|
|
|
|
|
|
by xamuel
4058 days ago
|
|
|
Very doubtful. Within the formalized version of mathematics, some sort of namespace mechanism will probably be necessary to deal with terminology collisions. And certain things commonly thought of as one notion might need to be split up into multiple notions (example: in some contexts, "function" means "set of pairs satisfying the vertical line test"; in other contexts, "function" means "set of pairs satisfying vertical line test, along with designated 'codomain'). But at most, this amounts to changing the wallpaper. The structure itself is sound unless something really shocking happens like a proof that PA is inconsistent or something. Such a shocking event needn't hinge on computer-aided proofs (although computer-aided proofs will surely help to convince mathematicians who would otherwise assume such a startling result must be mistaken). |
|
|
The critical parts in mathematics are the foundations and basics. These have been studied over and over again by many smart people. I think if there is something wrong, it would have been found. If you are familiar with mathematics you will probably agree with this assessment.
And, in the unlikely case that there is an issue, then it will be worked around, so no harm is done to the rest of mathematics. Luckily, in mathematics, we make the world (axioms and definitons) like we wish it to be. There is a famous quote I don't recall right now about mathematics being now in the garden of eden and mathematics shall never be forced to leave it again.
However, non-critical parts of mathematics like advanced research is often plagued by mistakes.
Computer-verified proofs are the necessary future of mathematics.
One little remark, people often assign more attributes to functions than just the mapping-tuples itself and optionally the codomain. Often the arithmetic formulation of the mapping is part of the function because we want to differentiate based on the way the function is written.