Is function a mathematical object?

Question

From Euler's Analysis book he gave the definition "A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities". When i was in highschool the teacher taught me that a function is a rule that we apply to number and get other number. Now when I read a modern analysis book, it tell that a function is a subset of $A \times B$ where A is domain and B is codomain sastifying if (a,b)=(a,d) then c=d . All of that refer to me as mathematician are trying to describe a mathematical object rigorously.

For example the equation $x^2 -4x +3=0$ has two roots $x=1, x=3$, it has nice property that when you know these roots, the term $x^2-4x+3$ could be represent as $(x-3)(x-1)$ but without be put in the equation $x^2 -4x +3=0$ the term $x^2-4x+3$ alone seem meaningless. Another example is consider the statement for two number $6^n -1$ and $6^n +1$ for all interger $n$ we put in at least one number we get is prime, it is not true and we can disprove that if we let $n$ be the form $2k$ where k is interger then $6^(2k) -1$ would be factored out as multiple of two intergers not 1 and itself. But how can we exactly tell what the "form" is? For the form $2k$ for each $k =1,2,3,...$ we get $2k =2,4,6,...$ and for each x is real number we get a number $f(x)=x^2-4x+3$ by the rule f, then we can write $f(x)=(x-3)(x-1)$. As mathematic developed it is generalized to other branches like we could consider $f(A)=A^2 -4A+3I$ where A and I are square matrix,also we get f(A)=(A-3I)*(A-I).

So the notion function is just mathemacian trying to describe an object after that all "factorization of roots of polynomial" or "all polynomials of degree n has n complex roots" things will make sense. Is it a mathematical object in the same sense of circle,line,quadrilateral,.. all of that plane figures which are favour of mathematicians back in time?.If so, in modern day people tend to study function and all that figures could be describe as a function. In higher mathematics, Do we study complicated mathematical objects and all that things are defined as a function?

Answer 1

When i was in highschool the teacher taught me that a function is a rule that we apply to number and get other number.

That's how we teach at schools. It's not necessarily bad, it is just a simplification. Plus you have to be careful about dosages, young mind won't necessarily be able to comprehend the abstract nature of true math.

The most important thing you need to know is: function and formula are different things.

Function is just a special set:

Now when I read a modern analysis book, it tell that a function is a subset of $A \times B$ where A is domain and B is codomain sastifying if (a,b)=(a,d) then c=d .

That's how one would approach it in Zermelo-Fraenkel (ZF) set theory. But that's not necessarily the only way. It is standard though. I will talk about formulas later.

without be put in the equation $x^2 -4x +3=0$ the term $x^2-4x+3$ alone seem meaningless.

It isn't meaningless. It may represent a polynomial, depending on context. Maybe there isn't much information in the example above, but it definitely is not meaningless. Context is important.

Is it a mathematical object in the same sense of circle,line,quadrilateral,.. all of that plane figures which are favour of mathematicians back in time?

Yes, it is a mathematical object. And no, not in the same sense as circles, line, etc. Why do you think that mathematical objects have to be geometric figures?

In higher mathematics, Do we study complicated mathematical objects and all that things are defined as a function?

Of course not every object is a function. Most things are sets. Functions are sets, complicated mathematical objects like rings, modules, schemes are sets. But there are things that are not sets, like: collection of all sets. Or linguistic objects, like formulas.

But indeed, the typical setup is to have "objects" (e.g. sets) and "arrows" (e.g. functions) between them. And typically at least being able to compose arrows is a must have. This observation lead to the development of the category theory.

But how can we exactly tell what the "form" is?

A mathematical expression is something that arises from language that we use. It is a finite combination of symbols that satisfy some semantic rules. It can be formally defined through lambda calculus for example. You may want to read this for more details: https://en.wikipedia.org/wiki/Expression_(mathematics)

Thanks to this approach given an expression we indeed can plug it into different contexts, which otherwise wouldn't be possible. For example if we have a function $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^2$ it is a correct observation that we can extend it to a function of matrices $F:M_n(\mathbb{R})\to M_n(\mathbb{R})$ with the same formula $F(x)=x^2$. That's because the "$x^2$" expression passes semantic rules in both worlds.

It seems contradictory to what we claimed earlier that $f$ is merely a subset of $\mathbb{R}\times\mathbb{R}$, and so where is the formula encoded? Well, a function itself does not carry a formula. At the moment you construct a function, formula is gone. Sometimes we can recover it, if we know enough about the function (e.g. linear homomorphisms between finite dimensional spaces are particularly simple examples), but for an arbitrary function this is not possible. We have to look at it from another angle: formula is a separate object and it induces a function $\mathbb{R}\to\mathbb{R}$ and a function $M_n(\mathbb{R})\to M_n(\mathbb{R})$ as well. So formula exists, but is somewhat separated from function itself. And a formula may generate a function, but once we do this some information is lost (unless we decide to keep it around of course).

Answer 2

If I understand what you're asking now, the short answer to your question is "yes." I think what is worth saying is that a function is the "minimal" kind of relation between objects that let's us understand consistent behavior. That is, if I know I start with $x$, then I know I will land on $y$. Without this consistency, it makes it very hard to make any conclusions about the behavior of our relationship. Maybe you have a relationship that is not a function in your context, something like $x=y^2$, so $(1,1)$ and $(1,-1)$. Now, I will say "you can't make any meaningful relationship between these numbers", and you will say, "well, I can actually tell that this does relate things to the positive and negative values, there is a relationship between the inputs and the distinct outputs", to which I will say, "you just extended this relationship so that instead of sending a number to a number, you send a number to its distinct square roots, $(1,\{-1,1\})$, which is now a function, just not from real numbers to real numbers, but from real numbers to the set of pairs of real numbers, which is not a function merely between 2 numbers." So, really, $x=y^2$ -is- a function, in a certain context, albeit not the one you were initially concerned with. The point being made here is that functions do not only exist from real numbers to real numbers, or even complex numbers, but from potentially any set to any set. So, to understand our objects in any context, it becomes helpful to understand what kind of "nice" functions can exist between them. What nice means depends on what category you're in, if you're in abstract algebra, a nice function should preserve operations so you can say meaningful things about its behavior on your objects (what is nice for a group may not be nice enough for a ring). If you're in analysis, preserving operations is nearly useless, and it takes a very different kind of "niceness" to say meaningful things. Even better, topology is the study of what properties of a space are preserved under continuous functions, where this definition of continuous exists as an abstraction of analysis's continuous on metric spaces, which is an abstraction of continuity under Euclidean space and real-world perceptions. In this way, you almost have two approaches of coming up with topology, either you ask "what properties do spaces preserve under certain functions?" or "how can I define a space so that it keeps certain properties under certain functions?"

In your example, in the xy-place, those objects you describe are certainly not functions in the algebra 2 sense, but you could easily parameterize those objects and trace them out in $\mathbb{R}^2$ in a continuous way. They are the image of continuous functions, just not in the same sense (or, space) that your $f(x)$ is. A circle can be seen as a function from $[0,2\pi]$ to $\mathbb{R}^2$, maybe given by $f(x)=(\cos(x),\sin(x))$. So, even when they are not nice in one context (from $\mathbb{R}$ to $\mathbb{R}$), they can be represented as continuous functions from $\mathbb{R}$ to $\mathbb{R}^2$. Now talking about "roots" of this function hardly even makes sense, yet it is still a very nice and reasonable function in most contexts.