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What is the nature of our mathematical knowledge? The great 19th century German philosopher/mathematician, Gottlob Frege,
famously attempted to show that the fundamental laws of classical arithmetic and analysis are theorems of pure logic. If this were true,
mathematical knowledge would be of a piece with knowledge of logic. Notoriously, Frege’s programme failed, his system collapsing
into inconsistency. He himself then renounced the whole approach and three generations of philosophers agreed that he was right to do so. The publication in 1983 of Crispin Wright’s Frege’s Conception of Numbers as Objects (FCNO), marked the start of a period of active reappraisal of Frege’s attempt. Wright argued that, at least as far as number-theory (the theory of the finite cardinals) is concerned, important formal and philosophical insights can be salvaged from the wreckage of Frege’s account. In view of the increasing interest in the main philosophical ideas in FCNO and growing recognition of their technical potential, we now launch a five-year collaborative project to explore the prospects of extending the neo-Fregean approach to real and functional analysis and to classical set theory, and to examine its philosophical significance and problems in greater depth.
The neo-Fregean thesis about arithmetic is that knowledge of the basic arithmetical laws (essentially, the Dedekind-Peano axioms)—and hence of the existence of a range of objects which satisfy them—may be based a priori on “Hume’s principle”, that (informally)
The number of F’s is the same as the number of G’s just if the F’s and G’s can be put into one-to-one correspondence.
More specifically, the thesis involves four ingredient claims:
i. That the language of higher-order logic plus the cardinality-operator, “the number of…”, as introduced by Hume’s principle, provides a sufficient definitional basis for an expression of the axioms of arithmetic;
ii. That those axioms, so expressed, may be derived within a consistent system comprising Hume’s principle and a standard higher-order logic;
iii. That someone who understood a higher-order language to which the cardinality operator was to be added would learn, on receiving Hume’s principle as an implicit definition of that operator, everything needed to understand any of the new statements that are then expressible.
iv. Finally and crucially, that Hume’s principle may be laid down without significant epistemological obligation beyond that of any implicit definition: that the principle may simply be stipulated as explanatory of the meaning of statements of numerical identity, and that—beyond the issue of the satisfaction of the truth-conditions thereby laid down for such statements—no competent demand arises for an independent assurance that there are objects whose conditions of identity are as it stipulates.
Claims (i) and (iii) concern the epistemology of the meaning of arithmetical statements, while (ii) and (iv) concern the recognition of their truth. Claims (i) and (ii) are proven: Frege himself established (i) and the first explicit demonstration of (ii) was given in FCNO. So the neo-Fregean thesis about arithmetic turns on the informal philosophical theses, (iii) and (iv). If all four theses can be sustained, then arithmetical knowledge emerges (not indeed as of a species with knowledge of pure logic, but) as derivable from logic and implicit definition —a sufficient basis to sustain Frege’s claim of the analyticity of arithmetic.
FCNO went a fair way towards justifying (iii) and (iv). But increasing interest in the neo-Fregean programme, and specific contributions, variously supportive and critical, by logicians and philosophers on both sides of the Atlantic now encourage a deeper-reaching exploration of the issues.
Aims and Objectives in more detail
The task naturally divide into philosophical and formal (mathematical) ones. The main philosophical issues divide into two groups:
a. On Abstraction principles
The type of principle, illustrated by Hume’s principle—wherein the obtaining of an equivalence relation (like one-to-one correspondence) on items of a familiar kind is nominated as the truth-condition for identity statements concerning a new kind of object (in this case numbers)—have come to be known as abstractions. Two major issues concerning abstraction principles in general are:
1. the question whether implicit definition can constitute a non-inferential source of a priori knowledge, and whether, if so, Hume’s Principle and other neo-Fregean abstractions can qualify as implicit definitions in the relevant sense.
2. an abstraction says nothing about the truth-conditions of identity statements in which only one of the related terms is of the kind it concerns. (Hume’s principle, for instance, says nothing about “the number zero = Julius Cæsar”.) Does this mean that such principles fail to fix any distinction between the abstracts they introduce and things of other sorts and hence do not genuinely explain what sort of things those abstracts are? Is it a serious objection if so? More specific standing problems concerning Hume’s principle and the other abstractions which the neo-Fregean is likely to want to invoke include:
3. the question of Impredicativity. The first-order variables occurring in the second-order definition of “F is one-to-one corespondent to G” must range over numbers if Hume’s principle is to provide a proof of the infinity of the number series. Some commentators have worried that this makes the principle explanatorily circular. Can this worry be allayed?
4. the Bad Company objection. Some abstractions—axiom V of Frege’s own historic system is the classic example—are logically inconsistent. Others, though apparently consistent, are inconsistent with each other. Can principled restrictions be found which explain which are the "good" abstractions while saving the ones the neo-Fregean wants?
5. on Higher-order logic
Even if all the outstanding questions concerning Hume’s Principle and other needed abstractions could be settled in neo-Fregeanism’s favour, the interest of its technical results will depend on friendly answers to a number of fundamental questions about the logic in which they are obtained. Primary among these is:
5.1 is higher-order logic logic? The significance of neo-Fregean reductions of mathematical theory depends on logic’s being interestingly set apart from other formal a priori disciplines—by its generality, its implication in anything recognisable as rational thought, and its distinctive conceptual repertoire. Is this “fundamentalist” conception of logic—and especially higher-order logic—sustainable?
5.2 a closely related concern fastens on the (apparent) ontology of higher-order logic. Reductions of classical mathematical theories to set-theory would be no news. Neo-Fregeanism needs to show that Quine’s famous quip-that higher-order logic is “set theory in sheep’s clothing”—is mistaken: that quantification through predicate and functional expressions involves no problematical ontological commitments. The development of a more satisfactory philosophy of higher-order logic than Quinean orthodoxy is a major objective of the project.
6. formal extensions
The principal technical tasks are largely a matter of devising abstractions to play the role in analysis and set theory which Hume’s principle plays in arithmetic, and demonstrating that they satisfy the relevant analogues of conditions (1) and (2) above.
6.1 Boolos has developed one consistent abstractionist theory of sets, but the key abstraction—based on the idea of limitation of size—proves to suffer from crucial weaknesses as a foundation for standard set theory. Is there an acceptable abstraction for the purpose which is significantly stronger? (One suggestion would be a set-abstraction restricted to concepts which are not “indefinitely extensible” in Dummett’s sense. Work will be devoted to the clarification of this promising but obscure notion.)
6.2 A variety of abstractions which may prove suitable as foundations for real analysis are emerging. The simplest proposal—to which early attention will be given—is to mimic Dedekind cuts by developing an abstractionist theory of the rationals and defining “less than” on properties of rationals F and G as: every instance of F is smaller than every instance of G. We may then identify the Cut of F with the Cut of G just if every H less than F is less than G and vice versa, taking the real numbers as these cuts. An immediate task will be to show that the basic laws of classical analysis—those for a completely ordered field, including the crucial axiom that every bounded class of elements has a least bound—are forthcoming on this basis.
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