Since mathematicians have embraced classical logic, as e.g. MacFarlane points out in his 2021 intro book to philosophical logic (§ 7.4), one needs to distinguish between [meta-]reasoning and argument/inference in mathematics in the following obvious sense: if a mathematician derives as contradiction e.g. Russel's paradox, they do no happily proceed (by EFQ) to conclude that every statement insofar thought as false is actually true. Instead they revise their axioms (as e.g. happened with the axiom of comprehension in the process of formalizing ZFC). MacFarlane (since it's an intro textbook) exemplifies this in-extension with something more every-day rather than mathematical thought, but the idea is roughly the same:

Let’s grant that no mathematician would react to the discovery that she had accepted inconsistent premises by concluding (say) that there are infinitely many twin primes. [...]

We can distinguish between two things that might be meant by ‘inference’:

  • Reasoning: Inference in the broad sense is “reasoned change in view:” revision of beliefs in light of new information or reflection.

  • Argument: Inference in the narrow sense is a process of drawing out the consequences of a given set of premises, in isolation from one’s other beliefs.

Reasoning is belief revision. It can involve both additions and subtractions to one’s body of beliefs. Argument is what is modeled by practices of formal proof. It is monotonic: adding premises cannot spoil conclusions we already have. [...]

For example, it’s a confusion to think that Modus Ponens is a rule of Reasoning. Once on a train I heard a young boy exclaim, with some alarm,

(5) I have no pulse!

The boy may have believed

(6) If I have no pulse, I am dead.

But he would obviously not be reasoning well if he applied Modus Ponens to (6) and (5) and came to believe

(7) I am dead.

Dead people cannot check their own pulses. He should instead reexamine his beliefs (6) and (5), and give up one (or both) of them. Modus Ponens, then, is not a norm for correct Reasoning. It is a rule of Argument.

MacFarlane then proceeds to discuss Harman's 1984 paper on the matter... which is basically full of negative results, i.e. the rejection of simple theories of how Reasoning and Argument might fit together (I won't get into the details here, because it's a somewhat long discussion.) So MacFarlane's section on the matter basically concludes with a negative result (or "future work" question) rather than any plausible answers:

Perhaps there is some connection between logical implication/entailment and belief revision, but it is not at all obvious.

So, are there some "workable" (philosophical) theories of how belief/axiom revision actually proceeds in mathematics?

  • I would also like to know the answer to this. You're asking what the innate logic of the human brain is. The brain's logic is fuzzy, non-monotonic, heavily reliant on heuristics. It evolved to help us select the right actions to propagate our genes, and thus is heavily goal-oriented. At a very vague high level we might represent the brain-state using a very large numeric vector, with some sort of update equations for how this vector changes over time. But the specifics of the update equations are largely a mystery. (Well, we do know something about neurons)
    – causative
    Apr 13 at 1:35
  • @causative: oh, I'm not asking in that absolute certainty sense. As many-a-position in philosophy have been argued for and against, by "workable" I simply mean some ideas that have also been argued for, instead of someone just publishing a work in which they say "here are some simple/putative ways [of addressing this] that are easy to refute."
    – Fizz
    Apr 13 at 1:50
  • 1
    Compared to most other sciences, pure math (ie, theorem proving) most relies on deductive system already, if any paradox found, naturally most likely it's the axiom(s) as definitions in disguise has problems. So mathematician use something like Harvey Friedman's "reverse mathematics" (en.wikipedia.org/wiki/Reverse_mathematics), this may count as one of workable philosophical theories for mathematicians to revise their beliefs/axioms... Apr 13 at 2:41
  • A mathematician's thinking involves a lot more heuristic and empirical thinking than simply deducing from axioms. Often mathematical investigation begins by looking at a bunch of examples and trying to guess some general law. Other times one will use mental pictures and tell himself different "stories" that he tries to use to understand parts of his subject. When a mathematician applies deductive inference, he does so according to heuristics we cannot teach to computers yet. It is only the final step - the writing of the proof - that is deductive.
    – causative
    Apr 13 at 3:26
  • 1
    May I suggest that axiom revision in mathematics is typically driven by expanding scope and problem solving capacity rather than by resolving paradoxes, and the latter play a minor role. See Mehrtens, T. S. Kuhn's theories and mathematics on applying the framework of paradigm changes to mathematics. Lakatos's Proofs and Refutations is a classical case study that fuses Kuhn and Popper, and chapters 7-8 of Corfield's Towards a Philosophy of Real Mathematics are a critical elaboration on it.
    – Conifold
    Apr 13 at 4:23

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