I1 The Development of Greek Mathematics: Detachment from a Practical Context

In spite of its theoretical focus, Greek mathematics had its origin in a practical context. Geometry derived from land-surveying and city-planning, as we can see already from the etymology of the English word ‘geometry’, which comes from the Greek word geometria and literally means ‘measuring the earth’ or ‘land-surveying’. Herodotus claims in his Histories II, 109 that it is from the equal division of the land in Egypt that the Greeks learned geometria. And arithmetic was originally closely tied to administration and commerce – a connection against which Plato still argues as not covering the most important aspect of mathematics in Republic 525c–d.Footnote ¹⁰

But very early on, Greek mathematics focused on mathematical characteristics as such, independent of any practical context. We will see in Section 3 in the body of this Element that it is only due to such a theoretical understanding of mathematics that irrationals were discovered – for practical purposes, the approximations we derive from measurements would have sufficed.

It seems that it was the abstraction of mathematical structure from the human practical context that made these structures also applicable to the understanding of nature and thus influenced the conceptualisation of the world – extending arithmetic and geometry to the universe as a whole. Thus, we also find the idea that mathematical structures are essential for the physical world early on. We will see in Section 1 later in this Element that, for the Pythagoreans, numbers and proportions constitute the universe, while for Plato geometrical forms as well as proportions are employed in the setup of the universe. And for Plato and many subsequent thinkers, the universe has to be spherical, because of the geometrical perfection of the sphere (which includes that all points on the surface have the same distance to the centre and that all the other Platonic solids can be inscribed in the sphere, in the way we see it in Kepler); the motions of the heavenly bodies accordingly have to be circular, which guarantees their intelligibility.

I2 Specific Demarcation of Mathematics in Ancient Greece

What did the Greeks in fact understand by mathematics? If we look at the Greek word from which our term ‘mathematics’ derives, mathemata, we see that it originally had a much broader meaning, indicating everything that is learnt, pieces or fields of knowledge. It encompasses all those subjects where we need to go through a certain course of steps in order to learn them and that hence may not be known by everyone.

It is in effect three generations of philosophers whose usage of the word eventually led to the meaning we are familiar with today: the Pythagoreans, Plato, and Aristotle. First, the Pythagoreans distinguish among their students between the akousmatikoi (the exoteric learners, who are only aware of the practical rules of conduct) and the mathematikoi (the esoteric learners who have gone through the full theory, or the advanced learners). And they also seem to have been the first to use ‘mathematics’ as a common name for arithmetic and geometry together. Secondly, Plato, in his depiction of the education of the future leaders of the ideal state in Republic VII, gives a curriculum that builds mainly on arithmetic, geometry, astronomy, and music for the advanced learners; which are thus the mathemata to be learned. Finally, Aristotle divides the theoretical sciences – the first division we find in Western thought – into first philosophy (metaphysics/theology), physics (natural philosophy), and mathematics (see, e.g., Metaphysics 1026a18–19).

This long process of coining the meaning of the word also reflects the fact that mathematics and philosophy are usually understood to derive from the same origin, which was in general seen as wisdom.Footnote ¹¹ It is not clear when mathematics separated off from this, but it was one of the first, if not the first, science to do so. Accordingly, it seems to be its own field already in the fifth century BCE, and could be seen as paradigmatic for other sciences, as we will see later in Section 2 in the main body of the Element. As such, mathematics seems to have influenced philosophy by the rigour of its argumentation, by building a system starting from a couple of basic axioms, and by its use of deductions. And it also became an important subject of philosophical reflections for the question of what is essential for a science.

Among the sixth-century Presocratics, we find several philosopher-mathematicians, like the Pythagoreans and perhaps also Thales. Likewise, in the fifth and fourth centuries BCE we find thinkers working in both fields, such as the atomist Democritus, who wrote on conic sections, and the Pythagoreans Philolaus and Archytas. And Plato’s Academy seems to have been a centre where not only were philosophers trained, but also mathematical research was performed by people like Eudoxus and Theaetetus. A sign reading ‘Let no one ignorant of geometry enter here’ was allegedly placed over the entrance to the Academy. Plato himself saw mathematics at times as paradigmatic, at others as propaedeutic for philosophy. Aristotle also took mathematics as paradigmatic for philosophy in its method in certain respects, but distinguished it from first philosophy and physics by its objects – mathematical objects are changeless, but have no existence independent of empirical things. The Hellenistic philosopher Epicurus, however, excluded mathematics from the sciences as having no practical relevance, and for the Stoics likewise it was not central.

Which mathematical sciences belong to mathematics in ancient Greece? For the most part the four disciplines arithmetic, geometry, astronomy, and music are seen as constituting the mathematical sciences, a quartet we find first in the Pythagorean Archytas and in Plato’s Republic.Footnote ¹² In the Middle Ages they became the quadrivium, which together with the trivium formed the basis for liberal education at universities. Aristotle, however, groups astronomy and harmonics, together with optics, as the more ‘physical’ branches of mathematics.Footnote ¹³ These latter sciences especially are close to natural philosophy for Aristotle.

An understanding of ancient mathematics faces the additional problem that the ancient Greeks distinguished between arithmetic and what they call logistikê – a distinction not known to modern mathematics. There are different interpretations of this distinction, but for the most part, arithmetic is understood as dealing with number theory and logistikê as the art of calculation.Footnote ¹⁴

Finally, it is an interesting question whether the ancient Greeks thought that there exists a general mathematical science or not – here it seems philosophers and mathematicians come apart. The first text where we find the idea of a general mathematics discussed comes from Aristotle, who claims that universal mathematics applies to all mathematical kinds (Metaphysics 1026a27); scholars have usually understood him to be referring to Eudoxus’ theory of proportion which holds for different mathematical sciences. What is left open is the question whether general mathematics possesses its own specific subject (perhaps something like pure quantity) or not.Footnote ¹⁵ But we do not seem to find any such universal mathematics in Euclid. Euclid’s concern for homogeneity indicates that he thinks of a proposition in Book V as a unified formulation for a number of analogous propositions concerning particular magnitudes like lines, planes, and so on. He does not seem to think of it as a single proposition about more abstract objects like magnitudes or quantities as such, and he treats proportions in geometry and in the arithmetical books separately. Euclid is, however, also not interested in discussing the metaphysical basis for the applicability of propositions, but only in reasons to believe them.

I3 Relationship between Geometry and Arithmetic

Although geometry, arithmetic, music, and astronomy were seen as the main mathematical sciences, we will concentrate on geometry and arithmetic in this Element, which is also what Euclid’s Elements focuses on. While we can clearly understand, for example, Books I–IV of Euclid’s Elements as plane geometry, and Books VII–IX as concerning arithmetic, it is less clear how exactly geometry and arithmetic were defined and in which relation they were seen to each other. We may think of modern geometry as a study of spatial structures, but ancient Greek mathematics seems to treat it as a study of magnitudes, more precisely of figures. And while we seem to get a neat division of Greek mathematics into the study of multitude or discrete quantity, namely arithmetic, and the study of magnitude or continuous quantity, namely geometry, it is unclear whether it was conceptualised like this before Aristotle’s Categories.Footnote ¹⁶

In spite of hints of a universal mathematics, Aristotle usually treats geometry and arithmetic as two different sciences that differ in their genus. Their exact relationship was a matter of intensive dispute, which is connected with the question of a possible superiority of one over the other. (This also concerns questions like whether the operations of addition, subtraction, multiplication, division, and so on are geometrical or arithmetical.)

In contrast to Babylonian and Egyptian mathematics, Greek mathematics sometime in the fifth century BCE started to geometrise mathematics, performing mathematical proofs from then onwards mainly in geometrical terms (presumably since certain topics, like incommensurability, could only be dealt with geometrically, and arithmetic could not handle fractions).Footnote ¹⁷ Thus at least since 400 BCE geometry was dominant vis-à-vis arithmetic in Western mathematics, a dominance that lasted until early modern times. It is also reflected in the fact that Euclid’s Elements shows a much more systematically developed geometry than arithmetic, giving many results and procedures in geometrical terms that we would give in arithmetical ones. Furthermore, Euclid represents numbers by lines, and what we understand as arithmetic operations, such as addition and subtraction, he seems to understand in geometrical terms (e.g., what we would think of as adding number x to number y Euclid represents as extending a line AB by the length of line CD).Footnote ¹⁸ Geometrical terminology is used for arithmetic, for example, when in Book VII, definitions 16 and 17, the result of two numbers having been multiplied is called a plane number, and that of three numbers having been multiplied a solid number.

In philosophical reflection, however, we find different assessments of the relationship between arithmetic and geometry. For the most part, arithmetic is seen as superior. The Pythagorean Archytas calls it superior to geometry and other sciences in DK47B4, because it treats its objects in a clearer way and ‘brings proofs to completion where geometry leaves them out’.Footnote ¹⁹ And for Aristotle arithmetic is prior, more exact, and simpler than geometry, since it is based on fewer things.Footnote ²⁰ His example in Posterior Analytics is a point in geometry, which, like a unit in arithmetic, is a basic being, but in addition also has a position, so that it involves more features than the arithmetical unit. The greater simplicity and accuracy of arithmetic brings it closer to first principles than geometry for Aristotle and thus demonstrates the superiority of arithmetic over geometry.

The Pythagorean Philolaus, however, calls geometry the source and mother-city of the other mathematical sciences in DK44A7a and thus seems to assume geometry to be superior to arithmetic. This raises the question whether the differences in assessment of their relationship derives from a different point of view or from a development within mathematics. The latter suggestion has sometimes been connected with the fact that within ancient Greek mathematics, arithmetic seems to have been more prominent in the beginning before the geometrisation of mathematics in the fifth century (for example, Szabó (Reference Szabó and Christianidis2004) understands Archytas’ claim of the superiority of logistikê as depicting an older Pythagorean attitude dominant before the geometrical turn). Against this developmental thesis, however, speaks the fact that Aristotle, quite some time after any geometrical turn, claims arithmetic to be superior. And also the Pythagoreans were not unified in this respect, since Archytas, living in the first half of the fourth century, claims arithmetic to be superior, while Philolaus, a generation older than him, claims this for geometry. So it is more likely that these different assessments are due to different points of view – the mathematicians tending to geometry since this is the more powerful discipline within ancient Greek mathematics, and philosophers tending to arithmetic since its basis seems to be closer to first scientific principles.Footnote ²¹ In any case, the tension between the status of arithmetic and geometry led to general ontological and epistemological questions, some of which we will see in the first two main sections.

I4 Specificities of Greek Mathematics

While much more abstract than Babylonian mathematics, ancient Greek mathematics is much less abstract than modern mathematics, and the abstraction we find in ancient mathematics is of a very different sort than our modern one. Thus, when we talk about the philosophy of ancient Greek mathematics, we should be aware that with mathematics we are talking about a somewhat different beast than when we refer to modern mathematics.Footnote ²²

Greek mathematics has a certain level of abstraction in that it is interested in general mathematical features as such; it is, for the most part, non-metrical, that is, it does not use numbers for general mathematical problems, and it is independent of any concrete practical context. But ancient Greek geometry is much closer to perception in its reliance on diagrammatic representations than its modern pendant. It is pre-algebraic, pre-structural, and pre-set theoretic.Footnote ²³

Ancient Greek mathematics does not employ functional variables, but only parameters (and proportions) – so while for certain operations we can freely chose a value, once it is chosen it always has to be the same value and cannot vary. Infinity is in part presupposed by ancient Greek mathematicians, without, however, taking up the fundamental discussions about this concept we find in the philosophers. And, as we can see in Euclid, Greek mathematics of classical and early Hellenistic times gives us only the minimal meta-language to connect arguments – it is only philosophers who deal with meta-reflections. Let us finally have a closer look at the Greek understanding of numbers.

I5 Notion of Numbers

Number theory started in what we may consider as a systematic way in ancient Greek mathematics. Books VII–IX of Euclid’s Elements provide us with a definition of primes, an algorithm for computing the greatest common divisor of two numbers (what we now call the Euclidean algorithm; VII, 2), and the first known proof of the infinitude of primes (IX, 20).

But there are three important differences in the ancient Greek understanding of numbers to modern accounts:

1. (1) First, a number is a collection of units; it is defined as ‘multitude composed of units’ (Elements VII, 2). The number seven, for example, is a collection of seven units.Footnote ²⁴

2. (2) Second, 1 is for the most part not seen as a number, but as the beginning or principle (archê) of number, what defines the unit. 1 is not a number in the way 3 and 4 are, since the 1 determines what we count (1 defines the unit with which we count, for example, book, and then we count how many books there are in my room).Footnote ²⁵ So there is a strong difference between 1 and numbers. We see that units thus serve double-duty – they are the constituents of each number and they determine the one that is the basis for counting.

3. (3) Finally, for the Greeks, numbers mean positive integers, so we are dealing with a theory of natural numbers; other numbers are not known. Irrational numbers, like $\sqrt{2}$ , are not numbers for the Greeks, but proportions between magnitudes, since they can deal with them only geometrically, not arithmetically. Number in itself implies rationality and countability for the Greeks.

While Babylonian and Egyptian mathematics used fractions, there were no real fractions in Greek mathematics up to Archimedes in the third century BCE (apart from ⅔ in a practical context). Greek mathematicians took the 1 as indivisible unity, so that there could not be any fractions;Footnote ²⁶ instead, they used proportions and submultiples. Submultiples allow us to treat fractions as multiples of a more basic unit; for example, if I have $\frac{1}{7}$, I can understand one as a basic unit which I have to take seven times to get to the old basic unit. The expression “$\frac{1}{7}$” thus gives me two different units and indicates a mathematical operation (the relation of the smaller to the bigger unit), not a number.

Numbers are not defined in terms of successor functions or by what they allow us to do and they are not sets. Instead they are understood as a quantified plurality of things of some sort. Numbers used in ordinary life are seen as tied to a concrete group of things (for example, a three of cups, if I am counting my cups). And what is sometimes called ‘mathematical number’, that is, the numbers mathematicians deal with, is tied to a monas, namely a mathematical unit (which is what Euclid refers to in his definition of numbers).Footnote ²⁷

The ancient Greeks also investigated which further features characterise numbers. This can be seen especially in the discussion about oddness and evenness of numbers and of prime numbers. The Greeks not only distinguished between odd and even numbers, but also even-times odd numbers,Footnote ²⁸ even-times even numbers, and so on. And they not only discuss prime numbers, but also numbers prime to each other.Footnote ²⁹ The number 2 is an especially interesting case, for it is an even number for Plato and Aristotle, and a prime number for Aristotle and Euclid.Footnote ³⁰ It is, however, neither even nor prime for the Pythagoreans, but constitutes, together with the 1, the first principles of all numbers; and for other mathematicians, like Nicomachus, being a prime number is a sub-division of odd numbers.

Early Greek mathematics represented numbers by pebbles,Footnote ³¹ while later on numbers were represented by lines, which is also what we find in Euclid’s arithmetical books. Like the Egyptians, the Greeks used a decimal system (in contrast to the Babylonians, who used also a sexagesimal system). And the Greeks used two numerical systems with different advantages and disadvantages: the Herodiadic and the ordinary Ionic alphabetical numerals.Footnote ³²

1.1 Numbers as the Ultimate Constituents of Things with the Pythagoreans

When we look at early Pythagoreanism, we will distinguish between those Pythagoreans that Aristotle discusses in his Metaphysics and the earliest Pythagoreans of whom we possess reliable fragments, namely Philolaus, a contemporary of Socrates, and Archytas, a contemporary of Plato.Footnote ³⁷ We will start with the Pythagoreans as reported by Aristotle and then continue on to Philolaus.Footnote ³⁸

Aristotle reports in his Metaphysics that the Pythagoreans assume mathematical objects to exist in perceptible objects and that their principles and elements are the principles and elements of the sensible world:

So for the Pythagoreans, mathematical properties (presumably properties like being quantifiable, being of odd or even number) can be found in sensible phenomena, in whatever we empirically observe. This seems to have led them to infer that sensible bodies either consist of numbers or are phenomena of numbers in some way. While Plato’s idealism and Aristotle’s abstractionism may strike us as not true but plausible or at least intelligible, some readers may simply be flabbergasted by this Pythagorean position. However, it seems to have been a very early (if very specialised) position in Greek mathematics and philosophy and features of this position have been taken up repeatedly in the history of thought, most recently perhaps by Baron (Reference Baron2021), who calls his suggestion that the important properties of mathematical entities are structural and as such also to be found in the physical world a ‘Pythagorean proposal’. Let us look at possible reasons for the Pythagorean position.

It may be seen as answering the puzzle how it can be that numbers follow their own rules (independent of the physical world), but yet are applicable to the sensible realm to the degree that mathematical statements are true of the world.Footnote ³⁹ This combination would be possible, according to a Pythagorean picture, if mathematical objects are in the world and constitute sensible bodies – as such constituents, numbers can be understood independently but are yet immediately connected with the perceptible world.

The applicability of mathematics to the physical world is shown with the example of music and astronomy. There seems to be one phenomenon in particular mentioned in our two quotations that led to the idea that numbers constitute things: musical scales and harmonies. It is, after all, not the relationship of this very string to that very string that constitutes an octave, but the relationship between every pair of strings whose lengths are in a relation of 1:2. So it ultimately seems to be the relationship of 1:2 (which can be realised in different materials that can vibrate so as to produce a sound) which constitutes the octave. But if numerical ratios underlie musical intervals, they may also underlie other perceptible things and phenomena which display mathematical features.

The preceding passages suggest a strong interpretation of the idea that numbers underlie the sensible world in the sense that numbers are indeed seen as the ultimate constituents and essence of sensible things.Footnote ⁴⁰ Aristotle claims that for the Pythagoreans, numbers are not only the principles of everything but also that the whole of nature is ‘modelled after numbers’.

But there is also a weaker interpretation available for the idea that mathematical structures underlie the physical worldFootnote ⁴¹ – an interpretation that may be displayed by some fragments of the Pythagorean Philolaus, who claims that ‘all things that are known have numbers. For it is not possible for anything to be thought of or known without this’ (DK44B4).

Philolaus here is not making a claim about the whole universe, but about all things that can be known (which may or may not have the same extension as the whole universe). And these things are not said to be number, but only to have number (in contrast to the picture in Plato’s Timaeus, they themselves possess numbers, rather than have numbers bestowed upon them). The basic idea seems to be that we can have knowledge of things, and our reason essentially works with numbers, so there has to be something numerical about the things known (for example, that they are quantifiable). Here numbers are a necessary condition for knowledge and because things we know have them, mathematical operations are presumably also applicable to these things. But it is left open how strong the ontological commitment is that is entailed in the idea of ‘having a number’ – it may simply mean that physical things have a quantitative aspect, or that they are a unity composed of a number of parts.Footnote ⁴² And while the fact that numbers are tied to the knowability of things suggests that they are not tied to some accidental feature, it is left open whether there may also be other features we can know about things, for example, some quality, that may be something over and above the fundamental knowledge we have based on numbers.

Three points seem to be in the background of the Pythagorean assumption that things either have numbers or are even constituted by numbers: first, the wide-spread understanding in ancient Greece that numbers are multiples of a certain unit, usually of concrete things,Footnote ⁴³ and thus closely linked to the perceptible world; secondly, a peculiarity of the Pythagoreans, that numbers are seen as generated;Footnote ⁴⁴ and finally, the Pythagorean idea that there is a cause for the separation and distinction of the number series – their discreteness is accounted for or grounded by the void.Footnote ⁴⁵ Thinking of numbers as requiring void to separate them suggests understanding numbers along the lines of bodily stuff. And it fits the fact that the Pythagoreans conceived of numbers with the help of pebbles. Accordingly, when they claim that physical things are constituted out of numbers, this must also be seen against the background of a more physical understanding of numbers. Aristotle, however, thinks that the Pythagorean position confuses the indivisibility of numerical units with the indivisibility of physical things and accordingly faces two problems: first, it leads to the assumption of atoms (indivisibles) of a sort, and Aristotle attempts to show in his Physics that atomism gets us into problems if assumed for the physical worldFootnote ⁴⁶ and destroys mathematics (for then a mathematician could not cut a line wherever she needs to, but only where atoms allow); second, it remains unclear how numerical units, which in themselves have no physical magnitude and weight, can make up something with a physical magnitude and weight.Footnote ⁴⁷

Moving on to Plato’s account of mathematics now, we will encounter two of his central ontological claims concerning mathematical objects: (a) the claim that mathematical objects are entities with their own independent existence, and (b) the idea in the Timaeus that geometrical bodies, which are themselves made up of perfect triangles, underlie the four elements and are thus the building blocks of physical things. The first claim (a) derives from a discussion about the question what ontological status mathematical objects have which Plato raises explicitly in his Republic, and to some degree also in his Phaedo. It seems to hold true of mathematical objects in general, but the emphasis seems to be on arithmetic. And so we learn of numbers not only that numbers as such exist independently of the physical world, but also that the many instances of each individual number which the mathematicians use for their calculations are intermediates between the sensible and the intelligible objects. The second claim (b) derives from the attempt in the Timaeus to explain the perceptible world as something that can be known, that is intelligible. The focus here is on geometry, but proportions are also understood to underlie the orbits of the motions of the heavenly bodies.

1.2 Mathematical Objects as Part of the Intelligible Realm in Plato’s Republic and Phaedo

Aristotle, Plato’s main opponent with respect to the ontology of mathematical objects, provides us with a rationale for the assumption of separately existing mathematical entities in his Metaphysics 1090a35–b1: those thinkers who make numbers exist separately, like Plato, do so because axioms do not seem to hold of perceptible things, but yet are true in themselves.

In his example of the line, that we will discuss in the next section, Plato assumes mathematical objects to exist separately as intelligible entities that are not derived from the perceptible world (even though mathematicians use sensible objects as examples). But his mathematical ontology is in fact more complex than that. For the main claim by Plato and many Platonists about mathematical entities, namely that they exist not only independently of physical entities but also separately in an intelligible realm, leaves it open how to think of this logical space. And this separate logical space is not necessarily the same as the logical space of Platonic Forms. We get a distinction among mathematical entities between those that behave very much like Forms, and those that are the tools of the mathematicians that have an intermediate status between Forms and sensible things. Let me explain this a bit more with respect to its most prominent example, the distinction between what is commonly called Form numbers and mathematical numbers.Footnote ⁴⁸

Forms are, roughly speaking, separately existing universals (Beauty as such) that are eternal, immutable, simple, and existing on their own.Footnote ⁴⁹ In the same way in which on a Platonic account there is a Form of Justice, and a Form of Equality, there seems to be a Form of twoness (Phaedo 101c) – the Form of two is what it means to be two. Every pair, and every 2 I use, participates in twoness. Being a Form also implies that there can be only one of it.Footnote ⁵⁰ No mathematical operation is defined on the Form number, since this would not fit with Forms existing independently. In addition to Form numbers, there are also the numbers the mathematicians work with when they multiply, for example, two times two. Of these there has to be a plurality and operations need to be defined on them, otherwise our arithmetical practises would not be possible. For the Platonists, we thus need to have one intelligible object that grounds twoness of which there can only be one, the Form number, and another one that can be an object of mathematical operations and of which there can be a plurality, mathematical numbers.Footnote ⁵¹

These mathematical numbers are also independent of the sensible two I write in the sand and of concrete pairs in the perceptible world; if, let us say, we were living in a world which contained only five items, a mathematician could still calculate what is six times six, even if no items corresponded to this operation in the perceptible world. Accordingly, mathematical numbers seem to be treated as so-called intermediates:Footnote ⁵² they possess an ontological status in between the individual intelligible single Forms and the many perceptible things, since they are many, like the perceptible things, but yet intelligible and not sensible, like the Forms.Footnote ⁵³ The full Platonic story in fact even seems to contain four kinds of ‘twos’: (1) the perceptible pairs in the world, (2) the mathematical numbers which are used in mathematical operations, (3) the Form number two, and finally (4) the indefinite two (or dyad) which Aristotle reports to function as the principle of plurality in Plato’s unwritten doctrine.Footnote ⁵⁴

1.3 Mathematical Objects as Underlying the Physical Realm in Plato’s Timaeus

We saw that with the Pythagoreans, mathematical objects constitute the physical world. And also according to Plato’s Timaeus, mathematical objects exist in the world – geometrical bodies underlie the physical phenomena we perceive. In contrast to the Pythagoreans, it is not numbers but geometrical bodies that Plato sees employed. Moreover, these mathematical objects used in the Platonic universe originally exist separately from the physical world and are thus independent of it; the geometrical solids are used to bestow order onto the originally given chaos and to transform it into a regular cosmos.

The Timaeus is Plato’s cosmology and most encompassing account of the natural world –from the microstructure of the atomistic particles underlying the basic elements of the world all the way up to the arrangement of the heavenly bodies. It gives an account of how our cosmos was shaped by a divine demiurge out of an independently existing space-like receptacle which is filled with traces of the physical elements. Our physical universe is seen as understandableFootnote ⁵⁵ because the demiurge builds it by imitating an intelligible model; and he does so by using mathematical structures in order to imbue the irregular traces of the chaotically moving elements with order and measure. It is in three areas that the demiurge uses mathematical structures: (a) proportions are used for connecting all the physical material there is in the world body, thus securing the unity of the world body, and for determining the order according to which the fabric of the world soul and thus also the orbits of the heavenly bodies are formed (Timaeus 31b–32c and 35b–36d); (b) the number series is employed for measuring the motions of sensible objects (39b–d); and finally (c) the material atoms are built from geometrical bodies (53a–55e);Footnote ⁵⁶ we will concentrate on this last point.

Before the divine demiurge starts his work, there are only traces of the four so-called elements (53b). With these four elements, fire, air, water, and earth, Plato takes up what in the physical investigations of his time was seen as the material foundation of the perceptible world, while he also wants to make clear that these are not really the most basic elements there are. For they themselves are made up of geometrical bodies. Originally there are solely unintelligible traces of the elements. It is only with the work of the demiurge that order and measure are introduced into the world, so that the presumably crooked surfaces of these elemental traces are formed into straight ones that make up the surfaces of geometrical bodies.Footnote ⁵⁷ The geometrical bodies the demiurge chooses as the basis for the physical ‘elements’ are the most regular (and thus most ‘beautiful’) solids, what we have come to call ‘Platonic bodies’Footnote ⁵⁸ – tetrahedron, cube, octahedron, icosahedron, and dodecahedron. Here Plato seems to have employed new mathematical research of his time, presumably undertaken by Theaetetus, that showed that there can only be five convex polyhedra that are fully regular in the sense that all faces are congruent regular polygons and the same number of faces meet at every vertex.Footnote ⁵⁹ This research showed the exhaustiveness of these regular polyhedra which Plato took up as a suitable basis for material ‘elements’.

Thinking of bodies in a mathematical way, Timaeus claims that these geometrical solids are composed of basic surfaces – of triangles. As there is no single kind of triangle out of which all the Platonic solids can be constructed, it is assumed that the demiurge uses two different kinds of triangles: the isosceles right-angled triangle out of which the surfaces of the cube are formed, and the half-equilateral triangle as a basis for the tetrahedron, the octahedron, and the icosahedron (see Figure 2). So the real ‘elements’ (in the sense of the basic foundation of everything there is in the sensible world) are not fire, air, water, and earth, but two kinds of triangles; and Plato leaves it explicitly open whether there may be some even simpler elements out of which these surfaces of the world are originally formed, such as lines.

Figure 2 Platonic solids made out of basic triangles.Footnote ⁶⁰

The different geometrical solids are ascribed to the four elements according to the following kind of similarity: as the cube has the most stable base (given its surface area), it is the basis of earth, which is the most immobile of the elements; whereas the body with the fewest faces must be the most mobile and the lightest, thus the tetrahedron underlies fire.

While a single geometrical solid underlying the physical elements is too small to be seen, an aggregated mass of them is perceptible. And, as in the case of other atomistic accounts, what we then perceive is not simply an aggregate of atoms, for example, not a bunch of mathematical tetrahedra, but the phenomenon we are used to, fire. In contrast to merely mathematical tetrahedra, the tetrahedra constituting fire have a tendency to move to other pieces of fire and are connected to the Form of fire in that they are an image of that Form (51b–52a).

Like mathematical solids, these building blocks of the elements can also be transformed into each other. The transformation rules follow their mathematical base: thus a particle of air can dissolve into two particles of fire, since an octahedron provides the surfaces required to build two tetrahedra out of it.

While the Plato of the Republic is concerned about the intelligible status of mathematical objects, in the Timaeus we find Plato focused on the usage of geometrical figures for the explanation of the universe. He does not discuss the ontological status of the geometrical solids, but we have also no reason to assume he has given up on his earlier idea of ‘the square itself’ (Republic 510d–e), only because he is dealing with many squares in the physical world. For him, the many geometrical solids are used to make the world understandable – and literally so, since they (and not just an approximation to them) are the constituents of the physical elements. They can keep their non-perceptible status due to the fact that they are too small for perception and the many geometrical solids that together form a phenomenon appear not as geometrical solids, but as physical elements.

This ‘geometrical atomism’Footnote ⁶¹ allows Plato to tie the plurality and changeability of the phenomenal world to intelligible mathematical structures. He traces physical and perceptual features, like the piercing experience of the heat of fire, back to mathematical ones, like the acuteness of the angles of the tetrahedra constituting fire (Timaeus 61d–62a). This mathematisation of the universe, however, also shows that the physical realm cannot be assimilated to the mathematical realm without further ado. Plato’s attempt to do so leads to at least three problems: first, for reasons of regularity and exhaustiveness, Plato assumes the Platonic solids to be the basis of the elements – but there are five Platonic solids, and only four physical elements in ancient physics; so the mathematical basis does not fit exactly the physical requirements. Plato is aware of this problem and accordingly gives an account of what happens to the fifth geometrical element eventually: the dodecahedron is assumed to be the form of the world as a whole (55c), as it seems to be close enough to a sphere, which was assumed as a form for the world body before because of its regularity and completeness (33b). Second, while Timaeus himself suggests that on the phenomenal level, it appears as if all four elements can be transformed into each other (49b–c), the mathematical basis in fact excludes earth from such transformations since cubes are made of a different kind of triangles than tetrahedra, octahedra, and icosahedra (54b–d).

In these two cases we see that the underlying geometrical structure leads to assumptions that are in tension with phenomenal observations or physical needs. By contrast, the third problem is a consequence from the mathematical realm that is in need of further explanation in the Timaeus: while the elemental changes between fire, air, and water are accounted for in terms of the number of faces (so that, for example, two tetrahedra which turn into one octahedron do indeed share the same number of faces), what is problematic are the changes of the volumes resulting from such an elemental change; for the volume of the two tetrahedra is only half the volume of the octahedron. Given that Plato excludes any void between the elements (58a, 59a), and elemental change is a constant feature of the universe, an explanation would be needed why these continuous elemental transformations do not lead to incessant changes of volumes of the universe.Footnote ⁶²

Furthermore, the transfer of mathematical features to the physical realm raises the question whether the consequences these mathematical features bring with them should be dealt with as they are in mathematics or not – for example, should we assume that three-dimensional bodies can be made up of planes, as they are in mathematics? Aristotle answers with a clear no, Plato with a clear yes. Plato makes the physical realm fit the mathematical and assumes mathematical rules to be valid for the physical world, a view which Aristotle contests.

1.4 Mathematical Objects as Abstractions in Aristotle

Aristotle develops a third position concerning the ontology of mathematical objects. While he does not give us a treatise laying out in detail his own position, we can reconstruct it from his refutation of positions that assume independently existing mathematical entities in Metaphysics M.

In contrast to Plato’s assumption that mathematics deals with independently and separately existing objects, for Aristotle mathematics deals with perceptible things, since they are the only things that exist for him. However, mathematicians do not deal with these sensible objects insofar as they are perceptible, for example, insofar as they have colours and are made from a certain material, but insofar as they have mathematically relevant properties, for example, extensions such as length. While the fact that a certain building, like the Palazzo Vecchio, has a square façade is accidental to its being a government building (it would also work as a government building if the façade was a rectangle), it is what the mathematician will take as object of investigation. And since it is the best way to investigate what really interests her, the mathematician will treat this feature accidental to the building as if it were separate (from its material, etc.). She thus posits what is accidental as independently existing and makes her object of investigation independent of the changes the physical thing may undergo (or its generation and corruption).Footnote ⁶³

Aristotle’s position has three aspects – abstraction, being separate in thought, and what is called a ‘qua-operator’. Let us briefly look at each of these.

Qua-operator: this operator is a tool commonly used by Aristotle but especially important for his account of mathematics. It makes it clear that we are looking at some x insofar as it is y, that is, qua y.Footnote ⁶⁴ And insofar as x is y, only certain properties are relevant (usually a sub-set of the properties of x). For example, an arithmetician looks at a turtle qua one indivisible unit (insofar as turtles can be considered as such units, they can be counted) and a geometer can look at the same turtle qua a solid (insofar as the turtle is, for example, shaped like a spherical segment, it will have the properties suitable for such spherical segments). Two understandings of the qua-locution have been suggested within Aristotelian scholarship: as looking at one specific aspect or property or as indicating a specific approach to the subject.

Being separated in thought: with the help of the qua-operator, the mathematicians can posit the mathematical object as separate in their thinking.Footnote ⁶⁵ This means that the mathematical object can be defined without referring to that from which it was abstracted (for example, the features of a sphere can be investigated without any reference to the physical ball from which the sphere was abstracted). For Aristotle, only substances can exist independently, but not quantities (which is what he regards mathematical objects to be) nor any of the features he considers as belonging to the other categories (such as qualities). Nevertheless, for all mathematical purposes, mathematical objects can be treated as if they were separate – separate from the substances they were abstracted from as well as from the motions and changes of these substances. Metaphysically, however, the basic things there are, are still substances, which are the only things that can in fact exist separately on their own for Aristotle. Treating mathematical objects as if they were separate when they really are not, has been seen as a mere façon de parler; or even as a form of fictionalism on Aristotle’s part.Footnote ⁶⁶

Abstraction: The starting point for Aristotle is the perceptible thing, from which we then abstract certain features. This abstraction is, however, not any old lack of attention to certain features that the mathematicians perform, but an intended inattentivenessFootnote ⁶⁷ – anything that is not of mathematical interest is disregarded; we only attend to certain features and intentionally leave out others.Footnote ⁶⁸

This still leaves it open, however, what exactly we take the things to be that we abstract from and what we take the result of the abstraction to be. Are we disregarding the matter of a physical thing or are we disregarding certain features, such as possessing colour or being moved? And is the result a certain object, for example, a square, or is it certain properties, such as extension or roundness? What about those mathematical features that do not seem to be instantiated (for example, a particular angle)? And how can abstraction work in the case of counting – can we really abstract away everything from a thing apart from the fact that it is one thing in order to gain a unit? Or is all we need in this case an appropriate sortal?

These points lead to two bigger questions – (a) is abstraction for Aristotle one unified process and (b) does it require some form of idealisation? (a) Do we in all cases abstract from the same kind of things in the same kind of process or does abstraction mean different things in different instances? The understanding of abstraction seems to differ with different cases, for example, if I count the turtles in the zoo, I seem to undertake a different kind of abstraction than if I abstract from the physical features of my turtle so as to derive a spherical segment. But if it is not always the same kind of process, how then do we know what is the right thing to abstract from in each case? Is it enough if this is decided by the person counting or abstracting? (b) Idealisation may be seen as required at two possible points in Aristotle’s account: on the one hand we may think that in order to derive a tangent that does indeed touch the circle at only one point or a triangle whose angles do indeed equal 180 degrees, it is not enough to abstract from matter and other properties of physical circles and triangles, but we may also have to idealise the slightly crooked lines to derive straight lines or perfectly curved ones and the mathematical properties mathematicians do indeed study.Footnote ⁶⁹ Furthermore, there may be cases where in fact no instantiation does indeed exist even though we can construct them mathematically.Footnote ⁷⁰ In both cases it would seem mathematicians may also have to add something to the physical world (a section of the straight line, for example, or the angle not instantiated in the world). But if the mathematician and thus her mind ‘adds’ something, is the thus gained triangle not mind-dependent? Accordingly, with respect to the ontological status of mathematical things, Aristotle’s account of mathematical objects has sometimes been understood as a version of psychologism, and mathematics as subjective (which would make the applicability of mathematical theorems to the natural world problematic and may call into question the position of mathematics as a model science).

Recent scholarship on Aristotle, however, as well as Neoplatonists, have pointed out that Aristotle nowhere claims mathematical objects to be mind-dependent.Footnote ⁷¹ And the Greek origin of our word ‘abstraction’, aphairesis, which Aristotle for the first time uses in a technical sense, literally means ‘taking away’.Footnote ⁷² Accordingly, several scholars have suggested that with Aristotle we should talk about subtraction rather than abstraction – for him everything is there already with the sensible things, certain features just have to be subtracted; there is no addition that an idealisation seems to suggest. For Aristotle, what mathematicians do is not to create something, but rather to actualize what exists already potentially (the line has the potential to be lengthened and its crookedness to be disregarded).Footnote ⁷³

Summing up, we can say that mathematical objects are not independently existing objects for Aristotle, rather they depend on the sensible objects we perceive; they are features of these objects that we separate in thought, but they do not depend on our mind for their existence. It may seem ironic then that Aristotle, for whom the objects of mathematics are much closer to sensible objects than for Plato, is very cautious to make it clear that the construction principles of mathematics, such as that from lines and surfaces we can construct solids, do not hold for physical bodies which Plato in his Timaeus seems to assume.

2.1 Mathematical Knowledge as a Model

Plato frequently uses the mathematical sciences as examples for secure knowledge or exact sciences.Footnote ⁷⁴ In his Meno mathematical knowledge seems to be employed as a model of real knowledge that we do not derive from the world but bring already with us.Footnote ⁷⁵ We find two passages there that use geometry as a paradigm for knowledge and inquiry, respectively. The first one (82b–86b) is better known and shows a young Greek slave coming to find a solution to the problem of how to double the area of a given square without any prior geometrical knowledge. Plato introduces it as a reaction to the so-called Meno Paradox (also called Paradox of Inquiry), the problem raised by Meno that inquiry seems to be impossible or useless: if we do not know beforehand what we inquire, inquiry seems to be impossible as (a) we will not know what to look for when starting our inquiry and (b) we will not recognise it once we have stumbled upon it. And if we do already know what we are looking for, inquiry seems unnecessary.

As a response to this seeming paradox, Plato wants to show that we do indeed already possess knowledge of what we learn beforehand, but we have forgotten and need to be reminded of it. Accordingly, his so-called theory of recollection claims that our inquiries do in fact not produce new knowledge but rather reawaken what we already know in some way; we become aware of knowledge we already possess. The kind of knowledge Plato seems to have in mind (or at least for which his theory seems most plausible) is what we would call a priori knowledge.Footnote ⁷⁶ And the example he gives for such rekindled knowledge is an instance of the geometrical doubling of the area of a square: a young boy who has not been taught any geometry is given a square with an area of four square feet, thus the length of each side being two feet. He is asked to find the length of the side of a square that is double this area, so eight feet. Not having any experience in mathematics, he first tries a square with double the length of the sides and then using sides of three feet length. Eventually, Socrates brings out the right answer from him with the help of a diagram: a square with an area of eight feet has a side with the length of the diagonal of the two feet square (see figure 3). This mathematical example is meant to show how somebody without experience or training can derive new knowledge simply by deduction.

Figure 3 Doubling the area of a square in the Meno.

In this geometrical example, we can look at the diagram of the square ABCD and its diagonals and from this ‘see’ how to proceed. We acquire geometrical knowledge via seeing the figure of the original square that is divided by the diagonals so that we then can ‘count the triangles’ constituting the squares without, however, depending on this particular drawing (we could as well simply imagine one in our head). There seem to be a couple of problems, however, with this example: for instance, we are explicitly told at the end that the young boy ends up with true opinion, not yet knowledge (85c); we seem to begin with an arithmetical question (how long is the side of a square with double the area of the two-feet square), but end up with the boy simply pointing to the line of the diagonal; and Socrates’ leading questions have been seen as in fact doing much more than simply help the boy discover the knowledge already within himself.Footnote ⁷⁷ All these objections, however, do not undermine the fact that Plato uses a mathematical example as a paradigm for knowledge that can be derived a priori. Socrates’ questions may be leading questions that prompt certain answers, but his careful layout of steps leading to the solution of the problem ends up showing the boy’s ability to follow a deduction or a kind of proof.Footnote ⁷⁸ And while the boy ends up pointing to a line, this move from what seems to be an arithmetical question to a geometrical answer (the length of the diagonal of the original square) simply mirrors the fact that the answer to this question has to be geometrical within the context of Greek mathematics. For the problem of doubling the square has a geometrical solution (i.e. the diagonal) in ancient mathematics, but no arithmetical one, since we cannot arrive at the side of the square in question numerically within the realm of positive integers.Footnote ⁷⁹ But the task of finding a square of double the area is nevertheless achieved: Socrates is satisfied when they have reached the new square (without any indication of irony here), and Meno and Socrates do not have to discuss first whether this is an instance of knowledge. That the boy is seen as ending up with true opinion rather than knowledge is explicitly claimed to rest on the fact that the insight gained is not yet tied down in his mind (85c), but frequent questioning will tie it down and thus turn it from true opinion to knowledge.Footnote ⁸⁰

While we are not given a general view of Plato’s account of mathematics here, it is remarkable that he uses this geometrical piece of reasoning as the prominent example of recollection in the Meno and the Phaedo and lets it serve as a paradigm for something that is indisputably knowledge (it is knowledge which Meno and Socrates themselves possess, so they can clearly see when the boy goes wrong and when he has reached the right answer). The geometrical construction allows him to show the process of inquiry and it provides the security that we are doubtlessly dealing with a piece of knowledge,Footnote ⁸¹ since the mathematicians agree on the fact that we gain a square with double the area from building it on the diagonal of the original square; an agreement that may be much harder to reach on philosophical questions.

In the second geometrical passage, 86c–87b, we are presented with a mathematical method that is used as a model for philosophical investigation, the hypothetical method.Footnote ⁸² The idea is that with problems we cannot solve straightaway, we may nevertheless make progress by spelling out possible consequences for the different possible solutions. So while we may not be able to give an answer to a particular problem, we may nevertheless say that if it is x, then y follows and if it is w then z follows. For the issue at hand in the Meno, this means that whereas it does not seem possible to answer the question directly whether virtue is teachable, it is possible to say that if it is knowledge, then it is indeed teachable, but if it is not knowledge it may not be. While this method does not as such lead to answer the question whether virtue is teachable in the Meno – for this Meno and Socrates would first have to inquire into what virtue is and attempt to find a definitionFootnote ⁸³ – it presents a way forward and a method of discovery.

Both geometrical passages are used for the overall interest of Plato’s dialogue in ethical knowledge, the question what virtue is and whether it is teachable. Geometrical knowledge is introduced in order to show how acquiring knowledge in general is possible and how to deal with questions that do not seem to be answerable directly. And these passages may be seen as suggesting that geometry and philosophy are similarly deductive.Footnote ⁸⁴ Thus these passages have been read by some as the first inkling of the idea that philosophy can work ‘in a geometrical manner’, more geometrico; the most literal and extensive understanding of this we probably find in Spinoza’s Ethics Demonstrated in Geometrical Order.Footnote ⁸⁵

Let us now move on to Aristotle’s Posterior Analytics, his theory of demonstration and account of how to organize and present the results of research as a systematic science.Footnote ⁸⁶ For this, Aristotle uses the mathematical sciences to provide him with crucial and often the most prominent examples; and it is evident that the mathematical sciences are not only a very clear case of scientific knowledge, but in many respects a model for him.Footnote ⁸⁷ While it has been argued that the model of a science Aristotle works with is in fact not mathematics, but biology,Footnote ⁸⁸ the two seem to be models in very different senses: biology is a model science for Aristotle in the sense that he wants to establish such a science in his own work, but it is not yet such a science when Aristotle enters the philosophical scene; rather, it becomes such a science only with him.Footnote ⁸⁹ By contrast, the mathematical sciences are already existing sciences that his audience would be familiar with.Footnote ⁹⁰ And we should not forget that the mathematical sciences were one of the few already established sciences outside of philosophy and probably the most rigorous among them. Aristotle can draw from the mathematical sciences in order to show to his audience, for example, the role principles play, the classification of principles, and, more generally, the way a deductive science works; they are a model he can refer to for how a science should be organized and set out.Footnote ⁹¹ This does not necessarily mean mathematics is a science that philosophy should emulate in its form (given that geometry works with constructions, this does not even seem possible), but in its systematicity and rigor.

At the very beginning of his Posterior Analytics, Aristotle takes up the idea from the Meno that learning requires prior knowledge: ‘All teaching and all learning of an intellectual kind proceed from pre-existent knowledge’ (71a1f.). Talking about ‘intellectual learning’ here, Aristotle makes room for knowledge by perception, while Plato talks about knowledge full stop.Footnote ⁹² For Aristotle, there are two ways in which we must already have knowledge at the beginning of an inquiry: of some things we must already believe that they are, of others we must grasp what the things spoken about are. The examples he gives for these kinds of prior knowledge are, as with Plato, mathematical, but also logical:

We see that Aristotle first gives a logical example – of the principle of the excluded middle we must know that it holds, in such a way that everything is either asserted or denied. And then he switches to two mathematical examples: of a triangle we must know what it means (but not that it exists, since it is not the most basic element of geometry and its existence can be proven by construction). And of the unit we must both know what it means and that it is since it is the most basic item in arithmetic, namely the element of numbers. For Aristotle, too, there must be some form of knowledge prior to any teaching and learning. Mathematical knowledge and logical principles, such as those referred to in the passage just quoted, demonstrate this. For Aristotle, we know certain things universally before we learn, but we do not know them simpliciter. As an example for knowing something universally but not simpliciter he gives us a mathematical example concerning triangles: ‘you already knew that every triangle has angles equal to two right-angles; but you got to know that this figure in the semicircle is a triangle at the same time as you were being led to the conclusion’ (71a19–21).

Both Plato and Aristotle use mathematics for showing that we do not get into Meno’s paradox. In Metaphysics 1025b4–7, Aristotle uses mathematics and medicine as examples with the help of which he generalizes his inferences for all sciences. And in De caelo 306a23 ff. he attacks certain atomistic accounts by claiming they assert that not all bodies are divisible and thus come into conflict with ‘our most accurate sciences, namely the mathematical’.

2.2 The Distinction between Mathematical Knowledge and the Highest Form of Philosophical Knowledge in Plato

While at least in some texts Plato and Aristotle understand mathematics as potentially an ideal science, this does not mean, however, that the two do not also sometimes criticise mathematics as it is practised at their time for falling short of such a science.Footnote ⁹³ And in his Republic Plato makes it clear that mathematics is not the highest form of knowledge. The mathematical sciences are very important to turn the soul towards the intelligibles there, but the philosophical discipline of dialectic, which deals with the first principles of what there is, is seen as a more fundamental science.

This is probably made most explicit in Plato’s so-called example of the line, which is part of a series of three well-known examples in the Republic to explain what is and can be known – the example of the sun, of the line, and of the cave. The line example shows a fourfold division of the realm of what is, as well as of the cognitive states with which we can access these different things; thus it closely links ontology and epistemology.Footnote ⁹⁴ And it places mathematics clearly below dialectics – the lower ontological status of mathematical objects goes hand in hand with a lower epistemological status of mathematical knowledge.

This example asks us to think of a line divided first into two sections – one standing for the visible realm, the other for the intelligible realm (509d8), the first is what is opinable, the second what is knowable. Each of the two sections is then divided again into two sections, so that we end up with four sections in total (see Figure 4).Footnote ⁹⁵ Of the visible realm, we are given objects first: to the first subsection belong all kinds of images, such as shadows and reflections in water; while to the second subsection belong all the originals of these images, that is, perceptible living things as well as artificial ones. Of the intelligible realm, the first section uses the originals of the visible world as images (in the way we use a diagram in geometry or a bronzen sphere as a visualisation of a sphere as such) and proceeds from hypothesis to conclusions; while in the second and last sub-section we proceed from hypothesis to first principles without using images; this is the realm of first principles and Forms. These two sections of the intelligible realm are not characterised by different objects so much as by different methods. In the following, we are told that the people using the method described in the first sub-section of the intelligible realm are those dealing with geometry and arithmetic (510c2–3), and that they use as images the lines they draw. Their claims are, however, not about those lines drawn, but rather about ‘the square itself’ and ‘the diagonal itself’ (510d7–8). Of the second-subsection, which goes to unhypothetical first principles, we are told that its objects – first principles and Forms – are grasped by dialectic. We access the mathematical things by thought (dianoia), but the Forms by understanding (noêsis); while belief (pistis) is of perceptible things and imagination (eikasia) of images (511c5–e4). Understanding and thought together form the realm of knowledge, belief and imagination the realm of opinion.

Figure 4 The divided line in Plato’s Republic.

While the example of the line does not discuss mathematics in any detail, we can derive at least four important points, epistemological as well as ontological ones, from it for our project:

1. 1) The true objects of mathematics are things like ‘the square itself’ and the ‘diagonal itself’ – the objects the mathematicians are concerned with do really exist, but only in the intelligible realm, not in the perceptible one. They are what it means to be a square, not this or that particular square.

2. 2) The mathematicians use what is perceptible only as images for the intelligible objects they are really talking about.

3. 3) The way mathematicians proceed is from hypothesis to conclusions without deriving first principles.

4. 4) Since the mathematicians do not turn to first principles, but start from given axioms, their activity is not concerned with the most fundamental things in the way dialectic is.

In the Republic, dialectic is understood to be more fundamental than mathematics, since it deals with the first principles of what there is. Mathematics is situated below dialectics because it cannot justify its principlesFootnote ⁹⁶ in the way dialectics allegedly can – and perhaps also because geometry relies on the perceptible world for its operations, as when diagrams are used for proofs.Footnote ⁹⁷ This also suggests that dialectic and mathematics do not provide knowledge of the very same typeFootnote ⁹⁸ and probably that knowledge of the principles and knowledge of what is derived from principles differ not only in status but also in kind; but Plato says very little here that would allow us to flesh out these epistemological differences in any detail.

While mathematical knowledge is not the highest form of knowledge, it is nevertheless crucial according to the Republic in order to turn the soul around, from the perceptible realm to the intelligible one (521d). This turning requires a very long education in mathematics for the future rulers of the state, the so-called philosopher queens and kings of the kallipolis (521c–535a): as small children, they are taught music, poetry, and are engaged in physical exercises; after which a ten-year period of training in different mathematical sciences ensues. In each case the real interest is for the pure version of this branch of mathematics. But in order to show that acquiring these different kinds of mathematical knowledge does not make students useless for practical tasks, the fact that there are applied branches of mathematics is also employed. First, arithmetic is taught; explicitly not for the purpose of commerce (525c–d), but as the basis for order and for all sciences. It is seen to be required for warfare; but most of all in order to lead the philosophical mind to the one and numbers as such and thus to intelligible things. Then geometry is introduced, not just so that the students will be prepared for practical purposes, like setting up a camp or dealing with different formations of an army, but most of all so that its students turn towards what truly is. Within the systematic layout of the sciences, solid geometry – stereometry – would be next, but according to the Socrates of the Republic this is a subject that has not really been explored so far. Given that we do find it in Euclid’s Elements, scholars have suggested that this claim in Plato’s middle period represents a time before Theaetetus had done his work on the Platonic solids that turn up in Plato’s later work, in the Timaeus.

With the last two subjects, astronomy and harmonics, Socrates turns to mathematical subjects that are per se practical and thus closer to the perceptible realm. But in the education of the future philosophers, they should be taken up in as pure a form as possible, so that astronomy ‘leaves the things in the sky alone’ (530b–c). True astronomy should not be done as it is ‘currently’ practised – that is, as concerned with the visible movements of the heavenly bodies – but as concerned with the intelligible model of what we see in the sky, that is, in the way geometry is.Footnote ⁹⁹ It is a kind of ideal kinematics that is only imperfectly expressed in the visible heavens in time and space (as the irregularities of the motions of the heavenly bodies show, such as the changing ratio of day and night). Similarly, harmonics should not be concerned with the audible tones so much as with the pure mathematics belonging to it (531a–c).

A systematic connection between these mathematical sciences – arithmetic, geometry, astronomy, and harmonics – is already claimed by the Pythagoreans, who call them sister-sciences. But in contrast to the Pythagoreans, Plato’s Republic is concerned with a pure form of these sciences that deals with the perceptible realm as little as possible and should thus guide the students from the perceptible to the intelligible realm. According to Plato, this path from the perceptible to the intelligible is very difficult for us, and mathematics has the power to lead us along this path. Learning to focus on intelligible things from mathematics is crucial for the ascent to philosophy for the middle Plato.

But these mathematical sciences are not the highest sciences in the Republic. The coping stone is dialectic. The Republic points out that we cannot expect a full account of dialectic here (532e–533a); nor will we get one in the other dialogues of Plato. But the Republic makes it clear that dialectic is the journey out of the cave, that is, the process through which we get to understand what each thing is in itself and what the Good itself is, the highest Form of Being. This is done solely by means of argument (dialegesthai). The difference between dialectic and mathematics, at least in its pure form that should be used in the education of the guardians, is not so much that the later uses perception, as the example of the line may seem to suggest. But rather their respective treatment of the first principles, which according to Plato is such that dialectic is the only

Dialectic is the only systematic knowledge inquiring the being of each thing, while almost all of the other so-called crafts deal with becoming (growing, construction). Mathematics has a special status in that it is at least ‘dreaming’ of what truly is; but the problem is that the mathematicians start from a hypothesis of which they cannot give an account. Thus they base their discipline on something ‘unknown’; while dialectic allegedly gives an account of these first principles. Plato here even goes so far as to claim that this means that the mathematical sciences are not sciences in the strict sense (533c–d); rather, they are something ‘clearer than opinion, but darker than knowledge’.

Also other authors have seen mathematics as a kind of essential propaedeutic. Plato’s contemporary, the rhetorician Isocrates, for example, recommends mathematics to make children quicker for learning other subjects and in order to get used to the fact that the process of acquiring knowledge demands hard thought and precision.Footnote ¹⁰⁰ But for Plato, mathematics achieves much more than this by training our mind to deal with the intelligible, instead of the perceptible. And some interpreters, like Burnyeat, have even claimed that mathematical structures for Plato are ‘a constitutive part of ethical understanding’,Footnote ¹⁰¹ which includes at least two ideas: first, central concepts of the mathematical sciences, such as unity, proportion, and concord, are also central for the political realm so that the structures studied in mathematics ‘are the very structures that the philosopher-rulers will seek to establish in the social order of the ideal city and in the souls of its citizens’. And second, since mathematical theorems are unqualifiedly true in every context, which is what Plato wants to show for the realm of ethical value, it articulates objective values.

Let us finally look at the kind of explanation mathematics provides according to ancient philosophers.

2.3 The Possibility of Explanation in the Mathematical Sciences

In contemporary philosophy of mathematics, one central epistemological question concerns the possibility of explanations in mathematics. More exactly speaking, it is debated whether we find within mathematics (a) real explanations (and not just non-explanatory demonstrations) and (b) explanations for the sciences.Footnote ¹⁰² We seem to get into similar worries if we look at Aristotle’s account of scientific understanding that closely ties knowledge to causes, as we can see from the following passage:

We see that knowledge is defined as understanding the cause of a thing to be this cause and to be necessarily this cause. So there are two conditions for understanding something simpliciter, (a) knowing the cause and (b) this being tied to some form of necessity.Footnote ¹⁰³ Given that knowledge and understanding is tied to causes here,Footnote ¹⁰⁴ we may be worried that such an account does not fit mathematics at all, as causes seem to belong to the area of the natural sciences only. How can mathematics explain anything if knowledge and understanding is of causes for Aristotle?Footnote ¹⁰⁵

First we need to point out that the Greek word usually translated as ‘causes’, aitia, is much broader than our usual understanding of a cause. For Aristotle, aitia are not only efficient causes, but explicitly include formal, material, and teleological causes, as he makes clear in Physics II, 3; they are better understood as reasons or ‘becauses’. Accordingly, Barnes (Reference Barnes1993) translates them as ‘explanations’ rather than ‘causes’. Aristotle’s main point here is that knowledge includes having a grasp of why something is the case.Footnote ¹⁰⁶

Aristotle claims that arithmetic, geometry, and optics carry out their demonstrations through what he calls the first figure in his syllogistic, which he takes to be especially scientific, and belong to the sciences that inquire into the reason why (Posterior Analytics I, 14, 79a17–24). Accordingly, he assumes that there is some explanation within mathematics itself.Footnote ¹⁰⁷ But does mathematics also provide explanations for the sciences? Here we need to remember that at the time in question, physics and the natural sciences were not mathematicised in the way they are today. However, Aristotle himself introduces mathematical concepts into the realm of physics in order to develop central concepts further.Footnote ¹⁰⁸ And that mathematics can provide knowledge of ‘becauses’ in the sciences for Aristotle, he makes clear with examples like the wound in his Posterior Analytics (79a14–16): ‘it is for the doctors to know the facts that curved wounds heal more slowly, and for the geometers to know the reason why’.Footnote ¹⁰⁹

While the doctor will know that a round wound will take longer to heal, a mathematician can explain the reason behind this, presumably, because a round-shaped wound has the biggest surface in relation to its circumference.

So mathematicians can know the reasons why and thus provide explanations also for more empirical sciences. But what makes mathematics a model of a science is its employment of deductions. While deductive structures are first found paradigmatically in mathematics, it is philosophers who then raise questions about the workings of deductions and develop the notion of deductions. As we will see in the next section, it is especially completeness and universality where philosophers started to question the exact workings of the deductive systems they gained from mathematicians. But they also asked more general questions, for example, what can be used as starting points for our deductions; what guarantees that truth is indeed preserved in such a deduction; what does their validity rest on; or whether the principle of non-contradiction is the only essential law for such deductions.Footnote ¹¹⁰ The mathematicians, by contrast, seem to take these features of deductions for granted; we do not find any meta-reflections on deductions in Euclid or earlier texts.

Philosophers also ask for reasons why deductibility works at all. For Aristotle, deductibility rests on the form of the system. For him, a deduction allows that from some premises a conclusion containing new information can be derived by necessity and nothing external has to be relied upon for the necessity to hold.Footnote ¹¹¹ The validity of the conclusion simply rests on the very form of the deduction. If we have two premises of the form ‘all As are Bs’ and ‘all Bs are Cs’, the inference that ‘all As are Cs’ is valid, no matter what the content.Footnote ¹¹² It is this independence from any concrete empirical content that makes deductive structures so central not only for mathematics, but also for logic. And it is in Aristotle’s logic that the basic working of deductions as we find it in mathematics is first explicitly investigated. So, it is to the role of deductions that we will now turn in the next section.

3.1 Mathematical Deductions as Paradigmatic for Philosophical Proofs

Ancient philosophers took mathematics as a paradigm for a strict demonstrative science, as may best be seen from Aristotle’s Posterior Analytics, which is the first attempt in Western thought to give an account of the way in which scientific results should be structured, organized into an intelligible whole, and presented. According to the Posterior Analytics, sciences should be properly expounded in deductive systems.Footnote ¹¹⁵ That is, the body of truth of each science should be exhibited as a sequence of theorems derived from a few postulates or axioms. The axiomatisation is to be formalised in the sense that it is formulated in a well-defined language, and its arguments should proceed according to a set of logical rules.Footnote ¹¹⁶ Today it is sometimes discussed whether mathematics is a science – in case the natural sciences are taken as paradigm and the lack of dealing with empirical evidence is held against mathematics being a science in this sense. But if we think of science as a systematic body of knowledge, mathematics can not only be seen as a paradigm for this, but also was the first developed science in Western thought. Accordingly, when Aristotle discusses how to structure and systematize knowledge as a science in his Posterior Analytics, he uses mathematical sciences as crucial examples.Footnote ¹¹⁷ Aristotle, who lived presumably roughly fifty years before Euclid, also provides us with a glimpse of the extent to which the idea of an axiomatized science had already been sketched by mathematicians before Euclid.Footnote ¹¹⁸

Two points are especially important for an axiomatised science: the principles or axioms which serve as starting points for proofs and the rules of deduction. We will deal with the starting points in Section 4, and with the rules of deduction here.

3.1.1 Rules of Deductions

Mathematics could be seen as paradigmatically displaying a deductive structure – both within a single proof and in the way in which the genre of mathematical Elements sets out the whole of mathematical knowledge in such a way that each proposition only builds on the basic principles and the preceding propositions (but not on succeeding ones). We can see the deductive structure in mathematics in several propositions in Euclid already from the surface grammar which give a general conditional along the following lines: ‘if x (and y), then z has to hold’.Footnote ¹¹⁹ So from a given starting point it is shown that something else follows necessarily. For example, in Elements I, 32 we find the following: ‘In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles.’Footnote ¹²⁰ So here the premises we start out with is that we have a triangle ABC, one side of which gets extended (the side BC is produced to D; see Figure 5). And from this it is then shown that something else follows by necessity, namely that the angle ACD equals the sum of the two interior and opposite angles CAB and ABC and that the sum of the three interior angles ABC, BCA, and CAB equals two right angles. Nothing external comes in here. Solely from what is given and done,Footnote ¹²¹ prior propositions (I, 13, I, 29, and I, 31), and some basic principles (such as the definition of a triangle), we necessarily derive at what was meant to be shown.

Figure 5 Euclid, Elements I, 32.

Scholars also talk about ‘deductions’ with respect to Parmenides’ poem, fragment 8, where Parmenides lists the characteristics (sêmata) of what truly is, before he ‘deduces’ them. They are probably the oldest ‘deductions’ we find in philosophy, and we may think that Aristotle could have looked at Parmenides as a model for deductions.Footnote ¹²² What seems to have made mathematical deductions more attractive than the Parmenidean one for Aristotle are two facts. First, Parmenides’ deductions work negatively: by showing that not-F does not fit with what has been established before, they are meant to demonstrate that F has to be true. For example, it is shown that none of the conditions hold that could make what-fundamentally-is inhomogeneous; thus what-fundamentally-is has to be homogeneous.Footnote ¹²³ We will talk about a specific form of such indirect proofs later. But for deductions as such, Parmenides’ fragments are not a straightforward example. Second, mathematical deductions are used in a systematic way to build a whole science, as we find it in Euclid’s Elements (and presumably to some extent also in the Elements writings available before Euclid). Parmenides’s deductions, by contrast, do not so much build a systematic science as give arguments for the different characteristics of what-fundamentally-is. Accordingly, it is no surprise that Aristotle in his Posterior Analytics refers to mathematics but not to Parmenides’ poem as a model for a deductive method.Footnote ¹²⁴ Hence, we may claim that our philosophical understanding and realisation of deduction was originally shaped by mathematics.

But it is a philosopher, again Aristotle in laying out his logical project, who is the first thinker attempting to define what a deduction is, when in his Topics I, 1, 100a25–27 he claims that ‘a deduction is an argument in which, certain things being laid down, something other than these necessarily comes about through them.’Footnote ¹²⁵ The word Aristotle uses here for what is translated as ‘deduction’ is syllogismos – and we are dealing with valid and informative deductions in Aristotle. The Greek term syllogismos was originally used for reasoning, and in its general form this Aristotelian definition also holds for mathematical inferences. Indeed, it seems that such deductions were first perfected by the mathematicians. Euclid does not talk about syllogismoi in his Elements, nor does he use the common verb syllogizesthai. But in the course of presenting a proof, he commonly uses inferential language – particles expressing consequence,Footnote ¹²⁶ and verbs for inferring or deducingFootnote ¹²⁷ – vocabulary used in ways that indicate what we can characterise as deductions according to Aristotle’s definition.

Aristotle’s general characterisation of syllogismos and his understanding of proofs as demonstrative arguments fits well with the mathematics of his time and Euclid. His narrower understanding of syllogisms, however, is subject to clearly defined restrictions,Footnote ¹²⁸ as his Analytics makes clear, that do not hold for mathematics. We also do not find full-blown attempts to reconstruct mathematical proofs syllogistically.Footnote ¹²⁹ This does not mean, however, that mathematics is not a model science for Aristotle. After all, also Aristotle’s own natural sciences, his physical and biological works in the way they have come down to us, are not as such presented in syllogistic form. And Aristotle’s Physics Book Z has been seen as structured according to a mathematical treatise.Footnote ¹³⁰

Deductive structures as developed in mathematics were seen as a model by philosophers.Footnote ¹³¹ And we may be tempted to call early Greek geometry and arithmetic not only deductive systems but full-blown axiomatized systems. We should, however, keep in mind that axiomatised systems today are deductively closed and rigorous, like the Peano system. Ancient geometry and arithmetic in the form we know them from Euclid are not completeFootnote ¹³² and not universal to the degree we usually assume: also Euclid’s Elements, which is a systematic compilation of the mathematical knowledge of his time, displays deductive gaps even at the very basic levelFootnote ¹³³ and is thus not complete – many things are simply presupposed but not stated, such as the fact that two lines do not enclose an area;Footnote ¹³⁴ and operations such as addition, subtraction, multiplication, and division of magnitudes are used without them, or the laws they follow, ever being characterised, let alone defined.Footnote ¹³⁵ And they do not display full universality, which can be seen, for example, in the fact that we find parallel proofs of cases which seem easily universalisable.Footnote ¹³⁶

Keeping this restriction in mind, we may even think that ancient mathematics of the fifth and fourth centuries BCE is a far cry from contemporary axiomatised systems. It is, nevertheless, a central predecessor to such systems,Footnote ¹³⁷ and it was the one ‘exact’ science that philosophers like Aristotle could look to for laying down rules for exact sciences.Footnote ¹³⁸ Given that mathematical propositions rely only on definitions, postulates, and common notions and the theorems established beforehand, ancient mathematics displays a degree of internal connectednessFootnote ¹³⁹ that is much harder to achieve in philosophy. While the discussion of mathematics in philosophical texts shows that mathematics is frequently taken as a paradigmatic science, it is the points of universality and completeness that philosophers ultimately either raise as concerns or attempt to improve on in their philosophy.

As for completeness, we find that Plato in his Republic suggests that, in contrast to dialectics, mathematics cannot give a foundation to its axioms and thus is incomplete in this sense (see Section 2.2). And for Aristotle it is his own logical system that is complete in the sense of accounting for all valid inferences and forms a closed system (see Prior Analytics I, 23); he seems to have used mathematics as a model for deductive systems so far, but attempted to improve it in this respect.

While the importance of universality is stressed for deductions by Aristotle with the help of a geometrical example, as we can see in Prior Analytics I, 24, 41b13–22, it seems to be put in doubt by the role of geometrical constructions, and that means necessarily imperfect drawings and diagrams, for mathematical proofs. The fact that ancient mathematical deductions work with the help of constructions is essential for mathematical proofs at the time we are investigating in Greece. They cannot immediately be used by other axiomatised sciences and philosophy.Footnote ¹⁴⁰ For example, Elements I, 4 starts the deduction with ‘Let there be two triangles, ABC, DEF, having the two sides, AB, AC, equal to the two sides, DE, DF, respectively … .’ and either proceeds with a drawing or expects us to do a drawing in order to follow the proof.Footnote ¹⁴¹ Accordingly, the question presents itself what role these constructions play exactly.

3.2 Reductio ad Absurdum Proofs and Philosophical Paradoxes

Of the different mathematical methods of interest for philosophers, we will look at only one in a bit more detail, namely the method of reductio ad absurdum. The Latin term ‘reductio ad absurdum’ is a translation of the Greek expression hê eis to adunaton apagôgê (reductio to the impossible), which we find first in Aristotle’s Prior Analytics 29b9. But Aristotle does not suggest that he is introducing this terminology for the first time.

In the kind of reductio proofs we will look at, the necessity of some p is proven by showing non-p to be impossible; accordingly, they require tertium non datur to hold.Footnote ¹⁵³ These kinds of reductio proofs are of special interest, since they are a powerful method that we find both in mathematics and in philosophy for cases where we cannot find a direct proof.Footnote ¹⁵⁴ They are central for the philosophical genre of paradoxes; most of Zeno’s paradoxes work with such a reductio structure, for example. And there are lots of reductio proofs in Euclid.Footnote ¹⁵⁵ Within a mathematical context, it seems likely that reductios were first developed for the proof of the development of the incommensurability of the side and the diagonal of the square – at least, this is a rather old Pythagorean proof, and if Aristotle wants to give an example for a reductio, he usually refers to the incommensurability of the diagonal and the side of the square.Footnote ¹⁵⁶ We will look at it as one relevant and important example that also raises interesting questions about the interaction between philosophers and mathematicians.

The incommensurability of the side and the diagonal was a special discovery in the development of Greek mathematics – a fact that is better understandable if we remind ourselves of its Pythagorean background. We saw in Section 1.1 that on a Pythagorean view, the world is set up as mathematically structured, according to (natural) numbers and relations that constitute them. Everything can be brought into a numerical relation to everything else. The discovery of the incommensurability shakes such a world view, for it shows that there are entities that cannot possibly be captured in terms of numerical ratios – the side and the diagonal of a square can never be brought into such a numerical relationship. So there are some things that seem to have no ratio, no logos; they are ‘irrational’. And this seems to put the very idea of a rational cosmos into doubt.Footnote ¹⁵⁷ Plato expresses the puzzlement about this finding in his Laws, 819c–820d, in a passage claiming that the Greeks assume we can measure all lengths and breadths against each other; thus most people are not aware that this is not possible for certain lengths and breadths.

Its effect on the mathematical community has seen different assessments by modern scholarship: Tannery (Reference Tannery1887) thinks it led to a foundational crisis of mathematics not unlike the foundational crisis in mathematics at the end of the nineteenth century. By contrast, Knorr (Reference Knorr1975), pp. 308 ff., takes it to be the background to the geometric style of number theory in Euclid’s Elements, Book VII and the development of greater rigor in proportion theory; but he does not see it as a crisis in Greek mathematics. However, also Knorr thinks that the problem of the incommensurability engaged the effort of the most notable mathematicians of the fourth century, Theodorus, Theatetus, Archytas, and Eudoxus. It was a crucial force motivating Eudoxus’s theory of proportion which is the foundation of Euclid’s proportion theory in Elements, Book V. And it seems to have been one important factor for the increasing dominance of geometrical proofs – it shows that there are some things in mathematics, like the relation of the diagonal of a square to its sides, that cannot be dealt with arithmetically, but only geometrically.Footnote ¹⁵⁸

Given that the notion of the incommensurability of the side and diagonal would in fact never come up when measuring an empirical square in that such a measurement will always produce some result,Footnote ¹⁵⁹ this proof shows that by the time it was discovered, Greek mathematics had clearly developed as a theoretical science independent of its practical applicability. The claim that such quantities are irrational is introduced through the very question posed and based solely on logical deduction.Footnote ¹⁶⁰ The fact that incommensurability cannot be detected by the senses, may also have made it a discovery of special interest for Plato, who, as we have seen in Section 2, is very concerned about ways in which to lead students in their inquiries from the perceptible to the merely intelligible. Thus, it may not be a big surprise that Plato’s texts are the first ones where we find the discovery of the incommensurable talked about.

Incommensurability is first mentioned in Plato’s Theaetetus, but the imagined conversation between the young mathematician Theaetetus and Socrates does not refer to the square root of the number two, but to that of three and so forth up to seventeen (147dff.); hence, we are dealing with an already advanced stage of this discovery (the imagined date of the conversation is 399 BCE, the year of Socrates’ trial and death). The discovery of the incommensurable has been suggested to be no later than the last quarter of the fifth century,Footnote ¹⁶¹ but this still leaves a lot of flexibility about the discovery in the fifth century. Given these problems of dating it is also hard to say whether the method of reductio was first developed by mathematicians and then taken up by philosophers or the route of influence was the other way round. Accordingly, there has been some debate in the scholarship about the relationship between the incommensurability proof and Eleatic method. Szabo has claimed that the mathematicians in fact could take up the logic of the method from the Eleatics.Footnote ¹⁶² Knorr, on the other hand, assumes that this is an old proof of the Pythagoreans,Footnote ¹⁶³ which would allow for Zeno’s paradoxical method to have been influenced through exposure to some version of the incommensurability proof. There is, of course, also the possibility that both the Pythagoreans and Zeno arrived at a similar method independently of each other at around the same time. Given that the Pythagoreans were involved in both what we would call mathematical and philosophical investigations simultaneously, and we have seen so far in several places the keen interest philosophers took in questions and problems raised by mathematicians, it seems, however, much more likely that there was indeed some intellectual exchange between mathematicians and philosophers.

We may think that the method is prepared in the indirect proofs we find in Parmenides’ poem, fragment 8 and that his indirect proofs were an inspiration for the form in which the incommensurability proofs were put. But Parmenides’s proofs only show that F does not fit with what has been established before by Parmenides; they do not show as such that not-F has to hold independently of previous assumptions. This is something we only find with Zeno’s paradoxes and the mathematical reductio ad absurdum proofs. Let us have a brief look at these.

Our best evidence for the incommensurability proof is preserved as an appendix to Euclid, Book X, Heiberg proposition 117. This version is an insertion into Euclid’s text and shows some later reworking. But the core of the proof is a good deal older, going back to the Pythagoreans, sometime in the fifth century BCE. One version is mentioned by Aristotle, and according to Knorr there were half a dozen versions of the incommensurability proofs around.

The basic idea of this proof is the following: if we assume of a square ABCD the side AB and its diagonal AC to be commensurable, then we have to assume that the same number is both odd and even. This is a contradiction; thus they are incommensurable.

The main steps are as follows: we assume that AC and AB are commensurable with each other, hence they have a ratio equalling two numbers to each other; called EF and g (see Figure 6).Footnote ¹⁶⁴ The version in Euclid also claims that ‘these numbers be the least numbers in this ratio’, but all that is in fact required is that they are not both even. It is then concluded that EF cannot be ‘a unit’, that is, 1, but has to be a number. ‘For if EF is a unit, and has the ratio to g that AC has to AB, and AC is greater than AB, then EF is greater than the number g,Footnote ¹⁶⁵ which is impossible.’Footnote ¹⁶⁶

Figure 6 Proof for the incommensurability of the diagonal and the side of a square.

In a second step it is shown that EF has to be even, since it has the same ratio to g as AC has to AB, and the square of AC is double that of AB; if it is double something, it has to be even:

$EF:g=AC:AB,$

${\left(AC\right)}^2=2*{\left(AB\right)}^2$

$=>EFiseven.$

In a third step, it is demonstrated that g has to be odd, for otherwise EF and g would both be even and could not be relatively prime. In a fourth and final step, it is assumed that EF is divided in half by H, which shows that g has to be even. For the square of EF has to be four times the square of EH (EF being double EH), but also double of that of g, which in turn means that the square of g is double that of EH. Accordingly, ‘the square of g is even, and thus g is even’ (if it were odd, its square would be odd).

$EF=2\times EH;$

${\left(EF\right)}^2=4*{\left(EH\right)}^2;$

${\left(EF\right)}^2=2*g^2$

$=>g^2=2*{\left(EH\right)}^2$

$=>giseven.$

But in the third step it was shown that g is odd, so g has to be both odd and even, which is impossible. Thus the assumption of AC and AB being commensurable leads to a contradiction.

We find the main idea of the way in which this proof works also in many paradoxes of Zeno: in order to show that motion, plurality, or topos (place/space) are problematic, it is first assumed that there is a plurality, motion, or topos before it is shown that this very assumption leads to inconsistencies. Like in the incommensurability proof, so also in Zeno’s paradoxes, what is used as a starting point is the opposite of what is shown to be the result in the end. In the incommensurability proof we start with the assumption that AB and AC are commensurable, while Zeno uses as a starting point one that is not his own, but widely held, the position of a pluralist. And from this starting point it is then shown that plurality, motion, and topoi are problematic in themselves.

Both Zeno and mathematical reductio proofs start from a position they do not share and then show that this position leads to contradictions – that the same number has to be odd and even, that the same distance has to be finite and infinite, and so on. And this contradiction shows that the position implying it is false absolutely. In comparison to Parmenides’s indirect proofs, such reductios can be seen to be much stronger because

1. (a) they make sure we do not introduce our own assumptions which the opponent may not share; and

2. (b) the opponent is refuting herself – in the very act of thinking she is undermining herself.

For the realm of mathematics, it is a good and strong proof, as it is for Zeno who is the founder of the genre of paradoxes.Footnote ¹⁶⁷ The reason these proofs work very well here is that they work within a framework in which it is clear that I have only two alternatives – either AB and AC are commensurable or they are incommensurable, either we can consistently think of a plurality of things or we cannot.

For later philosophers starting with Aristotle, however, this method of proof ultimately needed further refinement, since in many philosophical contexts showing that F cannot hold does not in itself demonstrate that not-F holds. Accordingly, it is not necessarily seen as a strong proof and can only be used in certain contexts.

For the mathematicians, the particular usage of this method to prove the incommensurability of the side and the diagonal of the square seems to have been one reason that led to a sharper distinction between arithmetic and geometry, to a geometric style of number theory, and to shifting the burden of proofs to geometry.

4.1 Principles and Starting Points

4.2 Mathematical and Philosophical Notions of Continuity

The most important conception of continuityFootnote ¹⁹² in ancient times is Aristotle’s account.Footnote ¹⁹³ It is developed as an account for physical magnitudes, but yet it explicitly takes up mathematical insights. By understanding continua as what is always further divisible into divisibles without ever getting to zero extension or indivisible atoms, Aristotle’s account of continuity is the philosophical alternative to atomism.

The first philosophical account of continuity in the history of thought can already be found in Parmenides in the fifth century BCE. For him, something is continuous if it is homogenous in every respect, which means displaying no differences of kind, quality, or quantity.Footnote ¹⁹⁴ As a result of this uninterrupted homogeneity, continua are indivisible according to Parmenides: divisions are only possible where there are differences; since there are no internal differences in a continuum, it is completely indivisible.

Drawing the inference from complete homogeneity to indivisibility is supported by some paradoxes of Zeno, Parmenides’ student, which demonstrate the problems we get into if we assume continua like magnitudes and motion to be divisible. For example, in one of his motion paradoxes, called the ‘dichotomy’ or ‘runner paradox’ (fragment DK29A25), Zeno raises the problem of how to conceptualize a finite motion, like that of a runner covering a finite race course AB in a finite time FG. To do so, she first has to cover half of the course AC, and then again the first half of the remaining half CD, then DE, ad infinitum (see Figure 7).

Figure 7 Zeno’s runner paradox.

Thus it seems a runner has to cover an infinite number of spatial parts in order to cover a finite distance in a finite time. For a spatial distance seems to contain infinitely many parts, all of which have to be covered. Given that we cannot pass infinitely many spatial parts in a finite time, the paradox seems to suggest that motion cannot be conceptualized.Footnote ¹⁹⁵

In one of his plurality paradoxes,Footnote ¹⁹⁶ Zeno shows that if we take some physical magnitude to be divisible at one point (without this point being legitimised by any internal differences), then the magnitude is divisible everywhere; but if divisible everywhere, it can also be divided everywhere,Footnote ¹⁹⁷ and we never reach consistent parts in this way. For either these parts have no extension, so that we face the problem of how some extended physical whole can be composed of extensionless parts. Or these parts have some extension, in which case either this shows we are not yet done with our division and our parts are indeterminate; or the parts seem to be infinitely many,Footnote ¹⁹⁸ in which case they seem to lead to an infinite extension. Zeno raises these and similar paradoxes for physical continua. Mathematical replies to them have often been suggested, based on modern mathematical developments such as the possibility of calculating the sum of an infinite convergent series, mathematical functions since Cauchy, and the limit of a function. But it remains a question whether a solely mathematical reply is sufficient for the problems they raise for the physical realm.Footnote ¹⁹⁹

Mathematical atomism of the kind we find in the pseudo-Aristotelian treatise On Indivisible LinesFootnote ²⁰⁰ seems to have responded to these paradoxes, as we can see from the fact that the need to avoid Zeno’s paradoxes is mentioned as one of the points in favour of the assumption of indivisible lines. The basic idea of atomism here is that we can only divide magnitudes up to a certain point, before we hit indivisibles. The parts derived by division thus will never be infinite.

Also Aristotle’s account of continuity is meant to show, among other things, how a physical theory of continuity can reply to these Zenonian paradoxes. Aristotle gives us two different characterisations of continuity:

1. (1) Continuous are those things whose limit, at which they touch, is one (Physics V, 3).

2. (2) Continuous is that which is divisible into what is always further (or infinitely) divisible (Physics VI, 2).

The first account of continuity is two-place, ‘A is continuous with B’, while the second is one-place, ‘A is continuous’. Aristotle does not explain why he gives us two different accounts, but he moves freely between the two, which thus seem to be closely related.

Understanding extended magnitudes as being divisible without end is one of the crucial notions Aristotle employs from a mathematical context in order to set up a science of locomotion in his Physics. At first glance, the suggestion that Aristotle takes up a mathematical understanding may not seem very plausible. For we do not find a discussion or explicit definition of ‘suneches’, the Greek term translated as ‘continuous’, in the mathematical texts handed down to us from the time up to Aristotle. And most of the occurrences of the term suneches in Euclid refer to a continued proportion.Footnote ²⁰¹

We have, however, clear indications that the mathematicians understood lines, surfaces, and solids as continua in the Aristotelian sense: there are a few passages in the Elements in which Euclid uses the term suneches as being successive in the way continuous lines are – so as continuous in a two-place sense (AB is continuous with BC).Footnote ²⁰² And On Indivisible Lines makes it clear that a one-place understanding of continuity is presupposed by geometers, since they treat their geometrical objects as being always further divisible for their mathematical activities (969b20ff.). When they assume crossing lines and similar constructions, there is no reflection of atomistic worries, such as that a line crossing another would need to go between two atoms; rather, infinite divisibility is simply assumed.Footnote ²⁰³ And Aristotle himself in Physics III claims the mathematicians to understand magnitudes in this sense (203b17–18).

Aristotle’s own work is central for establishing an account of continuity within natural philosophy and for contrasting it to what is discrete. His Categories (4b20) is the first text demonstrating that quantities can be divided into discrete and continuous ones. It gives an argument for understanding numbers as discrete and lines as continuous. The fact that Aristotle does not simply take it for granted in his Categories that numbers are discrete, but thinks he has to argue for there being no contact between the parts of numbers, shows that he is not taking over ready-made terminology. But Aristotle’s primary examples for demonstrating what being discrete and being continuous mean are from the realm of mathematics – numbers and lines. So Aristotle seems to take up a distinction that is implicitly there in mathematical thinking. By employing and expanding Eleatic terminology – the term suneches plays a crucial role with the Eleatics but the notion of discretenessis is not to be found – Aristotle introduces the distinction between continuity and discreteness explicitly into the philosophical realm.

From the mathematicians Aristotle can take up a one-place as well as a two-place understanding of continuous quantities, on which his two different accounts of continuity are based. For him continua are homogenous, but in contrast to Parmenides, they only need to be homogenous in the genus in question and can allow for differences in other respects. And in contrast to the Eleatic inference that what is continuous has to be indivisible, Aristotle agrees with the mathematicians that continua are divisible wherever we want, ad infinitum.

But Aristotle is not simply transferring the mathematical understanding of magnitudes to physics. Rather he adopts it for the physical realm, as becomes clear when he discusses the difference between continuity and contiguity. While for mathematicians, two lines in the same plane that touch are continuous and one,Footnote ²⁰⁴ in physical contexts, two things that are continuous do not become one thing simply by touching. Accordingly, for Aristotle continuous things (a) possess touching limits (which suffices for continuity in mathematics) and (b) these limits have to be one, so that these objects move as one whole and are not only neighbouring things (an additional requirement for the physical realm) (Physics V, 3).

Aristotle’s understanding of continuity is closely linked to his account of limits and of infinity: we will briefly look at these two in the following sections 4.3 and 4.4.

4.3 Limits – the Distinction between Inner and Outer Limits

What delimits one thing from another? This question has been raised as a problem for natural philosophy by Zeno, but Aristotle’s answer is important also for the philosophy of ancient mathematics. Zeno claims in fragment DK29B3 that any plurality of things seems to be the very number it is; let us say there are two things, A and B. At the same time this plurality has to be infinitely many because between every two things there has to be another thing. For in order to ensure that A and B are indeed two separate things, there must be some other thing C in between. But then we also need a thing between A and C, and C and B to separate them from each other, ad infinitum.Footnote ²⁰⁵ Natural philosophers reacted to this paradox by pointing out that what separates one thing from another need not be another thing. For example, with the atomists it is the void that guarantees the separateness of two things, between two atoms as well as between two phenomenal things.

Also Aristotle does not assume that we need a further thing to separate two things, but only what he calls a limit: each physical magnitude possesses limits that are not themselves things nor extended parts. Limits possess one dimension less than what they limit; for example, the limits of a one-dimensional line are points and thus without extension. Limits that delimit a magnitude from its surrounding are external limits. For Aristotle, continua also have what we can call ‘inner limits’, i.e. division points that mark off possible parts within a continuum. Usually these division points are not given, but they can be constructed wherever we like. For example, if we measure a long plank with a small ruler, we mark off parts of the plank with the help of such marks. Outer limits separate one whole from another; they guarantee the unity of a continuum vis-à-vis other continua. Inner limits, by contrast, guarantee the internal unity of a continuum by showing that we can divide it wherever we please without ever finding a gap.

This distinction between inner and outer limits of continua is the first predecessor of the modern distinction between points lying anywhere within an interval and the end points of such an interval. Furthermore, the modern distinction between closed and open intervals, that is, between intervals which contain their end points and those that do not, can be seen as prefigured in Aristotle’s analysis of the outer limits that are of most concern to him in his Physics, namely those between motion and rest. Aristotle makes it clear that if a moving thing slows down and comes to a standstill, there is no last point of motion nor a first point of rest; for every point within a motion we may choose, there will always be another one closer to the limit between motion and rest. Thus, he is the first to clarify that there is no first or last point within a continuous interval. The limit between motion and rest can be seen in analogy to the way we think of the limit of a continuous open interval nowadays: it is the point of an interval that in contrast to all other possible points does not have an ε-neighbourhood.Footnote ²⁰⁶

We will not be able to discuss the notion of a limit further here. Instead, we will now move on to infinity.

4.4 Philosophical and Mathematical Notions of Infinity

With the notion of infinity (apeiron) we are dealing with a notion that from early on was important for mathematics as well as for physics – being about quantities in general, it is relevant for mathematics, but it also plays a role in discussions about the world as a whole and thus for physics. The mathematicians also deal with one-dimensional infinite extensions, of which we can have a plurality, such as the infinite lines we find in Euclid. By contrast, the natural philosophers discuss infinite extension with respect to bodies and thus as something three-dimensional, so that there could at most be one of these. For if more than one of these infinite bodies were to exist, these infinite bodies would limit each other and thus not be infinite any longer, as, for example, Gorgias argues.Footnote ²⁰⁷

The philosophers seem to be aware of the mathematical usage there is of that notion and attempt to preserve it in their conception of infinity. But while the intensive philosophical discussion about infinity, which started at least with Zeno and finds a full reply in Aristotle, may make us expect to find a reflection of it with the mathematicians, there is none to be found in Euclid.

In philosophy, being apeiron is a central notion almost from the very beginning, starting with Anaximander. The Greek word apeiron is formed with the help of the alpha privative that is usually understood as added to the Greek word for limit or end, peras; so apeiron is that which is without limit or end; it is not quantifiable. There is no systematic distinction between what is infinite and what is unbounded with the ancient Greeks; and our modern distinction between countable (ℕ, ℚ) and uncountable (ℝ) infinities is not known to ancient philosophers.

The first philosophers view infinity either as a characteristic or as a bearer of characterisations. The latter usage is prominent in Anaximander, who understands apeiron as the basic substance out of which everything comes into being and into which it dissolves again. By contrast, for Aristotle apeiron is an attribute of number and magnitude and hence incapable of independent existence.

Zeno attempts to show in several of his paradoxes that conceptualising multitudes as well as magnitudes of finite things entangles us in the assumption that they are also infinite and thus characterised simultaneously by opposing features. As we saw earlier, Zeno shows in DK29B3 that things that we take to be of finite number are also of an infinite number. And things we take to be of a finite size, such as a certain physical body, also have to be assumed to be of infinite size (DK29B1), for we can always go on to divide it into further parts and each of these parts will have some ‘magnitude and thickness’. We saw earlier that in the runner paradox, Zeno shows that a finite extension to be covered also has to be conceived as being infinitely divisible,Footnote ²⁰⁸ and thus seems to leave the runner to cover infinitely many spatial parts. In all these cases, Zeno thinks we face the problem that something finite is also infinite, which is seen as paradoxical.

Aristotle reacts to Zeno’s motion paradoxes by trying to show that the seemingly dubious alliance finite things have with infinity is unproblematic if conceived in the right way. He first distinguishes explicitly different kinds of infinity – most importantly infinity of division from infinity of addition.Footnote ²⁰⁹ According to Aristotle, Zeno did not sufficiently distinguish between the two so that from an always continuing division of a continuous stretch Zeno wrongly infers an infinite extension.Footnote ²¹⁰ Aristotle attempts to show that assuming an infinite extension, and especially an infinite body, is inconsistent (he argues for the universe to be finite).Footnote ²¹¹ Infinite divisibility, by contrast, which he explicitly claims to come from the mathematicians (203b15–30), is the main reason we cannot exclude the idea of infinity tout court. Continuous magnitudes are indeed always further divisible, Aristotle assumes. But this does not mean that they can have been divided wherever they are divisible (among other things, several of the possible parts derived from division will overlap, so we cannot derive all these overlapping parts at any given time). And more generally, for Aristotle the process of division can always go on but will never lead to an end. For him the infinity of division is only a potential infinity, not one that can be fully actualized.Footnote ²¹²

Fitting with his assessment of mathematics as a secure science, Aristotle is eager to show that his finitist conception of the universe can preserve the infinity the mathematicians in fact need for their practise (207b27ff.). According to Aristotle, the mathematicians do not really need an actual infinity; all they need is that any given line can be further extended or further divided, and there can always be a greater number than any assigned number (for any integer n there can be an integer n + 1)Footnote ²¹³ – all of which for him belong to the potential infinite.

A final look at Euclid’s usage of the infinite in the Elements shows that the fundamental philosophical discussions about infinity are interestingly absent from mathematical thinking of the time. On the one hand, infinite divisibility, which Aristotle claims to be important for the mathematicians and which Zeno’s paradoxes put into question, is not explicitly formulated (though implicitly assumed in ancient mathematics, as shown by On Indivisible Lines).Footnote ²¹⁴ The place where we may assume infinite divisibility to be explicitly in play (and to find some affinity to philosophical discussions) is the method of exhaustion. It is used in several propositions in Elements, Book XII, and shows that a figure can be approximated by a sequence of other, smaller figures inscribed inside it. For example, we ‘square’ a circle by continuously inscribing a sequence of regular polygons inside it, starting with a square, seemingly going on ad infinitum. However, Euclid stops after a small number of steps here, just enough to show that the difference between the original figure and the inscribed figure decreases by at least half at each step of the sequence. Accordingly, there is no talk about infinity here; Euclid does not tell us to continue always further.

On the other hand, Euclid assumes infinity in multitude and magnitude in the Elements in a way that does not suggest any conceptual difficulties with it; for example, for Euclid there exist infinitely many commensurable and incommensurable straight lines (Elements X, def. 3). And he even assumes infinity in cases where strictly speaking it is not needed, such as in I, 12, where it is ‘required to draw a straight line perpendicular to the given infinite straight line AB from the given point C, which is not on AB’ – here the given line is infinite, but in fact we can use a line of whatever size we choose; we do not need an infinite line.

There is one place in the Elements, however, where the employment of infinity is indeed necessary. In Book I, definition 23, parallels are defined with the help of infinity: ‘Parallel lines are straight lines which, being in the same plane, and being produced to infinity in each direction, meet with one another in neither (of these directions).’ And the fifth postulate in Book I claims that ‘if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced to infinity, meet on that side on which are the angles less than the two right angles’ (we are dealing with geometries only of flat spaces here); see also I, 29.

This is the only context where Aristotle’s suggestion does not seem to work, that mathematicians do not really need lines of infinite length but only of arbitrarily long length. For if we assume that the lines are not extended to infinity but only as long as we please, too many pairs of lines will count as parallel and thus falsify postulate 5. Hussey has suggested a modal reworking for Aristotelian finitists.Footnote ²¹⁵ But in Euclid, we do not find any such reworking, nor any sign of the fundamental philosophical debate about infinity. While philosophers of the classical time take mathematics as the main example to discuss what makes for a secure science, mathematicians at least in part ignore their philosophical considerations.