Why does mathematics manage to represent a function of reality?

Question

Why does mathematics manage to represent a function of reality? My question concerns how your logic and its structure (like topology, or the very fields of advanced logic in mathematics) manage to represent reality?

In Penrose words:

Plato made it clear that the mathematical propositions—the things that could be regarded as unassailably true—referred not to actual physical objects (like the approximate squares, triangles, circles, spheres, and cubes that might be constructed from marks in the sand, or from wood or stone) but to certain idealized entities. He envisaged that these ideal entities inhabited a different world, distinct from the physical world. Today, we might refer to this world as the Platonic world of mathematical forms. Physical structures, such as squares, circles, or triangles cut from papyrus, or marked on a flat surface, or perhaps cubes, tetrahedra, or spheres carved from marble, might conform to these ideals very closely, but only approximately. The actual mathematical squares, cubes, circles, spheres, triangles, etc.,

Particularly, I think this is possible because all statements in mathematics are ultimately conditional statements on chosen axioms. It does not tell us which assumptions are true, but it tells us what follows if certain assumptions are true (and also, if certain assumptions cannot be supported). For mathematical realists (radical Platonism), theorems have physically real consequences and our world is only one among the infinite ocean of mathematically possible universes.

But I don't understand why this abstract logic manages to "tangential" reality... why does this happen? I accept any suggestion for reading, or an explanation that might shed some light on this tunnel.

Answer 1

Here is my way of looking at "the unreasonable effectiveness of mathematics".

In the real world, counting macroscopic objects like pebbles or people is useful because in the real world, neither of these two things can spontaneously pop into or out of existence. Counting them has meaning. Then we discover that we can represent those counts conveniently using numbers instead of pebbles, people, or anything else. So, object permanence allows a 1-to-1 correspondence between counting objects in the real world and a mathematical representation of those counts; as such, arithmetic provides a reasonably effective tool.

Then we discover that arithmetic lets us usefully quantify not just pebbles or people but durations of time. Why is this possible? Because in the universe we inhabit, the rate of flow of time we experience happens to be the same in all ordinary circumstances, so that what we measure as one minute on a tuesday remains one minute on a thursday, and there is nothing inherently "unreasonable" about that effectiveness either.

To me, then, the reasonableness of quantification is at the root of the reasonableness of mathematical descriptions of the real world.

Answer 2

You are asking about a very rich field called measurement theory.

The standard textbooks are the three volumes of Foundations of Measurement by Krantz, Luce, Suppes, and Tversky (Volume 1 here).

Answer 3

It would be very unreasonable if math couldn't capture real physical processes. These processes show themselves to us as being orderly and being the same in fixed situations. Perfectly fit for math! Math also shows us orderly structures. Thousands of books about thousands of structures are written. It would be very unreasonable if none of these could be used in the description of reality. This is the reasonable effectiveness of mathematics. The reason being that math itself is derived from reality.

The structures can go living on and leading their own life in our minds. We can discover new structures which inspire new lines of investigation. And like that we can discover new features of reality (not just their mathematical structure). The standard model in physics is the ultimate example. Math rules suppreme in this area and most attempts to beat it or undermine it are countered by reference to the "wonderfull math" in the model. Thhis is of course a holdback. Some (most) times math is not all.