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Keith Elliott Peterson (Aug 16 2021 at 00:26):I'm not sure whether to ask this here, #philosophy, or #practice: applied ct, however, here seems general enough.
Is there a sense in which we can speak formally about representations of mathematical objects in our physical reality? A classical example from group theory might be an actual (say 3×3×3) Rubik's cube, which is one (of many) physical representation of the underlying group of, well, 3×3×3 Rubik's cubes.
Also, as an aside question (and I'm sorry if I'm being handwavy): Would then theoretical science be kind of like how in classical mathematics, we only care if a proof holds or is at least satisfiable, but here of physical representations (whether or not it is possible Rubik's cubes can physically exist), while experimental science and engineering is a kind of "constructivist" version of that (ways to possibly show or outright create physical Rubik's cubes)?
Morgan Rogers (he/him) (Aug 16 2021 at 11:06):Sounds like a philosophical question to me. At any rate, Euclidean 3-dimensional space is usually assumed to be a very good model of physical space (at least under familiar, non-extreme conditions), and classical physics is typically modelled as living in affine 3-dimensional space, so what you're looking for is probably the geometry (and representation theory) of 3-dimensional spaces.
Of course, if we need a 4th dimension, time is available to fill that role. There was a big story about "time crystals" in popular science news recently which amounts to a physical representation of a theorized mathematical object.
David Egolf (Aug 16 2021 at 17:19):In the engineering research work I've done on imaging, it is common to create a model for a physical imaging system. In this setting it is common to approximate an "observation function", which sends objects to observations of them, by a linear map. This approximating map will never be perfect. We also approximate unknown objects by an estimated vector.
In this setting, one way to formally describe how good our estimate of an unknown object (assuming our estimate of the observation function is pretty good) is as follows:
where is the observation made of some unknown object , is our approximation of the observation function, and is our approximation of the unknown object. The operator is taking the 2-norm.
This "quality of approximation" term is often used for regularization in optimization-based approaches to estimating .
So, this is one example of how we create some mathematical objects that are approximately represented in reality. The "quality of approximation term" is one way to try and capture how similar the two are.
John Baez (Aug 16 2021 at 17:32):If you want to speak formally about defining "representations of mathematical objects in our physical reality", you'll probably need a formalization of "physical reality". For this, the usual approach is to pick a theory of physics, and work with that.
For example, people now study the computational power of various physical theories - i.e., what sort of functions can be computed by systems described by these theories. Since computation is understood mathematically and the physical theories are mathematical too, people can prove actual theorems about this stuff.
But there's no need to limit this idea to "computation".