How can math prove different dimensions?

Your question: "How can math prove different dimensions?"

Mathematical dimensions are declared, asserted, stated as premise in a system of mathematical reasoning, ad hoc, according to the interests or needs of the mathematician.

They need not be proven.

Further assertions based on some dimensional schema might be provable, but that is way beyond the scope of your question or your ability to frame reasonable questions on this site.

If you have some particular dimensions in mind, you should have stated to what you are referring.

Otherwise there is no reason for this or any other reader to carry out the rest of your mission.

Pure mathematics proves nothing.

mathematics may be used to show that a proposed hypothesis or model is mathematically valid.

Look into string theory.