The mathematics we use today is based on the decimal system. I was wondering if some or all of these proofs would hold true in other numeral systems. For example binary or hexadecimal. Is there any field which works with questions like this?
With rare exception, mathematics works with numbers, not numerals, so the choice of numeral system you use to represent numbers is completely irrelevant to the mathematics.
The typical exceptions are things that explicitly reference numerals; e.g. elementary school algorithms for arithmetic of numbers expressed in a numeral form.