Below is a reproduction of my response to someone claiming mathematics is binary in nature, that answers are always either wrong or right, that there is no ambiguity. For context, the claim was made as a comment on a Vlogbrothers YouTube video. The monologue is very to the point albeit inelegant, but hopefully it will prove enjoyable or educational.

This is not far off, I suppose, but it is not entirely true. Take, for instance, the question: Is there a cardinal number strictly between aleph-0 and the cardinality of the continuum? This is not answerable in the canonical axiomatic structure of mathematics; in math lingo it is said to be independent of ZFC. Moreover, the implications of mathematics can often be disputed. If you had told me that Gromov-Witten invariants would be important for science and presented me with the paper introducing them but nothing else, I probably would have said that they would not be, at least not for a long time. But almost immediately they were put to use in physics!And while it is true that proof is absolute in that given the axioms as premises, the conclusion (theorem) must follow due to the validity of the proof, many mathematicians and philosophers still argue about what our premises should be. Are our axioms even consistent? What about completeness? Furthermore, what is interesting is very subjective, and this drives mathematics. Few people still study constructions via algebra these days because it is not deemed all that important except for historical reasons perhaps, but there is still plenty to be done in that field if one so wished to pursue it. Also, what if mathematicians cannot follow a peer’s work? This has been an issue lately with Shinichi Mochizuki’s work on deformation theory (“Inter-Universal Teichmuller theory”).Finally, mathematicians are people, and people make mistakes, and sometimes those mistakes are missed. Nerdighteria’s resident mathematician Daniel Biss is actually best known in the math community for being an incredibly promising, bright, and hard-working student and young mathematician who published several massively influential papers which were later found to be false due to human error. (This is not to say he was a bad mathematician or anything of that sort. I chose him only because he is probably the best known modern example.)Similar gripes could be made with saying any discipline is so decisive. In a way, math shows us that such decisiveness can never occur.

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