Our picture of Greek antiquity is distorted. Only a fraction of the masterpieces of antiquity have survived. Decisions on what to preserve were made by in ages of vastly inferior intellectual levels. Aristotelian philosophy is more accessible for mediocre minds than advanced mathematics and science. Hence this simpler part of Greek intellectual achievement was eagerly pursued, while technical works were neglected and perished. The alleged predominance of an Aristotelian worldview in antiquity is an illusion created by this distortion of sources. The “continuity thesis” that paints 17th-century science as building on medieval thought is doubly mistaken, as it misconstrues both ancient science and Galileo’s role in the scientific revolution.
Opinionated History of Mathematics
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To praise Galileo is to criticise the Greeks. The contrast class of “Aristotelian” science is constantly invoked to explain Galileo’s alleged greatness, both in Galileo’s own works and in modern scholarship. But this narrative gets it all wrong, in my opinion. It is based on a caricature of Greek science that effectively ignores the Greek mathematical tradition.
Francis Bacon put it well: when “human learning suffered shipwreck” with the death of the classical world, “the systems of Aristotle and Plato, like planks of lighter and less solid material, floated on the waves of time and were preserved,” while treasure troves of much more mathematically advanced works were lost forever.
Aristotelian science is not the pinnacle of Greek scientific thought. Far from it. It is not the best part of Greek science, but the part of Greek science that was most accessible and appealing to the generations of mathematically ignorant people who populated the universities in medieval Europe for hundreds of years. And perhaps some generations who still do.
Mathematicians have always felt differently. “So many great findings of the Ancients lie with the roaches and worms,” said Fermat. They are lost, in other words, these mathematical masterpieces that once existed. That’s how Fermat put it, and all his mathematical colleagues agreed. And they were right.
In the 20th century a few such masterpieces were recovered. So these 17th-century mathematicians were proven right in their intuition that great works were forgotten and hidden away among “roaches and worms” indeed.
In 1906, a work of Archimedes that had been lost since antiquity was rediscovered in a dusty Constantinople library. The valuable parchment on which it was written had been scrubbed and reused for some religious text. But the original could still just about be made out underneath it. As one historian put it: “Our admiration of the genius of the greatest mathematician of antiquity must surely be increased, if that were possible,” by this “astounding” work, which draws creative inspiration from the mechanical law of the lever to solve advanced geometrical problems. If even this brilliant work by antiquity’s greatest geometer only survived by the skin of its teeth and dumb luck, just imagine how many more works are lost forever.
Also in the 20th century, divers chanced upon an ancient shipwreck, which turned out to contain a complex machine (the so-called Antikythera mechanism). Again historians were astonished: “From all we know of science and technology in the Hellenistic age we should have felt that such a device could not exist.” “This singular artifact is now identified as an astronomical or calendrical calculating device involving a very sophisticated arrangement of more than thirty gear-wheels. It transcends all that we had previously known from textual and literary sources and may involve a completely new appraisal of the scientific technology of the Hellenistic period.”
Another example. The Greeks appear to have been much further ahead than conventional sources would lead one to believe in a number of mathematical fields. One example is combinatorics. Of this entire mathematical field little more survives than one stray remark mentioned parenthetically in a non-mathematical work by Plutarch:
“Chrysippus said that the number of intertwinings obtainable from ten simple statements is over one million. Hipparchus contradicted him, showing that affirmatively there are 103,049 intertwinings.”
“This passage stumped commentators until 1994,” when a mathematician realised that it corresponds to the correct solution of a complex combinatorial problem worked out in modern Europe in 1870, thereby forcing “a reevaluation of our notions of what was known about combinatorics in Antiquity.” It is undeniable from this evidence that this entire field of mathematics must have reached an advanced stage, yet not one single treatise on it survives.
These are just a few striking examples illustrating an indisputable point: the Hellenistic age was extremely sophisticated mathematically and scientifically, and we don’t even know the half of it.
Scores of key treatises are lost, and we are forced to rely on later commentators and compilers for accounts of the works of Hellenistic authors. It’s like trying to understand modern science and mathematics from popularisations in the Sunday newspaper. It’s vastly oversimplified and dumbed-down. It reduces complex science to one or two simplistic ideas while conveying nothing whatsoever of the often massive technical groundwork that it is based on. That’s the state of our sources for much Greek science: all that has come down to use are some clickbait headlines and blurbs by people who are themselves not scientists and wouldn’t understand the first thing about the technical details of the works they are trying to summarise.
Actually this is a misleading analogy. The situation is even worse than this. Here is how one historian puts it:
“Nearly all that we know on observations and experiments among the Greeks comes from compilations and manuals composed centuries later, by men who were not themselves interested in science, and for readers who were even less so. Even worse, these works were to a great extent inspired by the desire to discredit science by emphasizing the way in which men of science contradicted each other, and the paradoxical character of the conclusions at which they arrived. This being the object, it was obviously useless, and even out of place, to say much about the methods employed in arriving at the conclusions. It suited Epicurean and Sceptic, as also Christian, writers to represent them as arbitrary dogmas. We can get a slight idea of the situation by imagining, some centuries hence, contemporary science as represented by elementary manuals, second- and third-hand compilations, drawn up in a spirit hostile to science and scientific methods. Such being the nature of the evidence with which we have to deal, it is obvious that all the actual examples of the use of sound scientific methods that we can discover will carry much more weight than would otherwise be the case. If we can point to indubitable examples of the use of experiment and observation, we are justified in supposing that there were others of which we know nothing because they did not happen to interest the compilers on whom we are dependent. As a matter of fact, there are a fair number of such examples.”
In previous episodes we have discussed the many ways in which Greek sources already showed full awareness of many things often attributed to Galileo. Taking this context of filtering and lost sources into account means that we should give all the more weight to those arguments.
Sadly, however, the lack of appreciation for science among these ignorant commentators continues among scholars today. I collected some quotes on this by some very respectable classicists of today.
“The state of editions and translations of ancient scientific works as a whole remains scandalous by comparison with the torrent of modern works on anything unscientific — about 100 papers per year on Homer, for example. An embarrassingly large number of classicists are ignorant of Greek scientific works.”
“Classicists include many who have chosen Latin and Greek precisely to escape from science at the very early stage of specialisation that our schools’ curricula permit: and often a very successful escape it is, to judge from the depth of ignorance of science ancient and modern that it often secures.”
It is remarkable how strongly these authors make this point. The first quote is from Lloyd, the Cambridge professor. It takes a lot for people like that to almost condemn their colleagues to their face. They wouldn’t do this if it wasn’t serious.
Little wonder then that Greek science is systematically misunderstood and undervalued, and that simplistic ideas of philosophical authors and commentators are substituted for the real thing.
Galileo’s relation to the preceding philosophical tradition has been systematically misunderstood because of this.
How did modern science grow out of mathematical and philosophical tradition? The humanistic perspective is that science needed both: it was born through the unification of the technical but insular know-how of the mathematicians with the conceptual depth and holistic vision of the philosophers. The mathematical perspective is that science is what the mathematicians were doing all along. Science did not need philosophy to be its eye-opener and better half; it merely needed the philosophers to step out of the way and let the mathematicians do their thing. So which is it?
Many historians have tried to stress commonalities between Galileo and the Aristotelian philosophers who preceded him. That is to say, they argue for the “continuity thesis” which says that the so-called “Scientific Revolution” was not a radical or revolutionary break with previous thought. Here is what they say:
“Galileo essentially pursued a progressive Aristotelianism [during the first half of his life—the period of] positive growth that laid the foundation for the new sciences.”
“A particular school of Renaissance Aristotelians, located at the University of Padua, constructed a very sophisticated methodology for experimental science; … Galileo knew this school of thought and built upon its results; this goes a long way toward explaining the birth of early modern science.”
“The mechanical and physical science of which the present day is so proud comes to us through an uninterrupted sequence of almost imperceptible refinements from the doctrines professed within the Schools of the Middle Ages.”
“Galileo was clearly the heir of the medieval kinematicists.”
I agree with these authors that “those great truths for which Galileo received credit” are not his. But the notion that they were first conceived in Aristotelian schools of philosophy is wrongheaded.
The argument of these historians is based on a simple logic. First they show that various concepts of “Galilean” science are prefigured in earlier sources. Then they want to infer from this that these sources marked the true beginning of the scientific revolution. But in order to draw this inference they need two assumptions: first, that Galileo was the father of modern science; and second, that the Greeks were nowhere near the same accomplishments. These two assumptions are simply taken for granted by these authors, as a matter of common knowledge. But in reality both assumptions are dead wrong, and therefore the inference to the significance of the Aristotelian sources is unwarranted.
It is interesting that the continuity thesis on the one hand devalues the contributions of Galileo, yet at the same time desperately needs to reassert the traditional view that “Galileo has a clear and undisputed title as the ‘father of modern science’,” as one of these historians puts it. They need to say this because this is what gives them the one point of connection they are able to establish between medieval and modern science. The entire argument stands and falls with this false premiss. Therefore, if one proves, as I have done before, that Galileo was a mediocre scientists of negligible importance to the mathematically competent people who actually achieved the scientific revolution, then the continuity thesis collapses like a house of cards.
The defenders of the continuity thesis are equally ineffectual in establishing the second false premiss of their argument, namely the alleged absence of these “new” ideas in Greek thought. In fact, even continuity thesis advocates make no secret of the fact that the medieval tradition was built on “remnants of Alexandrian science.” For example, “although we are left with few monuments from the profound research of the Ancients into the laws of equilibrium, those few are worthy of eternal admiration.” Obviously, “masterpieces of Greek science [such as the works of] Pappus, and especially Archimedes, are proof that the deductive method can be applied with as much rigor to the field of mechanics as to the demonstrations of geometry.” All of that are quotes form Pierre Duhem, a passionate advocate of the continuity thesis.
How can people like Duhem acknowledge these “masterpieces” “worthy of eternal admiration” from antiquity, yet at the same time attribute the scientific revolution to medieval or renaissance philosophers? Here’s how. By writing off those ancient works as minor technical footnotes to an otherwise thoroughly Aristotelian paradigm. Only if this picture is accepted can any kind of greatness be ascribed to the pre-Galileans, as is evident from passages such as these:
“Some philosophers in medieval universities were teaching ideas about motion and mechanics that were totally non-Aristotelian [and] were consciously based on criticisms of Aristotle’s own pronouncements.”
“Admittedly, most of these significant medieval mechanical doctrines were formed within the Aristotelian framework of mechanics. But these medieval doctrines contained within them the seeds of a critical refutation of that mechanics.”
“The medieval mechanics occupied an important middle position between Aristotelian and Newtonian mechanics. [Hence it was] an important link in man’s efforts to represent the laws that concern bodies at rest and in movement.”
“The impressive set of departures from Aristotelianism achieved by medieval science nevertheless failed to produce genuine efforts to reconstruct, or replace, the Aristotelian world picture.”
If Aristotle is taken as the baseline, this looks quite impressive indeed. But why should Aristotle be accepted as the default opinion? Aristotle was one particular philosopher who was a nobody in mathematics and lived well before the golden age of Greek science. Medieval and renaissance thinkers indeed mustered up the courage to challenge isolated claims of his teachings almost two thousand years later, while mostly retaining his overall outlook. This does not constitute great open-mindedness and progress. Rather it is a sign of small-mindedness that these people paid so much attention to Aristotle at all in the first place. In my view, it is not so much impressive that they deviated a bit from Aristotle as it is deplorable that they framed so much of what they did relative to Aristotle, even when they disagreed with him. This is very different from post-Aristotelian thought in Greek times, where there is no evidence that any mathematician paid any attention to Aristotle’s mechanics.
In any case, “extravagant claims for the modernity of medieval concepts” suffer from “serious defects.” One historian has summarised it well:
“There was no such thing as a fourteenth-century science of mechanics in the sense of a general theory of local motion applicable throughout nature, and based on a few unified principles. By searching the literature of late medieval physics for just those ideas and those pieces of quantitative analysis that turned out, three centuries later, to be important in seventeenth-century mechanics, one can find them; and one can construct a “medieval science of mechanics” that appears to form a coherent whole and to be built on new foundations replacing those of Aristotle’s physics. But this is an illusion, and an anachronistic fiction, which we are able to construct only because Galileo and Newton gave us the pattern by which to select the right pieces and put them together.”
The main piece of such precursorism is the so-called “mean speed theorem.” This is a completely trivial result. You can visualise it in terms of a graph with time on the x-axis and velocity on the y-axis. Suppose you plot the graph of a uniformly accelerated motion, such as a freely falling object. It makes a straight line going from the bottom left to the to right. It starts from no velocity and goes to a certain final velocity. How far did the thing travel? Distance travelled is the area under the graph. So it’s the area of a triangle. Base times height over 2. That is to say, the time of fall, times half the final velocity. Or another way of putting it is that half the final velocity is the same thing as the average velocity. The triangle has the same area as a rectangle with the same base and half the height. The “mean speed theorem” is just this. In terms of distance covered, a uniformly accelerated motion is equivalent to a constant-speed motion with the same average speed. A very simple thing to see.
Some people praise this as an “impressive” achievement of the middle ages—”probably the most outstanding single medieval contribution to the history of physics,” derived by “admirable and ingenious” reasoning, according to one historian. Even though these medieval authors did absolutely nothing with this trivial theorem and only deduced it to illustrate the notion of uniform change abstractly within Aristotelian philosophy. Later the theorem became central in “Galilean” mechanics since free fall is uniformly accelerated. But it “was, in fact, never applied to motion in fall from rest during the 14th, or even in the 15th century” (only in the mid-16th century there is a passing remark to this effect within the Aristotelian tradition, “without any accompanying evidence”).
Let us not radically inflate our esteem for the Middle Ages by anachronistically praising them for pointing out a trivial thing that centuries later took on a significance of which they had no inkling. Let us instead recognise the theorem for the trifle that it is. Then we shall also not have any need to be surprised when it turns out that Babylonian astronomers assumed it without fanfare thousands of years earlier still. The utterly trivial “mean speed theorem” was implicitly taken for granted in Babylonian astronomy. They were too good mathematicians to make a big fuss about something so evident, unlike the medieval philosophers who sat around a proved this at length. They were so bad at mathematics that this trivial thing was the cutting edge to them, in their ignorance.
Galileo owes other debts to previous philosophical tradition as well, according to many historians. For example, we are told that there are “unmistakeable Jesuit influences in Galileo’s work”: “Above all Galileo was intent in following out Clavius’s program of applying mathematics to the study of nature and to generating a mathematical physics.” That’s a quote from Wallace. The preposterous notion that this was “Clavius’s” program can only enter one’s mind if one only reads philosophy. It was obviously Archimedes’s program, except, unlike Clavius, he proved his point by actually carrying it out instead of sermonising about what one ought to do in philosophical prose. Philosophers (ancient and modern alike) have a tendency to place disproportionate value on explaining something conceptually as opposed to actually doing it. After all, that is virtually the definition of philosophy. Hence they praise certain Aristotelians for explaining some supposedly profound principles of scientific method even when “it is quite clear that [none of them] ever applied his advocated methods to actual scientific problems.”
Descartes—a mathematically creative person—knew better: “we ought not to believe an alchemist who boasts he has the technique of making gold, unless he is extremely wealthy; and by the same token we should not believe the learned writer who promises new sciences, unless he demonstrates that he has discovered many things that have been unknown up till now.” Unfortunately, such basic common sense is often lacking among historians and philosophers assigning credit for basic principles of the scientific method.
There is a contradiction in the way modern historians try to trace many aspects of the scientific revolution to roots in the middle ages. On the one hand these historians like to claim that the traditional view of the scientific revolution is ahistorical and based on an anachronistic mindset, whereas their own account that sees continuity with the middle ages is more sensitive to how people actually thought at the time itself. Ironically, however, their view, which is supposed to be more true to the historical actors’ way of thinking, is actually all the more blatantly at odds with how virtually all leaders of the scientific revolution thought of the middle ages. One historian summarises it accurately: “The scientific achievement of the Middle Ages was held in unanimous contempt from Galileo’s time onward by those who adhered to the new science. Leibniz’ scathing verdict ‘barbaric physics’ neatly encapsulates the reigning sentiment.” This was not for nothing. Leibniz was an erudite scholar well versed in the philosophy of the schools. But he was also an excellent mathematician. The latter enabled him to pass a sound judgement on the “barbaric” science of the middle ages.