What is "mathematics in programming?"

Some well-known facts in mathematics are difficult to prove using a computer.
For example, the Leibniz series

1 - 1/3 + 1/5 - 1/7 + ...

is exactly equal to π/4, but calculating this series directly is impossible because it requires infinite computation.
However, from the theorems of analysis, this equality can be shown. Thus, if all these theorems are formulated in terms of logic symbols, the proof is reduced to a finite algorithm.

Some might say that a "proof assistant" or "theorem prover" would make this possible.
However, the language of proof assistants is too complicated to describe proofs of non-fundamental theorems, so proof assistants are used only to check proofs that have already been solved.
This is far from doing mathematics.

My question is where do mathematical ideas come from and whether programming can help yield them?