Traditional Empiricism and the Philosophical Nature and Status of Mathematics

Every empiricist must face the problem of the nature and status of mathematics within their philosophical system. Not only that but he must provide a satisfactory philosophical account of mathematics. He must say why he accepts mathematical propositions. Only an extreme empiricist, if there is such a person today, would completely reject mathematical propositions if accounted for in a non-empiricist - or even non-Millian - kind of way.

Why was mathematics, or the nature of numbers, such a problem for empiricists? Because empiricists were, well, empiricists. How could they defend the knowledge-claims of mathematics? What is the problem with these claims? Firstly, mathematical propositions appear to be knowable a priori. And if they are knowable a priori, that is, if they are also analytic in nature, then clearly they could not – or must not - depend on anything empirical. (In Hume's terms, they are 'relations of ideas' not 'matters of fact'.) This was clearly be a problem for the traditional empiricist in the obvious sense that a priori propositions would have no empirical content. In addition, according to many traditional empiricists, the very idea of a priori propositions was always suspect, even when they are mathematical or logical in nature (i.e., not just metaphysical). In terms of empiricist epistemology, the idea of non-empirical knowledge, or knowledge-claims, expressed by a priori propositions was equally suspect to the traditional empiricist. Not only that, they believed that no non-empirical propositions - or a non-empirical knowledge-claims - would not even be seen as examples of knowledge (or of the epistemically safe). And because of the non-empirical and a priori nature of mathematical propositions, then they could never be victims of empirical refutation. So not only did traditional empiricists have problems with any kind of proposition that could not be refuted; but it would show again that this would be another example of mathematical propositions having nothing to do with anything empirical (in this case, refutations are empirical/observational/experiential in nature). Following on from this, we can also say that mathematical claims do 'not need empirical support'. This again showed the traditional empiricist that mathematics appears to have no relation whatsoever to anything empirical.

From all that has been said, it is obviously the case that mathematics would have a problematic relation with empiricism, at the least this must have been the case at a certain point in the history of empiricism. There was a specific problem that empiricists were concerned with. It was an epistemological problem about the nature of knowledge-claims. More precisely, how could we ever know anything at all about a discipline whose claims were completely non-empirical in nature? In traditional epistemology, not only of the empiricist kind, it was thought that all examples of knowledge must have some kind of relation to the world. According to empiricists, they must have a relation to the empirical - to experience of some kind. It seemed to follow from all this, therefore, that the claims of mathematics could not be taken as examples of knowledge. However, although they may not be knowledge-claims, empiricists still accepted the possibility that they were claims of another kind - a kind that had no relation to experience. In addition, if empiricists did not accept that the claims of mathematics were knowledge-claims, as said, they still accepted mathematical propositions. This, again, was still the case if propositions, or mathematical propositions, had no relation to anything empirical. If empiricists denied any knowledge status for mathematical propositions, and  therefore also claimed that it could not be a fit subject for epistemology, but accepted the possibility that there be other kinds of claims and non-empirical propositions, then after these denials, rejections and acceptances, how, precisely, did these empiricists make sense of mathematics, describe it and, more importantly, show that despite its non-empirical nature, it is not thereby denied a valuable role and could also prove useful, in many ways, to work carried out by empiricist philosophers (as well as physicists)? Not only that, they accepted the strange possibility, or its actuality, that something completely non-empirical could be used, in various ways, to refer to empirical and observable realities or, perhaps, be simply used in various primarily empirical contexts. This means that we should not make the logically-flawed assumption that just because two things are completely unlike in nature, it does not follow that they cannot - or must not - have any relations or connections with one another. How this could be so, of course, was highly problematic. So problematic, indeed, that many philosophers simply denied the fat that they are two different kinds in the first place. (Another example that illustrates the strange relations between the empirical and the non-empirical is the fact that causes are often not, or never are, similar to their effects. For example, a lit match, a partial causal condition, is not like a burning forest, which would be partly the effect of the causal conditions that is the lit match.)

A Wittgensteinian Digression

Wittgenstein accepted and justified, non-epistemically (that is, logically), mathematics’ non-empirical nature; or in Wittgensteinian terms, the fact that mathematical propositions are not world-directed - or connected to objects in the world - by stressing what he took to be their essentially tautological nature [1921]. Because mathematical propositions were things that either expressed the same thing twice or used symbols (i.e., numerical expressions) that could be mutually inter-substitutable, and that such propositions simply state identities, then in a sense it could not even be possible in principle that such tautologies could have empirical content. Not only that, but Wittgenstein believed that they didn’t actually say anything at all precisely because of their decisively non-world-directed nature and that they are only propositions that expressed or stated various identities. It follows that mere identities or propositions that ‘said nothing’ cannot be seen as any kind of threat to all the propositions that were world-directed (i.e., those of science). These mathematical propositions need not be epistemically justified or explained, Wittgenstein thought, simply because they were not genuine propositions in the first place! If we do accept or justify mathematical expressions, it would be in ways that would not accept that they had a genuine propositional status. They didn’t have such a status because they were not world-directed. If we accept and justify mathematical expressions, it must be for the properties they have that have nothing to do with world-directedness. Indeed Wittgenstein believed that they had such non-propositional properties. So he happily accepted or even (non-epidemically) justified these expressions.

It was the case that many empiricists did not feel threatened by mathematical ‘propositions’ and for many of the same Wittgensteinian reasons¹.

⁽¹⁾ However, there is one slight difference between Wittgenstein and the empiricists that we should note here. Whereas Wittgenstein talked in terms of world-directedness, e.g., ‘the world’, ‘logical objects’, ‘atomic facts’ and so on, empiricists, not being ontologically and, more importantly, logically-minded, talked instead of ‘empirical propositions’, the ‘empirical world’, ‘experience’, ‘observation’, etc.)

Further Reading

Hume, D – (1739) A Treatise of Human Nature

Kant, I – (1787) Critique of Pure Reason

Mill, J.S – (1843) System of Logic

Wittgenstein, L – (1921) Tractatus Logico-Philosophicus