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Comment by Koshkin | Hacker News Reader

One could say that mathematics does not study (abstract) objects; rather, it studies relations that do, in fact, exist between (real) objects. Even a circle is merely a relation - that between what lies on the circumference and, for example, what's at the center (or the axis, if we speak about a cylinder); this also manifests itself in the form of rotation, ostensibly a physical phenomenon.

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That's kind of a cop out, though. Relations are presumably abstract objects, too?

Relations often realize themselves as physical interactions.

They sure do, much like words correspond to physical things, but words are still "abstract objects" (if you subscribe to Platonism, that is).

But relations are physical; the fact that they are the sole focus of mathematics (unlike physics where focus is also on objects and processes) does not change that.

In what sense are they physical? Can you point to one?

In the sense that they exist in nature. Natural numbers (correctly understood as relations) exist, for example, as atomic valences which permit certain interactions between atoms while preventing others. (This has nothing to do with the Platonism BTW.)

If to you relations exist in nature therefore they are physical, there is a physicalist assumption in there which doesn't go without saying.

Now, I happen to have sympathy for that position, but it's hardly as straightforward as you seem to imply.

Where it does have a lot to do with Platonism is that positing that relations are not abstract things but physical entities presumably entails that you can show what a relation physically is. And an instance of a relation isn't the relation. As soon as you go into "for example", you're not talking about the object itself anymore.

(To better see that relationship, see: https://plato.stanford.edu/entries/abstract-objects/)

So what kind of physical object is a relation?

It pays to draw a distinction between objects and relations. Relations manifest themselves when more than one object is involved. A pair of objects is an object; it is a physical relation between pairs, the isomorphism if you will, that we call "number 2"; it manifests itself in cases when pairs can physically "snap" together solely due to the fact that their cardinalities match. I think it is the fact that a number (a count, in this case) is not tied to a single object (e.g. a pair) but rather expresses the potential of a certain physical interaction between several objects that makes understanding this difficult.

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