When does something become a "mathematical object"?

Question

A mathematical object is an abstract object arising in philosophy of mathematics and mathematics.

Abstract object:

Abstract and concrete are classifications that denote whether a term describes an object with a physical referent or one with no physical referents.

Denoting, specifying, describing an "object" (wasn't the abstract object the "object"?).

What is an "object" in this sense? There seems to be no canonical explanation of what makes a mathematical object an "object", whatever that implies.

An object has multiple meanings, and the mathematical one doesn't do the justice of explaining it.

"Math scholars" have told me that "mathematical objects" are used to represent numbers in general, matrices, etc. However, a matrice is considered an "array". Where does this premise reach a concrete definiton if arrays can be in fact numerical?

Why is an "array" considered different than a number, if they can both be the exact same thing?

It seems math is quite shaky in my introspection of varying angles of the subject.

Therefore, I ask this ... why should I take something serious beyond elementary logic participles if it has no clarity, distinction of properties, or common sense?

Answer 1

An object is a thing.

A mathematical object is a thing that arises in mathematics. There are lots of them. Some of them are numbers, some are sets of numbers, some are of sets of other things, some are functions, some are ... .

I'm not sure about what your last sentence is really asking.

Answer 2

All of your criticisms are equally valid when applied to.. well, anything. How does a football coach know what a "formation" is, and whether it really applies to football? How does a software engineer know the difference between a "program" and the instructions executed by a computer? How does a dog know that a "frisbee" is something that you can catch in your mouth? How does a general use little flags to signify troop positions, when they are really just flags?

None of this is to say that these are not interesting questions—I personally find them quite fascinating. But saying that they are reasons not to take something seriously is rather antisocial. If a lover stares into your eyes on a moonlit night and professes his or her adoration, do you start measuring oxytocin concentrations?

I do think that many mathematicians are a bit too attached to the Cantorian or Platonist views, and have incorrectly made mathematics out to be about things which are more than what they are—and that starts many arguments unnecessarily (for example, when someone claims that a theorem is true "in all possible universes", as if that meant anything). In my opinion, topos theory provides a better foundation for mathematics in this sense, because it is easier to understand the relationship between semantics, syntax, and the ever-elusive ontology. One speaks of this topos or that topos (or "topic", if you prefer), and never needs to worry about whether something "is" this or "is" that.

One relatively recent paper which I think has helped advance this more enlightened way of thinking is the quantum mechanics paper (heavily inspired by the philosophical work of Heidegger) What is a Thing?. There it is argued that set theory has not quite succeeded in providing the proper background for interpreting the world as it appears to us. The "state space" of physics professes to arrange possible worlds into a set, and runs headfirst into various paradoxes as we realize that our experimental equipment itself changes what is being measured, blurring our picture of how things really work and necessitating the continual introduction of new concepts and interpretations.

In short: perhaps truth, in the pragmatic sense, is more sheaf-like than set-like. But I digress.

If anybody tells you that you should take math seriously because it has figured out, once and for all, the correct way to divide the abstract from the concrete, and has firmly established the foundations for rational thought, then they are too caught up in their subject and you really shouldn't pay attention to them. And, if you really want, you can simply walk away, shaking your head in disappointment that mathematicians have failed to live up to their promise.

But, however seriously you take it, mathematics remains a powerful force in the world. While we're not particularly better than anybody else at explaining what we're talking about, what we are good at is bringing disparate things together under the same semantical umbrella—to a large extent, precisely because we are given the freedom not to explain ourselves. Measure theory, for example, has allowed us to shuttle insights between discrete phenomena and continuous phenomena. Algebra has, for hundreds of years, improved our speed of numerical reasoning by a billion-fold, by knowing when to compute and when to encode. Algebraic geometry has provided a language that is equally at home with basic arithmetic, encryption, signal processing, causality, and phylogenetic trees. And so people keep finding it useful, however many students will stand up angrily in our classes and insist that they don't think it could possibly be useful because something something.

In short, mathematics saves time for certain kinds of projects. If you don't do any of those projects, then of course you don't need to take it seriously. But it's under no obligation to explain itself, particularly not to somebody who thinks he is entitled to answers and "justice". If you find the foundations lacking, then we would love for you to come make a career of improving those foundations. If you are mostly complaining however, then pardon us while we focus on our other students.