From my “watched a YouTube video” understanding of Gödel’s Incompleteness Theorem, a consistent mathematical system cannot prove its own consistency, and any seemingly consistent system could always have a fatal contradiction that invalidates the whole system, and the only way to know would be to find the contradiction.
So if at some point our current system of math gets proven inconsistent, what happens next? Can we tweak just the inconsistent part and have everything else still be valid or would we be forced to rebuild all of math from basic logic?
There is an MO thread about this:
Basically “our mathematical system” for mathematicians usually (though not always) refers to so-called ZFC set theory. This is an extremely powerful theory that goes far beyond what is needed for everyday mathematics, but it straightforwardly encodes most ordinary mathematical theorems and proofs. Some people do have doubts about its consistency. Maybe some inconsistency in fact could turn up, likely in the far-out technical fringes of the theory. If that invalidates some niche areas of set theory but doesn’t affect the more conventional parts of math, then presumably the problem would get fixed up and things would keep going about like before. On the other hand, if the inconsistency went deeper and was harder to escape from, there would be considerable disruption in math.
See Henry Cohn’s answer in the MO thread for the longer take that the above paragraph is cribbed from.
I’m not going to dive in there at the moment, correct me if I’m wrong (the ‘the problem would get fixed up and things would keep going about like before’ case I suppose).
To answer OP’s question, basically the same thing that happened last time an inconsistency was found, Russel’s paradox, which, to massively simplify, was add a new axiom that says you can’t do that and carry on. (Which gave rise to the aforementioned ZFC set theory). Working math is still going to work in any case.
Yes, that’s Frege’s system mentioned in Henry Cohn’s post. But that happened in a very naive time compared with today. So it would be more of a surprise if something like that happened again.
Well the last time that happened (Russels paradox) we just banned people from using sets in a manner that would cause a paradox. Soo, probably something like that
It would be hugely impactful to the high levels of academic math, but I don’t think we’d see any meaningful effects elsewhere. Consistent or not, math works—it performs perfectly for finance, engineering, statistical analysis, and a finite but practically uncountable number of other things. Some abstruse inconsistency won’t suddenly break all that, and if it were discovered we would just keep on using the same “broken” math because it does the job.
Mathematics is just a language to describe patterns we observe in the world. It really is not fundamentally more different from English or Chinese, it is just more precise so there is less ambiguity as to what is actually being claimed, so if someone makes a logical argument with the mathematics, they cannot use vague buzzwords with unclear meaning disallowing it from it actually being tested.
Mathematics just is a language that forces you to have extreme clarity, but it is still ultimately just a language all the same. Its perfect consistency hardly matters. What matters is that you can describe patterns in the world with it and use it to identify those patterns in a particular context. If the language has some sort of inconsistency that disallows it from being useful in a particular context, then you can just construct a different language that is more useful in that context.
It’s of course, preferable that it is more consistent than not so it is applicable to as many contexts as possible without having to change up the language, but absolute perfect pure consistency is not necessarily either.
Realistically, that doesn’t happen. At least not if we consider mathematics to be the useful kind of mathematics. The stuff that helps you do statistics/finances/engineering. Because that type of mathematics is based on reality and for the mathematics to be inconsistent would mean that reality is inconsistent, and then we have much much bigger problems.
As for the higher abstract maths (which is closer to philosophy anyways) yeah such things have happened many times and led to revolutionary insights each time. An example is when we started basing everything on set theory (which in medieval times did not really exist). But that’s a philosophical question, not one that concerns daily life.
