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

210 michael hallettinvestigation, and might be used to describe, say, Pasch’s work on projectivegeometry in his (Pasch, 1882), an important influence on Hilbert’s work, andalso Frege’s work on arithmetic, for Frege attempted to show that naturalnumber arithmetic can be derived from purely logical principles alone, withthe help only of appropriate definitions. Both Frege’s and Pasch’s projectsattempt to show that, when properly reduced or reconstructed, the respectivemathematical theories represent knowledge of a certain, definite kind, logicaland empirical respectively. The projects thus have conscious epistemologicalaims, and are very broadly ‘Euclidean’ in the general sense that they attemptto demonstrate that certain bodies of knowledge can be deduced from a stockof principles circumscribed in advance, a demonstration which can only beeffected by the actual construction of the deductions.While these are very important elements in Hilbert’s treatment of Euclideangeometry, they are by no means exhaustive.For one thing, the description of Hilbert’s geometrical work as an extensionof the Euclidean project does nothing to explain its entirely novel contributionto foundational study. The novelty comes in treating the converse of theEuclidean foundational investigation, asking in addition the question: Can weshow that a given proposition P (or some theoretical development T ) cannot bededuced (carried out) solely using the precisely specified ‘restricted methods’?¹⁴ In other words, as Hilbert puts it in the Grundlagen (pp. 89–90), weseek to ‘cast light on the question of which axioms, assumptions or auxilliarymeans are necessary in the proof of certain elementary geometrical truths’. Itis important to see that Hilbert’s investigations are not just complementary tothe Euclidean ones; indeed, to describe them so would be to underemphasizetheir novelty. The very pursuit of this new line of investigation involvedHilbert in a radical transformation of the axiomatic pursuit of mathematics. AsHilbert expressed it in some lectures from 1921/1922 (Hilbert, ∗ 1921/1922,pp. 1–3):The further development of the exact sciences brought with it an essentialtransformation of the axiomatic method. On the one hand, one found that thepropositions laid down as axioms in no sense could be held sublimely free fromdoubt, where no difference of opinion is possible. In particular, in geometry the¹⁴ There are two important variations to this:1. Show that P (or some theoretical development T ) can be deduced (carried out) using , butnotusing − (slight weakening)2. Show that P (or some theoretical development T ) cannot be deduced (carried out) using , butcan with + (slight strengthening).