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One of my colleagues argues that everything in math proves something in the physical world. For instance, he claims that the existence of math to describe fractals proves the infinite divisibility of matter (which is a non sequitur conclusion to make).

What is some simple mathematical concept that absolutely does not map to the physical world, contrary to this belief?

Jesan Fafon
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    I think we are in the physical world, so, at worst, anything made up by people (e.g., artificial mathematics, as opposed to natural), is still in the world, so is already mapped to the world as much as any human conception. A more penetrating question would be about whether a given mathematical (or other human...) idea clarifies (still, to human perception) anything about the world, perhaps other than itself, etc. – paul garrett Dec 06 '13 at 00:39
  • I'm not sure I like this style of philosophical arguing. Semantic (as in vocabulary) arguments are only interesting when the arguers realize that it is a semantic argument and know how to make a valid semantic argument. Anyway, since it occurred to me, maybe negative integers? How can you have a negative amount of something? – DanielV Dec 06 '13 at 00:39
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    IMO math does not prove anything in the physical world, suppose one knows all the math, but knows nothing about the physical world, he cannot prove anything about the physical world. What happens is that we observe some physical phenomena or properties can be modeled using mathematical concepts. Without observing the world we cannot chose what part of math to use. – hhsaffar Dec 06 '13 at 00:50
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    Possible counterargument: If we assume a world in which matter is not infinitely divisible (and whether or not our world has that property, it is certainly something that can be imagined), would anything necessarily prevent the inhabitants of that world from thinking up mathematics that describe fractals? – hmakholm left over Monica Dec 06 '13 at 00:51
  • Protons and electrons? The conservation of energy? @DanielIV – LASV Dec 06 '13 at 00:52
  • @Luis As I said it is a semantic argument, but one could just say that you can only have a positive number of proton and a positive number of electrons, and even if you assume energy is discrete, one could say that it only exists in positive amounts and that any negative amount is simply our interpretation. I'm not saying this is the only way to view it, but it is a way. – DanielV Dec 06 '13 at 00:56
  • @DanielIV I realised it probably sounded like I was challenging your opinion. I was not, rather asking what you think of those. You are right,there are too many ways to view it, and if something is correct is ill-defined. Sorry if it sounded as I was challenging your opinion. – LASV Dec 06 '13 at 00:59
  • So for example in the case of fractals one should first show (or assume for some practical reason) that something is infinitely divisible and then model it by fractals, fractals does not prove divisiblity. One can look it like that, discrete math proves everything is discrete while continuous math proves every thing is continuous! – hhsaffar Dec 06 '13 at 01:02
  • IMO math just helps us to find logical implications of the mathematical model we chose – hhsaffar Dec 06 '13 at 01:05
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    When our theories about the world fail, the math related to them is not refuted, if just the math proved the theory, then math must get refuted when the theory fail. – hhsaffar Dec 06 '13 at 01:12
  • With tongue in cheek: Any number $>10^{80}$ – Hagen von Eitzen Dec 06 '13 at 19:49
  • @HagenvonEitzen: But that would leave out googol (which is what 'Google' is based on), and googolplex, and Graham's number, and, for that matter, the number of arrangements of chess pieces on the chess board (which, I'm given to understand is about 10^112). –  Nov 05 '18 at 23:48
  • BTW, the converse of the notion that every mathematical result, no matter how abstruse, is reflected in the real world somehow, is Catherine Attard’s thesis, that one should have the capability to understand life experiences from a mathematical perspective. https://eric.ed.gov/?id=EJ978128 –  Nov 09 '18 at 01:57

2 Answers2

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No one knows the true nature of the physical world. However, recent work seems to suggest that it is quite discrete. That is, things like real numbers may not represent anything real. Also, it seems likely that the entire observable universe can exist in only finitely many possible states. If this is so, then likely nothing infinite in mathematics represents reality.

If you don't want to go to such extremes, I think you will still find it extremely difficult to find anything in the real world relating to even fairly small ordinal numbers, like $\omega_1$, or even fairly small cardinal numbers like $\aleph_2$.

dfeuer
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  • Very good points about highly infinite things... and, yet, I seriously suspect that human perceptions about various "infinities" do admit finitistic interpretations. E.g., AxCh as an over-the-top umbrella justification for finite choice schemes. To my mind, the point/issue is about human interpretation... e.g., we are in the "finite" world, and pretend to conceive of hugely infinite things... how is this already possible? :) – paul garrett Dec 06 '13 at 00:49
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Points and lines are not physical objects because a point has zero volume and a line has zero thickness.

lhf
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