A better reference is to section 65a of the Prior Analytics, where
Aristotle discusses whether parallel lines exist. He knows that if an
exterior angle of a triangle can be smaller than an opposite interior
angle, or if the sum of the angles of a triangle can be more than two right
angles, there can't be parallel lines. This of course was written just a
few decades before Euclid wrote, and shows how axiomatics was proceeeding.
>
>We don't know whether quantum mechanics would have developed from some other
>scientific tradition because Chinese mathematics, i.e., got displaced by
>Western mathematics after Ricci and the Jesuits. Needham suggests that the
>Chinese tried to leap to the worldview of a quantum relativistic universe
>without going through Newtonian mechanics first.
All kinds of metaphysical views can be conceived without any grounding in
experiment. Without the high technology to test quantum mechanics, it also
would be just airy speculation. But I must say, I fail to see how the
representation of the fundamental concepts of position and momentum as
self-adjoint operators on Hilbert space could be constructed out of
intuition alone.
>
>Also, what does the axiomatic method have to do with quantum mechanics?
Sure,
>von Neumann axiomatized it after the fact, but the work of Bohr,
Schroedinger,
>etc was all non-axiomatic. Newton in the early calculus was hardly using
>axiomatic approaches. Nor was most classical physics axiomatic, but rather
>bunches of problem solutions.
>
>Your claim is pure chauvanism.
You are making a link I didn't. I said that axiomatization led to
non-Euclidean geometry. I stand by that claim. I didn't link
axiomatization and quantum mechanics, but the whole point of von Neumann's
work was to make mathematical sense of the more physical reasoning of the
earlier physicists. To do that, he needed Hilbert space, and in particular
unbounded self-adjoint operators and the spectral theorem. You can't just
make these things up. You have to be guided at every step by rigorous
proof. The distinction between a symmetric operator and a self-adjoint one
is crucial, for example, even though they look intuitively very similar.
The need for rigor is a guide to discovery, as it was in the case of
non-Euclidean geometry.
>
>Val Dusek
>