The Genealogy of Knowledge

Analytical Essays in the History of Philosophy and Science

Stephen Gaukroger

• Publisher: Routledge
• Year: 2018
• ISBN:
  9780429776380
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

First published in 1997, this volume expands the analytical philosophical tradition in the face of parochial Anglo-American philosophical interests. The essays making up the section on 'Antiquity' share one concern: to show that there are largely unrecognised but radical differences between the way in which certain fundamental questions – concerning the nature of number, sense perception, and scepticism – were thought of in antiquity and the way in which they were thought of from the 17th century onwards. Part 2, on early modern thought, explores the theoretical characterisation of the role of experiment in early modern physical theory through Galileo's embracing of experiments, along with Descartes' automata and issues in a relatively neglected but especially intractable part of Descartes' philosophy: how he conceives of what a successful inference consists in and what it is that makes it successful. The final section deals with the philosophical foundations of physical theory, the distinction between the human and the natural sciences, the philosophical-cum-scientific foundations of Marx's idea of socialism, and Nietzche's criticisms of the very notion of science, concluding that Nietzsche's probing questions cannot be dismissed, as he has opened up some genuinely challenging issues which we ignore at our peril.

Part One
Antiquity

1
Aristotle on intelligible matter

In Metaphysics K, Aristotle speaks of mathematical objects as having matter: ὅλως δ' ἀπορήσεέ τιςaἂυnποίαςeἐστὶυmἐπιτήμηςtτὸSδιαπορñσαιrπερὶ tτñςtτῺυ(μαληματικῶυ ΰλης(1059^b14-1;). There can be only one kind of matter that he has in mind here. Sensible matter is ruled out because this is the proper object of physics whereas, as he points out (1059^b2021), the matter of mathematical objects is the proper study of first philosophy. Nor can prime matter be at issue here, since this is a purely limiting notion introduced to account for changes between contraries in sensible matter, such as generation and corruption. This leaves us with what Aristotle calls ";ληoυοητή—noetic or 'intelligible' matter—and, in Metaphysics Z, we are told explicitly that 'intelligible' matter is present in the objects of mathematics: ὕλη δὲ ἡ μὲυ αἰσλητή ἐστιυ ἡ δὲ υοητή, αἰσλητὴ μὲυ οου χαλκὸς κ:αὶ ξύου καὶ ὃλη, υοητὴ δὲ ἐυ τοῖς αἰσλλητοῖς ὑπάρχουσα μὴ αἰσλητά, ο ου τὰ μαληματικά (1036^a9-12).

Insofar as this claim relates to geometry it has occasionally caused puzzlement, but there is a relatively straightforward solution to the puzzle. Insofar as it relates to arithmetic it is acutely problematic, but Aristotle nowhere even suggests that it does not cover the whole of mathematics. My main concern in this paper is to make sense of the idea of numbers having intelligible matter but, for reasons that will become clear below, this requires that we first elucidate what is involved in the doctrine that geometrical figures have intelligible matter.

I: Geometrical figures

The subject matter of mathematics is characterised by the fact that its objects do not change and are dependent (Metaph E, 1025^b1f). These objects are noetic and we come by them through 'abstraction' (ἀφίρεσις) from the numbers and shapes of sensible things. Because these latter numbers and shapes are the numbers and shapes of sensible things we can call them sensible numbers and shapes. The matter of sensible numbers and shapes is clearly sensible matter, and not intelligible matter. Intelligible matter comes into consideration when we ask for the matter of noetic numbers and shapes, and Aristotle thinks that the matter of noetic numbers and shapes is noetic or intelligible matter. Numbers and shapes are always properties, or attributes, of something and when we abstract them from sensible objects they must still be the numbers and shapes of something. Since they cannot be the numbers and shapes of sensible matter they must be the numbers and shapes of something else: intelligible matter.

Put in another way, we come by the objects of mathematics by 'abstracting' the forms of sensible objects, the forms we abstract being those that come under the category of quantity. Because we abstract these forms, the form of what we abstract from and the form that we abstract are the same: the sensible circle and the noetic circle have the same form. This leads us to ask what it is that distinguishes the noetic circle from the sensible circle. The noetic circle is not pure form because geometrical abstraction, in Aristotle's view, does not yield pure forms; moreover, he is too wary of Plato's doctrine of the Forms of mathematical objects to talk of pure forms in this context. There is only one pure form, God as defined in Metaphysics A, ch. 9, and God is not a geometrical figure. Geometrical figures are always figures of something—just as numbers are always the numbers of something (cf. Ph 221^b14)—and what they are the figures of in the noetic case is a purely noetic substratum, intelligible matter.

In the case of geometry, Aristotle employs two quite different kinds of abstraction. The first kind involves disregarding the matter of sensible objects so that we are left with properties like being triangular and being round. Geometry investigates "being round' not qua a form in its own right since this is the province of first philosophy which alone can investigate the essence of roundness, something that geometry has to take as given (Metaph E, 1025^b10-16). Nor does it investigate it qua form of sensible objects, which is the province of physics: De Caelo 297^a9 ff., where it is a physical proof of the sphericity of the earth that is given, makes this clear. It investigates it qua form of whatever most generally speaking is round. And whatever most generally speaking is round is something that we arrive at through the second kind of abstraction,¹ in which we disregard the properties of sensible objects so that it is that has these properties becomes the object of investigation. What we are left with is the substratum of indeterminate extension characterised solely in terms of its spatial dimensions: length, breadth and depth. Such a substratum cannot be sensible since it has been deprived of the properties that would render it sensible; nor can it be something which is independent since it is simply an abstraction. It is this substratum that Aristotle calls 'intelligible' matter (Metaph Z, 1036^a1-12; also 1037^a2-5).

As Mueller² has pointed out, geometry can be seen to require an intelligible matter upon which purely geometrical properties are imposed, the intelligible matter and the geometrical properties being things that we come by through abstraction. Intelligible matter plays an important role in this respect since matter must be part of any definition: ἀεὶ τοῦ λόϒου τὀ δὲ ἐνέργειά ὰστιν, νίον ὁ κὑκλος σχ῅μα ἐπίπεδου (Metaph H, 1045^a34-35). Intelligible matter figures here as genus, a point reinforced by the discussion in ch. 10 of the first book of the Posterior Analytics where, in the case of arithmetic, Aristotle gives units as the genus and odd, even, square and cube as the properties; and in the case of geometry he gives spatial magnitude (jj£YE0oq) or simply points and lines as the genus (76^a35-36 and 76^b5-6), and incommensurable, inflection and verging as the properties. Aristotle considers that all spatial magnitudes can be generated from lines and points (he quotes the Pythagoreans with approval on this at de An 430^b20) and this allows him to carry the abstraction that yields the three-dimensional substratum further so that it yields planes and finally lines and points. Hence, as we might expect, lines, planes and solids have substrata of different dimensions.

In sum, noetic geometrical figures have form and matter, and both are abstracted from sensible objects yielding geometrical properties imposed upon a noetic substratum of the appropriate dimension.

II: Arithmetic

Aristotle's implicit claim that numbers have intelligible matter is much more problematic because, quite simply, it is difficult to make much sense of the idea, and the problems are compounded by the fact that the examples of intelligible matter that he presents are always geometrical. This could mean one of four things: (1) he gives us an account of the intelligible matter of numbers but in some lost work; (2) he expects us to draw some analogy from the geometrical case; (3) he is being disingenuous in that he is committed to the idea of numbers having intelligible matter but since he cannot provide any account of how this is possible he simply leaves us with the bald statement that they do: (4) he gives us an account of the intelligible matter of numbers but we have failed to recognise it as such.

The first two of these possibilities are implausible. As far as (1) is concerned, not only have we no reason to suspect that such an account is given in any of the works we think may be genuine but lost, but the fact that he provides several examples of the intelligible matter of geometrical figures, particularly in the Metaphysics, suggests that if there were an account then this is the place we should find it. As regards (2), it is highly unlikely that Aristotle expects us to draw arithmetical analogies, at least in any straightforward sense, from the geometrical case because he himself is always the first to try to clarify things by analogy, yet in this instance he makes no attempt to do so. And what analogies could we conceivably draw? What could we abstract from what has properties other than a substratum of the kind that he invokes in the geometrical case?

This leaves us with (3) and (4), and clearly whether we accept or reject (3) will depend on whether we reject or accept (4). Our best line of enquiry will therefore be to determine whether any case can be made for holding (4). I suggest that the crucial question here is that of the relation between geometry and arithmetic, for Aristotle begins with statements about mathematical objects having intelligible matter and immediately proceeds to a discussion of the intelligible matter of geometrical objects. One thing this might mean—and so far as I know this possibility has not been taken up before—is that the discussion of geometry is meant to cover the whole of mathematics, that is, geometry and arithmetic. If, for example, Aristotle had identified geometry and arithmetic then we would have a clear solution to our problem since the intelligible matter of both would be the same. If, on the other hand, he had kept the two absolutely distinct then no solution along the lines that I am suggesting would be possible. He does neither of these, however, and the problem is to determine what exactly the relation is and whether it is such as to allow us to move from his statements about the intelligible matter of geometrical objects to numbers. In pursuing this line of enquiry there is one important consideration that we must bear in mind, namely that there is reason to suppose that Aristotle's doctrine on this question is not completely idiosyncratic, Aristotle himself does not go through his normal procedure of criticising other views on the question, nor do the early commentators remark on the doctrine. These two considerations, together with the fact that he does not even remark on the (postulated) move from geometry to arithmetic, suggest that the doctrine that he is advocating on the relation between geometry and arithmetic is reasonably commonplace, commonplace enough not to deserve comment.

Let us begin by considering arithmetical operations involving noetic numbers. For Aristotle, and for the Greeks generally, these operations—addition, multiplication, subtraction, division, finding roots and taking powers—are performed using geometrical entities: points or lines, planes and solids. In Aristotle's work, as in Greek mathematics generally, the basic entity that we operate with is the line length, so that arithmetical operations are performed by manipulating line lengths. (For present purposes, it does not matter whether these are continuous line lengths or linear arrangements of points since the latter effectively function as lines in arithmetical operations.³) What is involved in such operations requires careful consideration. The traditional view is that the geometrical manipulations merely represent arithmetical operations in such procedures and this view gives a large degree of autonomy to arithmetic with respect to geometry. The kind of representation involved here is what we can call 'notational representation'. In this notational sense, we can represent noetic numbers and figures by numerals, letters, lines, figures or whatever. It is in this sense that we now 'represent' noetic numbers by numerals or letters, and noetic geometrical figures by geometrical figures drawn on paper. Aristotle, in this sense of 'represent', represents numbers by line lengths and sometimes by letters, and he also represents times and motions in this way (e.g. Ph 237^b34 f.)

If we consider that geometry simply provides a notation for arithmetical operations then we are clearly free to substitute a different and less restrictive notation for the geometrical one. This is what many commentators have traditionally done, 'reformulating' the geometrical proofs given by the Greeks algebraically.⁴ There is, however, no justification for this procedure in the Greek texts and in fact it seriously distorts the concerns of Greek arithmetic and geometry, as recent work which has taken these concerns seriously, rather than simply trying to find in them anticipations of more recent developments in mathematics, has shown.⁵ It would take us too far afield to go into the failings of the thesis the Greeks had a "geometrical algebra' here, but perhaps an illustration will serve to pinpoint more exactly what the problem of the relation between arithmetic and geometry is in Greek mathematics. Consider the arithmetical operation of multiplication and, in particular, the dimensional change involved in this operation. Since the dimensional aspects of geometry are retained in multiplication the product is always of a higher dimension. When we multiply two line lengths together, for example, the product is a rectangle having those line lengths as its sides. This is not a notational constraint, it is inherently connected with the idea that numbers, for the Greeks, are always the numbers of something. A consequence of this is that when we multiply, we must always multiply numbers of something: we cannot multiply two by three, for example, we must multiply two somethings by three somethings. It is in this sense that Klein has called numbers 'determinate' for the Greeks: they do not symbolise 'general magnitudes' but always a plurality of determinate objects.⁶ Moreover, the dimensional aspects of geometry are retained in arithmetical operations, as are the physical and intuitive nature of these dimensions so that, for instance, no more than three lines can be multiplied together since the product here is a solid, which exhausts the number of available dimensions.⁷

This in itself should alert us to the close relation between arithmetic and geometry, but we should not conclude from this that noetic numbers were identified, with noetic lines by the Greeks. One thing Aristotle is very clear about is the fact that we cannot identify arithmetic and geometry. This is ruled out by his division of the category of quantity (<οσóν) into two kinds: plurality (πλῆθος), which is numerable, and size or spatial magnitude (μέγεθος), which is measurable (Metaph ▵, 1020^a7 f.) Central to this distinction is the idea that a plurality is potentially divisible into discontinuous parts and a spatial magnitude into continuous parts (1020^a10-11). The line, for example, is divisible into continuous parts in that it is infinitely divisible (Ph 232^b4 and Cael 268^a7, 29), so noetic numbers cannot be identified with noetic lines.

Even if we were able to identify noetic numbers and lines, this would in fact not be ...

Table of contents

• Cover
• Half Title
• Title
• Copyright
• Contents
• Acknowledgements
• Introduction
• Part One: Antiquity
• Part Two: Early modern thought
• Part Three: The Enlightenment and beyond
• Index of names