Reviel Netz, New History of Greek Mathematics [end of year Aristotle series]
How Aristotle recorded maths history; Theories of ratio & proportionality; Was Aristotle the intended audience? Discovery of irrationality; Diagrams and letters..
Reviel Netz wrote:
Chapter 1
To the Threshold of Greek Mathematics
… I have already mentioned that early Greek civilization was all based on public performance, and what comes to mind, first of all, is the public performance of recitation and song. Alongside the bard, the early Greeks also had the sage: the man respected for his words of wisdom and advice, words that might ring paradoxical and yet impress for their kernel of truth, often touching on the political life of the community but sometimes reaching beyond that, to speculation about the cosmos, about the human condition, about truth itself. Such wise words could sometimes be noted and commemorated, and eventually – in the fifth century – a wise man even could decide, occasionally, to write down a book. But it was all about the public performance of wisdom – as it would still be all the way down to Socrates. Thales and Pythagoras certainly put nothing down in writing, and it is not clear that they had a “something”– a clearly articulated body of doctrines – for which “putting down in writing” would be the appropriate exercise.
The sense that they had such a thing is a later construct that we can see emerging very clearly from the works of Aristotle. One of Aristotle’s favorite techniques was to go through past views on a particular topic so as to see them as approximations – but no more – of Aristotle’s own views. Thus, past philosophers, according to Aristotle, only looked for the material cause, “the stuff out of which things are made”, unlike Aristotle, who developed a complex, nuanced understanding based on different kinds of causes. And so past sayings attributed to past figures were shoehorned into the model of material cause, and so also, whatever sage words were attributed to Thales concerning the cosmos got transformed, in Aristotle’s telling, to “Water is the Material Cause”, which still survives as the first thing one usually reads in a general history of philosophy: “Thales was the first philosopher, and he said that all is water.” [Footnote: The argument that Aristotle is an unreliable narrator of the earlier history of philosophy was made forcefully already by H. F. Cherniss, Aristotle’s Criticism of Pre-Socratic Philosophy. Most scholars today believe that Aristotle aimed to be faithful to his sources while being completely shaped by the assumptions and agendas of his own time and place. Is this not true of all historians?] …
… Aristotle had very few sources to work from as regards Thales and Pythagoras, but the point, concerning the nature of the evidence, is not merely methodological. The point has to do with the underlying historical reality itself. The world of Thales and Pythagoras had relatively little use for extensive writing, and so whatever knowledge they uttered would have to be oral. … Whatever the Greeks knew, back then, in the field of mathematics, would have to be understood along the terms of oral knowledge. This is not a matter of scientific doctrine but of shared cultural lore. …
[20th century misguided histories of ancient mathematics]
… in the 1930s and for many decades hence, the scholarly consensus was … to accept, at face value, the testimonies, such as those of Proclus and Aristotle, concerning the earliest Greek mathematics. … The implication of that consensus would be, essentially, that even early on, the Greeks had the equivalent of professionalized mathematics, authors whose goal was to promote theoretical mathematical understanding. …
… It has been central to later reconstructions of early Pythagoreanism that this philosophy somehow involved an early mysticism of number, as if somehow, integer numbers underlay the very structure of the cosmos. It is evident that such a mysticism would sit awkwardly with the discovery of irrationality. … But it is clear that very early on, the Greeks did discover irrationality! So this should be difficult for Pythagoreanism, if indeed we believe in its existence as an early mathematical doctrine based on rational numbers. Such a belief began in antiquity itself – perhaps as early as Aristotle – and by Late Antiquity, some authors speculated that the discovery of irrationality could cause a crisis for Pythagoreanism.
… [It] was tempting and natural to project a crisis of foundations, with its consequences, on the earliest history of mathematics. Babylonians came up with the algebraical study of numerical relations. But then their Greek followers discovered the phenomenon of irrationality and therefore realized that it is impossible to describe all relations in purely numerical terms. One therefore needed to formulate algebra in a strictly geometrical way – hence the mature geometry of Euclid’s Elements.
This account, then, in outline, was shared by scholars for decades. It came under pressure only very gradually. This happened in many stages [beginning in the early 1960s]. …Here was a more careful, professionalized classical philology, keen to understand the authors we read not as mere parrots, repeating their sources, but instead as thoughtful agents who shape and retell the evidence as suits their agenda.
Pythagoras, under such a reading, crumbles to the ground: almost everything – as noted earlier – comes to be seen as the making of later authors from Aristotle on. …
Chapter 2
The Generation of Archytas
[Archytas b.435/410 BC, d.360-350 BC]
… We recall: Thales of Miletus is said to have argued, very early on (around the beginning of the sixth century?), something that posterity remembered as all is water. In the ensuing generation, Anaximander of Miletus would argue (as transcribed by posterity) that all is the unlimited; Anaximenes of Miletus, that all is air. This is the gist of the information we have based on Aristotle, and it is probably false in many ways, but one thing is clear: here was a powerful voice – that of Thales – and a generation of local, immediate responses. And then – silence. For a long while, we hear of no more philosophers in Miletus, and elsewhere, wise people seem to be engaged with rather different concerns. A powerful voice – a set of echoes – and then, more or less, silence. This pattern seems to be repeated very often in ancient cultural life. A prominent master is followed by a prominent disciple but with no obvious continuation into a third generation. …
… One of the important questions for us … is whether mathematics, at this point, is even distinct from philosophy. But what we see already is that those who engage in mathematics, in this era, are as likely to contribute to philosophy and certainly as likely to communicate with philosophers – so much so that the entire network of authors could have had, at its center, not a mathematician but rather the mathematically curious philosopher, Plato. …
[PROPORTIONALITY]
… Let us first look, then, at how the Greeks would have gone about finding the single mean proportional … This involves perhaps the key insight of Greek geometry, namely: there is a tight connection between the two separate fields of quantity and geometrical form. Specifically: “proportion”, a quantitative relation, is almost the same as “similarity between figures”, a statement of geometrical form. …
… Archytas’s solution is a fantastically complicated three-dimensional exercise in extending the simple, plane result, which is now known as Elements VI.13 … you take the two sides, produce two right-angled triangles, and stick them within a third, to get three lines in continuous proportion. In Archytas’s solution, you take the two sides, produce right-angled triangles, and stick them together with a third and a fourth, to get four lines in continuous proportion. It is a reasonable story: Archytas’s solution was found as he was looking to extend a solution he already knew for a single mean proportional – finding how one can fit triangles in space to get the same kind of configuration. It is, in fact, hard to do – hence the spatial gymnastics. …
So, is it possible that something similar takes place in Archytas’s music? … many spurious works circulated in antiquity under the name of Archytas … In a lot of this, music does play a role: Pythagoreanism … was closely related to the scientific study of music. However, such spurious works tend to be more metaphysical or, indeed, almost mystical, and the very dry, technical character of this proof inspires confidence. This, then, is another significant piece of mathematics we may quite confidently ascribe to Archytas: “A superparticular proportion cannot be divided into equal parts by the interpolation of a mean proportional number.” …
… [Chords, harmonics] Archytas must have produced a complicated mathematical edifice, with both abstract arguments concerning numerical ratios as such and concrete numerical calculations, all in the service of a physical or even a cosmological theory of music. And it is indeed intriguing that this all ends up with the division of the tetrachord [Dict. a scale of four notes, the interval between the first and last being a perfect fourth] – quite literally, the division of one bigger ratio by the insertion of two numerical values in between. The division of the tetrachord is not at all the same as the finding of two mean proportionals between two given lines. The motivation and the mathematical tools are entirely different. One gives rise to sophisticated geometry; the other gives rise to sophisticated number theory. But the two do resonate, quite directly, in fact – they are the discrete and continuous solutions to the same problem of finding four terms in proportion. Both also combine sophistication with conceptual simplicity: this is what one can accomplish, starting with very little – other than enormous ingenuity. We have seen geometry in some detail, and we have good indications for harmonics, even for some number theory. …
… The interest in more abstract structures is perhaps to be seen in Eudoxus’s contributions to pure mathematics, as well. Eudoxus was the first to prove that the cone was one-third of the cylinder containing it. This achievement – now extant as a series of propositions in Book XII of Euclid’s Elements – would loom large, in particular, for the work of Archimedes. … I therefore just note now that we will see how Eudoxus’s measurement of the cone seems to have been motivated by an interest in rigorous proof: seeking a way to bypass, so to speak, infinity. …
The evidence is weak – once again, merely a scholium in the transmitted text of Euclid, stating that this book was the discovery of Eudoxus – but there is no special reason to deny it.
Aristotle seems to be aware of something like this book – and seems to know it as a fairly recent accomplishment. In general, it is the kind of book that should come relatively late, not because it takes so much for granted but, to the contrary, because it assumes so little. This is a study in foundations, taking terms that one could easily use without reflection – but then, insisting on defining them and demonstrating their properties. This is what mathematicians do, after much else has been done already.
And this is, to the Greeks, the most foundational topic of all: proportions, in full generality. We have seen proportions applied in music, geometry, and arithmetic.
[Euclid’s] Book V defines proportion and shows that all those properties – which previous mathematicians took for granted – were in fact correctly assumed.
The key definition (usually given as number 5) is astonishing:
magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
This is so abstract that it can only be legible to a modern reader with the aid of notation. …
… I am allowed to say, based on Eudoxus’s definition, that the four terms are in proportion.
You might be inclined to say: of course 5:4 and 10:8 will have the same pattern of relations of size under multiplications by various factors. This is because they have the same ratio! Eudoxus perhaps will not dispute this (it is not clear that his definition is taken ontologically, as the identification of the essence of what makes a proportion).
The point is that he has provided a mathematically determined equivalent to the pre-theoretical expression “being in proportion”. This term has been analyzed into the simple terms of multiplication, equality, greater than, and smaller than. At some abstract level, “proportion” and a certain set of simpler terms are equivalent.
Because the terms used in the definition have clear mathematical significance – we know what it means to multiply and to state equalities and inequalities between mathematical magnitudes – the definition can, in fact, be applied. For instance, to state that four terms are not in proportion is to state that there is at least one couple of factors where the order of size between the terms in the two ratios is not the same: we know what this means, then. … It is … possible, of course, that Eudoxus has joined a more robust field of research into the foundations of proportion, and that his contribution is simply the only one that is now known in detail.
There is a passage in Aristotle where he states that proportion was once defined – before Eudoxus, then? – through a different mechanism. The earlier contribution implied by Aristotle seems to imply a narrower, purely geometrical study of proportion. …
We see the peak alone of an ancient range – but we notice, once again, the proximity of mathematics to philosophy. The logical, almost philosophical motivation is palpable. … Theaetetus’s classification of irrationals, is so cumbersome, so little motivated – why would a mathematician, capable of finding charming results such as the golden section in the pentagon, devote so much effort to such a dead end? Perhaps because of the sheer metaphysical fascination of bringing order to the seemingly unordered. [Euclid’s] Book V is less tedious, but its results are even less surprising. Do we even care about the results? Are they there because, ultimately, the working mathematician needed them?
I am not so sure that there was even an urgent sense of need to prove such basic assumptions. This is because in the practice of Greek mathematics, an important set of results involves ratios’ inequality: for instance, that from a:b > c:d, one can deduce a:c > b:d. This is harder than the results concerning ratios’ equality or proportion. Once we get to this complexity, it does indeed become crucial to distinguish the valid results. And yet, [Euclid’s] Book V, as it now stands, is almost entirely confined to the very basic, the level that requires proof only if one is truly pedantic, and it does not engage with ratios’ inequalities at all. And yet, future mathematicians seem to have been content to take the results concerning proportion inequality for granted! After Eudoxus, mathematicians are content to take for granted that if a:b > c:d, then a:c > b:d. I therefore suspect that prior to Eudoxus, mathematicians were content to take for granted that if a:b::c: d, then a:c::b:d.
The definition is its own point, and its point is philosophical. In fact, the definition plays with philosophical fire. It is nonconstructive. That is, there is no finite, doable procedure that can establish that four terms are in proportion.
A single counterexample will establish that the four terms are not in proportion, but to verify that they are, one needs to check all equimultiples – all the possible ms and ns, all infinitely many of them, twice over – in order to conclude that a, b, c, and d are, in fact, in proportion. This does not mean that the definition cannot be applied by the mathematician – but the very willingness suggests an explicit assumption of the existence of a mathematical reality, preceding human construction. I do not believe that a mathematician conversing with Plato would fail to notice this, and therefore I suspect that Eudoxus, in fact, meant this. The definition is not the tool to derive the results (we trust the results regardless). To the contrary: the definition is the point. The function of the results is to verify the definition. (We find that the definition allows us to prove the basic results we expect to be true for proportion, and so we conclude that the definition is indeed correct.)
The definition seems to carve out, with supreme analytical precision, something deep about the nature of proportion – proportion, as it were, is a kind of extension of the concept of equality. But it is, precisely, an infinite extension, one that, for better or worse, escapes verification – which is necessary because the definition is required to cover finite as well as infinite cases, rationals as well as irrationals. The quest is for supreme generality and abstraction. As far as we can tell, later mathematicians did not extend or attempt to revise [Euclid’s] Book V. …It is, for the Greeks, something of a mathematical dead end.
But philosophers paid attention immediately. Aristotle seems to refer to Eudoxus’s definition, and he does this not as an aside but to make a fundamental claim: definitions should be produced in their full generality. As far as Aristotle was concerned, Eudoxus made a contribution to the theory of definition as such. Was this a correct interpretation? Was this Eudoxus’s intention?
At a more basic level: Was Aristotle, in a sense, the intended audience? …
… [This] much is clear: Euclid and Autolycus, writing a few decades later, already assume a very fixed genre; even earlier, many of the references by Aristotle to mathematics are explicit enough to get a sense of a form of writing that is almost identical to that of Euclid. It thus makes sense to read Euclid as an indication of what mathematics could have been like, already, in the mid-fourth century.
Let me quote, now, a simple proposition in its original form. Because it is relatively simple, it is unrepresentative (many propositions will have a more complex structure). I chose one that, although simple, already displays some important structures. This is [Euclid’s] Elements 1.35 …
Parallelograms that are on the same base and in the same parallels are equal to one another.
Let ABCD, EBCF be parallelograms on the same base BC and in the same parallels AF, BC.
I say that ABCD is equal to the parallelogram EBCF.
For, because ABCD is a parallelogram, AD is equal to BC. For the same reason, EF is equal to BC, so AD is also equal to EF.
And DE is common; therefore, the whole AE is equal to the whole DF.
But AB is also equal to DC; therefore, the two sides EA, AB are equal to the two sides FD, DC, respectively, and the angle FDC is equal to the angle EAB, the exterior to the interior; therefore, the base EB is equal to the base FC, and the triangle EAB will be equal to the triangle FDC.
Let DGE be subtracted from each.
Therefore, the trapezium ABGD that remains is equal to the trapezium EGCF that remains.
Let the triangle GBC be added to each.
Therefore, the whole parallelogram ABCD is equal to the whole parallelogram EBCF.
Therefore, parallelograms that are on the same base and in the same parallels are equal to one another, which it was required to prove.
This is very distinctive writing. Not only is it unlike Homer or Euripides, but it is also, evidently, very much unlike Plato. When a Greek wrote in this precise genre, then, even if his audience might have consisted of philosophers, they had no doubt that what they read was, specifically, mathematical.
The most distinctive thing is, obviously, the combination of diagrams and letters.
This is pervasive in all extant Greek mathematical writing and is one of the clearest things reflected in Aristotle’s references to mathematics.
Mathematical texts come as brief, distinct passages, each making an individual claim. Each is followed by a drawing, which is a simple network of lines, some straight, some curved. The drawing is interspersed with letters of the Greek alphabet, typically (with some exceptions) set next to some of the points of intersection in the drawn network. … These letters act as the written equivalent of a hand gesture. To write down “BC” is equivalent to pointing your hand at the particular line segment bounded by the points of intersection marked in the diagram by the letters B and C. The diagram thus has two faces. On the one hand, it makes the text concrete, and its reading becomes a reenactment, in the reader’s mind, of the author’s direct presence, pointing to a piece of drawing. On the other hand, it is consciously literate, not merely a written form but, indeed, one that explicitly refers to letters as its tool of choice.
Chapter 3
The Generation of Archimedes
[Archimedes b. c.287 BC, d. c.212]
[change of approach generations after Aristotle]
… Archimedes was a scientist – as opposed to a natural philosopher – not because he pursued the distinctive career path of a scientist, defined by the institutions of science. In this institutional sense, indeed, science did not exist. There were no royal societies, no journals of scholars, no university departments of science, no science chairs to fill.
[His choice of] literary genre was written with a particular audience in mind – that of fellow mathematicians and, to a much lesser extent, the elite audience as a whole … written, always, against the background of wider literary currents, emphasizing subtlety and surprise.
And it is those motivations, this sociology – and not the ideas of an Aristotle or a Plato – that account for the specific scientific route taken by Archimedes. He picked up a particular technique … because its subtlety (required, ultimately, by its subject matter) made a certain kind of surprise especially satisfying. Hence the infinitary methods. And he saw the possibilities of applying geometry to a seemingly unrelated field – the study of centers of the weight in solids, balancing outside and inside liquids – because there was a particular payoff of subtlety and surprise to be obtained by the bringing together of apparently irreconcilable, maximally distinct fields of study. …
… The objects discussed by Archimedes – and by the following generation as a whole – are hardly motivated by concrete applications. There seems to be a special value placed, if anything, precisely on somewhat outré objects: a conchoid, a spiral; above all, again and again, the conic sections. Other cultures simply did not consider such objects as part of their mathematics: Why should they? One studies, normally, things such as rectangles and triangles, squares and circles, perhaps cubes and spheres. This is natural when mathematicians are at the service of a society. You refer to those objects that are already known as part of more or less established practical usage.
The study of, let us say, the paraboloid of revolution [circular paraboloid] can arise only when mathematicians can make up their own objects to study and essentially no longer even worry about an outside audience. Of course, one does not just invent objects for study on a whim. The typical route we see – and one that is found time and again in the later history of mathematics – is that the thing that was first studied as a tool for other purposes gradually becomes the subject of research for its own sake. Study the finding of proportions, and therefore invent the tool of conic sections – until, at some point, you are no longer studying proportions as such, but rather, you simply study the conic sections themselves. Why is this possible? Because your audience shares, with you, the interest in the tools: because it is an audience of fellow mathematicians.
And so, this is what the generation of Archimedes did – to the point that, even when studying physics, the preferred object of study was, precisely, conic sections: the center of the weight of a parabolic slice; the stability of a segment of a paraboloid; a paraboloid mirror. Something crucial took place in the making of science in the generation of Archimedes: the making of a well-defined field, pursued apart from philosophy, based on its own concerns and techniques. The manner in which this happened favored, precisely, an engagement with a ludic realm, of toy objects of no obvious practical significance, pursued purely for the aesthetic value of their study. … This mathematization was based not on Plato’s conceptual preference for the abstract over the material. It was based, instead, on the autonomy of mathematical research.
Chapter 4
Mathematics in the World
… Writing about theoretical mechanics (as opposed to concrete machines) was simply not a recognized genre. One only wrote about it in the rare case that one explicitly strove … to write about everything. This brings us to our final example of ancient theoretical mechanics, coming from yet another context where an author – or better put, a group of authors – sought universal scope.
The treatise in question is titled Mechanics and is ascribed, in our manuscripts, to Aristotle. It is likely that the attribution is not precise and that the philosopher Aristotle did not write this treatise. (Quite simply, this treatise does not carry the imprint of Aristotle’s genius; this view is widely shared, and it is usually said to be by “pseudo-Aristotle”.) The language and the overall attitude of wide-ranging curiosity are indeed reminiscent of Aristotle, and the consensus is that this is the work of an author trained in the Aristotelian school, close in time to the master. An early text, then, perhaps roughly contemporary with Euclid’s Elements – and surely preceding Archimedes.
The Aristotelian project was, indeed, universal. A follower of Plato, Aristotle shared his master’s fascination with the sciences. Unlike Plato, Aristotle went on to expand this project systematically, striving to survey a wide array of phenomena and to provide them with rational accounts.
The Aristotelian corpus is like a running commentary to the universe: it discusses logic, animals, speeds and motions, sense perception, comets, rainbows, ethics and rhetoric, sleep, the Athenian constitution, tragedy, chemical mixtures, predicate logic .. . Individual works very often devolve into nearly unstructured lists of observations, followed by explanations. (Why is it the case that .. . ? Maybe because .. . ?)
The series of observations and explanations in Ps.-Aristotle’s Mechanics all involve the working of various human artifacts (notably, including no military devices). They are mostly accounted for on the model of the lever – no surprise there. Much more surprising to us, the lever is understood not on the basis of anything such as Archimedes’s geometrical proportions but, instead, in terms of the geometry of the circle. The central model of the Aristotelian mechanics is of a rotation around a fulcrum point, and most of the results reduce to the fact that longer circles cover greater distances. (This is supposed to explain, essentially, why the weight further away from the fulcrum exerts the greater force; such texts remind us that Archimedes was indebted to a tradition – and how much he improved upon it.)
Once again: the Aristotelian Mechanics is, on the whole, more wrong than right.
And once again, it appears to be an isolated text. The latter is true, in fact, for many other works within the Aristotelian corpus. Aristotle produces a comparative study of the animal kingdom; a predicate logic; a theoretical study of poetics. In all such cases, it is difficult to find any authors, away from Aristotle’s immediate disciples, who add anything new to these uniquely Aristotelian projects. …
… Ancient civilization was large, and even the things that happened very rarely would happen from time to time. The exceptions occurred, and they mattered: Ps. Aristotle’s mechanics, Archimedes’s works in mathematical physics, Pappus’s Book VIII – all those works were not merely composed but were also copied and preserved. In time, they would provide an important impetus to the scientific revolution. …
The Source:
Reviel Netz*, A New History of Greek Mathematics, Cambridge University Press 2022
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