In this paper I attempt to convey an idea of Hao Wang's style as a philosopher and to identify some of his contributions to philosophy. Wang was a prolific writer, and the body of text that should be considered in such a task is rather large, even if one separates off his work in mathematical logic, much of which had a philosophical motivation and some of which, such as his work on predicativity, contributed to the philosophy of mathematics. In a short paper one has to be selective. I will concentrate on his book From Mathematics to Philosophy [1974], since he considered it his principal statement, at least in the philosophy of mathematics. It also has the advantage of reflecting his remarkable relationship with Kurt Gödel (which began with correspondence in 1967) while belonging to a project that was well under way when his extended conversations with Gödel took place. Although Gödel's infiuence is visible, and in some places he is documenting Gödel's views by arrangement with their author, the main purpose of the book is to expound Wang's philosophy. Although I will comment on the Wang-Gödel relation, a discussion of Wang's work as source for and interpretation of Gödel, work which included two books written after Gödel's death ([1987] and [199?]) will have to be deferred until another occasion.

Style, convictions, and method

Wang's writings pose difficulties for someone who wishes to sort out his philosophical views and contributions, because there is something in his style that makes them elusive. FMP, like other writings of Wang, devotes a lot of space to’ exposition of relevant logic and sometimes mathematics, and of the work and views of others. Sometimes the purpose of the latter is to set the views in question against some of his own (as with Carnap, 381-384); in other cases the view presented seems to be just an exhibit of a view on problems of the general sort considered (as with Aristotle on logic, 131-142). The presence of expository sections might just make Wang's own philosophizing a little harder to find, but it is not the most serious difficulty his reader faces. That comes from a typical way Wang adopts of discussing a philosophical issue: to raise questions and to mention a number of considerations and views but with a certain distance from all of them.

The best known and most widely discussed aspect of Kurt Gödel's 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 Russell's report in his autobiography of one or more encounters with Gödel is well known:

On this Gödel commented:

One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.

A feature of Gödel's 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.

The year 2006 marked the centennial of Kurt Gödel, who was born on 28 April 1906. The importance of Gödel's work for nearly all areas of logic and foundations of mathematics hardly needs to be explained to our readers.

The year 2006 saw several centennial observances. In particular, the program committee for the 2006 Association for Symbolic Logic annual meeting, which took place on 17–21 May at the Université du Québec à Montréal, commissioned a subcommittee to arrange a portion of the program that would commemorate the Gödel centennial. The subcommittee arranged three one-hour lectures, by Jeremy Avigad, Sy-David Friedman, and Akihiro Kanamori. It also arranged a two-hour special session on Gödel's philosophy of mathematics, with lectures by Steve Awodey, John Burgess, and William Tait. All of the lectures have led to papers in this volume. The volume contains one other new paper, “The Gödel hierarchy and reverse mathematics,” by Stephen G. Simpson. Other papers included in the volume are reprinted, in all but one case from The Bulletin of Symbolic Logic. We have included the papers presented at the 2004 ASL annual meeting at Carnegie-Mellon University, in a special session organized by the editors of Gödel's Collected Works, by Martin Davis, John W. Dawson, Jr., Cheryl A. Dawson, Solomon Feferman, Warren Goldfarb, Donald A. Martin, Wilfried Sieg, and William Tait. These appeared in the June 2005 Bulletin. Also reprinted are papers by Mark van Atten and Juliette Kennedy and by Charles Parsons that appeared in earlier issues of the Bulletin, as well as a paper by Peter Koellner that appeared in Philosophia Mathematica.

An objection

In the last section, we put aside the question whether sets should simply be an exception to a general structuralism about pure mathematical objects. Structuralism about sets is not immediately implied by what has been argued up to now, even the argument against eliminative structuralism in §17. Indeed we would save eliminative structuralism in application to other kinds of objects (such as those of pure geometry) by denying it here, although, to be sure, we would not thereby save eliminative structuralism as a program for eliminating all mathematical objects or even all pure mathematical objects. We still have to consider whether the universe of sets consists of more than a domain of “objects” related by relation called “membership” satisfying conditions that can then be stated in the language of set theory.

We can see the difficulty from our examination of how attempts to use the structuralist idea to eliminate reference to mathematical objects of various kinds founder when applied to higher set theory. Such attempts still require that one make out that a structure of the kind required is in some way possible. I argued in §17 that in the present state of knowledge, this possibility can be made out for higher set theory only by domains of pure abstract objects, of which pure mathematical objects are a paradigmatic instance. That this possibility be made out is, however, still required by the noneliminative structuralism of §18.

The structuralist view of mathematical objects

In order to focus on problems concerning mathematical objects, I turn now to what I call the “structuralist view” of them. By this I mean the view that reference to mathematical objects is always in the context of some background structure, and that the objects involved have no more by way of a “nature” than is given by the basic relations of the structure. The idea is succinctly expressed by Michael Resnik:

Views of this kind can be traced back to the end of the nineteenth century, and the structuralist view in the sense I intend is an ontological conception that particularly fits the abstract mathematics that came into being in the late nineteenth century and has flourished in the twentieth. The statement that mathematics is about, or primarily about, structures is a frequently offered rough description of this character of modern mathematics. With reference to mathematics, the term ‘structuralism’ is often used to refer to no more than this rough description, or to a conception that expands on it in a different direction from what I have in view.

What are the natural numbers?

The discussion of intuition in the last chapter naturally leads to two further inquiries. Perhaps the most obvious one is about natural numbers: How are natural numbers given to us, and, in particular, are they objects of an intuition of the kind described in §28 or something similar to it? The second concerns intuitive knowledge: We gave, particularly in §29, some examples of propositions about the strings of our little language that are intuitively known. Since strings are evidently a model of arithmetic, the question arises how far intuitive knowledge in arithmetic extends, when we understand arithmetic by reference to this model. That formulation of the question sidesteps the first question, whether numbers properly speaking are objects of intuition.

Because our first question is probably the more urgent for most readers, I will take it up first, in this chapter. The other question will be the subject of Chapter 7. The question: “What are the natural numbers?” is motivated independently of questions about intuition, and we will examine it in general as well as answering the question about intuition of numbers. But both chapters will explore the limits of intuition as understood in Chapter 5, first by considering in this chapter whether intuition extends beyond quasi-concrete objects to some pure abstract objects such as numbers, and second by inquiring in Chapter 7 how far intuitive knowledge extends, referring to a domain that we have taken to be intuitive.

Reason and “rational intuition”

It will be painfully obvious to the reader that what has been said so far about the epistemology of mathematics, even arithmetic, is very incomplete. In §45, it was argued that very likely intuitive knowledge has rather narrow limits. Many would hold that that is due to the very restricted character of the conception of intuition developed in Chapter 5. Occasional reference has been made to conceptions of intuition of a quite different nature, which might promise to lead further than the one we have developed. I propose in this section and the next to approach the problem of mathematical knowledge from a different direction, not beginning with intuition at all, but leading very quickly to a place where some other conceptions of intuition, in particular that of Kurt Gödel, can be situated. I will begin with some very general considerations about Reason, not at all specific to mathematics.

The relevance of Reason to our inquiry is indicated by a long tradition of regarding mathematics as rational knowledge, echoed specifically by Kant in a remark at the beginning of one of the discussions of mathematics in the Critique of Pure Reason. He describes mathematics as “rational cognition from the construction of concepts” (A713/B741). Construction of concepts is construction in intuition, and it is here that intuition takes its place in Kant's account of mathematics.

Mathematical modality

In §§11–12 the use of modal notions appeared attractive as a way of formulating eliminative structuralist views of mathematical objects so as to deal with the vacuity problem that such views faced. Before I pursue that idea, I shall consider modal notions more directly in relation to mathematics. Although this provides necessary background for continuing the previous line of argument, it is also of interest in its own right and will be relevant also to other parts of the present work.

Modality is what is expressed primarily by the words ‘necessary’ and ‘possible’ and by the modal auxiliaries ‘can’ and ‘must’. These words have such a variety of uses, some of quite different character, that we could not hope to survey them in a brief compass (and probably to do so at all would be a major linguistic undertaking). However, two distinctions made between uses of modal words or between modalities are of general importance and important for our purposes: (a) that between epistemic and nonepistemic and (b) that between absolute and nonabsolute.

(a) In Naming and Necessity, Kripke stresses the difference between the epistemological notion of the a priori and the notions of necessity and possibility with which he is concerned, which he says belong to metaphysics (pp. 35–36). In this discussion, he does not emphasize the idea of epistemic modalities or epistemic uses of modal words, but of course such can be found.