Timeline for answer to How might mathematics have been different? by JF Meier
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| when toggle format | what | by | license | comment | |
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| Mar 31 at 23:12 | comment | added | Robert Furber | There's a difference between type I computability (that works by acting on numbers coding computable reals, functions between reals etc) and type II computability (that computes convergent approximations even when given a non-computable real as an argument). In fact Type I and Type II are not the only types, there are many notions of computability for functions in general (the regular user Andrej Bauer knows a lot about this) which would seem to necessitate a general notion that didn't use computability to start with. | |
| Jul 10, 2025 at 13:00 | comment | added | Simon Henry | I don't think there is a real difference between logical rules and axioms ( for example, some system like type theory don't have this distinction at all), so I'm not sure I agree with the begining of your sentence (unless when you say that constructivism is a philosophical paradigm, you are refering to the fact that there are various flavour of constructivism). In any case my point can be phrased without talking about axioms: you can talk about "computable" things in classical ZFC set theory and you can work in a constructive system without having the assumption that everything is computable. | |
| Jul 10, 2025 at 12:14 | comment | added | Timothy Chow | @FedorPetrov A plausible axiomatic system for "computable mathematics" is $\mathsf{RCA}_0$, which is based on classical logic (in particular, the law of the excluded middle), and therefore different from, say, Heyting arithmetic, which is arguably closer to what people mean by "constructivism." | |
| Jul 10, 2025 at 10:02 | history | edited | JF Meier | CC BY-SA 4.0 |
added 2 characters in body
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| Jul 10, 2025 at 9:49 | comment | added | Fedor Petrov | @SimonHenry but philosophical paradigm (like constructivism) precedes an axiomatic system. Thus, if we were thinking about reals as of computable reals, the mainstream axioms also could likely be different. | |
| Jul 10, 2025 at 8:08 | comment | added | Simon Henry | @FedorPetrov Not really. One is about dropping axioms like AC or excluded middle, the other about changing what you think is the important object of studies from the set of reals number to the set of computable real numbers. The only link between the two is that without excluded middle the statement "every real is computable become consistent". But you can believe excluded middle holds and that computable real and computable functions are more important that general ones, or not believe in excluded middle and still think that non-computable real exists and are important. | |
| Jul 10, 2025 at 7:34 | comment | added | Fedor Petrov | Is not it less or more the same alternative as constructivism? | |
| S Jul 10, 2025 at 6:48 | history | answered | JF Meier | CC BY-SA 4.0 | |
| S Jul 10, 2025 at 6:48 | history | made wiki | Post Made Community Wiki by JF Meier |