Quote of Last Century
From Wigner's The Unreasonable Effectiveness of Mathematics in the Natural Sciences:
It is not the intention of the present discussion to refute the charge that the physicist is a somewhat irresponsible person.
posted by Kea | 1:45 PM
4 Comments:
"A possible explanation of the physicist's use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection. It is not the intention of the present discussion to refute the charge that the physicist is a somewhat irresponsible person. Perhaps he is. However, it is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena."
- Eugene P. Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, 1960.
"It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of spacetime is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities."
- R. P. Feynman, The Character of Physical Law, BBC Books, 1965, pp. 57-8.
It's just a matter of curve fitting. If you generalize the definition of "curve" sufficiently wide, you can always find a curve that fits the data amazingly well. That doesn't mean that you know anything about the subject ontologically.
Math is big.
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Carl, I agree with you that curve fitting by equations in itself is not very advanced physics.
Is anyone really sure if there are any really continuous curves in nature? Because everything is made up of particles, if you magnify a curvy line or anything physical that looks curved, eventually you'll come to a series of atoms arranged in what (on larger scales) looks like the shape of a curve. The illusion of continuous curvature will of course disappear on on the scale where you can see the individual molecules and particles.
Similarly, I don't see how there can be any curved trajectories, because if fields are quantized, the field quanta will only approximate to curves on large scales. On sufficiently small scales, motion will be more erratic.
When Newton's apple fell, presumably it was accelerated by gravitons interacting with it. That's not a truly continuous acceleration. Presumably according to quantum gravity, only at the instants when gravitons are being exchanged, do accelerations occur as impulses.
Maybe this is why differential geometry describing curvature was recognised by Einstein as a problem for quantum field theory:
"I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures. In that case nothing remains of my entire castle in the air, gravitation theory included, [and of] the rest of modern physics."
- Einstein, 1954 letter to M. Besso, quoted by Abraham Pais in his biography Subtle is the Lord: The Science and the Life of Albert Einstein, Oxford University press, 1983, p. 467.
There is a problem on both sides of the differential field equation of general relativity: firstly, you can't fundamentally model (except as an approximation valid statistically only for large scales) the distribution of particulate matter using the energy-momentum tensor T_{nm}, and second, you can't model field quanta interactions accurately by the Ricci tensor for curvature R_{nm}.
There are no smooth geodesics of curved trajectories in quantized fields, just a lot of impulses from field quanta, gravitons. Einstein was exaggerating the problems of quantum field theory, however, since calculus is a useful approximation on large scales where the flux of field quanta involved in the interactions between particles is large. The real problem is that the differential geometry of tensors provides the wrong framework mathematically for making progress on the fundamental problem of quantum gravity, ad when the mainstream is in a hole, it keeps digging instead of trying alternatives.
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