The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.

Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

_Thoughts and Utterances_ is the first sustained investigation of two distinctions which are fundamental to all theories of utterance understanding: the semantics/pragmatics distinction and the distinction between what is explicitly communicated and what is implicitly communicated.

The interaction between intuitions about inference, and the normative constraints that logical principles applied to mechanically-recognizable derivations impose on (informal) inference, is explored. These intuitions are evaluated in a clear testcase: informal mathe­matical proof. It is argued that formal derivations are not the source of our intuitions of validity, and indeed, neither is the semantic recognition of validity, either as construed model-theoretically, or as driven by the subject-matter such inferences are directed towards. Rather, psychologically-engrained inference-packages (often opportunistically used by mathematicians) (...) are the source of our sense of validity. Formal derivations, or the semantic construal of such, are after-the-fact norms imposed on our inference practices. (shrink)

In _Realistic Rationalism_, Jerrold J. Katz develops a new philosophical position integrating realism and rationalism. Realism here means that the objects of study in mathematics and other formal sciences are abstract; rationalism means that our knowledge of them is not empirical. Katz uses this position to meet the principal challenges to realism. In exposing the flaws in criticisms of the antirealists, he shows that realists can explain knowledge of abstract objects without supposing we have causal contact with them, that numbers (...) are determinate objects, and that the standard counterexamples to the abstract/concrete distinction have no force. Generalizing the account of knowledge used to meet the challenges to realism, he develops a rationalist and non-naturalist account of philosophical knowledge and argues that it is preferable to contemporary naturalist and empiricist accounts. The book illuminates a wide range of philosophical issues, including the nature of necessity, the distinction between the formal and natural sciences, empiricist holism, the structure of ontology, and philosophical skepticism. Philosophers will use this fresh treatment of realism and rationalism as a starting point for new directions in their own research. (shrink)

When ordinary people - mathematicians among them - take something to follow from something else, they are exposing the backbone of our self-ascribed ability to reason. Jody Azzouni investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians. One claim of the book is that, despite our apparent intuitive grasp of consequence, we do not introspect rules by which we reason, nor do we grasp the scope and range of the domain, as it were, (...) of our reasoning. This point is illustrated with a close analysis of a paradigmatic case of ordinary reasoning: mathematical proof. (shrink)

Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent (in an informal sense) and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

In _Realistic Rationalism_, Jerrold J. Katz develops a new philosophical position integrating realism and rationalism. Realism here means that the objects of study in mathematics and other formal sciences are abstract; rationalism means that our knowledge of them is not empirical. Katz uses this position to meet the principal challenges to realism. In exposing the flaws in criticisms of the antirealists, he shows that realists can explain knowledge of abstract objects without supposing we have causal contact with them, that numbers (...) are determinate objects, and that the standard counterexamples to the abstract/concrete distinction have no force. Generalizing the account of knowledge used to meet the challenges to realism, he develops a rationalist and non-naturalist account of philosophical knowledge and argues that it is preferable to contemporary naturalist and empiricist accounts. The book illuminates a wide range of philosophical issues, including the nature of necessity, the distinction between the formal and natural sciences, empiricist holism, the structure of ontology, and philosophical skepticism. Philosophers will use this fresh treatment of realism and rationalism as a starting point for new directions in their own research. (shrink)

In this 1992 book, Professor Koslow advances an account of the basic concepts of logic. A central feature of the theory is that it does not require the elements of logic to be based on a formal language. Rather, it uses a general notion of implication as a way of organizing the formal results of various systems of logic in a simple, but insightful way. The study has four parts. In the first two parts the various sources of the general (...) concept of an implication structure and its forms are illustrated and explained. Part 3 defines the various logical operations and systematically explores their properties. A generalized account of extensionality and dual implication is given, and the extensionality of each of the operators, as well as the relation of negation and its dual, are given substantial treatment because of the novel results they yield. Part 4 considers modal operators and studies their interaction with logical operators. By obtaining the usual results without the usual assumptions this new approach allows one to give a very simple account of modal logic minus the excess baggage of possible world semantics. (shrink)

A distinction between the epistemic practices in mathematics and in the empirical sciences is rehearsed to motivate the epistemic role puzzle. This is distinguished both from Benacerraf's 1973 epistemic puzzle and from sceptical arguments against our knowledge of an external world. The stipulationist position is described, a position which can address this puzzle. Methods of avoiding the stipulationist position by using pure logic to provide knowledge of mathematical abstracta are discussed and criticized.

Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...) pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics both sensitive to mathematical practice, and to the ontological and epistemological issues that concern philosophers. (shrink)

In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents genuine (...) meaning-dependent mathematical characteristics that cannot be captured by formal calculi. ‘…there is a conflict between mathematical practice and the formalist doctrine.’ [Kreisel, 1969, p. 39]. (shrink)

Some philosophers plaintively wonder why there is something rather than nothing. Others refuse to wonder: Explaining has its field of application outside of which the activity makes no sense.