In this paper I argue that mathematics should be interpreted realistically – that is, that mathematics makes assertions that are objectively true or false, independently of the human mind, and that something answers to such mathematical notions as ‘set’ and ‘function’. This is not to say that reality is somehow bifurcated – that there is one reality of material things, and then, over and above it, a second reality of ‘mathematical things’. A set of objects, for example, depends for its existence on those objects: if they are destroyed, then there is no longer such a set. (Of course, we may say that the set exists ‘tenselessly’, but we may also say the objects exist ‘tenselessly’: this is just to say that in pure mathematics we can sometimes ignore the important difference between ‘exists now’ and ‘did exist, exists now, or will exist’.) Not only are the ‘objects’ of pure mathematics conditional upon material objects; they are, in a sense, merely abstract possibilities. Studying how mathematical objects behave might better be described as studying what structures are abstractly possible and what structures are not abstractly possible.

The important thing is that the mathematician is studying something objective, even if he is not studying an unconditional ‘reality’ of nonmaterial things, and that the physicist who states a law of nature with the aid of a mathematical formula is abstracting a real feature of a real material world, even if he has to speak of numbers, vectors, tensors, state-functions, or whatever to make the abstraction.

Carnap's attempt to construct a symbolic inductive logic, fits into two major concerns of empiricist philosophy. On the one hand, there is the traditional concern with the formulation of Canons of Induction; on the other hand, there is the distinctively Carnapian concern with providing a formal reconstruction of the language of science as a whole, and with providing precise meanings for the basic terms used in methodology.

Of the importance of continuing to search for a more precise statement of the inductive techniques used in science, I do not need to be convinced; this is a problem which today occupies mathematical statisticians at least as much as philosophers.

But this general search need not be identified with the particular project of defining a quantitative concept of ‘degree of confirmation’. I shall argue that this last project is misguided.

Such a negative conclusion needs more to support it than ‘intuition’; or even than plausible arguments based on the methodology of the developed sciences (as the major features of that method may appear evident to one). Intuitive considerations and plausible argument might lead one to the conclusion that it would not be a good investment to spend ones own time trying to ‘extend the definition of degree of confirmation’ it could hardly justify trying to, say, convince Carnap that this particular project should be abandoned. But that is what I shall try to do here: I shall argue that one can show that no definition of degree of confirmation can be adequate or can attain what any reasonably good inductive judge might attain without using such a concept.

Craig's observation

‘Craig's theorem’ (Craig, 1953), as philosophers call it, is actually a corollary to an observation. The observation is that (I) Every theory that admits a recursively enumerable set of axioms can be recursively axiomatized.

Some explanations are in order here: (1) A theory is an infinite set of wffs (well-formed formulas) which is closed under the usual rules of deduction. One way of giving a theory T is to specify a set of sentences S(called the axioms of T) and to define T to consist of the sentences S together with all sentences that can be derived from (one or more) sentences in S by means of logic. (2) If T is a theory with axioms S, and S' is a subset of T such that every member of S can be deduced from sentences in S', then S' is called an alternative set of axioms for T. Every theory admits of infinitely many alternative axiomatizations – including the trivial axiomatization, in which every member of T is taken as an axiom (i.e. S = T). (3) A set S is called recursive if and only if it is decidable – i.e. there exists an effective procedure for telling whether or not an arbitrary wff belongs to S. (This is not the mathematical definition of ‘recursive’, of course, but the intuitive concept which the mathematical definition captures.) For ‘effective procedure’ one can also write ‘Turing machine’ (cf. Davis, 1958.).

Let us make up a logic in which there are three truth-values, T, F, and M, instead of the two truth-values T and F. And, instead of the usual rules, let us adopt the following:

1. (a) If either component in a disjunction is true (T), the disjunction is true; if both components are false, the disjunction is false (F); and in all other cases (both components middle, or one component middle and one false) the disjunction is middle (M).

2. (b) If either component in a conjunction is false (F), the conjunction is false; if both components are true, the conjunction is true (T); and in all other cases (both components middle, or one component middle and one true) the conjunction is middle (M).

3. (c) A conditional with true antecedent has the same truth-value as its consequent; one with false consequent has the same truth-value as the denial of its antecedent; one with true consequent or false antecedent is true; and one with both components middle (M) is true.

4. (d) The denial of a true statement is false; of a false one true; of a middle one middle.

These rules are consistent with all the usual rules with respect to the values T and F. But someone who accepts three truth values, and who accepts a notion of tautology based on a system of truth-rules like that just outlined, will end up with a different stock of tautologies than someone who reckons with just two truth values.

Philosophers and logicians have been so busy trying to provide mathematics with a ‘foundation’ in the past half-century that only rarely have a few timid voices dared to voice the suggestion that it does not need one. I wish here to urge with some seriousness the view of the timid voices. I don't think mathematics is unclear; I don't think mathematics has a crisis in its foundations; indeed, I do not believe mathematics either has or needs ‘foundations’. The much touted problems in the philosophy of mathematics seem to me, without exception, to be problems internal to the thought of various system builders. The systems are doubtless interesting as intellectual exercises; debate between the systems and research within the systems doubtless will and should continue; but I would like to convince you (of course I won't, but one can always hope) that the various systems of mathematical philosophy, without exception, need not be taken seriously.

By way of comparison, it may be salutory to consider the various ‘crises’ that philosophy has pretended to discover in the past. It is impressive to remember that at the turn of the century there was a large measure of agreement among philosophers – far more than there is now – on certain fundamentals. Virtually all philosophers were idealists of one sort or another. But even the nonidealists were in a large measure of agreement with the idealists. It was generally agreed any property of material objects – say, redness or length – could be ascribed to the object, if at all, only as a power to produce certain sorts of sensory experiences.

It has been maintained by such philosophers as Quine and Goodman that purely ‘extensional’ language suffices for all the purposes of properly formalized scientific discourse. Those entities that were traditionally called ‘universals’ – properties, concepts, forms, etc. – are rejected by these extensionalist philosophers on the ground that ‘the principle of individuation is not clear’. It is conceded that science requires that we allow something tantamount to quantification over non-particulars (or, anyway, over things that are not material objects, not space-time points, not physical fields, etc.), but, the extensionalists contend, quantification over sets serves the purposes nicely. The ‘ontology’ of modern science, at least as Quine formalizes it, comprises material objects (or, alternatively, space-time points), sets of material objects, sets of sets of material objects, …, but no properties, concepts; or forms. Let us thus examine the question: can the principle of individuation for properties ever be made clear?

Properties and reduction

It seems to me that there are at least two notions of ‘property’ that have become confused in our minds. There is a very old notion for which the word ‘predicate’ used to be employed (using ‘predicate’ as a term only for expressions and never for properties is a relatively recent mode of speech: ‘Is existence a predicate?’ was not a syntactical question) and there is the notion for which I shall use the terms ‘physical property’, ‘physical magnitude’, ‘physical relation’, etc., depending on whether the object in question is one-place, a functor, more than one-place, etc.

I hope that no one will think that there is no connection between the philosophy of the formal sciences and the philosophy of the empirical sciences. The philosophy that had such a great influence upon the empirical sciences in the last thirty years, the so-called ‘logical empiricism’ of Carnap and his school, was based upon two main principles:

1. (1) That the traditional questions of philosophy are so-called ‘pseudoquestions’ (Scheinprobleme), i.e. that they are wholly senseless; and

2. (2) That the theorems of the formal sciences – logic and mathematics – are analytic, not exactly in the Kantian sense, but in the sense that they ‘say nothing’, and only express our linguistic rules.

Today analytical philosophers are beginning to construct a new philosophy of science, one that also wishes to be unmetaphysical, but that cannot accept the main principles of ‘logical empiricism’. The confrontation with the positivistic conception of mathematics is thus no purely technical matter, but has the greatest importance for the whole conception of philosophy of science.

What distinguishes statements which are true for mathematical reasons, or statements whose falsity is mathematically impossible (whether in the vocabulary of ‘pure’ mathematics, or not), or statements which are mathematically necessary, from other truths? Contrary to a good deal of received opinion, I propose to argue that the answer is not ‘ontology’, not vocabulary, indeed nothing ‘linguistic’ in any reasonable sense of linguistic.

What logic is

Let us start by asking what logic is, and then try to see why there should be a philosophical problem about logic. We might try to find out what logic is by examining various definitions of ‘logic’, but this would be a bad idea. The various extant definitions of ‘logic’ manage to combine circularity with inexactness in one way or another. Instead let us look at logic itself.

If we look at logic itself, we first notice that logic, like every other science, undergoes changes – sometimes rapid changes. In different centuries logicians have had very different ideas concerning the scope of their subject, the methods proper to it, etc. Today the scope of logic is defined much more broadly than it ever was in the past, so that logic as some logicians conceive it, comes to include all of pure mathematics. Also, the methods used in logical research today are almost exclusively mathematical ones. However, certain aspects of logic seem to undergo little change. Logical results, once established, seem to remain forever accepted as correct – that is, logic changes not in the sense that we accept incompatible logical principles in different centuries, but in the sense that the style and notation that we employ in setting out logical principles varies enormously, and in the sense that the province marked out for logic tends to get larger and larger.

The two statements that Donnellan considered in his paper (Donnellan, 1962) are both more or less analytic in character. By that I mean that they are the sort of statement that most people would consider to be true by definition, if they considered them to be necessary truths at all. One might quarrel about whether ‘all whales are mammals’ is a necessary truth at all. But if one considers it to be a necessary truth, then one would consider it to be true by definition. And, similarly, most people would say that ‘all cats are animals’ is true by definition, notwithstanding the fact that they would be hard put to answer the question, ‘true by what definition?’

I like what Donnellan had to say about these statements, and I liked especially the remark that occurs toward the end of his paper, that there are situations in which we are confronted by a question about how to talk, but in which it is not possible to describe one of the available decisions as deciding to retain our old way of talking (or ‘not to change the meaning’) and the other as deciding to adopt a ‘new’ way of talking (or to ‘change the meaning’).

In this paper I want to concentrate mostly on statements that look necessary, but that are not analytic; on ‘synthetic necessary truths’, so to speak. This is not to say that there are not serious problems connected with analyticity.

The Margenau and Wigner ‘Comments’ (1962) on my ‘Comments on the Paper of David Sharp’ (Putnam, 1961; Sharp, 1961) is a strange document. First the authors say, in effect, ‘had anything been wrong (with the fundamentals of quantum mechanics) we should certainly have heard’. Then they issue various obiter dicta (e.g. the ‘cut between observer and object’ is unavoidable in quantum mechanics; the – highly subjectivistic – London-Bauer treatment of quantum mechanics is described, along with von Neumann's book, as ‘the most compact and explicit formulation of the conceptual structure of quantum mechanics’). My assumption 2 (that the whole universe is a system) is described as ‘not supportable’, because ‘the measurement is an interaction between the object and the observer’. The ‘object’ (the closed system) cannot include the observer.

The issues involved in this discussion are fundamental ones. I believe that the conceptual structure of quantum mechanics today is as unhealthy as the conceptual structure of the calculus was at the time Berkeley's famous criticism was issued. For this reason – as much to emphasize the seriousness of the present situation in the foundations of quantum mechanics as to remove confusions that may be left in the mind of the general reader upon reading the Margenau and Wigner ‘Comments’ – I intend to restate the main points of my previous ‘Comments’, and to show in detail why the Margenau and Wigner remarks fail completely to meet them.

(1) Grünbaum's last memorandum in defense of ‘conventionalism’ with respect to geometry seemed to me to mark a definite shift in his position since our Princeton conference. I hope that this shift was conscious and intentional, and that ‘polemical’ considerations will not lead Grünbaum to repudiate this ‘shift’ if I point it out!

The shift I have in mind is this: at the Princeton conference Grünbaum emphasized that his ‘conventionalism’ is a logical (or perhaps metaphysical?) thesis, but certainly not an empirical one. I specifically put to him the following question, which I regard as of central importance for this area: what he would say if the world were ‘Newtonian’ (e.g. the Michelson-Morley experiment had the ‘other’ result, the velocity of light relative to the ‘ether’ were measurable, etc.); and he replied that his position would be unchanged. That is, in a Newtonian world, even one in which the notion of ‘absolute velocity’ has obvious physical meaning, length is still not an intrinsic (Grünbaum's word, not mine!) property of physical objects, and simultaneity is still not an intrinsic relation between physical events. The choice of a metric is still a matter of convention; although (in such a world) one conventionf is immensely superior to all rivals, from a practical standpoint. (This was Reichenbach's standpoint in his Space-Time book as I pointed out; and it is precisely this notion of a ‘convention’ which cannot be told from a bona-fide physical law‡ that I find so puzzling.) Now, however, it appears that the strongest arguments for conventionalism are from its alleged empirical import; and this empirical import of ‘conventionalism’ is linked by Grünbaum to the special theory of relativity.

Sir Karl Popper is a philosopher whose work has influenced and stimulated that of virtually every student in the philosophy of science. In part this influence is explainable on the basis of the healthy-mindedness of some of Sir Karl's fundamental attitudes: ‘There is no method peculiar to philosophy’. ‘The growth of knowledge can be studied best by studying the growth of scientific knowledge.’

These attitudes are perhaps a little narrow (can the growth of knowledge be studied without also studying nonscientific knowledge? Are the problems Popper mentioned of merely theoretical interest – just ‘riddles’?), but much less narrow than those of many philosophers; and the ‘obscurantist faith’ Popper warns against is a real danger. In part this influence stems from Popper's realism, his refusal to accept the peculiar meaning theories of the positivists, and his separation of the problems of scientific methodology from the various problems about the ‘interpretation of scientific theories’ which are internal to the meaning theories of the positivists and which positivistic philosophers of science have continued to wrangle about (I have discussed positivistic meaning theory in chapter 14 and also in chapter 5, volume 2).

These essays were written over a fifteen-year period. During that time my views underwent a number of changes, especially on the philosophy of mathematics and on the interpretation of quantum mechanics. Nevertheless they have, I believe, a certain unity.

The major themes running through these essays, as I look at them today, are the following: (i) Realism, not just with respect to material objects, but also with respect to such ‘universals’ as physical magnitudes and fields, and with respect to mathematical necessity and mathematical possibility (or equivalently with respect to mathematical objects); (2) the rejection of the idea that any truth is absolutely a priori; (3) the complementary rejection of the idea that ‘factual’ statements are all and at all times ‘empirical’, i.e. subject to experimental or observational test; (4) the idea that mathematics is not an a priori science, and an attempt to spell out what its empirical and quasi-empirical aspects really are, historically and methodologically.

Realism

These papers are all written from what is called a realist perspective. The statements of science are in my view either true or false (although it is often the case that we don't know which) and their truth or falsity does not consist in their being highly derived ways of describing regularities in human experience. Reality is not a part of the human mind; rather the human mind is a part – and a small part at that – of reality. But no paper in this collection is entirely devoted to the topic of realism, for my interest in the last fifteen years has not been in beating my breast about the correctness of realism, but has rather been in dealing with specific questions in the philosophy of science from a specific realist point of view.