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Suppose it is proven that in the physical universe all magnitudes are finite:

  1. there are no infinitely long magnitudes.

  2. there are no infinitely small magnitudes.

Then:

Would we get a mathematic contradiction?

If we assume this to be true, then: would all the notions of infinite, and consequently, integrals, derivatives, limits, etc will have to be rewritten using finite operations?, e.g., sums instead of integrals?

Under this assumption, what kind of applicability would have (lets say) the concept of infinite and maths with infinite magnitudes?

Well, one can say, what's the problem, we can preserve math as it is, however, the implications of doing this seems to be risky...

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By the way, have you ever seen $-3$ apples in the physical universe? –  Rahul Jan 25 '13 at 11:47
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@RahulNarain or even $+3$ electrons. Someone (Feynman?) once suggested that there is only one real electron, but it travels back and forth in time a lot (looking like a positron on it's ways back). –  Hagen von Eitzen Jan 25 '13 at 12:20
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@Hagen: It was John A. Wheeler, Feynman's thesis advisor, who came up with it. –  Rahul Jan 25 '13 at 12:27
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Yuri Gurevich (a prominent computer scientist) once told me that the infinite is an approximation to the finite. –  Andreas Blass Jan 25 '13 at 15:16
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4 Answers

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The usefulness of mathematics (be it discrete or continuous) is in its modeling ability. Whether or not the real world is infinitely divisible or not does not matter to the usefulness of the calculus to solving real world problems.

It's important to remember that all models are wrong. Otherwise, they won't be models. The art is to take a complex situation and make some simplifications. The simplifications are aiming at producing a workable models that gives relatively plausible predictions. It turns out that taking a very complex world composes of a huge but finite numbers of things and modeling it by a mathematical model that is infinitely divisible is a powerful simplification giving rise to the calculus.

The success of the mathematical theory and its usefulness in physics does not mean that the world is the model. So, even if tomorrow you will find that the world really is infinite and infinitely divisible discrete mathematics will still be interesting and useful. For the exact same reasons that continuous ideas are useful for modeling discrete situations - they yield good models.

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E.g., it is much easier and well justifiable to model, describe and compute the macroscopic behaviour of a few litres of gas as a continuous fluid instead of some $10^{23}$ particles bouncing around ... –  Hagen von Eitzen Jan 25 '13 at 12:18
    
@HagenvonEitzen, lattice gas (finite cellular automata) models are also justifiable and easier than PDEs for certain types of problems. Even PDEs are simulated by discretizing them, though obviously nowhere near that exponent. –  alancalvitti Jan 25 '13 at 13:50
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The answer to your first question is no, which renders all your other questions moot. Mathematics is not a branch of physics.

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There's a quote by Hilbert that I read somewhere but can't source it: "there is no difference between the methods of physics and the methods of geometry". That doesn't mean one is a branch of the other but it points out the close interconnection. –  alancalvitti Jan 25 '13 at 14:18
    
According to Max Tegmark, physics is a branch of mathematics. –  MatthieuW Jan 25 '13 at 16:41
    
According to Vladimir Arnold, math is the branch of physics where experiments are cheap –  alancalvitti Jan 26 '13 at 3:44
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No. I think you have a fundamental misconception of the relationship between physics and mathematics. Mathematics can be very useful in physics, and the development of mathematics has very often been guided by what might be useful in physics, but it doesn't rely on physics. You can't get a mathematical contradiction by showing anything about the physical world. That's not specific to infinity, but a very general consequence of the relationship between the two disciplines. Many branches of mathematics that make use of infinite sets are tremendously useful in physics, and that wouldn't change if it turned out that this is due to some approximation to an even more successful finite model.

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"Suppose it is proven that in the physical universe all magnitudes are finite:"

It's not really an issue of proving it as much as observing and measuring it:

(1) there are no infinitely long magnitudes.

Not just magnitudes, but there appear to be no infinite extents, whether length, volume, time, energy: The observable universe is ~14 billion years old, there's a cosmic background beyond which we literally can't see (NASA WMAP); it's estimated that there are $10^{80}$ atoms in the universe; The most energetic particles detected are extreme energy cosmic rays at $10^{19}$ eV (AGASA), and so on. Although strangely the universe is accelerating, it is also "cooling down" - I read that the rate of star formation is only 1% of what it was when the universe was young.

(2) there are no infinitely small magnitudes.

There are Planck scales: $10^{-35}$ m, $10^{-44}$ sec, $10^{-8}$ kg, below which nature is very different than even nanoscale. That does not mean it's discrete however. There is evidence that only wavelike structures exist at this scale, but it's a controversial topic, eg Clifford's space postulates.

There's two points to be made here: one is that math helps us engineer increasingly accurate instrumentation to probe both large and small scales of the universe. The second is that as we accumulate increasingly accurate measurements, we need to develop math models that fit the measurements, fit newly discovered relationships.

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Re downvote, please explain rationale –  alancalvitti Feb 24 '13 at 0:35
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