Figures of Thought

Mathematics and Mathematical Texts

David Reed

• Publisher: Routledge
• Year: 2003
• ISBN:
  9781134892532
• Edition: 1
• Topic: Filosofía
• Subtopic: Historia y teoría filosóficas

Rarely has the history or philosophy of mathematics been written about by mathematicians, and the analysis of mathematical texts themselves has been an area almost entirely unexplored. Figures of Thought looks at ways in which mathematical works can be read as texts, examines their textual strategies and demonstrates that such readings provide a rich source of philosophical issues regarding mathematics: issues which traditional approaches to the history and philosophy of mathematics have neglected.
David Reed, a professional mathematician himself, offers the first sustained and critical attempt to find a consistent argument or narrative thread in mathematical texts. In doing so he develops new and fascinating interpretations of mathematicians' work throughout history, from an in-depth analysis of Euclid's Elements, to the mathematics of Descartes and right up to the work of contemporary mathematicians such as Grothendeick. He also traces the implications of this approach to the understanding of the history and development of mathematics.

Part I

The subject matter of geometry in Euclid, Descartes and Hilbert

1
The opening of the Elements

1.1
Orientation

A vast amount of commentary has accumulated since ancient times around the Definitions, Postulates and Common Notions with which Euclid commences the Elements. In certain cases (one thinks immediately of the notorious fifth or ‘Parallel’ Postulate) much of the subsequent history of mathematics can be viewed as one extended commentary on this text! For the English reader, Heath's edition of the Elements surveys much of this commentary up to its publication date (2nd edn, 1925) and has the benefit of being easily available.¹ The analysis conducted below does not require an extensive familiarity with this material as it refers to the lines of commentary that have been generally pursued only to clarify the points at which it diverges from the various traditional approaches. At certain points, however, the analysis does turn on specific words or phrases in the Greek original. Where this is the case the appropriate philological or linguistic background is provided.

The reader may find the nature of the commentary more in line with the methods which might be more typically employed in the analysis of a poem or other ‘literary’ texts rather than in the examination of a mathematical treatise. The feeling of disorientation that results is likely to be productive. To facilitate the initial stage of the analysis the content of the first seven Definitions of the Elements (together with Definitions 1 and 2 of Book XI) are laid out schematically in Table 1.1.

1.2
Points and lines

Euclid's Definition 1², ‘A point is that which has no part’, frames the entire sequence of Definitions and provides a ‘principle’ in the sense of ‘a beginning’.3 Euclid does not have reference here either to spatial

Table 1.1

Objects	Definition	Limits	Regularity
Point	That wich has no part	None	None
Line	Breadthless length	Points	Straght lines lie evenly with points on themselves
Surface	Length and breadth only	Line	Plane surfaces lie evenly with straight lines upon themselves
Solid	Lenght and breadth and depth	Surfaces	None

location or to ‘quantity’ as such. It will be seen in Part II that quantities are derived terms in the Elements, arising from specific ways to analyse specific things. Instead of these more frequently found approaches, Euclid employs the term ‘part’ (Greek

) as he will again at other turning points in his argument, and a negative grammatical construction (the only other Definition to use negation in Book I being the last Definition, that of parallel lines, which shares a framing function with this Definition) to indicate a limit or an extremity.

The limit in question is not a limit of magnitude or position, rather it is a limit of intelligibility, an extreme of discourse. In this opening phase of the Elements ‘parts’ are things in terms of which other things can be defined, discussed and understood. That which has no part merely has existence–completely undifferentiated existence. One does not have to construct or imagine such things; anything can be considered to be a point if it is considered without parts.⁴ All such things are undifferentiated: one cannot speak of singular and plural, of here and there, or indeed of any other type of qualification.⁵ Hence this definition marks the limit of that which can be discussed or analysed. Everything else which will enter into the argument of the Elements will have ‘parts’ of some kind, and determining the nature of these parts and their relationships to each other and to the wholes which they constitute is the method of Euclid's science. Euclid's choice of this rather strange formulation for defining ‘point’ will be seen to be reflective of his entire approach.

It should be superfluous to add that Euclid is not here attempting to describe (well or poorly) the visual appearance of a point, nor his, our or the ideal mathematician's intuition or perception of what a point is. Any reading of Euclid along these lines (Heath's notes contain a number of examples) omits all of the aspects of this definition which differentiate it from other similar definitions used by mathematicians ancient and modern. While it may have been traditional then as now to begin geometry texts with the definition of a point, whatever the tradition demanded Euclid has provided a definition which provides its own justification for commencing his argument. It is sufficient to note that he begins his argument with that which indicates the beginning of discourse; to refer to extraneous reasons for the formulation or positioning of this definition is unnecessary.

The orientation provided by Definition 1 clarifies the sequence which follows. Unlike a point, a line does have parts, one part in particular which is labelled length.⁶ By referring to something with one part Euclid shows clearly that he does not have in mind a sense of ‘part’ which has to do with division or with ‘sub-objects’. Parts are the ways in which things may be known or described. As far as their definition is concerned, lines are things about which one can know only one thing, their length. For convenience this type of definition will be referred to as definition by specification of a measureable (in this case length). It is inappropriate to think of length in this context in geometric, metric or measure theoretic terms. All of these approaches require previous specification of some type of measure or some kind of line. Definition 2 merely states that lines (a) are distinguished from points by having parts, (b) are distinguished from other geometric things by having only one part and (c) can be compared amongst themselves by this part, their length. To repeat, anything can be considered to be a line, provided it is considered as having only one characteristic or term of comparison. Clearly there is no distinguishing (at this stage in the argument) between ‘two’ lines of the ‘same’ length. The grammar of singular and plural is limited precisely to distinctions of length.

It is also inappropriate to think of this definition as embodying some (primitive) type of ‘dimensional’ analysis. As can be seen from Table 1.1, each item in the sequence ‘point’, ‘line’, ‘surface’, ‘solid’ is defined in a (slightly) different manner. The distinguishing characteristic of dimensional definitions is that the things to be defined are defined in the same way except for the number of dimensions involved. Euclid's sequence by contrast has a form or shape and each term occupies the place it does for a particular reason. It therefore cannot have a ‘dimensional’ character, even a primitive one. The Elements opens with the definition of point because point is the extreme of discourse. It continues with lines and surfaces because these have parts through which they can be known. Eventually we will see why it terminates with solids.

Now if lines are those things which differ from one another in precisely one way how does one actually go about making comparisons between lines? The answer is provided by Definition 3 which, as far as most commentators are concerned, does not define anything at all. This Definition states that points are the extremities⁷ of lines, i.e. the comparisons of lines as lengths is effected by means of points as their extremities. One might say therefore that Definition 3 defines delimited lines by specifying the means by which this delimitation takes place. There are two aspects to this:

• points can function in this way without further specification just because there is nothing further to say about a point other than that it is a point;

• by functioning as the extremities of lines, points acquire a further characteristic which allows them to be differentiated, i.e. one can now speak of singular and plural, point and points.

Once again it should be clear that Euclid is not attempting to describe how lines and points ‘look’, nor is he asserting that lines are ‘made up of points’ or foreshadowing some notion of ‘incidence’. The purpose of these Definitions is to define terms and establish a subject matter, not to describe already known or existing things.

Definition 4 is of ‘straight lines’ and is, on most readings of the Elements, virtually unintelligible. The approach proposed here, however, suggests a simple and clear interpretation. As points define the delimiting of lines, straight lines are precisely those for which no additional specification of the relationship between points-as-extremities and lines-as-delimited is either necessary or possible. For straight lines this relationship is always the same. No metric or measure theoretic specification is implied here. The delimitation happens in the same manner throughout the line, but nothing is said about what this manner may be.⁸

Definitions 2, 3 and 4 taken together provide the paradigm in Euclid's mathematics of a ‘measured thing’:

• that which is measured (in this case lines) is defined in terms of a measurable (in this case length);

• the measurable is further determined by specifying how the measuring or delimitation is to be performed (in this case by points-as-extremities) and this specification requires nothing further (there is nothing further to say about points);

• a special type or kind of measured thing (in this case straight lines) is defined by requiring that the delimitation of the measurable does not vary with what is delimited and that no further specification of this relationship is necessary.

This must be understood as a formal procedure which moves mechanically step-by-step to define and determine the measured thing. Once the starting point (!) is adopted there is no opportunity for variation or deviation. This is Euclid's approach to ‘rigour’ in mathematics and failure to understand his method leads to the view that some ‘selection’ process based on tradition or philosophical inclination led Euclid to pick the particular formulations and sequence of definitions that we find in the Elements.⁹ The importance of a clear understanding of the nature of Euclid's argument goes beyond an appreciation of the clarity and cogency of his thinking; it is fundamental to understanding his subject matter. As has been shown, points, lines and straight lines can be completely and satisfactorily related to one another through the definitions themselves. Because of this they do not pose by themselves interesting mathematical problems for Euclid and do not constitute the subject matter of the Elements. None of this can be said for surfaces.

1.3
Surfaces

Definition 5 of surfaces mimics Definition 2 of lines with two parts, length and width, instead of length alone. The remarks above on the definition of line carry over with no substantial modifications. Clearly, once again, spatial intuition is not at issue. Nor should it be thought that a specific combination of length and breadth in a notion of area is referred to. The two measurables, length and breadth, exist independently and, at least at this stage of the argument, there is no way to put them together.

Definition 6 mimics Definition 3 in specifying lines as the extremities of surfaces as points are of lines. However, the grammar of the plural ‘lines’ differs from that of points. As noted above, it was precisely in functioning as the extremities of lines that any distinctions between points could be drawn. From their definition alone there are no characteristics which would permit either numerical or generic distinctions among them. But lines do have a variety of ways in w...