Was Galileo “the father of modern science” because he was the first to unite mathematics and physics? Or the first to base science on data and experiments? No. Galileo was not the first to do any of these things, despite often being erroneously credited with these innovations.
Opinionated History of Mathematics
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Galileo is “the father of modern science,” people would have you believe. But why? What exactly did he do that was so new that he fathered the entire concept of science? Was Galileo the first to bring together physics and mathematics? Was he the first to base science on data and experiments, or to give practical experience more authority than philosophical systems?
The answer to these questions is: no, no, no. Galileo was nowhere near the first to do any of these things. But he is still often credited with these innovations, even in scholarly sources. So I’m going to run down the list and prove point by point why these people are wrong.
The notion that Galileo was somehow “the father of modern science” remains a standard view among modern historians. For instance, the Oxford Companion to the History of Modern Science published in 2003 flat out says that Galileo “may properly be regarded as the ‘father of modern science’.” This view is considered so unassailable that even the very Pope once conceded that Galileo “is justly entitled the founder of modern physics.” Pope John Paul II said this is 1979.
But there is less agreement on what exactly Galileo did to deserve this epithet. As Dijksterhuis says in his classic history of mechanics: “No one indeed is prepared to challenge [Galileo’s] scientific greatness or to deny that he was perhaps the man who made the greatest contribution to the growth of classical science. But on the question of what precisely his contribution was and wherein his greatness essentially lay there seems to be no unanimity at all.”
So let’s go though all major attempts at capturing Galileo’s alleged greatness, and criticise them one by one.
First: Mathematics and nature.
It is a common view that Galileo was the first to bring together mathematics and the study of the natural world. I could give you long list of scholars who have said exactly this. For this to make sense, one must obviously maintain that, before Galileo, mathematics and natural science were fundamentally disjoint. This assumption is plainly and unequivocally false. In Greek works by mathematically competent authors, there is zero evidence for this assumption and a mountain of evidence to the contrary. “We attack mathematically everything in nature” said Iamblichus of Greek science, and he was right. This is a commonplace, explicit methodological program in Greek science, as the The Cambridge Companion to the Hellenistic World points out: “Hellenistic natural philosophers often took mathematics as the paradigm of science and sought to mathematize their study, that is, to ground all its claims in mathematical theorems and procedures, a goal shared by modern scientists.” This is the exact opposite of the claim that the ancients were unable to conceive the unity of mathematics and science.
How can so many historians get it exactly backwards? By ignoring the entire corpus of Greek mathematics and instead relying exclusively on philosophical authors. Thus we are told that, following “the classification of philosophical knowledge deriving from Aristotle,” a sharp division prevailed among “the Greeks” between “natural science (or ‘physics’), which studied the causes of change in material things,” and “mathematics, which was the science of abstract quantity.” Well, this was perhaps a problem for philosophers who spent their time trying to classify scientific knowledge instead of contributing to it. But I challenge you to produce one single piece of evidence that this division had any impact whatsoever on any mathematically creative person in antiquity.
The alleged divide doesn’t exist in Aristotle’s own works either, for that matter. Aristotle lived well before the glory days of Greek science, and he was clearly no mathematician. But even Aristotle lists mechanics, optics, harmonics, and astronomy as fields based on mathematical demonstrations. He even explicitly calls them “branches of mathematics.” How can anyone infer from this that Aristotle saw the very notion of mathematical science as a conceptual impossibility? That’s nuts. But historians in fact do so, by insisting that these fields are mere exceptions. Here’s a typical quote, from A Short History of Scientific Thought published by Palgrave Macmillan in 2012:
“Previous assumptions [before Galileo], encouraged by Aristotle and scholastic philosophers, held that mathematics was only relevant to our understanding of very specific aspects of the natural world, such as astronomy, and the behaviour of light rays ([that is to say] optics), both of which could be reduced to exercises in geometry. Otherwise, mathematics was just too abstract to have any relevance to the physical world.”
The implausibility of this view is obvious. If, as Aristotle himself clearly states, mechanics, optics, harmonics, and astronomy are four entire fields of knowledge that successfully use mathematics to understand the natural world, who in their right mind would then categorically insist that, nevertheless, other than that mathematics surely has nothing to contribute to science. It makes no sense. If mathematics has already given you four entire branches of science, why close your mind to the possibility of any further success along similar lines? It is hard to think of any reason for taking such a stance, except perhaps for someone who themselves lack mathematical ability and want to justify their neglect of this field.
The strange habit of writing off the numerous branches of mathematical science in antiquity as so many exceptions is necessary to maintain triumphalist narratives of the great Galilean revolution. For example, we are told that “it was Galileo who first subjected other natural phenomena to mathematical treatment than the Alexandrian ones.” In other words, except mechanics, astronomy, optics, music, statics, and hydrostatics, Galileo was *the very first* to take this step. That is to say, if you ignore all previous mathematicians who did this exact thing in great detail, Galileo’s step was completely revolutionary.
Another strategy for explaining away the obvious fact of extensive mathematical sciences in antiquity is to discount them as genuine science on the grounds that they were abstractions. Thus some claim that, despite ostensible applications of mathematics in numerous fields, “mathematical theory and natural reality remained almost entirely separate entities” due to the “high level of abstraction” of the mathematical theories, which meant that they were “barely connected with the real world.”
Supposedly, Galileo broke this spell — an absurd claim since this critique is all the more true for his science: even Galileo’s supposedly “best” discoveries are often way out of touch with reality: his law of fall, his law of parabolas, they obviously fail experimentally. Not to mention Galileo’s many erroneous theories, which were even more disconnected from reality for obvious reasons. Meanwhile, Greek scientific laws of statics, optics, hydrostatics, and harmonics concern everyday phenomena that can be verified by anyone in their own back yard using common household items. Indeed, they are still part of modern physics textbooks — and high school laboratory demonstrations — to this day. Take optics, for example. Heron of Alexandria proved the law of reflection, which anyone with a mirror can readily check, using the distance-minimisation argument still found in every textbook today. Light travels along the shortest path from point A to point B via the mirror. Diocles demonstrated the reflective property of the parabola and used it to “cause burning” by concentrating the rays of the sun with a paraboloid mirror: a principle still widely applied today, for example in satellite dishes and flashlights. Ptolemy demonstrated the magnifying property of concave mirrors, such as modern makeup mirrors. These kinds of results, which are not atypical, are clearly not disconnected from reality by any means.
The false notion of a divide between mathematics and science also rests on a conception of mathematics itself as a purely abstract field. Here’s a quote expressing a typical view:
“Traditionally, geometry was taken to be an abstract inquiry into the properties of magnitudes that are not to be found in nature. Dimensionless points, breadthless lines, and depthless surfaces of Euclidean geometry were not traditionally taken to be the sort of thing one might encounter while walking down the street. Whether such items were characterized as Platonic objects inhabiting a separate realm of geometric forms, or as abstractions arising from experience, it was generally agreed that the objects of geometry and the space in which they are located could not be identified with material objects or the space of everyday experience.”
This is again a view expressed by philosophers only. Nothing of the sort is ever stated by any mathematically competent author in antiquity. On the contrary, mathematicians routinely take the exact opposite for granted. Allegedly “abstract” geometry is constantly applied to physical objects in Greek mathematical works without ado. The long list of Greek mathematicians who studied the natural world always took for granted the identification of geometry with the space and material objects around us. And why shouldn’t they? For thousands of years geometry had been used to delineate fields, draw up buildings, measure volumes of produce, and a thousand other practical purposes — exactly “the sort of thing one might encounter while walking down the street.” Every single theorem of Euclid’s geometry can be verified by concrete measurements and constructions with physical tools and materials. So why would mathematicians suddenly insist that their field is completely divorced from reality? What could possibly be their motivation for doing so? It accomplishes nothing and creates tons of obvious problems when one wants to apply mathematics far and wide in numerous areas, as mathematicians always did. The only people with any motive to take such an extremist stance are philosophers with an axe to grind.
Only those ignorant of the vast tradition of Greek mathematical science can maintain that the unity of mathematics and science in the 17th century was in any way revolutionary. However, even if one accepts this completely wrongheaded view, credit still should not go to Galileo. Some recent historians have begun to stress that “the mathematization of the sublunary world begins not with Galileo but with Alberti,” who wrote on the geometrical principles of perspective in painting in the 15th century.
“The invention of perspective by the Renaissance artists, by demonstrating that mathematics could be usefully applied to physical space itself, [constituted] a momentous step toward the general representation of physical phenomena in mathematical terms.”
These historians correctly challenge the narrative of Galileo as the heroic visionary who united mathematics and the physical world, but they retain the erroneous underlying assumption that this unification was revolutionary to begin with. Perspective painting is fine mathematics, but it wasn’t a “momentous step” “demonstrating” that mathematics could be applied to the world, because that had already been demonstrated over and over again thousands of years before. Vitruvius, to take just one example, had pointed out the obvious: “an architect should be instructed in geometry,” which “is of much assistance in architecture.” Certainly a strange thing to say if the “momentous” insight that geometry is relevant to “the space of everyday experience” is still more than a thousand years in the future! No, the absurd notion that the application of geometry to physical space was somehow a Renaissance revolution can only occur to those who spend too much time reading philosophical authors pontificating about the divisions of knowledge instead of reading authors actually active in those fields.
The restriction to “the sublunary world” in the above quotation is also telling. The allegedly profound conceptual divide between heaven and earth in this period is a standard trope among historians, as we have discussed before. Of course, the Greeks mathematised the sublunary world too, but you have to read specialised works to find out much about that. Astronomy, on the other hand, is such an obvious example of an extremely successful and detailed mathematisation of one aspect of reality that even philosophers and historians cannot ignore this elephant in the room. Hence they rely on the qualifier that the allegedly revolutionary step was “the mathematization of the *sublunary* world.”
Aristotle did indeed make much of the difference between the earthly, sublunary world and the world of heavenly motions. But this is one particular dogma of one particular school of philosophy. There is no reason for any mathematician to accept it, nor is there any evidence that any mathematically competent person in the golden age of Greek science did so. The Aristotelian dichotomy is far from natural or necessary: in fact, “Aristotle argues, *against his predecessors*, that the celestial world is radically different from the sublunary world,” as one historian has observed. For that matter, even if Aristotle’s dogmatic and arbitrary dichotomy is accepted, it would still be madness to acknowledge the undeniable success of mathematics on one side of the divide, yet consider its application on the other side of the divide a conceptual impossibility.
Ptolemy, the ancient astronomer, speaks in Aristotelian terms when he contrasts astronomy with physics. The subject matter of astronomy is “eternal and unchanging,” while physics “investigates material and ever-moving nature situated (for the most part) amongst corruptible bodies and below the lunar sphere.” This is arguably more of a fact than a philosophical commitment: planetary motions are regular and periodic, whereas falling bodies, projectile motion, and other phenomena of terrestrial physics are inherently fleeting and limited to a short time span. It is conceivable that someone might seize on this dichotomy to “explain” why mathematics is suitable for the heavens only, and not for the sublunary world. This, however, is definitely not Ptolemy’s stance. He unequivocally expresses the exact opposite view: “as for physics, mathematics can make a significant contribution” there too.
In sum, the Aristotelian dichotomy between heaven and earth was never an obstacle to mathematicians. And this with good reason. The whole business of emphasising the dichotomy in the context of the mathematisation of the world is a figment of the imagination of historians, who find themselves having to somehow explain away astronomy as irrelevant when they want to claim that there was a mathematical revolution in early modern science. We do not need to resort to such fictions if we instead accept that the unity of mathematics and science had been obvious since time immemorial.
Another argument for Galileo as the unifier of physics and mathematics consists in stressing that other mathematicians of his day were often more concerned with pure geometry than with projectile motion and the like. For instance, in France there were highly capable “new Archimedeans” like Descartes, Roberval, and Fermat, but their focus differed from that of Galileo. Here’s a quote from a recent book expressing this view:
“They were indeed good mathematicians, but they did not consider mathematics as a method for understanding physical things. Mathematical constructions were only abstractions to them, with which it was fun to play, but which were not to be confused with what really happened in nature. Moreover, they were not interested in the ways in which motion intervened in natural processes.”
In my view, Galileo would have loved to have been this kind of “new Archimedean” too if only he had been capable of it. And it is not true that these Frenchmen ignored motion and the mathematisation of nature. We have already noted that Descartes studied the law of fall, and that Fermat corrected Galileo on the path of a falling object in absolute space. Both Descartes and Fermat also wrote on the law of refraction of optics, deriving it from physical considerations regarding the speed of light in different media. Also, Descartes explained the motion of the planets, and the fact that they all revolve in the same direction about the sun, by postulating that they were carried along by a vortex. So these mathematicians were clearly not ignorant of or averse to studying how “motion intervened in natural processes.”
So it is not attention to motion per se, but the study of projectile motion specifically, that sets Galileo apart from these mathematical contemporaries. Does Galileo deserve great credit in this regard? I don’t think so. Why is projectile motion important? With Newton, projectile motion took on a fundamental importance because he saw that planetary motion was governed by the same principles. Galileo had no inkling of this insight. With Newton, projectile motion is also fundamental as a paradigm illustration of the principles — such as inertia and Newton’s force law — that govern all other mechanics. In Newtonian mechanics this is the basis for understanding phenomena such as pendulum motion. Galileo, however, got this wrong, so he cannot be celebrated for this insight either.
Thus we see that praising Galileo for studying projectile motion is anachronistic. Galileo got lucky: the topic he studied later turned out to be very important for reasons he did not perceive, so that in retrospect his work seems much more prescient and groundbreaking than it really was. He himself in fact motivates the theory of projectile motion almost exclusively in terms of practical ballistics — a nonsensical application of zero practical value, which one cannot blame other mathematicians for ignoring.
So those are my rebuttals of the various ways in which Galileo has been praised for mathematising nature in innovative ways.
Another way in Galileo was supposedly innovative is in his emphasis on an empirical scientific approach.
The Cambridge Companion to Galileo expresses this view clearly: “Galileo became (and still is) the model for the empiricist scientist who, unlike the natural philosophers of his day, sought to answer questions not by reading philosophical works, but rather through direct contact with nature.” This is an image Galileo eagerly (but dishonestly) sought to promote, as we have seen. Recall the story of the Babylonian eggs cooked in a sling for example, and also Galileo’s rhetoric against Aristotle on the law of fall.
Praise for Galileo in this regard naturally goes hand in hand with “the verdict that Greek science suffered from an overdose of rash generalizations at the expense of a careful scrutiny, whether experimental or observational, of the relevant facts.” In other words, “Greek thinkers generally overrated the power of unchecked, speculative thought in the natural sciences.” So many people have claimed.
In reality, an empirical approach to the study of nature is not a newfangled invention by Galileo but just common sense. It was obviously adopted by the Greeks, especially the mathematicians. Even Aristotle, who practiced “speculative thought in the natural sciences” to a much greater extent than mathematicians, was a keen empiricist, and his followers insisted on this as one of the key principles of his philosophy. Aristotle’s zoology largely follows a laudable empirical method quite modern in spirit, such as braking open lots of bird eggs at different stages to study the development of the embryo and many other things like that. The same approach was applied by his immediate followers in botany and petrology, including for example cataloging extensive empirical data on how a wide variety of minerals react to heating.
This was far from forgotten in Galileo’s day, where one often encounters passages like these from committed Aristotelians:
“We made use of a material instrument to establish by means of our senses what the demonstration had disclosed to our intellect. Such an experimental verification is very important according to [Aristotelian] doctrine.” That’s Piccolomini, an Aristotelian philosopher, writing well before Galileo, in the 16th century.
Not infrequently, Galileo’s Aristotelian opponents attacked him for being too speculative while they saw themselves as representing the empirical approach. For example, one critic writes to Galileo:
“At the beginning of your work, you often proclaim that you wish to follow the way of the senses so closely that Aristotle (who promised to follow this method and taught it to others) would have changed his opinion, having seen what you have observed. Nonetheless, in the progress of the book you have always been so much a stranger to this way of proceeding that all your controversial conclusions go against our sense knowledge, as anyone can see by himself, and as you expressly say yourself, speaking of the theory of Copernicus, which was rendered plausible and admirable to many by abstract reasoning although it was against all sensory experience.”
It is true that there were also many spineless “Aristotelians” in Galileo’s day who preferred hiding behind textual studies rather than engaging with actual science. But this was one perverse sect of scholasticism, not the overall state of human knowledge before Galileo. A contemporary colleague of Galileo put is well:
“The Science of Nature has been already too long made only a work of the Brain and the Fancy: It is now high time that it should *return* to the plainness and soundness of Observations on material and obvious things.”
That’s Robert Hooke. Note that word choice: “return” — “return to observation.” Not: Galileo invented this new thing, empiricism. Rather: empiricism is the natural and obvious way to study nature, and the departure from it in certain philosophical circles is a corrupt aberration.
The misconception that the Greeks were anti-empirical stems from a foolish reading of the mathematical tradition. Galileo fan Stillman Drake put it like this:
“Archimedes never appealed to actual measurements in any of his proofs, or even in confirmation of his theorems. The idea that actual measurement could contribute anything of real value was absent from physics for two millennia.”
“The mathematics of Euclid and the physics of Archimedes were necessary, but not sufficient, for Galileo’s science. They leave unexplained Galileo’s repeated appeals to sensate experience.”
On a superficial reading this may indeed appear so. Open, say, Archimedes’s treatise on floating bodies and you will find no mention of any measurement or experiment or data of any kind, only theorems and proofs. It may seem natural to infer from this that Archimedes was doing speculative mathematics divorced from reality, and that he had no understanding of the importance of empirical tests. This is what it looks like to historians who insist on an overly literal reading of the text and lack a sympathetic understanding of how the mathematical mind works. The fact of the matter is that Archimedes’s theorems are empirically excellent. It makes no sense to imagine that Archimedes was reasoning about abstractions as an intellectual game, and that his extremely elaborate and detailed claims about the floatation behaviour of various bodies given their shapes and densities just happened to align exactly with reality by pure chance. Archimedes doesn’t have to point out that he made very careful empirical investigations, because it is obvious from the accuracy of his results that he did.
Here is a better way of putting the relation between mathematics and empirical data, from The Oxford Handbook of the History of Physics:
“Mixed mathematics were often presented in axiomatic fashion, following the Archimedean tradition. In this tradition, experiments were often conceived of as inherently uncertain and therefore they could not be placed at the foundation of a science, lest that science too be tainted with that same degree of uncertainty. To be sure, experiments were still used as heuristic tools, for example, but their role often remained private, concealed from public presentations.”
So the point is not that empirical data is neglected, but that it is a mere preliminary step. Anyone can make measurements and collect data. Self-respecting mathematicians do not publish such trivialities. Instead they go on to the really challenging step of synthesising it into a coherent mathematical theory. Galileo did not have the ability to do the latter, so he had to stick with the basics, and pretend, nonsensically, that this was somehow an important innovation. Then as now, there were enough non-mathematicians in the world for his cheap charade to be successful.
What about the experimental method? Was that Galileo’s special contribution and insight?
Some say so. Empiricism, which we just discussed, is mere passive observation. The real innovation was active experiment. A famous supporter of this view is Immanuel Kant, who wrote as follows in the Critique of Pure Reason:
“When Galileo caused balls to roll down an inclined plane, a light broke upon all students of nature. Reason must approach nature in order to be taught by it. It must not, however, do so in the character of a pupil who listens to everything that the teacher chooses to say, but of an appointed judge who compels the witnesses to answer questions which he has himself formulated.”
Modern historians have expressed the same idea. Here is one example:
“The originality of Galileo’s method lay precisely in his effective combination of mathematics with experiment. The distinctive feature of scientific method in the seventeenth century, as compared with that in ancient Greece, was its conception of how to relate a theory to the observed facts and submitting them to experimental tests. [This feature] transformed the Greek geometrical method into the experimental science of the modern world.”
In reality, the use of experiment in Greek science is abundantly documented to anyone who bothers to read mathematical authors.
Greek scientists knew perfectly well that “it is not possible for everything to be grasped by reasoning, many things are also discovered through experience,” as Philon said. This quote refers to the precise numerical proportions needed for the spring in a stone-throwing engine. The same author also offered an experimental demonstration that air is corporeal. Ptolemy experimented with balloons (or “inflated skins” as he says) to investigate whether air or water has weight in their own medium. Does a balloon full of water sink in water, or float or what? Indeed, Ptolemy “performed the experiment with the greatest possible care,” according to Simplicius. Heron of Alexandria gives a detailed description of an experimental setup to prove the existence of a vacuum. He explicitly states that “referring to the appearances and to what is accessible to sensation” trumps abstract arguments that there can be no vacuum. Such arguments had been given by Aristotle, but here we have a mathematically minded author saying “no way, that’s nonsense” and proving as much with experiment. In optics, Ptolemy explicitly verified the law of reflection by experiment. He also studied refraction experimentally, giving tables for the angle of refraction of a light ray for various incoming angles in increments of 10 degrees for passages between air, water, and glass.
Archimedes caught a forger who tried to pass off as pure gold a crown that was actually gold-coated silver. By an experiment based on hydrostatic principles, he was able to expose the crown as a knock-off without damaging it in any way. This discovery was the occasion for him to reportedly run naked through the streets yelling “eureka” in excitement. Such was his love of empirical, experimental science — yet many scholars keep insisting that, like a second Plato, all he really cared about was abstract geometry. Evidently, even running naked through the streets and screaming at the top of one’s lungs is not enough for some people to open their eyes. It is hard to imagine what else one can do to draw their attention to the obvious: namely that Greek mathematicians embraced experimental method through and through.
Ok, so I have argued that Galileo wasn’t the first to apply mathematics to nature, nor the first to base science on data, nor on experiment. So we’ve ruled out those three but we’re still only halfway down the list of things that Galileo supposedly pioneered. We will have to go through the other ones next time.