Why the function concept is so important in all branch of mathematics,e.g algebra,analysis,applied math,...?

The term "function" has to meanings. One is more of an intuitive (applied) understanding. The other is a more theoretic (pure) understanding.

The applied understanding is basically a function is a formula. For instance, the total money spend for gas going x miles/hour. This provides us was a rule. The input is x and the output is f(x) the result we want to know. One of the goals of applied mathematics is to find functions that solve a specific problem. Because that will provide us with a formula that will describe the process.

The pure understaning is hard to explain elementary. But basically a function between to sets X and Y (sets are just another meaning for "collection") is a rule which sents elements from X into elements of Y. This one of the foundations of pure math. A simple but important concept. So the role of a function in pure mathematics is a type of relationship between two things.

Historically the term "function" was coined and used by Gottfried Wilhelm von Leibniz. But Leibniz used the intuitive approach. The mathematician who first used the formal approach was Lejunne Dirichelt. He

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"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran

That's like asking why the camera is important to the photographer

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In many branches of mathematics ,concept of function appears. It seems to be naturally. But I was searching more recondite reasons for this matter.
Many of applications and also facts deal with mathematics, may expose with
respect to philosophy by philosophers .Time ,motion ,point ,line ,energy, etc are
a few examples .

Above mentioned matters may regard as cause and effect ,matter and form ,and
such dual concept which are important in philosophy . On the other hand a function may express such concepts .

A function usually contains two sets , Domain and co-domain .These are
ordered as first and second . That implies precession of cause. Each member of domain relates to only one member of co-domain that may be similar to effect
uniqueness .

Such ideas may be reason of function importance. Considering function as this point of view can help us to get more insight to this concept .

I appreciate introducing reference books about that.

I am not exactly sure what you are asking because your English is too good for me. But it seems that you want to take the concept of function in math and use it in philosophy. If so this is again one of those pathetic things philosophers do (like I mentioned before on this forum), they take concepts from math - which is pure, and perfect, and corrupt it with the impurity of philosophy and its ugliness in reasoning. I absolutely hate it when a philosopher, in particular one who knows almost no math, thinks he is so smart and so clever to take math and use it in philosophy. But that never-ever works. Because what these philosophers do not understand is how math works. It has rules to follow, very very strict rules that if obeyed properly it always works. But when it is used like philosophers use it then it is absolute garbage which will break down and not work. So if that is your intention then there is not point in doing it.

One of the things I like to do is to watch sub-humans on the internet make thier video respones on math. I do that because I love to see how philosophers (mathematically challenged ones) argue. It is so absolutely pathetic. One of the most common videos are: '.99999.. = 1' and 'is 00 a number'. And they have this idea in this mind as those this is some unsolved math problem.

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"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran

Oh if you think this is a cool question to consider, think about units. There is a distinct relationship between functions and unit systems because, just as functions "connect" two different sets (domains and ranges), but they also "connect" to different units. (That's the Physics connection you were talking about.)

Frankly such considerations have driven me half-mad over the years. (Some would delete the word "half" from my previous statement.)

-Dan

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"I must not fear. Fear is the mind killer. Fear is the little death that brings total obliteration. I will face my fear. I will permit it to pass over me and through me. And when it has gone I will turn the inner eye to see its path. Where the fear has gone there will be nothing. Only I will remain." - The Litany Against Fear, "Dune" by Frank Herbert

I wondered .We owe philosopher mathematicians such as Descart ,Newton and Leibniz to mathematics of today . Philosophy and mathematics both have
powerful own structures and thousands of specialists . So it is not possible
these two branches corrupt each other . On the other hand ,history of both
shows that their relation always have been bidirectional .

In addition today, as I know , philosophy is one of important university courses
in modern countries as U.S.A and European countries and philosophy of science is
teaching as well as other lessons.


My intention is clear .I don't want to take the concept of function in math and use it in philosophy. I mean ,a philosopher who is not necessarily a math. specialist usually survey sciences e.g. mathematics and consider their general facts and rules .He or she gets results from these facts and rules then others study and use these results to improve concepts and definitions. I'd like to provide an example of above mentioned .A philosopher sees function concept in several mathematics branches with almost unique definition, e.g. analysis ,geometry ,algebra, etc. He see function of sets as soon as he opens a probability book too. So he think it is not just a chancy matter. Understanding function definition generally is not so difficult and a philosophy educated can do it. If he is also a math. specialist ,that will be great.Thanks

No, these people were not philosophers when it came to math, they were mathematicians. So mathematicians do not own anything to philosophers.
Let us assume that Descrates was in fact a mathematician. Now he was certainly a philosopher. But when it came to math we was not a philosopher, he did not think like a philosopher, he thought like a mathematician. So philosophly is of completely no importance in math at all, i.e. philosophers do not contribute anything to math.

I am not sure what you mean by that.

I find it just as important as Art History (which to me is not very important).

One thing about philosophers is that they have excellent English. I just cannot follow them. I have to re-read my philosophy text so many times and still hardly understand anything.

So I am not exactly sure what you are saying. The problem with many philosophers is that they no idea how math works. What they should do is pick up a book on math and start learning from it. And not just a basic book. But a book that has proofs in it. In that way a philosopher will understand how math reasoning works.

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"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran

Almost everything that you wrote above is completely wrong.
You ignorance can be put down to your youth.
Did you know that for much of human history there was no difference in being a mathematician and being a philosopher?
Did you know that Descartes and Liebnitz did not know what a mathematician was?
Do you know why the Newton wrote more on the book of Revelations in the Christian Bible by a factor 75% than he did on mathematics?
Do you understand that Bertram Russell, Kurt Godel, A Church, Paul Cohen all considered themselves to be philosophers?

There is no need to express you own weakness.

I am not allowed to argue with you because you are much older than I am. So I am going to try to be careful.

I know this. That the term "scientist" used to be natural philosopher. Thus, it seems that a philosopher was a person who thought about the world, whether naturally (like a scientist), or as a mathematician, or in some other way. Back in those days a thinker in any area was a philosopher. Today a natural philosopher has a name, namely, scientist. Today a mathematical philosopher has a name, namely, a mathematician. Today being an alchemist has a name, namely, a chemist. Thus, even though they were called philosophers they were really mathematicians.


Let me tell you why I hate a lot of philosophy, maybe you will understand me better. I hate the philosophy when the person "reasons" about some strange not-well-defined idea and then believes in what he deduced. For example, in my philosophical textbook there was a "proof" by Immanuel Kant on how he "disproved" the existence of reality. When I looked at that I said to myself "wow, is this person serious, it is such pathetic reasoning". It bothers me because in mathematics the reasoning is so elegant so formal so well-defined but in philosophy (certain areas in philosophy) the reasoning is so pathetic. Especially the Ontological Argument. I find that to be a complete waste of time, because such reasoning cannot be justified so we have no idea what he deduce is true or false. That is the type of philosophy I hate with a love.

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"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran

Did you know that it was Kant who drove the nail into the heart of the Ontological Argument and put it to rest?