# the word ' function'

#### apple2357

##### Full Member
"the word ‘function’ probably expresses the most important idea in the whole history of mathematics,…”

This comes from a book called ' Mathematics and the Imagination' by Kasner and Newman.

It is quite a claim. Do others agree?

#### Powerful Wizard IRL

##### New member
I don't know about the MOST important, but the idea of a function is absolutely fundamental to all of mathematics.

A function is a "black box" which accepts an input at one end and supplies an output at the other end, and is defined by the relationship between the output and the input. Or in other words, the function performs operations on the input to "transform" it into the output. So a function is essentially a set of mathematical operations to be performed in a certain order.

Take for example the simplest possible function $$f(x) = 1$$. This describes an operation where we input any number $$x$$, and receive the constant $$1$$ as an output. The mathematical operation performed by this function is to multiply $$x$$ by its reciprocal $$x^{-1}$$.

Anyway I'm just kinda rambling here, but in general I agree. Mathematics as we know it would not exist without the concept of functions.

#### Dr.Peterson

##### Elite Member
"the word ‘function’ probably expresses the most important idea in the whole history of mathematics,…”

This comes from a book called ' Mathematics and the Imagination' by Kasner and Newman.

It is quite a claim. Do others agree?
I think there are a lot of other candidates. "Number" could be one ... and "proof" might be another.

And, of course, the explicit concept of function came after a lot of history had already passed.

But "function" is fundamental at least to calculus as we know it (because without it, you can't even talk about what differentiation acts on and produces). And a lot of other mathematics followed. So I'm not sure about "the whole history of", but I can agree with "modern".

#### JeffM

##### Elite Member
The word “probably” in that quotation actually means “arguably.” In any case, it makes no assertion of certainty.

#### apple2357

##### Full Member
Yes good thoughts..

I'd agree - Number ( “God made the integers, all else is the work of man” Leopold Kroneker (1823-1891)) is up there. As is proof. Pattern? Generality? Are other thoughts

#### HallsofIvy

##### Elite Member
I would argue for "set".

#### Jomo

##### Elite Member
Take for example the simplest possible function $$f(x) = 1$$. This describes an operation where we input any number $$x$$, and receive the constant $$1$$ as an output. The mathematical operation performed by this function is to multiply $$x$$ by its reciprocal $$x^{-1}$$.
Be careful there! The way you defined f(x) =1 as f(x) =x*x-1 is not defined at x=0 while f(x) = 1 is.

#### HallsofIvy

##### Elite Member
Please say more!
One can argue that the "set" concept precedes that of "number" as illustrated by Peano's axioms,
"The whole numbers consist of a set of objects, N, called "numbers", together with a "Successor function", S, from N to N, satisfying these axioms.
1. There exist a unique member of N, called "0", that is NOT the image, under S, of any other number in N.
2. If a subset, M, of N contains 0 and, for any n in M, S(n) is also in M them M= N. (induction axiom)."

• apple2357 and JeffM

#### JeffM

##### Elite Member
One can argue that the "set" concept precedes that of "number" as illustrated by Peano's axioms,
"The whole numbers consist of a set of objects, N, called "numbers", together with a "Successor function", S, from N to N, satisfying these axioms.
1. There exist a unique member of N, called "0", that is NOT the image, under S, of any other number in N.
2. If a subset, M, of N contains 0 and, for any n in M, S(n) is also in M them M= N. (induction axiom)."
This suggests that set and function are of equal logical importance.

#### Dr.Peterson

##### Elite Member
This suggests that set and function are of equal logical importance.
Of course, functions are defined (a) on sets (domain and range) and (b) as sets (of ordered pairs). So sets logically come before functions.

Of course, the original question was about "the most important idea in the whole history of mathematics", which doesn't necessarily mean most fundamental. If we're looking for the idea that produced the most wide-ranging effect, one candidate would be symbolic notation! It didn't take long from the invention of x and y to the invention of calculus, and I think there may be a direct connection. Then, as I see it, the idea of function became necessary to explain what in the world calculus was about (deriving new functions from old), and that led to further new ideas.

#### JeffM

##### Elite Member
Of course, functions are defined (a) on sets (domain and range) and (b) as sets (of ordered pairs). So sets logically come before functions.

Of course, the original question was about "the most important idea in the whole history of mathematics", which doesn't necessarily mean most fundamental. If we're looking for the idea that produced the most wide-ranging effect, one candidate would be symbolic notation! It didn't take long from the invention of x and y to the invention of calculus, and I think there may be a direct connection. Then, as I see it, the idea of function became necessary to explain what in the world calculus was about (deriving new functions from old), and that led to further new ideas.
I obviously agree that functions are defined on sets and represent sets, but, in the context of what I wrote, namely the Peano postulates, you can go nowhere without the succesor function.

I agree as well with the importance of literal symbolism: the amount of math developed in the century after Viete read Diophantus exceeded everything done in the preceding millenium.

However, I was speaking a bit tongue in cheek. I also agree with you that logically prior does not mean more important in any general sense. To me, the most important idea in mathematics is quantity.

#### jonah2.0

##### Full Member
Beer soaked recall follows.
... If we're looking for the idea that produced the most wide-ranging effect, one candidate would be symbolic notation! It didn't take long from the invention of x and y to the invention of calculus, and I think there may be a direct connection. Then, as I see it, the idea of function became necessary to explain what in the world calculus was about (deriving new functions from old), and that led to further new ideas.
Another candidate (a necessary development for symbolic notation to have taken place) would be the positional system of numeration; one might say it's the collective unconscious' expression of an elusive intuitive concept expressed explicitly. 