A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.

Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.

In this paper, we are concerned with the arithmetical definability of certain notions of integers and rationals in terms of other notions. The results derived will be applied to obtain a negative solution of corresponding decision problems.In Section 1, we show that addition of positive integers can be defined arithmetically in terms of multiplication and the unary operation of successorS(whereSa=a+ 1). Also, it is shown that both addition and multiplication can be defined arithmetically in terms of successor and the relation (...) of divisibility | (wherex|ymeansxdividesy). (shrink)

The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues.

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying second-order logic—and (...) investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. (shrink)

This book is well known for its proof that many mathematical systems - including lattice theory and closure algebras - are undecidable. It consists of three treatises from one of the greatest logicians of all time: "A General Method in Proofs of Undecidability," "Undecidability and Essential Undecidability in Mathematics," and "Undecidability of the Elementary Theory of Groups.".

The book is an attempt at explaining to the nation the ideas of Frege's Grundlagen. It is wordy and trite, a paradigm case of a redundant piece of writing. The reader is advised to steer clear of it.

Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege (...) was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many. (shrink)

In this paper we show how to interpret Robinson’s arithmetic Q and the theory R of Tarski, Mostowski, and Robinson as theories of cardinals in very weak theories of relations over a domain.

Rips et al. argue that the construction of math schemas roughly similar to the Dedekind/Peano axioms may be necessary for arriving at arithmetical skills. However, they neglect the neo-Fregean alternative axiomatization of arithmetic, based on Hume's principle. Frege arithmetic is arguably a more plausible start for a top-down approach in the psychological study of mathematical cognition than Peano arithmetic.

The conceptual building blocks suggested by developmental psychologists may yet play a role in how the human learner arrives at an understanding of natural number. The proposal of Rips et al. faces a challenge, yet to be met, faced by all developmental proposals: to describe the logical space in which learners ever acquire new concepts.

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)