Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.

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English contains singular terms, quantifiers and predicates (e.g. ‘it’, ‘something ’ and ‘... is an elephant’). But it also contains plural terms, quantifiers and predicates (e.g. ‘they’, ‘some things ’ and ‘... are scattered on the floor’). 1 Philosophers have become increasingly interested in plurals over the past couple of decades. The purpose of this paper is to explain why plurals might be thought to have philosophical importance, and why they have led to philosophical debate. 1

Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider many-valued functions, since they too bring in plural terms—terms such as ‘�4 ’ or the descriptive ‘the inhabitants of London ’ which, like plain plural descriptions, stand for more than one thing. Logicians need to take plural reference seriously if only because mathematicians take many-valued functions seriously. We assess the objection (by Russell, Frege and others) that many-valued functions are illegitimate because the corresponding terms are ambiguous. We also assess the various methods proposed for getting rid of them. Finding the objection ill-founded and the methods ineffective, we introduce a logical framework that admits plural reference, and use it to answer some earlier questions and to raise some more. 1. Russell’s theory of plural descriptions Everybody knows that Russell had a theory of definite descriptions. Not everybody realizes that he had two: one for singular descriptions, another for plural descriptions. The contents of ‘On Denoting ’ have blinkered the popular conception of his agenda. 1.1 The Principles of Mathematics In the Principles class talk is plural talk: ‘so-and-so’s children, or the children of Londoners, afford illustrations ’ of classes; ‘the children of Israel are a class ’ (1903c, pp. 24, 83). Readers brought up on modern set theory must beware. Russell’s plural descriptions each stand for many things, and accordingly his classes are ‘classes as many’: they are many things—the children of Israel are a class—not one. (Unless, of course, they only have a single member. Throughout this paper we use the plural idiom inclusively, to cover the singular as a limiting case. Purists should read ‘the Fs ’ as ‘the F or Fs ’ and adjust the context to suit.) Russell first investigates plural idioms in the chapter on ‘Denoting’. His exposition is complicated, however, by his insistence that distribu-

Abstract. Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this extended system.

More and more enterprises are currently undertaking projects to integrate their applications. They are finding that one of the more difficult tasks facing them is determining how the data from one application matches semantically with the data from the other applications. Currently there are few methodologies for undertaking this task -- most commercial projects just rely on experience and intuition.

When prompted to consider Frege’s views about definite descriptions, many philosophers think about the meaning of proper names, and some of them can cite the following quotation taken from a footnote Frege’s 1892 article “ Über Sinn und Bedeutung.”2

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