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

202 michael hallettto something important and deep. Thus, the first lesson which might be drawnis that a standard ‘purity’ question (‘Is this means necessary to this end?’) is oftenan occasion for a foundational investigation, and this is not carried out to showthat a certain kind of preferred knowledge is or is not sufficient, but ratherfor reasons of mathematical productivity and logical clarity, and that answeringsuch possibility and impossibility questions was frequently the occasion foropening up ‘new and fruitful domains of research’, as happened, say, withAbel’s investigation.The heuristic value of unsolved problems was stressed in a powerful wayby Hilbert in his famous lecture on mathematical problems (Hilbert, 1900c)held a little over a year later. Hilbert stresses that the lack of a solution to aproblem might well raise the suspicion that it is indeed insoluble as stated, orwith the expected means, and that one might then seek a demonstration ofthe relevant unprovability. Such a demonstration for Hilbert counts as ‘a fullysatisfactory and rigorous solution’ of the problem at hand (for example, theproblems concerning whether we can prove the Parallel Axiom or square thecircle), ‘although in a different sense from that originally intended’; see Hilbert(1900c, p.261), English translation, p. 1102. Settling such problems in this wayis, as Hilbert says, in no large part responsible for the legendary mathematicaloptimism he displays, i.e. for his ‘conviction’ of the solvability of everymathematical problem. Hilbert says of the achievement of impossibility proofs:It is probably this remarkable fact alongside other, philosophical, reasons whichgives rise in us to the conviction (shared by every mathematician, but which, atleast hitherto, no one has supported by a proof) that every definite mathematicalproblem must necessarily be susceptible of exact settlement, be it in the formof an answer to the question as first posed, or be it in the form of a proof ofthe impossibility of its solution, whereby it will be shown that all attempts mustnecessarily fail. ...This conviction of the solvability of every mathematical problem is a powerfulspur in our work. We hear within us the perpetual call: There is the problem; seekits solution. You can find it by pure thought, for in mathematics there is no ignorabimus!(Hilbert, 1900c, pp. 261–262)This is not merely a peculiarity characteristic of mathematical thought alone,but rather what he calls a ‘general law’ (or an ‘axiom’) inherent in the natureof the mind, that all questions ‘which it [the mind] asks must be answerable’.This is an important remark, and I will return to it. For the moment, suffice itto say that this is why Hilbert seeks ‘projection onto the conceptual level’, aterm we will elucidate in Section 8.3.1 (see especially p. 217).These remarks suggest why Hilbert says that seeking insight into apparentcases of unprovability is the ‘most fruitful and deepest principle in