Is it possible to build math backwards?

[daniel_i_l]

In many areas of math doing something "backwards" is much harder that doing it "forwards", for example - inverting a matrix is harder than multiplying it, taking a derivative is easier that an integral...
But why should this be, I mean, there's nothing fundamental in the way that we happened to set things up. Is it possible to "build math backwards" so that all the things that are hard fo us now are easy and vice versa? If not then why not?
Thanks.

 

[HallsofIvy]

I think you are talking about the "inverse" problem. A "direct" problem, means that we are given a specific forumla, such as "f(x)= x⁶- 2x⁵+ 3x+ 4" and asked to evaluate f(2). An "inverse" problem is the other way: "If f(x)= 10, what is x" does not give us a formula. It is more difficult for several reasons: there may be more than one solution (if you did the original problem one of them is easy!), for some polynomials there may be no (real) solution, and, in fact, it may not be possible to write a solution in terms of roots or other standard ways of writing numbers.

 

[symbolipoint]

In many areas of math doing something "backwards" is much harder that doing it "forwards", for example - inverting a matrix is harder than multiplying it, taking a derivative is easier that an integral...
But why should this be, I mean, there's nothing fundamental in the way that we happened to set things up. Is it possible to "build math backwards" so that all the things that are hard fo us now are easy and vice versa? If not then why not?
Thanks.

That question reminds one of the arithmetic axioms, which rely ONLY on addition and subtraction. They DO NOT specifically include subtraction or division.

 

[HallsofIvy]

I think you meant "rely ONLY on addition and multiplication".

 

[uart]

But why should this be, I mean, there's nothing fundamental in the way that we happened to set things up

Yes quite possibly so. Imagine for example that (in some perpendicular universe) things were reversed and integration just happened to be substantially easier than differentiation. Then it's highly likely that integration would then be taught prior to differentiation and differentiation would commonly be thought of as "anti-integration" rather than the other way around. See what's happened, even when we've reversed the relative difficulties your "inverse processes seem harder" observation still holds.

 

[symbolipoint]

I think you meant "rely ONLY on addition and multiplication".

You are exactly correct; I did mean "Addition and Multiplication". The laws of arithmetic which we learn formally during the first two years of "high school algebra" rely only on addition and multiplication.