X X I Î M odifying the structuralist account
It might seem that with all the problems presented in the last Chapter, there would be no point in trying to present a philosophical account of structure. Such problems include those facing structuralism in general — that the epistemological advantages it offers are illusory, that its ontological reduction is contestable and its solution to problems of referential access circular — as well as problems facing both abstract-structuralism — that it overgeneralises on the office/object distinction, at the expense of a faithful representation of this distinction in mathematical practice, while also offering no real explanation of the applicability of the account — and pure-structuralism — that the initial plausibility of the account is eroded as the intuitive notions used in the account are replaced by sophisticated and artificial formal notions.
But it should be remembered that the structuralist’s involvement in these issues, and her claim that all mathematics is structural, arose from a desire to replace objects- based accounts with structure-based accounts. Recall that philosophical structuralism, as a quite general position, was presented as a combination of two tenets: that philosophy of mathematics should relate to the working practices of professional mathematicians, and that those practices give an insight into how to solve certain philosophical problems; where ‘practices’ is inteipreted as the methods of mathematical sti'ucturalism.
Clearly it is the second of these two tenets which motivates the desire to do away with the objects-based accounts; dropping this extreme doctrine will lead to a more modest structuralism, one which accepts that as well as structural areas of mathematics, there are parts of mathematics not best described in structural terms, such as mathematical systems. This gives a very different picture of the role of structures in a philosophy of mathematics; it implies that an account of structure will be a burden additional to, and not in place of, an account of the objects occurring in mathematical systems.
Therefore, the task for the philosopher is to give an account of the nature of mathematical objects, especially in relation to reference to such objects and knowledge of them; to delineate the scope of structural mathematics, and harking back to Benacerraf’s insight, to give an account of why reference — in the usual sense — lapses in the case of structural mathematics. Moreover, a general framework will be required, spelling out just how the structural and the non-structural areas of mathematics relate.
The rest of this first section looks at the distinction between structures and
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systems, and places this distinction on a firmer footing. Frege’s views on arithmetic form the core of an account for mathematical systems; the remaining sections offer alternatives as to how the remaining tasks might be completed, that is, to give a characterisation of structure and explain how structural and non-structural areas of mathematics relate. The second section (§XXIII) deals with the first of three ways that such a structuralist position might develop, based quite closely on extending the Fregean account of systems; §§XXIV-V deal with two more ways of doing this, based on modifying the abstract- structuralist and pure-structuralist accounts.
i Structures and Systems
The difference between structures and systems — as pointed out in §XIV, i and
§XIX, ii — is that systems are particular instantiations of structures; or rather, structures
are the underlying basis of systems. If a distinction is to be drawn in such a fashion, it might be drawn in terms of those parts of mathematics where the items are particulars — individuated in some way — contrasted with those areas where the items are entirely general, as in group theory, where the places in a structure stand for arbitrary objects. On this interpretation of the structure/ system distinction, familiar systems such as natural numbers, integers, reals, rationals and complex numbers are all systems, with groups,
topologies, varieties, etc. being examples of structures.
This characterisation relies purely on the semantic content of the mathematical theories in question, and the extent to which the items featuring in the theory are deteiTninate or vague. Another characterisation might be given in terms of epistemological differences between those parts of mathematics where grasp of individual objects is structurally mediated — as in groups, rings and vector fields — and those areas where it is not: such as in arithmetic and set theory. Recall Wright’s argument that the concept ‘equinumerous’ is prior to the concept ‘progression’, and that grasp of the natural numbers need not depend on a prior grasp of the structure of the natural number system. ‘
However (contra Wright (1983) §xv) on the basis of such epistemological differences, I
claim real analysis falls on the structural side of the divide: unlike the arithmetical case, grasp of an individual real number — as a real number — does depend on a prior grasp of the structure in which it occurs. If the objects of a system are taken to be something like a natural kind — they are, after all, individuated by sortal concepts as natural kinds
‘ See §XX
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are — then taking a number to be a natural number, a member of a paiticular mathematical kind, does not depend upon knowledge of the stmcture of that kind. On the other hand, to appreciate a number as a real number (and not just its rational approximation) requires
structural knowledge not required in the arithmetical case, e.g. knowledge that the reals
are arranged as to be continuous, dense and connected." This is a major difference between the reals (and complex numbers) and the natural or rational numbers, and suggests that there may be two different ways of distinguishing systems from structures. In general, it will be the semantic distinction that is taken as primary; the epistemological differences will be returned to at the end of Chapter 5, where a fuller treatment of real and complex analysis will be given.
ii Frege on mathematical systems
To develop the sort of modest structuralism which has been suggested, at least three component accounts are required: one for mathematical systems and the objects of such systems; one for structures and for the places in those structures, and finally an explanation of how these two areas relate and exactly what the difference in status is, between objects proper and places in a stiucture.
Given the treatment of Frege’s arithmetical platonism in Chapter 2, it is an obvious choice to take this as a basis for an explanation of mathematical systems. This will supply an account of arithmetic, and with minor modifications should also give an account of set theory. Dealing with other examples of systems — such as the rationals, reals and complex numbers — may prove more difficult. Frege did try to develop an account of real analysis based on his treatment of arithmetic: generally the prospects for success of this endeavour are less than enthusiastic. For the moment however, assume that there is an account available for the reals and complex numbers, in the spirit if not the letter of Frege’s arithmetic. As was mentioned above, this issue will be returned to at the end of the next Chapter.
Meanwhile, recall that the revitalisation of Frege’s logicist project was glossed in Chapter 2, touching on the work of Dummett, Wright, Hale, Boolos and Heck. The main thought was that by interpreting the Context Principle as applied to reference, the following thesis emerges: that singular terms in true statements, in appropriate (indicative)
^ This may explain some of the problems caused by the discovery by the Pythagoreans of real numbers. Without the grasp of concepts such as continuity and denseness they could not appreciate, for example, Vi2
as part of a system — the reals — and hence took it as an isolated aberation. - 124 -
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contexts, refer to objects. As numerals are such singular terms, and as numerical identities are true and offer the appropriate context, numbers are objects. Hume’s Principle (N=) was introduced to define Number, and to generate arithmetic.
iii B enacerraf s insight and some desiderata
In the previous Chapter, two claims were made concerning Benacerraf’s work. Firstly, that the interpretation of Benacerraf (1965) as arguing for the indeterminacy of reference to mathematical abstracta, not only cannot be sustained, but that this misrepresents Benacerraf’s actual argument. Rather he should be taken as arguing that as singular reference — were it to occur in a discourse — would be determinate, there can be no singular reference occuning in arithmetic, due to the multiple instantiations of co- sequences which are possible. Secondly, that Benacerraf appeals to an analysis of the obviously structural areas of mathematics — to a theory of modest snucturalism in effect — in order to conclude that the statements of mathematics do not have the logical structure that their grammar suggests — they are quantified hypotheticais.
Using Benacerraf’s first insight — that reference is not singular reference in those areas of mathematics where reference is not to unique and determinate objects — in the light of the overall failure of the Extension Argument to sustain radical structuralism, requires some further work.^ If Frege is right — and I take it that Chapter 2 supports the claim that he is — then because numerical identity statements provide the appropriate semantic contexts, numerals refer determinately to objects, to numbers. To hold onto both the Fregean and Benacerrafian insights, required that there be some gap between co- sequences and arithmetic: that perhaps while co-sequences capture all of the mathematical uses of numbers, they fail to capture the full concept of Number.
Benacerraf writes:
“Objects” do not do the Job of numbers singly; the whole system performs the Job or nothing does. I therefore argue, extending the argument that led to the conclusion that numbers could not be sets, that numbers could not be objects at ail; for there is no more reason to identify any individual number with any one particular object than with any other"*
^ The failure of the Extension Argument to secure extreme structuralism is briefly discussed in §XIX, iii\
Wright’s criticisms, discussed in §XX, ii block the arguments in favour of radical structuralism. "* Benaceiraf(1965), pp290-I
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This conclusion to a Benacerraf-style argument, in the light of the distinction between co-
sequence and arithmetic becomes: in structural contexts, although there appear to be singular terms occurring in true statements, this does not guarantee their singular reference: instead the ternis are features of generality.
Obviously, there are different ways of developing an account of modest
structuralism: Benacerraf’s in re strategy is but one way of articulating modest
structuralism. There may be ways of refining Shapiro’s ante rem account, concentrating
on that feature that when fewer linguistic resources are used to describe an office than are used to pick out an object, that leaves the office open for a kind of indeterminacy or vagueness that is not otherwise possible. Alternatively, one might take a third route, and think of the terms referring to places in a structure not as proper names, but as arbitrary names.^
iv Some desiderata
Already, three aspects of an overall philosophy of mathematics have been identified: an account of systems and the objects occurring in them, along the lines of
Frege’s treatment of arithmetic, i.e. supported by the key thought that singular terms in
true statements occurring in the appropriate contexts, refer to objects. In addition, an account of structures and places in a structure should be given; this should be compatible with the basic Fregean insight, but should accommodate the thought that the terms referring to places in a structure are features of generality: perhaps such terms might be called arbitrary or general names. Thirdly, an account is required that will explain how these two areas relate, why Frege’s thesis breaks down or applies only in a weakened form in structural contexts, and what the difference in status is, between objects of a system and places in a structure.
At the end of §XIV, a possible amendment of Shapiro’s ante rem structuralism
was suggested, based on a notion of narrow reference — a stratified account of reference, somewhat similar in strategy to Dummett’s use of thin reference — which takes places in a structure to be only fully determinate in a narrow range of cases (those involving other
structural terms) and are indeterminate in the wider case. Interpreting Shapiro’s ante rem
structuralism in this way will result in the following view: that offices are a certain kind of
This may be a promising route, based on notions given in Lemmon (1965) and Fine (1985); see §XXIII, / below.
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object — ones which cannot be fully identified, and can only be distinguished with respect to the other elements of the structure in which they occur. Once more information is available — once the quantity of the linguistic resources is increased — these offices can be distinguished and discriminated from ordinary objects. This way of meeting these desiderata is considered in §XXIV.
Rather than take reference as the weak link in the chain, it is possible to concentrate upon another aspect of Frege’s semantic thesis. Benacerraf’s conclusion is that singular reference does not occur because structural statements are not indicative. He argues that they are quantified conditionals, and many have sought to perform reductionist rephrasing of mathematical statements as a result. Heilman’s modal structuralist account is an attempt to make the logical structure of these expressions explicit in a different fashion, using a primitive notion of possibility rather than quantification.
Explicit rephrasing may not be necessary. If the statements are not indicative, then they may be straightforward subjunctive expressions; not only does this meet the desiderata above, it will also sidestep most, if not all, of the criticisms raised against the more usual forms of pure-structuralism. §XXV is devoted to developing such an account
of modest in re structuralism.
XXIII Substitution & Divided Reference
Before trying to modify the structuralist accounts presented in Chapter 3, it is worth briefly considering whether there is a more natural extension of Frege’s arithmetical platonism. The simplest tale will be that abstraction principles — or some other means of fixing truth conditions — determine a base ontology of numbers, sets, classes and so on. Given this base, structures could be seen as no more than the various combinations of those elements; Putnam’s phrase describing a structure as a “possible combination of objects”® springs to mind.
i Systems, structures and substitution
Systems, on this view, are primary — they are constituted by the various
independent mathematical objects of a particular type, e.g. sets, natural numbers, and
* Putnam (1967)
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rational numbers. Structures are to be seen as the possible combinations of the objects of such systems. A structure will then be a family of system-like arrangements, and reference to a place in a structure would be divided among the possible substitution instances from the family of systems in that common arrangement.
This would hold the spirit of Benacerraf’s comments to the effect that structures characterise what a family of systems have in common. There will be no more to the structures than the systems — and the objects of those systems — which constitute them. Reference to a place in a structure is shorthand for reference to any of the objects which occupy that position in the systems underlying the structure. Talldng in terms of reference to a place in a structure will be shorthand for the class of objects which can fill that place.
There are two ways of interpreting this use of the substitution class; either to take the places in a structure as representative of all the possible substitutions, in the sense that reference to the places in a structure could be replaced by reference to any of the various instantiations of those places, or to take the places in a structure as an intermediary between the terms of the theoiy and the set of instances taken collectively.
The first of these options involves treating the places in a structure as representational or arbitrary objects, and taking the terms referring to such places as arbitrary names. Articulation of this view would draw on the following thought:
Think of what Euclid does when he wishes to prove that all triangles have a certain property; he begins Met ABC be a triangle’, and proves that ABC has the property in question; he then concludes that all triangles have the property. What here is ‘ABC’? Certainly not the proper name of any triangle, for in that case the conclusion would not follow. ... It is natural to view ‘ABC’ as the name of an arbitrarily selectedtriangle,B.
particular triangle certainly but any one you care to pick. For if we can show that an arbitrarily selected triangle has F, then we can certainly draw the conclusion that all triangles have F. ... introduce ... names of arbitrary selected objects in the universe of discourse, and call them for short arbitrary names?
Euclid’s proofs work on a simple approach to axiom schemes and to the notion of representational objects; mathematical structuralism, heavily influenced by Klein and Hilbert’s structural treatment of geometry, is a refinement of this proof technique: places
in a structure are arbitrary objects par excellence. Fine describes arbitrary objects in the
following manner:
Lemmon (1965), pp 106-7
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In addition to individual objects, there are arbitrary objects: in addition to individual numbers, arbitrary numbers ... With each arbitrary object is associated an appropriate range of individual objects, its values; with each arbitrary number, the range of indvidual numbers ... An arbitrary object has those properties common to the individual objects in its range.®
However, despite the appeal — and appropriateness — of explaining places in a structure in terms of arbitrary objects, the accounts which Lemmon and Fine give are entirely general, and need not be restricted to the structural case. According to Fine, what a theory of arbitrary objects shows is that there are genuine alternatives to the ‘Frege-Tarski
analysis of quantifiers’, i.e. alternatives to the standard semantic analysis of objectual
quantifiers. If he is correct, then there would be nothing unique about a description of