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Occam's Razor and mathematical beauty appear to be compatible when reviewing Michael Atiyah's video.

But are the high levels of complexity associated with mathematical physics compatible with Occam's Razor?

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    $\begingroup$ I migrated this from Meta, but I wonder why this question is asked at MSE and not at, say, Physics.Stackexchange... $\endgroup$ – Willie Wong Apr 11 '13 at 15:48
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    $\begingroup$ This is an interesting question, but I agree with Willie Wong that it's not on topic for this site. $\endgroup$ – Grumpy Parsnip Apr 11 '13 at 15:52
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    $\begingroup$ Michael Atiyah talks about mathematics not physics. Also should this question be closed as not constructive? $\endgroup$ – Qmechanic Apr 11 '13 at 16:55
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    $\begingroup$ @Qmechanic this question offends me much less than that one, which should in my opinion have been closed as not constructive because in its current form it is stated in a way too confrontational manner. I have not idea why this got so highly upvoted ... $\endgroup$ – Dilaton Apr 11 '13 at 17:04
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    $\begingroup$ @Qmechanic I'm not sure about this. I can definitely see how it could be considered non constructive, but it doesn't strike me as an obvious candidate for closure. It's not the sort of open-ended question we really need to shut down. It's just soft, i.e. not a problem that takes an application of physics to solve (or something like that). $\endgroup$ – David Z Apr 11 '13 at 23:29
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The classical formulation of Occam's razor is: "Entia non sunt multiplicanda praeter necessitatem", or "one should not multiply things without a good reason".

Complex though many of the theories considered in mathematical physics are, their goal is nevertheless to find a description of the world that is simpler than alternative ways of explaining the same features of the world -- not in the superficial sense of being quicker to state, but in limiting the number of arbitrary choices that go into making a world. These theories may or may not eventually succeed in that goal, but simply because that is goal, the endeavor is perfectly compatible with Occam's Razor.

(Here, as everywhere, "simplicitly" is of course a matter of viewpoint, and open to discussion).

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Complicated things require complicated answers. There are a lot of very different phenomena out there, and considering the relatively small number of assumptions made in mathematical physics, I think they do pretty well simplifying things. YMMV.

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  • $\begingroup$ What phenomena has mathematical physics hypothesis actually explained or predicted prior to discovery that could not be explained simpler? $\endgroup$ – Jamahl Peavey Apr 18 '13 at 13:09
  • $\begingroup$ @JamahlPeavey: are you kidding? The bending of light by the sun. The anamolaous magnetic moment of the muon, the positron, the Higgs Boson, etc.,etc., ad nauseaum. $\endgroup$ – Jerry Schirmer Apr 18 '13 at 13:24
  • $\begingroup$ The Standard Model predications are theoretical physics not mathematical physics. $\endgroup$ – Jamahl Peavey Apr 18 '13 at 13:26
  • $\begingroup$ General Relativity predictions are also not mathematical physics. $\endgroup$ – Jamahl Peavey Apr 18 '13 at 13:28
  • $\begingroup$ @JamahlPeavey: it's all very closely related. Mathematical physics researchers work in theoretical physics and vice versa. When i think of "mathematical physics", i think of taking physical results and cleaning them up, axiomatically. Almost all of theoretical physics, this is done to various levels of rigor. $\endgroup$ – Jerry Schirmer Apr 18 '13 at 14:59
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Occam's razor is not a concept as well defined as you usually assume. Or at least, it is relative to what looks parsimonious or simple for your particular taste. For instance, the simplest hypothesis to explain the universe would be: "every consistent structure that can exists, exists", and then you have the multiverse theory of Max Tegmark, in which our physical universe is just an example of an infinite number of possible mathematical structures. On the other side of the coin, you can see the simplicity not in the description, but in the number of objects and interactions assumed to exists. This is the example of modern physics, you have very complicated mathematical derivations, but in the end they describe a universe consisting of as few objects and interactions as possible (the complexity is in deriving the predictions, but it doesn't matter because you chose to apply the razor to the number of objects and interactions).

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  • $\begingroup$ In modern physics complexity limits solvability so Occam's Razor is a real tool for simplifying equations in order to increase solvability. Mathematical physics is not modern physics but a branch of mathematics. Theoretical physics is a branch of modern physics such that it is the subject of experiment regardless of complexity. $\endgroup$ – Jamahl Peavey Apr 16 '13 at 0:08
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There is no fundamental contradiction between mathematical physics -- which is simply a rigorous way of formulating theory within the verified results discovered via experiment -- and Occam's razor.

But the question seems to stem from the idea that Occam's razor isn't precise enough to be serious. This is not really true, though -- Occam's razor is clearly required, for example in choosing the theory "general relativity" over "general relativity holds true until May 2027, then replaced with Newtonian gravity", even though both theories agree completely with experimental results so far.

Occam's razor can be made precise via Bayesian reasoning -- what the razor provides is a way to determine prior probabilities (more complicated theories have lower prior probability), then experimental results interfere with this probability distribution and produce a new one. This idea is formalised, e.g. in Solmonoff's theory of inductive inference (although I prefer the term interference, because you don't actually infer a definite conclusion, you just have experimental results to interfere with your probability distribution making it more and more precise and accurate), where complexity is formalised in terms of Kolmogorov complexity.

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