When does something become a "mathematical object"?

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.

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.

Object

The term object in mathematics is a loose term. In most every case, it's something you can act on or do something to. They are the "nouns" in the language of mathematics.

A good rule of thumb, is a mathematical object is something you would label. The point $p$, the line $AB$, the number $n$, the function $f$, the matrix $A$, the group $G$, the manifold $M$, ...

Not everything is an object. When you say $2 \cdot 3 = 6$, the multiplication isn't really an object. Here it's more like an action - a verb. However the functions $f(x) = 2\cdot x$ or $g(x,y)=x\cdot y$ which encapsulate the same multiplication would be considered objects. It's a matter of context or perspective. In the first example of multiplication, you can't really do anything to the $\cdot$ operator. We can do things to the functions. We can add them: $f(x)+g(x,y)=2x+xy$, we can multiply them: $f(x)\cdot g(x,y)=2x^2y$, we can raise one to power of the other: $g(x,y)^{f(x)}=(xy)^{2x}$. Note that $g(2,3)=2\cdot 3=6$ is just like the first example, but encapsulating the multiplication (verb) in a function (noun) allowed us to treat it as an object and perform actions on it.

Representation

It is not accurate to say that all mathematical objects represent numbers. While there are many that do, and there are many that behave the same ways numbers do, there are many objects that don't represent anything numerical. The group $D_{2\cdot 3}$ is a set of elements $\{i,r,r^2,f,fr,fr^2\}$. (A group is a set whose elements relate to each other in special ways with respect to an operation.) This group is not representing a number, but more to the point, it's elements do not represent numbers. They are the rotational and flip symmetries of an equilateral triangle. Those flips and rotations don't represent numbers. However, we do need a way to represent and use the operation that makes this set a group. We could define any operation we like, but it's convenient to repurpose multiplication to represent that operation. Thus we could say $f$ times $r$ is $fr$. This isn't the same multiplication used with numbers (you're not going to start carrying digits), but it behaves similarly, so it's easy to use the same way.

It's also not accurate to say matrices represent numbers. For one thing, with numbers $a$ and $b$, $ab=ba$, but with matrices $A$ and $B$, it's usually the case that $AB \ne BA$. There are no real numbers that these matrices could be representing, because you can always change the order of multiplication with real numbers. Oftentimes matrices will represent processes. If you have vectors $b$ and $x$ and a matrix $A$, then in the equation $b=Ax$, $x$ could represent the locations of all the parts of a bridge before a force was applied, $A$ could represent the application of the force, and $b$ could represent the locations of all the parts after the force was applied. In another context, someone else might use a different equation $bx^T \approx A$ where the matrix $A$ represents some data that was taken (a picture, or a fingerprint) and $x$ and $b$ would be the compressed version that get stored in the computer (this would be a very poor compression unless there were many more vectors on the left hand side). In both of these examples, the objects did not represent numbers, but data (sets of numbers) and a process. The objects had numbers inside of them, and that is the most common way these objects are used, but just as with the group of symmetries of a triangle, vectors and matrices can contain elements that are not numbers in any way.

Array vs Matrix vs Number

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?

I will show you an example of matrices that will hopefully clarify. Let's multiply

$$A = \left[ \begin{array}{cc} 3 & 8 \\ 7 & 4 \\ 5 & 6 \end{array} \right], \text{ and } x = \left[ \begin{array}{c} 1 \\ 2 \end{array} \right].$$

A vector is a matrix with a width of 1 column, or a height of 1 row.

$$Ax = \left[ \begin{array}{cc} 3 & 8 \\ 7 & 4 \\ 5 & 6 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ 2 \end{array} \right] = \left[ \begin{array}{c} 3 \cdot 1 + 8 \cdot 2 \\ 7 \cdot 1 + 4 \cdot 2 \\ 5 \cdot 1 + 6 \cdot 2 \\ \end{array} \right] = \left[ \begin{array}{c} 18 \\ 15 \\ 17 \end{array} \right]$$

When we multiply matrices (and vectors), the dimensions matter. Note in this case:

$$(3 \text{ x } 2)\cdot (2 \text{ x } 1) = (3 \text{ x } 1).$$

The dimensions on the inside of the product have to match, and they disappear, and the dimensions on the outside become the dimensions of the result. It always matches this pattern:

$$(m \text{ x } n)\cdot (n \text{ x } p) = (m \text{ x } p).$$

Now consider this matrix product:

$$\left[ 2 \right] \cdot \left[ \begin{array}{cc} 3 & 8 \\ 7 & 4 \\ 5 & 6 \end{array} \right] = ???$$

I can't do that. The multiplication isn't defined because the dimensions of the first matrix and the second matrix don't match on the inside of the product:

$$(1 \text{ x } 1) \cdot (3 \text{ x } 2) = ???$$

However, I can do a different type of multiplication and multiply numbers times matrices:

$$2 \cdot \left[ \begin{array}{cc} 3 & 8 \\ 7 & 4 \\ 5 & 6 \end{array} \right] = \left[ \begin{array}{cc} 2 \cdot 3 & 2 \cdot 8 \\ 2 \cdot 7 & 2 \cdot 4 \\ 2 \cdot 5 & 2 \cdot 6 \end{array} \right] = \left[ \begin{array}{cc} 6 & 16 \\ 14 & 8 \\ 10 & 12 \end{array} \right].$$

I don't have to worry about the dimensions, because a number doesn't have dimensions. To further illustrate the difference between numbers and matrices, this is the matrix product that would be the equivalent to multiplying by the number $2$ as above:

$$\left[ \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array} \right] \cdot \left[ \begin{array}{cc} 3 & 8 \\ 7 & 4 \\ 5 & 6 \end{array} \right] = \left[ \begin{array}{cc} 2 \cdot 3 + 0 \cdot 7 + 0 \cdot 5 & 2 \cdot 8 + 0 \cdot 4 + 0 \cdot 6 \\ 0 \cdot 3 + 2 \cdot 7 + 0 \cdot 5 & 0 \cdot 8 + 2 \cdot 4 + 0 \cdot 6 \\ 0 \cdot 3 + 0 \cdot 7 + 2 \cdot 5 & 0 \cdot 8 + 0 \cdot 4 + 2 \cdot 6 \end{array} \right] = \left[ \begin{array}{cc} 6 & 16 \\ 14 & 8 \\ 10 & 12 \end{array} \right].$$

This shows that numbers and matrices are NOT the same: $2 \ne [2]$. Similarly, a number and the set that contains that number as an element are not the same: $3 \ne \{3\}$. These two sets can be equal: $\{b\}=\{3\}$ and these two numbers (which are elements of the sets) can be equal: $b=3$. Matrices, numbers, sets, and elements of sets are all different objects, and an object can only ever be exactly equal to another object of the same type.

Philosophy

Mathematics is exactly the opposite of shaky. There are difficult concepts, that can take some time to fully understand, but the foundations of mathematics are very concrete and precise. Some things, like the word object are not rigorously defined because the meaning varies so much from one context to the next. However, it's not strictly important what the meaning of that word is. There is no mathematics that is based on the meaning of that word. It's an informal term, that allow mathematicians to convey a general notion. Things that do matter are always clearly and precisely defined. That doesn't mean that they're all immediately clear to you (or me) or anyone else that hasn't been taught the particular definitions and concepts and notations required to precisely interpret the meaning. One thing that will serve you well is to make sure your understanding of something is crystal clear. If you think you understand something, keep studying it and asking questions until you know you understand it. One thing that has a huge impact on your ability to make sense of things is who is teaching them to you, and how they're teaching them. Keep at it, and keep asking questions. :-)