[FOM] Unreasonable effectiveness
vladik at utep.edu
Sat Nov 2 17:54:07 EDT 2013
I think Wigner somewhat exaggerated the claim, because in most applications available at his time,
* it was NOT the case that first some mathematics was invented and then it turned out to be useful for natural science,
* the reality is that in both great successes of mathematics, mathematical methods were invented specifically to describe physical phenomena.
These two examples are general relativity, where Riemannian geometry helped, and quantum mechanics, where Hilbert spaces helped.
General relativity: Riemann spaces were invented way before general relativity, true, but Riemannian spaces as they were known in the 19 century describe only proper metric (with triangle inequality) and proper space, not space-time. Their extension to pseudo-Riemann metrics which describe proper time (and which have signature +---) was done by Einstein (with Michel Grossman) and, simultaneously, by Hilbert with an explicit purpose of describing space-time.
This generalization may sound easy now, but it took a lot of efforts at that time.
Quantum physics: while most ideas and results behind what is now known as a Hilbert space was indeed developed by Hilbert, the abstract notion of a Hilbert space was, if I remember correctly, introduced by John von Neumann for a specific purpose of describing the foundations for quantum physics.
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Timothy Y. Chow
Sent: Saturday, November 02, 2013 9:41 AM
To: fom at cs.nyu.edu
Subject: [FOM] Unreasonable effectiveness
In 1960, Wigner argued for the unreasonable effectiveness of mathematics in the natural sciences, and his thesis has been enthusiastically accepted by many others.
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