mathematicians from Pythagoras through Dedekind had absolutely no problem talking about numbers in the absence of the Peano axioms.

So, they were lucky. It could have been that that-which-Pythagoras-calls-number was not that-which-Fibonacci-calls-numbers.

Are there absolutely no examples of cases where mathematicians disagreed about some theorem about some not-yet-axiomatized subject, and then it turns out the disagreement was because they were actually talking about different things?

(I know of no such examples, but I would be surprised it none exist)

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Consider also the parallel 'postulate', reluctantly introduced as an axiom in Euclid. People tried to prove it as a theorem for two thousand years, until it was realized that its negation defined entirely different kinds of geometry.

0[anonymous]7yVarious theorems, lemmas, and other principles equivalent to the Axiom of Choice (e.g. Zorn's lemma) were argued over until it was established (by Kurt Gödel and Paul Cohen) that the AoC is entirely independent of the ZF axioms, i.e. ZFC and ZF!C are both consistent systems. I think this is the canonical example. "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona
1magfrump7yA not-quite-but-close fit: the distinction between prime and irreducible elements of a number field became necessary because unique factorization into primes failed in simple number fields.

[LINK] Steven Landsburg "Accounting for Numbers" - response to EY's "Logical Pinpointing"

by David_Gerard 1 min read14th Nov 201247 comments


"I started to post a comment, but it got long enough that I’ve turned my comment into a blog post."

So the study of second-order consequences is not logic at all; to tease out all the second-order consequences of your second-order axioms, you need to confront not just the forms of sentences but their meanings. In other words, you have to understand meanings before you can carry out the operation of inference. But Yudkowsky is trying to derive meaning from the operation of inference, which won’t work because in second-order logic, meaning comes first.

... it’s important to recognize that Yudkowsky has “solved” the problem of accounting for numbers only by reducing it to the problem of accounting for sets — except that he hasn’t even done that, because his reduction relies on pretending that second order logic is logic.