Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

purity of method in hilbert’s grundlagen 215formulation of the congruence axioms. Linear congruence in geometry, boththe idea of congruence, and the central propositions governing it, was originallymotivated by simple observations about movement of rigid bodies in space,but Hilbert’s axioms are no longer to do with movement itself, though theconnection between them and the movement of rigid bodies is not hard todiscern. Rather, Hilbert’s view (see Hilbert, ∗ 1899, pp. 59–60) is that a propermathematical analysis of spatial movement requires an independently establishedand neutral notion of congruence. Thus the abstract notions are to be appliedin the analysis of movement, but geometry with Hilbert’s congruence axiomsis not dependent upon the purely empirical matter of whether or not thereare in fact rigid bodies, and thus whether bodies can indeed ever be congruentin the intuitive sense. In pursuing mathematical investigations of geometry,one is investigating ‘possible forms of connection’ and not necessarily ‘actualconnections’, an emancipation which reflects the ascent to what Hilbert callsthe conceptual.It follows from this account that even if the underlying notion of intuitionwere strong enough to guarantee the ‘apodeictic certainty’ of the axioms, itwould still be possible to drop axioms, or modify them, or replace them, and itwould still be part of the task of mathematics to investigate the consequencesof so doing. In other words, a strong notion of intuition would not restrainHilbert’s axiomatic programme.There is another important consequence of this, namely that theoriescannot be straightforwardly true or false through correctly representing somefixed subject matter, or failing to represent it. This is stated clearly in the1922/1923 lectures, but it is already clear in the 1898/1899 lectures and the1899 correspondence with Frege. Theories, variously interpretable, are either‘possible’ or not, and what shows their ‘possibility’ is a demonstration ofconsistency, in which case the mathematical theory ‘exists’. Thus, for Hilbert,the correct account of truth and falsity with respect to mathematical theoriesis that of consistency/inconsistency. Derivatively a mathematical object exists(relative to the theory) if an appropriate existence statement can be derivedwithin the consistent theory. As Hilbert puts it in his 1919 lectures:What however is meant here by existence? If one looks more closely, one findsthat when one speaks of existence, it is always meant with respect to a definitesystem taken as given, and indeed this system is different, according to the theorywhich we are dealing with. (Hilbert, 1919, p.147, book p. 90).Hilbert sums this up in his 1902 lectures on the foundations of geometry:We must now show the freedom from contradiction of these axioms taken together;... .In order to facilitate the understanding of this, we begin with a remark: