FOM: more on Hilbert's "Axiomatishes Denken"
neilt at mercutio.cohums.ohio-state.edu
Fri Jun 11 07:10:53 EDT 1999
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.
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