How do mathematical objects relate to the real world?

Question

I am just going to give an example of what I mean using Skolem's Paradox. I DO NOT want to get into Skolem's Paradox itself or its "resolution."

Skolem's showed that countability is relative in models of first-order formulations of ZFC (assuming ZFC has a model).

For example, take a model 𝔐 of ZFC. Let 𝔐 satisfy the statement "S | S is uncountable." So, there is no bijection B ∈ 𝔐 from S to ℕ. Now, let's just add B to 𝔐 and let 𝔑 be 𝔐 ∪ B. Thus S ∈ 𝔑 is countable.

When there's a mathematical result such as Skolem's Paradox indicating the relativity of countability (at least with first-order formulations of ZFC, this problem does not come up in second-order formulations), we can ask, "Well, is countability actually relative?" Or some derivative questions: "What are the natural numbers, or what does it mean to say there is a bijection between the natural numbers in the real world?" (whatever you take to be "actual" or the "real world")?

More generally, what is the relation of math-objects/results to the real world (however construed)?

Also see the same question asked and answered on the Mathematics site.

Answer 1

There were at least three different idealistic positions regarding the existence of mathematical objects:

• Mathematical concepts like geometry or logic exist before any (physical) experience in our mind and are what enables us to have and interpret experiences. Mathematical objects exist a priory as categories of our mind.
• Mathematical objects exist in a stronger and more real sense than physical objects. Physical objects only exist as shadows of ideal objects, while mathematical objects exist as ideal objects.
• The existence or non-existence of mathematical objects tells us something about the existence or non-existence of actual or possible physical objects with corresponding properties.

Common to all three position is a strong commitment to the existence of mathematical objects and to the falsifiable consequences of that existence. These strong forms of idealism have actually been falsified theoretically and practically by progress on the foundations of mathematics and the tremendous success of abstract mathematics. (But the strong forms of formalism have also been falsified.)

You might want to consider some potential commitments related to strong forms of mathematical idealism: Foundations of mathematics are neither necessary nor possible. Axioms have to be intuitively true, like Euclid's axioms for geometry. Mathematical object are at least as real as any object in the physical world. The question of whether objects like zero, infinity, square root of two, or square root of minus one actually exist is non-trivial and must be answered for each of these objects separately.

Note The above description is neither historically correct nor does it give full justice to the corresponding positions. However, I preferred to give a description of idealistic positions which still includes its rough edges, as a contrast to the "why should we care" position of fictionalism.

Answer 2

There are a number of schools of thought on this:

Mathematical Platonists argue that mathematical objects exist in the same way that other Platonic ideals exist.

Intuitionists, on the other hand, argue that mathematical objects exist only as psychological constructions.

Fictionalists argue that it doesn't really matter if mathematical objects exist or not-- that one can bypass ontological commitments altogether by treating mathematical objects as convenient fictions.

Personally, I'd suggest you start with the question raised by the Fictionalists, and ask yourself: why does it matter?

Answer 3

Numbers on their own represent nothing, one always represents an entity, two and beyond are all groups of entities. It is a matter what entity the number represents. If that entity is for example millimeters then it is an objective unit of measurement, if that unit is some arbitrary mental unit it relates to nothing and the sum is nonsense.

Answer 4

"How do mathematical objects relate to the real world?" The question seems to presuppose that (1) the real world is a determinate thing, and (2) mathematical objects are similarly determinate, in order to ask whether the latter relate to the former. Assuming for the sake of the argument that (1) is correct, I would argue that the answer would have to be negative, by providing evidence against (2). If, as I argue, mathematical entities are indeterminate, it would follow that they can't "relate to the real world" in any simple fashion.

My argument is based on Hamkins' theory of the multiverse. Here each instance of a universe U^+ can be viewed as vastly larger than other instance U^-. This applies, in particular, to the integers Z. To put it another way, we don't have an absolute notion of "finitude". Therefore Z could not correspond in any meaningful way to anything in "the real world".