No, LLMs just can't do math.

If your math does not involve multiplying 20 digit numbers, modern LLMs can "do" math even without a Python tool despite the counterintuition of next token prediction.

They can definitely recognize the problem class and build programs to do math. So what's the difference?

It's like saying that people can't turn high torque nuts on machine bolts, because you can't use your fingers to do it. But you can use a wrench, so effectively, we can turn high torque nuts on machine bolts even though it isn't something we can natively do unaided.