Philosophy of mathematics

David Hilbert (1862-1943) is best known as the founding father of mathematical formalism. This philosophical school has been primarily interested in form –or use of signs– as opposed to meaningful content of mathematical statements, i.e. what is actually said. Accordingly, its preoccupations have been with keeping under control the development of mathematical knowledge rather than with furnishing it with an undubitable foundation. In between Hilbert's mathematical research, two phases of philosophical concern can be discerned, one in the early twentieth century, in the wake of his most famous publication, Grundlagen der Geometrie (Foundations of geometry; 1899), and a second in the 1920s.

In the Grundlagen, Euclid's foundational account of geometry, which had come under severe pressure at the time, was revised, by establishing a new conception of what is axiomatics. It does no longer appeal to any intuitive understanding of the basic concepts, which are implicitly defined by the axioms, but is utterly free of any prior interpretation. With this, traditional philosophical problems (what are mathematical objects, and how are we to reach reliable knowledge of them) were dismissed, and what remained were mathematical (or logical) concerns about the axiom system itself: completeness and consistency. In order to secure these for classical mathematics, soon, Hilbert's attention was directed to arithmetic, as he intended to show that the real numbers were a model of his geometrical axiom system. As indeed the latter's truths appeared to be translatable into (or could be mapped onto) algebraic ones, this effort yielded Hilbert's famous relative consistency proof: if algebra is consistent, so is geometry. But now of course the search for a direct consistency proof for real number arithmetic was still in order, and this would colour Hilbert's further philosophical excavations. In the 1910s, he was briefly fascinated by the logicist programme, but finally held on to his earlier idea that mathematics and logic had to be developed simultaneously. However, without defending the possible reduction of the former to the latter, he did come to appreciate logic's contribution to what would soon stand out as his proper 'metamathematical' programme: the formal (logico-axiomatic) study of mathematical proof systems.

During the 1920s, Hilbert thoroughly developed the main ideas of his programme, viz., formalism and, especially, finitism. He did this largely in collaboration with the Swiss Paul Bernays (1888-1977), a student of Ernst Zermelo who had joined him in Göttingen as his assistant in 1917. It was not so much the mathematical genius as the clarifying power of Bernays, a trained philosopher, that turned out to be of vital importance in this respect. In this period, Hilbert opposed himself explicitly to the other foundational programmes. Quite firmly, if not hostilely, in the case of intuitionism (Brouwer), which was fuelled by his student Hermann Weyl (1885-1955) joining that camp in 1921. With considerably more mildness in the case of logicism. Both contending programmes indeed exerted a particular kind of pressure on Hilbert's labour: logicism because of the foundational deadlock Frege had been driven into, intuitionism by its powerful attack on the notion of (actually) infinite collections, and the accompanying denial of excluded middle. Although Hilbert was impressed by this latter attack, in contrast to Brouwer et al., he sought to safeguard the whole of classical mathematics nevertheless. Hence his famous dictum: Cantor has created a paradise from which no one shall expel us. To secure this he established a number of distinctions: between finite and transfinite axioms, inferences and statements, and between real and ideal elements; distinctions to be seen relative to a particular system. Real mathematics, for Hilbert, was constituted by 'genuine' a priori judgements about shapes, i.e. forms of concrete signs or figures, and their combinatorial features. Ideal mathematics, on the other hand, proceeded by 'pseudo'-judgements, which did not pertain to (our knowledge of) real things at all. For Hilbert, only in the latter realm, there is room for meaningful or interpretative reasoning.

In the end, the problems turned out too persistent. In real mathematics, one always deals with a finite number of elements or observations. To prove the consistency of the axioms, however, requires a passing from this level to that of infinite sets. Finite models could do the trick, but the axiom systems of the major mathematical branches do not appear to allow for this. As such turned out to be the case for elementary arithmetic, Hilbert's target, he was driven to devise for it an absolute consistency proof. Here he exploited the possibilities of formalization to the maximum, which led into his famous distinction of meaningless mathematical vs meaningful metamathematical talk, separating syntax and its interpretation. For Hilbert, to claim that arithmetic is consistent, as a statement about arithmetic, clearly belonged to the latter or interpretative realm. An absolute consistency proof for arithmetic would show, by strictly finitist means, that the structural properties of its axiom system did not allow for the formulation of inconsistencies. In 1931, Kurt Gödel, a then 25 years old Austrian mathematician, would show why this project had to fail.

Bart Van Kerkhove