Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of (...) the Continuum Hypothesis, is a classic of modern mathematics. The continuum hypothesis, introduced by mathematician George Cantor in 1877, states that there is no set of numbers between the integers and real numbers. It was later included as the first of mathematician David Hilbert's twenty-three unsolved math problems, famously delivered as a manifesto to the field of mathematics at the International Congress of Mathematicians in Paris in 1900. In The Consistency of the Continuum Hypothesis Gödel set forth his proof for this problem. In 1999, Time magazine ranked him higher than fellow scientists Edwin Hubble, Enrico Fermi, John Maynard Keynes, James Watson, Francis Crick, and Jonas Salk. He is most renowned for his proof in 1931 of the 'incompleteness theorem,' in which he demonstrated that there are problems that cannot be solved by any set of rules or procedures. His proof wrought fruitful havoc in mathematics, logic, and beyond. (shrink)

We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically (...) different: the latter is a major fragment of the former. (shrink)

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) is about things that really exist. (shrink)

The problem of absolute generality has attracted much attention in recent philosophy. Agustin Rayo and Gabriel Uzquiano have assembled a distinguished team of contributors to write new essays on the topic. They investigate the question of whether it is possible to attain absolute generality in thought and language and the ramifications of this question in the philosophy of logic and mathematics.

Despite some discomfort with this grandly philosophical topic, I do in fact hope to address a venerable pair of philosophical chestnuts: mathematical truth and existence. My plan is to set out three possible stands on these issues, for an exercise in compare and contrast.' A word of warning, though, to philosophical purists (and perhaps of comfort to more mathematical readers): I will explore these philosophical positions with an eye to their interconnections with some concrete issues of set theoretic method.

The paper begins with an examination of Gödel's views on absolute undecidability and related topics in set theory. These views are sharpened and assessed in light of recent developments. It is argued that a convincing case can be made for axioms that settle many of the questions undecided by the standard axioms and that in a precise sense the program for large cardinals is a complete success “below” CH. It is also argued that there are reasonable scenarios for settling CH (...) and that there is not currently a convincing case to the effect that a given statement is absolutely undecidable. (shrink)

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

The unfolding of schematic formal systems is a novel concept which was initiated in Feferman , Gödel ’96, Lecture Notes in Logic, Springer, Berlin, 1996, pp. 3–22). This paper is mainly concerned with the proof-theoretic analysis of various unfolding systems for non-finitist arithmetic . In particular, we examine two restricted unfoldings and , as well as a full unfolding, . The principal results then state: is equivalent to ; is equivalent to ; is equivalent to . Thus is proof-theoretically equivalent (...) to predicative analysis. (shrink)

In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the freedom (...) provided by Cantorian set theory was purchased at a heavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support. -/- Beginning with a previously unpublished lecture for a general audience, Deciding the Undecidable, Feferman examines the famous list of twenty-three mathematical problems posed by David Hilbert, concentrating on three problems that have most to do with logic. Other chapters are devoted to the work and thought of Kurt Gödel, whose stunning results in the 1930s on the incompleteness of formal systems and the consistency of Cantors continuum hypothesis have been of utmost importance to all subsequent work in logic. Though Gödel has been identified as the leading defender of set-theoretical platonism, surprisingly even he at one point regarded it as unacceptable. -/- In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. -/- This volume of important and influential work by one of the leading figures in logic and the foundations of mathematics is essential reading for anyone interested in these subjects. (shrink)

Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical logic or entirely on intuitionistic logic. But a natural conception of the settheoretic universe is as an indefinite (or “potential”) totality, to which intuitionistic logic is more appropriately applied, while each set is taken to be a definite (or “completed”) totality, for which classical logic is appropriate; so on that view, set theory should be axiomatized on some correspondingly mixed basis. Similarly, in the (...) case of predicative analysis, the natural numbers are considered to form a definite totality, while the universe of sets (or functions) of natural numbers are viewed as an indefinite totality, so that, again, a mixed semi-constructive logic should be the appropriate one to treat the two together. Various such semiconstructive systems of analysis and set theory are formulated here and their proof-theoretic strength is characterized. Interestingly, though the logic is weakened, one can in compensation strengthen certain principles in a way that could be advantageous for mathematical applications. (shrink)

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly formalize (...) almost all, if not all, scientifically applicable mathematics in a formal system that is justified simply by Peano Arithmetic (via a proof-theoretical reduction). It is argued that this substantially vitiates the indispensability arguments. (shrink)

From 1931 until late in his life (at least 1970) Godel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Godel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might be (...) a uniform (though non-decidable) rationale for the choice of the latter. Despite the intense exploration of the \higher in nite" in the last 30-odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting set-theoretical consequences. (shrink)

• Definite totalities are set-like. If definite totalities are sets then the totality of all sets is indefinite (Russell).

Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions.

John Searle began to discuss his recently published book `The Construction of Social Reality' with Anthony Freeman, and they ended up talking about God. The book itself and part of their conversation are introduced and briefly reflected upon by Anthony Freeman. Many familiar social facts -- like money and marriage and monarchy -- are only facts by human agreement. They exist only because we believe them to exist. That is the thesis, at once startling yet obvious, that philosopher John Searle (...) explores in The Construction of Social Reality. (shrink)

• This comes from my general view of the nature of mathematics, that it is humanly based and that it deals with more or less clear conceptions of mathematical structures; for want of a better word, I call that view conceptual structuralism.

With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the set-theoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist” have become commonplace. Mathematics is seen as the investigation, by more or less rigorous deductive means, of “abstract structures”, systems of objects fulfilling certain structural relations among themselves and in relation to other systems, without regard to the particular nature of (...) the objects themselves. Geometric spaces need not be made up of spatial or temporal points or other intrinsically geometric objects; as Hilbert famously put it, items of furniture suitably interrelated could satisfy all the relevant axiomatic conditions as far as pure mathematics is concerned. A group, for instance, can be any multiplicity of objects with operations fulfilling the basic requirements of the binary group operation; indeed the very abstractness of the group concept allows for its remarkably wide applicability in pure and applied mathematics. Similar remarks can be made regarding other algebraic structures, and the many spaces of analysis, differential geometry, topology, etc. Of course, mathematicians distinguish between “abstract structures” and “concrete ones”, e.g. made up of familiar, basic items such as real or complex numbers or functions of such, or rationals, or integers, etc. (For example, the space L2 of square-integrable functions from R (or Rn) to C, with inner product (f, g) =. (shrink)