The connection between mathematics and physics in the actual infinity. You can organize an infinite loop of conversion of limited items, for example.

An infinite loop could be only as self interaction, which should be renormalised (canceled)  in final equations (in quantum electrodynamics etc.)

Andrey Kolmogorov, 1969. Finite creatures in a finite world may be

forced to invent infinity.

"... let us imagine an intelligent creature living in a world of finite

complexity, [a world] taking only a finite number of physically distinct

states and evolutioning in a "discrete time". ... It is possible to explain

plausibly how such a creature, because of its structure, being unable to

cover all the complexity of the world around it and confronted with

systems of growing complexity and consisting of very large numbers of

elements, will create during the process of its entirely practically and

reasonably oriented activities the concept of an infinite sequence of natural

numbers." (Sorry, my own English translation, KP.)

The original text:

A. N. Kolmogorov. Scientific foundations of the school mathematics

course. First lecture. Contemporary views on the nature of mathematics,

Matematika v Shkole, no. 3 (1969), 12–17 (in Russian). Reprinted in 1988

as pp. 232-233 in http://ilib.mccme.ru/djvu/bib-kvant/maths.htm 

Infinity, eternity are human words, they don't exist (Kant, and many)

so these words are not actual nor potential, there are fictions (see Vaihinger)

Michel

There are some approaches, where infinity and/or eternity is equal to the God. I agree, that infinity (and God) is fiction (human) and could be linked with some border, similar to horizon, moving away during our movement to it. Seems, that Petr Vopenka (and A. N. Kolmogorov!?) has such interpretation of infinity. 

I believe that infinity is only an artifact of our mathematics. We extrapolate whatever is uncountable as being continuous, and that will lead us to the infinite divisibility. In real world we cannot have infinite divisibility but only as a mathematical construct of calculus.  

The reality problem of infinity is considered seriously by proponents of digital physics (see Wikipedia for a concise overview). See also ““REAL” ANALYSIS Is A DEGENERATE CASE of DISCRETE ANALYSIS” (http://users.uoa.gr/~apgiannop/zeilberger.pdf) by Doron Zeilberger. The crucial problem seems to be the continuous character of physical symmetries, see https://en.wikipedia.org/wiki/Digital_physics#Physical_symmetries_are_continuous

Two important physical effects could be linked with the infinity: nonlocal effects (quantum entanglement or EPR (Albert Einstein, Boris Podolsky, and Nathan Rosen) paradox (see https://en.wikipedia.org/wiki/Quantum_entanglement)) and spontaneous symmetry breaking (SSB) (see https://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking).

My idea is that the possible places (“coordinates” in some generalized sense) for the SSB could be continuous, but the realization (in some general sense, may be including the creation of virtual particles and the polarization of vacuum) is discrete. Existing particles could be stabilized by the discrete, quantized self-interaction (similar to the quantization of the possible electronically excited states of atoms).   

Quantum entanglement could create continuous link between particles, which could be (unpredictably) broken in a discrete way.

In both cases the “digital physics” fail to predict the precise way of the development of the world, but the “real” content of the world is finite.

In my philosophy, the mathematical continuum, digital physics etc. are, first of all, more or less successful means of model-building. Is the world, “in fact”, purely digital, or not? Answering this question is infinitely “far behind”, or impossible at all... , unless we translate it in terms of model-building: will the future physical models always contain continuous structures (pi, e, sinuses, Laplacians, etc.) at their BOTTOM level?

 We should go back to the principle that “A physical law must possess mathematical beauty.” (P.Dirac)

The models are effective, if they are internally connected and self - consistent and, therefore, beautiful (minimum basic principles give maximum consequences). If the finite mathematics (Petr Vopenka, A.A. Markov) could give good (effective and beautiful) physical models, they are preferable.

Some questions arise:

Are the mixed (“connected”) models (finite for particles and infinite for spontaneous symmetry breaking (SSB) and EPR paradox) enough beautiful and effective?

May be pairs (complementary) of models (finite for particles and infinite for SSB) are necessary?

What is the hidden reason of “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” mentioned by Eugene Wigner?

My contribution to Wigner's problem from my paper "Fourteen Arguments":

As put by Stephen W. Hawking (2002): “... we are not angels, who view the universe from the outside. Instead, we and our models are both part of the universe we are describing.”

Argument 6. Any formal system, after its definition is put on paper, becomes part of the physical universe. Therefore, asking about the “unreasonable effectiveness of mathematics in the natural sciences” (E. P. Wigner (1960)) is, in fact, asking about the applicability of a particular fragment of the physical universe to other fragments. This rebuts the “applicability argument” raised by Gottlob Frege against game formalism (for details, see A. Weir (2011)).

In fact, the above idea was proposed already by Kenneth Craik in his book "The Nature of Explanation" published in 1943. For the corresponding quotes, see my working paper "Demystified Theory of Truth and Understanding", look at 2 pages after the phrase "Craik's philosophy of mathematics". Craik was unique in asking and answering the question: why, at all, is modeling/paralleling possible in this world?