September 26, 2006

Anticipation of Feynman's path integral

Eddington's Turing Test
In the (brief, I swear it!) breaks I take from work here at the University of British Columbia, I have been skimming through an old copy of Arthur Stanley Eddington's Space, Time and Gravitation I found in the office I was assigned. It is a fascinating book, written in 1920 and explaining -better than many more recent efforts- the central concepts of the then brand-new Theory of General Relativity in a fairly non-technical way, sprinkled with interesting conceptual and philosophical dicussion. I will quote two paragraphs, one worth reading for philosophers and the other one for physicists.The first one: Eddington stresses constantly the formal nature of scientific laws and the way they capture structure, not content. E.G., in a discussion of Weyl's theory of electromagnetism (which failed in its original form but introduced into physics the inmensely fruitful concept of gauge symmetry) he says:
The geometrical potentials (k) obey the recognised laws of electromagnetic potentials, and each entity in the physical theory -charge, electric force, magnetic element, light, etc.- has its exact analogue in the geometrical theory; but is this found correspondence a sufficent ground for identification? The doubt which arises in our minds is due to a failure to recognise the formalism of all physical knowledge. The suggestion "This is not the thing I am speaking of, though it behaves exactly like it in all respects" carries no physical meaning. Anything which behaves exactly like electricity must manifest itself as electricity. Distinction of form is the only distinction that physics can recognise; and distinction of individuality, if it has any meaning at all, has no bearing on physical manifestations. [Emphasis added]
I have added the emphasis to remark what seems to me a bold conjecture, which Eddington ventures here but does not pursue in other places. returning to a conventional distinction between "form" and "content", which of course entails together with the formality of physics that the content or intrinsic structure of the physical world is unknowable. The temptation of believing that in our conscious thoughts we capture definite content, in turn, leads then naturally to panpsychism, or proto-panpsychism, or perhaps phenomenalism. But if we resist it and hold, despite initial implausability, that content has no meaning independent from a particular highly complex form (in Douglas Hofstadter's words, content is fancy form) then the obstacles for a naturalistic metaphysics disappear, the Turing Test is embraced as trivially valid, and zombies become an absurdity.
The second one: after remarking that in Weyl's theory curvature is no longer an absolute, gauge independent quantity, but action is, he says that action can be expressed as a pure number which, however, can take fractional values (though he mentions Planckian quantization of action he does not give it a fundamental importance. I don't understand very well why, but I forgive him; after all, it was 1920 and full quantum mechanics was still a thing of the future). He speculates then:
I can only think of one interpretation of a fractional number which can have absolute significance, though doubtless there are others. The number may represent the probability of something, or some fuinction of a probability. The precise function is easily found. We combine probabilities by multiplying, but we combine the actions of two regions by adding; hence the logarithm of a probabiliy is indicated. Further, since the logarithm of a probability is necessarily negative, we may identify action provisionally with minus the logarithm of the statistical probability of the state of the world which exists.The suggestion is particulary attractive because the Principle of Least Action becomes the Principle of Greatest Probability. The law of nature is that the actual state of the world is that which is statistically most probable.
Isn't this a remarkable anticipation of Feynman's path integral? Okay, Eddington uses a real exponential intead of an imaginary one, and doesn't hit upon the quantum principle of summing over all paths and making the least action path the one with greatest weight only in the classical approximation. But still, I find the connection between action and probability, made on intuitive grounds years before quantum mechanics was formulated, to be a great insight bearing the mark of genius. Don't you agree?
UPDATE: Upon further reading, Eddington would definitely not endorse the Turing Test. He says in the last chapter: "The matter of the brain in its physical aspects is merely the form; but the reality of the brain includes the content". What a pity. posted by Alejandro at 10:55 PM Thursday, September 21, 2006

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