Mathematics, foundations of

6. Later developments

Along with the modifications of the major post-Kantian viewpoints noted above, two other developments in the second half of the twentieth century are important to note. One of these is the shift towards empiricism that was brought about by Quine’s (following Duhem’s) merging of mathematics and the empirical sciences into a single justificatory unit governed by a basically inductive-empirical method. On this view, mathematics may on the whole be less susceptible to falsification by sensory evidence than is natural science, but this is a difference of degree, not kind.

This conception of mathematics dispenses with a ‘datum’ of mathematical epistemology that philosophers of mathematics from Kant on down had struggled to accommodate: namely, the presumed necessity of mathematics. It puts in its place a general empiricist epistemology in which all judgments – those of mathematics and logic as well as those of the natural sciences – are seen as evidentially connected to sensory phenomena and, so, subject to empirical revision.

To accommodate the lingering conviction that mathematics is independent of empirical evidence in a way that natural science is not, Quine introduced a pragmatic distinction between them. Rational belief-revision, he said, is governed by a pragmatic concern to maximize the overall predictive and explanatory power of one’s total system of beliefs. Furthermore, predictive and explanatory power are generally aided by policies of revision which minimize, both in scope and severity, the changes that are made to a previously successful belief-system in response to recalcitrant experience.

Because of this, beliefs of mathematics and logic are typically less subject to empirical revision than beliefs of natural science and common sense since revising them generally (albeit, in Quine’s view, not inevitably) does more damage to a belief-system than does revising its common sense and natural scientific beliefs. The necessity of mathematics is thus accommodated in Quine’s epistemology by moving mathematics closer to the centre of a ‘web’ of human beliefs where beliefs are less susceptible to empirical revision than are the beliefs of natural science and common sense that lie closer to the edge of the web.

In Quine’s view, merging mathematics and science into a single belief-system also induces a realist conception of mathematics. Mathematical sentences must be treated as true in order to play their role in this system, and the world is to be seen as being populated by those entities that are among the values of the variables of true sentences. Mathematical entities are thus real because mathematical sentences play an integral part in our best total theory of experience (see Quine 1948, 1951; Putnam 1971, 1975).

Quine’s views have been challenged on various grounds. For example, Parsons (1980) argues that treating the elementary arithmetical parts of mathematics as being on an epistemological par with the hypotheses of theoretical physics fails to capture an epistemologically important distinction between the different kinds of evidentness displayed by the two. Even highly confirmed physical hypotheses such as ‘The earth moves around the sun’ are more ‘derivative’ (that is, roughly, more theory-laden) than is an elementary arithmetic proposition such as ‘7+5=12’. It is therefore not plausible to regard the two claims as based on essentially the same type of evidence.

Others have challenged different aspects of Quine’s position. Field (1980) and Maddy (1980), for example, both question his merging of mathematics and natural science, though in different ways. Field argues that natural science that utilizes mathematics is a conservative extension of it and, so, has no need of its entities. The mathematical part of natural science can thus, in an important sense, be separated from the rest of it. (See Shapiro (1983) for an apt criticism of Field’s arguments.). Maddy investigates the possibility that our knowledge of at least certain mathematical objects might not be so diffuse and inextricable from the whole scheme of our natural scientific knowledge as Quine suggests. She argues that perceptual experience can be tied closely and specifically to certain mathematical objects (in particular, to certain sets) in a way that seems out of keeping with Quine’s holism.

In addition to Quine, others have suggested different mergings of mathematics and natural sciences. Kitcher (1983), for instance, presents a generally empiricist epistemology for mathematics in which history and community are important epistemological forces. Gödel, on the other hand, argued that mathematics, like the natural sciences, makes use of what is essentially inductive justification ([1947] 1964: 477, 485) when it justifies higher-level mathematical hypotheses on the grounds of their explanatory or simplificatory effects on lower-level mathematical truths and on physics. He allowed, however, that only some of our mathematical knowledge arises from empirical sources and regarded as absurd the idea that all of it might do (1951: 311–12).

Another important influence on recent philosophy of mathematics is Benacerraf’s ‘Mathematical Truth’ (1973), in which he argues that the philosophy of mathematics faces a general dilemma. It must give an account of both mathematical truth and mathematical knowledge. The former seems to demand abstract objects as the referents of singular terms in mathematical discourse. The latter, on the other hand, seems to demand that we avoid such referents. There are mathematical epistemologies (for example, various Platonist ones) that allow for a plausible account of the truth of mathematical sentences. Likewise, there are those (for example, various formalist ones) that allow for a plausible account of how we might come to know mathematical sentences. However, no known epistemology does both. Towards the end of the twentieth century a great deal of work has been devoted to resolving this dilemma. Field (1980, 1984), Hellman (1989) and Chihara (1990) attempt anti-Platonist resolutions. Maddy (1990), on the other hand, attempts a resolution at once Platonist and naturalistic. To date there is no general consensus on which approaches are the more plausible.

An earlier argument of Benacerraf’s (see Benacerraf 1965, but also Dedekind 1888: §73; Hilbert 1900; Weyl 1927; Bernays 1950) was similarly influential in shaping later work. It is the chief inspiration of the position known as ‘structuralism’ – the view that mathematical objects are essentially positions in structures and have no important additional internal composition or nature (see Resnik 1981, 1982; Shapiro 1983). Apart from the desire for a descriptively more adequate account of mathematics, the chief motivation of structuralism is epistemological. Knowledge of the characteristics of individual abstract objects would seem to require naturalistically inexplicable powers of cognition. Knowledge of at least some structures, on the other hand, would appear to be explicable as the result of applying such classically empiricist means of cognition as abstraction to observable physical complexes. Structures identified via abstraction become part of the general framework of our thinking and can be extended and generalized in a variety of ways as the search for the simplest and most effective overall conceptual scheme is pursued.

Structuralism as a general philosophy of mathematics has been criticized by Parsons (1990) who argues that there are important mathematical objects for which structuralism is not an adequate account. These are the ‘quasi-concrete’ objects of mathematics – objects that are directly ‘instantiated’ or ‘represented’ by concrete objects (for example, geometric figures and symbols such as the so-called ‘stroke numerals’ of Hilbert’s finitary arithmetic, where these are construed as types whose instances are written marks or symbols or uttered sounds). Such objects cannot be treated in a purely structuralist way because their ‘representational’ function cannot be reduced to the purely intrastructural relationships they bear to other objects within a given system. At the same time, however, they are among the most elementary and important mathematical entities there are.