Here are just some thoughts I had on attempts to prove mathematics, can you review it please?

Question

When attempting to formulate a 'proof of mathematics' you look to the simplest aspects of it, and try to prove those. The simplest aspects of mathematics would be numbers and basic functions, but the basic functions can be defined after we define addition. Of numbers, imaginary, complex, irrational, fractional, and large numbers can all be defined by these basic functions, and other numbers, specifically starting with 1 and 0.

So we have addition, 1, and 0 to prove. It might even be possible to hold out on 0.

Does this make sense? In an attempt to prove mathematics, we should start by trying to prove addition and '1', and building on that?

Well of couse you will need to add ideas as you go on, but you need a foudnation to start with.

Answer 1

You are oversimplifying a very complicated subject.

"Math" cannot be proven simply by proving addition, one, and zero. I am sure there are many many counterexamples, but I will just state one that is a direct result of your thoughts.

Negative numbers:
In order to prove that a negative number exists, you must prove there is a number less than 0. If you prove that addition, one, and zero exist, you are essentially proving that all numbers are greater than or equal to 0. Without defining negative numbers, we do not have subtraction and thus we don't have a lot of important concepts in math. So there is one important counterexample to your argument.

Answer 2

in group theory, we call the element "g" generator if we can write all elements of the group as a positive multiple of g or negative multiple of g or 0.

for example the group Z of integeres can be generated by 1:
and element is a finite sum of 1-s or its negative
Even Q can be defined based on 1:
n/m = (1+1+...+1)/(1+1+...+1), where at numerator 1 appears n-times and at denominator 1 appears m-times

With irrational numbers, you cannot generate in this way, maybe you can if you allow infinite sums.
For complex numbers, that is a problem: how do you define
i, the imaginary in terms of i.
Also there are countless objects that are not based on numbers.
how do you define the polynomial x in terms of 1?
how do you define e^x in terms of 1?

You are right that some sets can be generated by 1 but far from truth that 1 has a role in proving mathematics.
On the other hand you noticed the important role that 1 and 0 has in mathematics:
in fields and unitary rings, you always have an 1 and 0 on the same time
They have specila properties: 1 is the only number such that
x*1 =x and 0 is the only number such that x+0=x
and they are used to define the multiplicative and additive inverses. You can't have 1/x if you don't have 1 in first place
. And like the king in chess, they have small moves but they are surprisingly important in game.