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Ah that makes sense, but do they really matter? If I want to use calculus to design the thickness of a bridge, does the selection of fundamental axioms matter at all?

Don’t all roads lead to Rome? It’s like an engineer deciding to use SI or Imperial units, people have their opinions but at the end of the day both work.

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I think in theory, there could be very different axioms that both have so many open questions that competing schools vie for the best talent to explore 'their' space. Whether that's a situation likely to ever occur I have no idea.

In applied maths, probably not.

In theoretical maths, yes.

I may just be a biased engineer, but until you find a way to apply your theoretical maths, surely it doesn’t matter if there are theoretical differences between approaches?

In the same way some heterodox political philosophies may result in new moral systems, but it doesn’t really matter that these moral systems are different to the status quo, until someone uses it as an ideology for their revolution.

It’s probably good to have a variety of theory to choose from, except when your different theories result in different practical outcomes.

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