Use of infinity in physics

Question

There have been lots of questions on this site about the use of infinity in different ways in physics.

• Infinitely big - Physics near null infinity

• Infinitesimals - Using differentials in physics

• Renormalization - A simple, easily understandable example of infinities making trouble in physics?

This is all fine. But one mathematically ordinary use of infinity bothers me. You can show the counting numbers and the even numbers have the same size by mapping 1->2, 2->4, and so on. This obviously fails if you try it with the counting numbers and even numbers below n. You run out of even numbers around n/2. No problem - make n bigger. Now it takes longer to run into trouble. And when you do the trouble is bigger. But for the full set, you never have to settle the accounts. You sweep the infinitely big error under an infinitely distant rug.

This makes me squirm a little. You can use this idea to prove unphysical results like the Banach-Tarski theorem. Vsauce demonstrates how this idea leads to the ability to cut a sphere into 5 "pieces" and put them back together into two spheres. Of course, you can't really cut out the infinitely detailed shaped needed to do this.

But he says that some physicists think this is physical. Some paper have been written using this idea. For example to explain things about quark confinement. He doesn't explain the application to physics. Can anyone enlighten me? Hopefully with a reasonably simple explanation.

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There is also the use of

$$1+2+3 + ... = -\frac{1}{12}$$

in string theory. But this might not be the same thing. This isn't true if you just add natural numbers and see what it converges to.

This has something of the flavor of how p-adic numbers work. For p-adics, the norm is different. Numbers that differ in bigger digits are closer together, as explained in The Most Useful Numbers You've Never Heard Of.

To sum 1+2+3+⋯ to −1/12 outlines several ways of showing this sum, including one by Euler.

For more about the use in string theory, see

• Why are extra dimensions necessary?
• A simple and rigorous proof of 26/10 dimensions in string theory …does not exist.

Answer 1

Fourier analysis is a much more routine example: the Fourier transform is defined by an integral with infinite limits. It gets used all over physics, as well as in other sciences, and even more in engineering.

Everything we do with Fourier analysis could be done without infinite limits, but it would require carrying inner and outer scale parameters through calculations. This tedious process would add confusion but no value.

Infinity is a useful concept in the construction of mathematical models. You should remember George Box's aphorism: "All models are wrong, but some are useful."

Answer 2

You sweep the infinitely big error under an infinitely distant rug.

You lose me a bit here - there is no error to sweep. Two sets have the same cardinality if and only if the elements of each can be put into one-to-one correspondence with the elements of the other. That is true of $\mathbb N$ and $2\mathbb N$, as you say. No problem.

Now, if two sets $A$ and $B$ both have a finite number of elements, then they have the same cardinality if and only if they have the same number of elements. In this way, cardinality reduces to "number of elements" for finite sets. For non-finite sets, "number of elements" is not a meaningful notion.

This makes me squirm a little. You can use this idea to prove unphysical results like the Banach-Tarski theorem. [...] But he says that some physicists think this is physical. Some paper have been written using this idea. For example to explain things about quark confinement.

I found only one paper linking Banach-Tarski to hadron physics. Its central thesis is that there is a connection between the two, because it can be shown that the so-called minimal decomposition required to implement Banach-Tarski is a single sphere being split into 5 pieces and then reassembled into two spheres, one consisting of 2 pieces and the other consisting of the remaining 3. We are to imagine that the "pieces" are quarks, and that a 2-piece sphere and a 3-piece sphere are a meson and baryon, respectively. The author observes that if this connection exists, then quark confinement is the statement that there does not exist a decomposition in which one of the resulting spheres consists of only one piece.

Personally, this seems ... unlikely to bear much fruit. I suspect the reason for this apparent correspondence can be attributed to the strong law of small numbers.