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

238 michael hallettthat words like equal, greater, smaller be replaced by arbitrary word formations,like a-ish, b-ish, a-ing, b-ing. That is indeed a good pedagogical means forshowing that an axiom system only concerns itself with the properties laid downin the axioms and with nothing else. However, from a practical point of viewthis procedure is not advantageous, and also not even really justified. In fact, oneshould always be guided by intuition when laying things down axiomatically, andone always has intuition before oneself as a goal [Zielpunkt]. Therefore, it is nodefect if the names always recall, and even make it easier to recall, the content ofthe axioms, the more so as one can avoid very easily any involvement of intuitionin the logical investigations, at least with some care and practice. (Hilbert, ∗ 1905,pp. 87–88)The emancipation from intuition and the ‘origin’ of the theorem concernedis clearly indicated in this passage, but Hilbert’s remark that ‘one alwayshas intuition before oneself as a goal’ is also very important. Although theachievement of the results necessarily requires the deliberate suspension ofintuition, crucially they yield important information about intuition. One thingwhich Hilbert stresses in the passage quoted on p. 237 is that adopting the fullTriangle Congruence Axiom allows us to avoid any continuity assumption indemonstrating the ITT. That is certainly correct, and this fact belongs alongsideothers concerning congruence and continuity, a prime example being Hilbert’sreconstruction of the Euclidean theories of proportion and surface area withoutinvoking continuity. But there are other subtle conclusions to be drawn. Forexample, as explained above, the (apparently planar) full Triangle CongruenceAxiom appears to contain some hidden spatial assumption, since it licenses‘flipping’ arguments. But now Hilbert’s independence results seem to show thatwe can get the result without the spatial assumption, and thus with genuinelyplanar congruence axioms, if we accept some modest continuity assumptionsabout the plane. So one has a choice between spatial assumptions in the planarpart of geometry, or continuity assumptions. However, what makes this a rathermore complicated matter is that the main continuity assumption involved, theArchimedean Axiom, is quasi-numerical, something one might think shouldbe avoided as far as possible in a purely ‘geometrical’ axiomatization.But whatever the right conclusion to be drawn about our geometricintuitions, two very important things seem to follow from Hilbert’s analysis. Onthe one hand, the results seem to strengthen Hilbert’s holism about geometry,at least in the sense that the axioms are very intricately involved with eachother, and that there might be more than one way to achieve many importantresults. Secondly, the adumbration of the intuitive picture here, and perhapsits correction or adjustment, follows from the high-level logico-mathematicoinvestigation which Hilbert engages in. The information is obtained only