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*To*: fom@math.psu.edu*Subject*: FOM: more on Hilbert's "Axiomatishes Denken"*From*: Neil Tennant <neilt@mercutio.cohums.ohio-state.edu>*Date*: Fri, 11 Jun 1999 07:10:53 -0400 (EDT)*Cc*: neilt@mercutio.cohums.ohio-state.edu*Sender*: owner-fom@math.psu.edu

Mention was recently made of Hilbert's "Axiomatisches Denken". Here is a translation of the final paragraph (p.415). I think Steve will like what Hilbert says here: "I believe: every possible object of scientific knowledge succumbs, as soon as it is ripe for theorizing about it, to the axiomatic method and thereby indirectly to mathematics. By pressing ahead to ever-deeper layers of axioms in the sense described above we attain also ever deeper insights into the nature of scientific thought itself and become ever more aware of the unity of our knowledge. In the form of the axiomatic method, mathematics appears to be called upon to play a leading role in science at large." Interestingly, Hilbert had noted on p.413 that of all the questions he had listed about mathematics, the following is the "most frequently discussed, because it goes to the heart of mathematical thought": namely, the "problem of the decidability of any mathematical question by means of a finite number of operations". As thus formulated, it is not quite the question of whether there is a single effectively decidable system of mathematics. It could instead be construed as the question whether every mathematical question can be decided by means of a finite proof in some system or other. Neil Tennant

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