical arguments for which the Greek mathematicians were universally admired. All those brilliant feats of ingenuity that Archimedes and his friends had blessed us with, it just wasn’t happening for Galileo. Perhaps out of frustration, Galileo turned to the failed mathematician’s last resort since time immemorial: trial and error. Unable to crack the cycloid with his intellect, he attacked it with his hands. He cut the shape out of thick paper and got his scales out to have this instrument do his thinking for him. As best as he could gather from these measurements, Galileo believed the area under the cycloid was somewhere near, but not exactly, three times the area of the generating circle.1 This was no way to audition for the pantheon of geometers. Galileo was left red-faced when mathemat- ically competent contemporaries solved the problem with aplomb while he was fumbling with his cutouts. These actual mathematicians proved that the cycloid area was in fact exactly three times the area of the gen- erating circle,2 even though Galileo had explicitly concluded the contrary on the basis of his little cardboard diorama. When Galileo heard of others working on the cycloid challenge, he sought help on this “very difficult” problem from his countryman Bonaventura Cavalieri, a competent mathematician. “I worked on it fruitlessly,” 1 [Drake (1978), 19, 406]. 2 Roberval in 1634. [Struik (1969), 232–238], [Whitman (1943)], [Kline (1972), 350–351]. 2 Figure 1: The cycloid. lamented Galileo. “It needs the mind of a Cavalieri and no other,”3 he pleads, tacitly acknowledging his own unmistakably inferior mathematical abilities. It is interesting to contrast this with the very different reaction to the same problem by Galileo’s contemporary René Descartes, the famous philosopher who was also a vastly better mathematician than Galileo. When Descartes heard of the problem he immediately wrote back to his correspondent that “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.”4 He then immediately goes on to give his own proof of the result composed on the spot. Descartes is not famous for his humility, but the fact of the matter is that a number of mathematicians solved the cycloid problem with relative ease, while Galileo was fumbling about with scissors and glue. In the case of the cycloid, it is an unequivocal fact that Galileo used an experimental approach because he lacked the ability to tackle the problem as a mathematician. If Galileo could have used a more mathematical approach he would unquestionably have done so. It is my contention that what is so glaringly obvious in this case holds for Galileo’s science generally. Galileo’s celebrated use of experiments in science is not a brilliant methodological innovation but a reluctant recourse necessitated by his pitiful lack of mathematical ability. The cycloid case also makes it clear why the mathematically able prefer geometrical proofs to experiments: the latter are hopelessly unreliable. By relying on experiments unchecked by proper mathematics, Galileo got the answer wrong, and not for the first time nor the last. “Do not think that I am relying on experiments, because I know they are deceitful,”5 said Huygens, and all other mathematicians with him. It had always been obvious that mathematics and science can be explored using experiment and observation. As Galileo says: “You may be sure that Pythagoras, long before he discovered the proof . , had satisfied himself that the square on the side opposite the right angle in a right triangle was equal to the squares on the other two sides”6—presumably by making numerical measurements on various concretely drawn triangles. But able mathematicians had always known that haphazard trial and error had to be superseded by rigorous demonstration for a treatise to be worth the parchment it is written on. This and only this is why you don’t see experimental and numerical data defiling the pages of masterpieces of ancient mathematics and science such as those of Archimedes.7 In this case as in so many others, Galileo’s inglorious contribution to the history of thought is to cut off mathematical reasoning at the training-wheels stage, and to air in public what true mathematicians consid- ered unworthy scratch work. He experiments because his intellect comes up short.