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I think they mean consensus about which axioms to accept. There could be competing schools of mathematics along those lines, I think there just happens not to b

I think they mean consensus about which axioms to accept. There could be competing schools of mathematics along those lines, I think there just happens not to be.

defrost 3y ago
Hardly 'competing', more like (say) Commodore-64 demo Vs nVidia GPU demo scenes - you pick your arena and push what can be done with constraints (axioms). In mathematics you might compare groups that work with Vs without the Axiom of Choice, you can can have one group satisfied by an existence proof that asserts X exists otherwise a contradiction must exist and another group that only accept existence proofs that demonstrate a means to construct an example of X. The habeas corpus divide :-)
- FieryTransition 3y ago
  But is a Hilbert system easier to run in a computer since there are many axioms which clarifies the cases where contradictions or some inference rules are used which might be hard to program? For example, on the top of my head, in predicate logic, how would you encode an inference rule like exists elimination? It's tricky enough for me to think about, but to be able to encode a computer with it, is harder, and I can't remember if something like this falls under the undecidable problems category or not.
brutusborn 3y ago
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.
- suoduandao2 3y ago
  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.
- rgbgraph 3y ago
  In applied maths, probably not. In theoretical maths, yes.
  - brutusborn 3y ago
    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.
pjacotg 3y ago
If you did write a paper that assumed different axioms to the norm you would just state that you had done so! Because mathematics is axiomatic, mathematicians are happy to play around with axioms as long as it leads to something interesting.

I think they mean consensus about which axioms to accept. There could be competing schools of mathematics along those lines, I think there just happens not to be.

defrost 3y ago

Hardly 'competing', more like (say) Commodore-64 demo Vs nVidia GPU demo scenes - you pick your arena and push what can be done with constraints (axioms). In mathematics you might compare groups that work with Vs without the Axiom of Choice, you can can have one group satisfied by an existence proof that asserts X exists otherwise a contradiction must exist and another group that only accept existence proofs that demonstrate a means to construct an example of X. The habeas corpus divide :-)

FieryTransition 3y ago
But is a Hilbert system easier to run in a computer since there are many axioms which clarifies the cases where contradictions or some inference rules are used which might be hard to program? For example, on the top of my head, in predicate logic, how would you encode an inference rule like exists elimination? It's tricky enough for me to think about, but to be able to encode a computer with it, is harder, and I can't remember if something like this falls under the undecidable problems category or not.

brutusborn 3y ago

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.

suoduandao2 3y ago
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.
rgbgraph 3y ago
In applied maths, probably not. In theoretical maths, yes.
- brutusborn 3y ago
  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.

pjacotg 3y ago

If you did write a paper that assumed different axioms to the norm you would just state that you had done so! Because mathematics is axiomatic, mathematicians are happy to play around with axioms as long as it leads to something interesting.