X Ï V Introduction
In the first Chapter, I hinted that the major problem motivating this dissertation, was finding the point of contact between logicism and structuralism, and finding the exact shape of the disagreement. Hopefully, the development of Frege’s theory of arithmetic — one of the threads in this knotty tangle — in the previous Chapter was persuasive. The aim of this and the next Chapter is to consider a quite distinct aspect of the tangle: the notion of structure. For the past century, mathematical activity has concentrated largely upon the notion of structure and mappings which preserve structure. Therefore, any philosophy of mathematics which takes mathematical practices seriously will need to give some account of structure.
What is meant in mathematics by structure will be explained in more detail shortly, but the following is a helpful first stab at capturing the notion:
A system is a particular, a structure repeatable. The set-theoretic hierarchy, the real numbers and the natural numbers are all mathematical systems. However, many systems share the same structure — for instance the natural numbers, the finite von Neumann ordinals and the Zermelo ordinals all share a particular structure.*
i The Structuralist Strategy
Very few philosophers take the explanation of structure to be a additional burden to the task of giving a philosophical account of arithmetic or set theory;" either structural concepts are disregarded, as they are taken to be only of marginal philosophical importance; or structure is taken to be the predominant notion, and used to give an account of arithmetic as well as of algebraic structures. Non-foundationalist writers, drawing their inspiration from current mathematical practice, have tended to concentrate on the notion of structure, opting for the second horn of this ‘dilemma’; traditional philosophies of mathematics opt for the first.
In an interesting article, Awodey has proposed a further distinction — between mathematical and philosophical structuralism:
[Mathematical structuralism] has already met with considerable success through a century of work by mathematicians pursuing a structural approach to their subject. * Melia (1995), pl27, Melia attributes this distinction to Corcoran; see Corcoran (1980)
’ One who does accept the additional burden is Charles Parsons; see for example. Parsons (1990). ~ 83 ~ ~
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Indeed this success is reflected in the current prominence of the notion of structure in mathematics.®
Philosophical structuralism, on the other hand, is a position which claims that philosophical mileage can be made by adopting the approach of (mathematical) structuralism and applying it, as it were, to problems in philosophy. Such an approach is an obvious way to take the methodology and practice of professional mathematicians seriously, by concentrating on the types of item which they study, rather than by concenti'ating on more familiar ‘High School’ disciplines such as geometry or arithmetic. This thought will often be summed up by the slogan ‘taking mathematical practice seriously’; it does not imply that the traditional positions ignore mathematical practice, but rather that the structuralist places a paiticularly strong emphasis on mathematical research and the methods of mathematical structuralism. A typical expression of structuralism might be this:
Reference to mathematical objects is always in the context of some background structure, and that the objects involved have no more to them than can be expressed in terms of the basic relations of the structure.*
or like this:
structuralism is the doctrine that mathematics in générai is solely concerned with structures in the abstract sense, that is, with systems left no further specified than as exemplifying the structure in question.®
ii Structures and Systems
The distinction between structures and systems will appear at various times thought the rest of this and the following Chapters. Although the distinction will be examined in closer detail later on, a fuller explanation than that offered briefly above will be helpful at this stage.
Mathematical systems — such as the natural number system N, the rational number system Q or the real number system R — have underlying structure: for example,
® Awodey (1996), p209 * Parsons (1990), p272 ® Dummett (1991), p295
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R has the same structure as any open interval of itself, e.g. (0, 1). This structure is the
complete Archimedean field, also known as the real closed field (RCF). It can be described — by way of an axiom scheme — independently of the objects which can form such fields. Any collection, such as R or (0,1), which satisfies these axioms defining the structure, will be an example of a real closed field. The structure can be studied independently of any of the instances or examples of the systems with that structure, and the results will hold for any of the systems with that structure; investigating structures is therefore a powerful methodological technique.
To the extent that every system has an underlying structure, there have been attempts to reduce all mathematics to the study of structure; for example, as was attempted by Bourbaki,® who took it that structure was the only mathematically salient feature of a system. Bourbald set about expressing every area of mathematics in explicitly structural terms, by concentrating on the first order models of mathematical systems. The power and elegance of these techniques influenced other academics, especially in France, such
as those working in fields such as theoretical linguistics {e.g. Saussure) and anthropology
{e.g. Lévi-Strauss). The rise — and fall — of structuralism as a literary theory, despite the early connection with mathematics, is entirely separate from the recent growth of stiucturalism in mathematics
The Bourbaki project constitutes an incredible mathematical achievement; however, it was not an unmitigated success — largely due to the expressive inadequacy of the first order logic that Bourbald relied upon.
More recently, category theory has been developed; this considers structures and
structure preserving mappings in order to characterise ‘the structure of structures’; i.e.
categories seem to be to structures, what structures are to systems. Some have suggested that category theoiy be used to provide a foundation of mathematics, while others have taken it as a pure distillation of the best methods of mathematical structuralism.
This Chapter explores the notion of structure and various positions which are forms of philosophical structuralism — on the whole, the account is critical, leading to the conclusions that the various forms of philosophical structuralism not only fail to give
Nikolas Bourbaki was the name adopted by a group of young French mathematicians, originally from Nancy. The original Bourbaki was one of Napolean’s generals — there is an old statue of him in the garden next to the Mathematics Institute in Nancy.
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solutions to familiar philosophy puzzles, but that it also does not capture the nature of mathematical structuralism. In the next Chapter various modest accounts of structure are developed, in order not only to give a philosophy of structure but also to expose the false dichotomy between the objects-only perspective of traditional philosophy of mathematics and the structures-only view espoused by modem structuralists.
The first section of this Chapter (§XV) introduces philosophical structuralism in more detail, and is followed by a section which looks at one of the main forms of structuralism, abstract-structuralism. The third section looks at pure-structuralism, while the fourth is more critical, dealing with some of the problems with the structuralist strategy. The fifth section leads on from these considerations, to offer an amended argument for structuralism about mathematics. Finally, the chapter concludes with a review of some of the arguments between logicists and structuralists.
X V Philosophical Structuralism
The structuralist hopes to make headway on the platonist on three important counts: she claims to have solutions to the platonist’s problems with epistemology, ontology and with reference.
It is usually thought that the objects-platonist is committed to a realm of abstract objects, and Benacerraf, as mentioned in the first Chapter, is usually thought to have caused trouble for any view of mathematical objects as acausal. The structuralist claims that it is possible to have knowledge of a structure from instances of the patterns which exemplify that structure; moving from the concrete to the abstract in one step, thus giving a cleaner account not only of the epistemology of mathematics, but giving rise to a simpler ontology too.
Michael Resnik presents his form of structuralism as a response to the platonist’s plight when confronted with this sort of epistemological problem. By arguing that the entities of mathematics are being misdescribed — by mistakenly thinking of them as objects, rather than as structures — he contends that the problem of knowledge of abstract entities can be resolved. The objects of mathematics are for him, not items such as numbers or sets, but patterns. He takes patterns to have a fairly obvious, non-technical meaning. As there is causal contact with patterns, Resnik is able to conclude that a reliabilist account can be given of mathematical knowledge, relying on this notion of
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pattern recognition.’
Resnik’s argument needs some preliminaries to get it going. He argues that
mathematical knowledge is not sui generis, but that it is of a kind with music and
language.
Of course, no one fully understands the mechanism behind any of these skills, but by putting mathematics in the same epistemological context as music or language, we remove some of the mystery enveloping standard platonism. The standard platonist has no way of convincing a skeptic that knowledge and experience of this type exists. The structuralist by contrast, can point outside of mathematics in order to demonstrate the possibility of mathematical experience and knowledge. He can even indicate the place to look for the mechanism involved*
Several steps are involved: first, recognition of patterns, then abstraction from those patterns. Resnik claims that:
At the last stage we leave experience far enough behind that our theories are best construed as theories of abstract entities.’
He is sensitive to the question of whether beliefs about patterns formed in this way are the source of genuine mathematical knowledge, but while he shows some sensitivity in his discussion of this question, his final response is dogmatic: he takes it as given that there are mathematical truths, and that we have mathematical knowledge, and that beliefs about patterns will be involved in this loiowledge.
This sounds very like Mill’s empirical epistemology for mathematics, where the mathematical is taken to be constituted of high level explanations of empirical regularities of a very general kind.*° From the various instances of patterns, the more general form is adduced. Although Resnik claims to be a realist about mathematics — possibly he is even a platonist — he solves Benacerraf’s dilemma by opting for a causal epistemology. (To a certain extent Shapiro also plumps for this option.) Although abstract, there is contact between mathematical items and the subjects of knowledge, by way of various intermediary patterns. While this at first sounds plausible, no mechanism is given to
’ The account of pattern recognition is developed in Resnik (1981) and (1982) * Resnik (1981), p35
’ ibid., p35
See Frege (1879), §16 for a concise treatment of Mill’s philosophy of arithmetic. — 87 —