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discontinuum physics

Posted 8 years ago

in a recent column (http://blog.stephenwolfram.com/2015/12/what-is-spacetime-really/) Stephen discusses his idea of a network model of the universe (originally presented in chpt. 9 of NKS and in his earlier blog (http://blog.wolfram.com/2007/09/11/my-hobby-hunting-for-our-universe/). It is worth noting that Albert Einstein had also thought in tihs direction. here are some comments of Einstein that i have located:

"But you have correctly grasped the drawback that the continuum brings. If the molecular view of matter is the correct (appropriate) one, i.e., if a part of the universe is to be represented by a finite number of moving points, then the continuum of the present theory contains too great a manifold of possibilities. I also believe that this too great is responsible for the fact that our present means of description miscarry with the quantum theory. The problem seems to me how one can formulate statements about a discontinuum without calling upon a continuum (space-time) as an aid; the latter should be banned from the theory as a supplementary construction not justified by the essense of the problem, which corresponds to nothing "real". But we still lack the mathematical structure unfortunately. How much I have already plagued myself in this way!" - letter to Walter Dallenbach, Nov 1916

"In any case, it seems to me that the alternative continuum-discontinuum is a genuine alternative; i.e. there is no compromise here. In [a discontinuum] theory there cannot be space and time, only numbers [...]. It will be especially difficult to elicit something like a spatio-temporal quasi-order from such a schema. I can not picture to myself how the axiomatic framework of such a physics could look [...]. But I hold it as altogether possible that developments will lead there [...]."

"I consider it entirely possible that physics cannot be based upon ... continuous structures. Then nothing will remain of my whole castle in the air including the theory of gravitation, but also nothing of the rest of contemporary physics"

"The problem seems to me how one can formulate statements about a discontinuum without resorting to a continuum (space-time) ... But for this we unfortunately are still lacking the mathematical form. How much I have toiled in this direction already!"

Its worth ponting out that the first comment was made in 1916 at the same time that Einstein created his theory of GR and not later, when he was in what many physicists consider (incorrectly, and often condescendingly) to be in his intellectual dotage.

note: if you are interested in other discrete models of the universe, you might look at the publications of Raphael Sorkin (http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/) related to the causal set approach.

finally, let me just point out that whether one prefers a continuum or discontinuum approach, these are MODELS of realty, not representations of reality. in the words of John von Neumann:

"The sciences [...] make models [that are] mathematical constructs which, with the addition of certain verbal interpretations, describes observed phenomena."

note that In this statement, von Neumann accepts the Popperian falsifiability criterion for science, rather than the post-empirical confirmation criterion nonsense recently proposed by Dawid and finding some support amongst theoretical cosmologists who have been unable to develop experimentally testable models such as string theory and GQL theory.

POSTED BY: Richard Gaylord
3 Replies

See also Noncommutative geometry and the spectral model of space-time by Alain Connes where he says (p. 183)

The traditional notions of geometry all have natural analogues in the spectral framework. We refer to [9] for more details. The dimension of a noncommutative geometry is not a number but a spectrum, the dimension spectrum (cf. [14]) which is the subset ? of the complex plane C at which the spectral functions have singularities.

and also The Spectral Model (A. Connes & A. Chamseddine), as you see do the mathematicians refer correctly to it as a model.

POSTED BY: Udo Krause

Feynman pointed out that a continuous model of space-time would require an infinite amount of information to describe an arbitrarily small volume. There's been a lot of work on loop quantum gravity, which is a discrete model.

POSTED BY: Frank Kampas

i know about Feynman's statement. of course. as for LQL theory, i think its a dead end because it makes no testable predictions. - see (https://www.youtube.com/watch?v=jEr038WOKFI) . i like Sorkin's causal set approach much better (i have no idea what, if anything it predicts but i like its aesthetics) but Stephen's trivalent network model seems best (i.e. appeals most) to me so far. even so, it hasn't as far as i know, produced any results that we didn't already have - e.g. Einstein's equations which we've had for exactly 100 years. personally, i don't have a preference for either continuum or discontinuum approaches or even combinations of the two. i think there is no way to know what reality IS. all we do is build models and we can chose between models using a variety of criteria such as the ability to agree with experimental results, conceptual intuitiveness and mathematical simplicity. i worked in soft matter physics where i and others had competing models describing the phenomena of the rubber elastic behavior of polymer networks, all of which give equally good in matching a wide variety of data. the models aren't right or wrong - they're simply diffeent representations of the effect of polymer chain entanglement. actually, caricatures of the effect, not even caricatures of the cause of the effects themselves - that would require representing the various entanglements which is impossible becuase there is no way to either control or characterize them (i'll note that my model would give the same result regardless of the nature of the entanglements because mathematically, it uses a universal scaling form. I basically agree with the statistician, George Box, who said "All models are wrong but some are useful". i just wanted to post Einstein's comments becuase they are prescient and they demonstrate what a profoundly deep thinker Einstein was - developing a continuous field model of GR while simultaneously considering the possibility of a discrete alternative (which he never developed).

POSTED BY: Richard Gaylord
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