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This is a question about the philosophy of physics. If one takes a glance at the philosophy of mathematics its easy to see that the idea of number is filled with philosophical niceties and is a much more subtil beast than practical common-sense would give it credit for.

Yet, the physics literature or polemic is rife with the idea of simple physical laws. It seems to me simply by analogy that this view must be mistaken.

What are the standard conceptual schemes that show that they are in fact deeply subtle? Presumably Humes critique of causality and Kants riposte by centering it in consciousness.

Are mathematicians trained to develop what looks complicated to others an aesthetic that is simple - and similarly for physicists?

Does Husserls phenomenology offer further clues to this subtlety?

EDIT

There is another way that mathematics can be simple & subtle at the same time. For example a prime number is simple enough to define, yet to say that one understands a prime number merely from its definition is to miss the point - it is how it manifests itself in many different ways that displays ones understanding of what primeness means.

Another example in mathematical physics: Coordinates were introduced in Greek Antiquity as a device - it took two millenia before Descarte applied it to mathematics, and Newton picked up on it to develop the calculus. It appears thus - that coordinates are a simplifying device. However, after the invention of tensor calculus and its successful use in General Relativity by Einstein which prompted the development of smooth manifold theory - Weyl declared 'coordinates are a violence visited on nature' and in modern differential geometry the extrinsic coordinate view is banished. This is the modern expression of calculus which is intrinsic and coordinates are seen as being a vulgar (but sometimes neccessary) imposition. In post-modern differential geometry the smooth structure is made non-unique ie relative; this is accomplished by stating what possible coordinate systems are functionals are allowed. So coordinates are allowed back to allow this profound relativisation of smooth structure. In modern descriptions of geometry - algebra and geometry are seen as dual descriptions. This is reminiscent of the dual description of wave-particle duality in QM.

Through this rather brief historical exegesis - one notes that the idea of coordinates is not simple, although when one looks at a map it certainly appears as though it is.

  • Interesting question--I'm especially interested to see if there are any answers that suggest that simple physical laws are not simple without claiming that absolutely everything objective is not simple. – Rex Kerr May 29 '13 at 5:47
  • @Kerr: Good point. Set theory was a simple theory of sets - but no longer if one examines the large cardinal hierarchy, or more modernly topos theory. Is simplicity in the eye of the beholder? – Mozibur Ullah May 29 '13 at 5:55
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    Can I get a better idea of this? You speak of how you can define a prime number (divisible by 1 and itself), yet you can look at it and find amazing features that are not in that definition? Somewhat an example of this is my short research into primes. So is your question; we can easily define some (possibly, hopefully, all) physical laws simply but like the primes, as I suggest, there are more complex but subtle events that come from our simple explanations of these laws? – KDecker May 29 '13 at 21:59
  • You are probably overstating the importance of the hack to introduce coordinate systems and smooth structure in smooth manifold theory. The hack is done because it works good enough for all practical purposes (assuming smooth manifolds), not because it would be the only correct way to introduce coordinates or embody any especially important concept. Most manifolds come with a differentiable structure inherited from their construction, quite independent of any coordinate systems. – Thomas Klimpel May 30 '13 at 21:19
  • @Klimpel: Its been some time since I've touched manifold theory - is there a construction for smooth manifolds that avoids coordinates completely? I can certainly believe them as mathematical objects without coordinates. My point was historical that it was via the invention of coordinates that the idea of manifold manifested themselves. Unless of course you trace a different historical trajectory? Should euclidean * non-euclidean geometry be counted in manifold theory? – Mozibur Ullah May 30 '13 at 21:27
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Simplicity actually depends mostly on the point of view. In physics, this becomes obvious when choosing coordinate system which are most suitable to the problem at hand, or simplifying equations by introducing suitable notations. Before people in Europe started to write numbers with arabic decimal digits, division was its own subject in university education. Today, it is taught in primary schools, because we found a more useful notation...

The other component of simplifying the world is abstraction: disregarding any information which is not seen as important at the moment.

So: there are simple laws of physics, if you find the right way to look at the world, and disregard everything that would make it too complicated for your taste...

  • I like that division bit! Could you back that up with a reference? – user3164 May 30 '13 at 18:27
  • @Gugg: I heard it from a university teacher in my maths for physicists classes, but I can hunt around a bit for more evidence... – hbarck May 30 '13 at 18:51
  • +1: This is the usual point of view in physics. I'm asking to examine it. I've edited my question to give more of an idea of what I'm trying to get at vis-a-vis with your example of coordinates. – Mozibur Ullah May 30 '13 at 18:58

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