Charles Parsons

Mathematical thought and its objects

In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim of navigating between nominalism, which denies that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a “nature” than that confers on them. Parsons also analyzes the concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. An intuitive model witnesses the possibility of the structure of natural numbers. However, the full concept of number and knowledge of numbers involve more that is conceptual and rational. Parsons considers how one can talk about numbers, even though they are not objects of intuition. He explores the conceptual role of the principle of mathematical induction and the sense in which it determines the natural numbers uniquely. Parsons ends with a discussion of reason and its role in mathematical knowledge, attempting to do justice to the complementary roles in mathematical knowledge of rational insight, intuition, and the integration of our theory as a whole.

On a Number Theoretic Choice Schema and its Relation to Induction

Publisher Summary This chapter investigates the Relation of a number theoretic choice schema to induction. The usual first order number theory Z can be viewed in a number of ways as the union of an infinite sequence of subtheories whose axioms and rules are of ascending complexity. The chapter discusses the elementary relations of several hierarchies of subsystems based either on restricted induction principles or on others of a set theoretic character. The result is an independence theorem for finite axioms of choice which implies that for every n > 0, the axiom schema of induction with ≤ n nested quantifiers is stronger than the rule of induction with ≤ n nested quantiliers. A formula is Herbrand interpretable if it has a no-counter-example interpretation by functionals explicitly defined from function arguments and recursive functions. It is natural to expect such an axiom to be non-Herbrand interpretable (if the induction predicate is not recursive), because the Godel interpretation of such an axiom involves the universal recursion functional with values of type 0.

What is the Iterative Conception of Set

I intend to raise here some questions about what is nowadays called the ‘iterative conception of set’. Examination of the literature will show that it is not so clear as it should be what this conception is.

Platonism and mathematical intuition in Kurt Go¨del's thought

The best known and most widely discussed aspect of Kurt Godels philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russells report in his autobiography of one or more encounters with Godel is well known: Godel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians might hope to meet it hereafter. On this Godel commented: Concerning my “unadulterated” Platonism, it is no more unadulterated than Russells own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it. One of the tasks I shall undertake here is to say something about what Godels platonism is and why he held it. A feature of Godels view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish.

Kant’s Philosophy of Arithmetic

The interest and influence of Kant’s philosophy as a whole have certainly been great enough so that this by itself would be enough to make Kant’s philosophy of arithmetic of interest to historical scholars. It is also possible to show the influence of Kant on a number of important later writers on the foundations of mathematics, so that Kant has importance specifically as a figure in the history of the philosophy of mathematics. However, my own interest in this subject has been animated by the conviction that even today what Kant has to say about mathematics, and arithmetic in particular, is of interest to the philosopher and not merely to the historian of philosophy. However, I do not know how much of an argument the following will be for this.

Arithmetic and the Categories

On its conceptual side, mathematics as Kant understands it involves in an essential way the categories of quantity. This much should be obvious to readers of the Critique of Pure Reason. To trace this connection in more detail, however, has not been a main concern of interpreters of Kant’s philosophy of mathematics, at least recent ones. No doubt it has been thought that the connection is bound up with traditional logic and with a conception of mathematics more restrictive than what has come to prevail since the rise of set theory and abstract mathematics. The questions concerning Kant’s conception of intuition and of construction of concepts that have dominated the literature on Kant’s philosophy of mathematics are more directly connected with philosophical debates of recent times.

Reason and Intuition

In this paper I will approach the subject of intuition from a different angle from what has been usual in the philosophy of mathematics, by beginning with some descriptive remarks about Reason and observing that something that has been called intuition arises naturally in that context. These considerations are quite general, not specific to mathematics. The conception of intuition might be called that of rational intuition; indeed the conception is a much more modest version of conceptions of intuition held by rationalist philosophers. Moreover, it answers to a quite widespread use of the word intuition in philosophy and elsewhere. But it does not obviously satisfy conditions associated with other conceptions of intuition that have been applied to mathematics. Intuition in a sense like this has, in writing about mathematics, repeatedly been run together with intuition in other senses. In the last part of the paper a little will be said about the connections that give rise to this phenomenon.

Structuralism and the Concept of Set

The term “structuralism” with reference to mathematics has perhaps a basic meaning as referring to the idea that mathematics is particularly concerned with structures or structure. My concern is a more specific one, with what I call the structuralist view of mathematical objects. In its general lines, it is familiar: It holds that reference to mathematical objects is always in the context of some structure, and that the objects involved have no more to them than can be expressed in terms of the basic relations of the structure. The idea is well expressed by Michael Resnik: In mathematics, I claim, we do not have objects with an “internal” composition arranged in structures, we have only structures. The objects of mathematics, that is, the entities which our mathematical constants and quantifiers denote, are structureless points or positions in structures. As positions in structures, they have no identity or features outside of a structure.1

Developing arithmetic in set theory without infinity: some historical remarks

In this paper some of the history of the development of arithmetic in set theory is traced, particularly with reference to the problem of avoiding the assumption of an infinite set. Although the standard method of singling out a sequence of sets to be the natural numbers goes back to Zermelo, its development was more tortuous than is generally believed. We consider the development in the light of three desiderata for a solution and argue that they can probably not all be satisfied simultaneously.

Kurt Gödel : essays for his centennial

Part I. General: 1. The Godel editorial project: a synopsis Solomon Feferman 2. Future tasks for Godel scholars John W. Dawson, Jr, and Cheryl A. Dawson Part II. Proof Theory: 3. Kurt Godel and the metamathematical tradition Jeremy Avigad 4. Only two letters: the correspondence between Herbrand and Godel Wilfried Sieg 5. Godels reformulation of Gentzens first consistency proof for arithmetic: the no-counter-example interpretation W. W. Tait 6. Godel on intuition and on Hilberts finitism W. W. Tait 7. The Godel hierarchy and reverse mathematics Stephen G. Simpson 8. On the outside looking in: a caution about conservativeness John P. Burgess Part III. Set Theory: 9. Godel and set theory Akihiro Kanamori 10. Generalizations of Godels universe of constructible sets Sy-David Friedman 11. On the question of absolute undecidability Peter Koellner Part IV. Philosophy of Mathematics: 12. What did Godel believe and when did he believe it? Martin Davis 13. On Godels way in: the influence of Rudolf Carnap Warren Goldfarb 14. Godel and Carnap Steve Awodey and A. W. Carus 15. On the philosophical development of Kurt Godel Mark van Atten and Juliette Kennedy 16. Platonism and mathematical intuition in Kurt Godels thought Charles Parsons 17. Godels conceptual realism Donald A. Martin.