About mathematics and the physical world

Question

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...

Answer 1

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.

Answer 2

The answer to your first question is no, which renders all your other questions moot. Mathematics is not a branch of physics.

Answer 3

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.

Answer 4

"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.