4 ms·
Here's what looks like the original paper; it's a mere 3 pages long: https://arxiv.org/abs/2002.01653 https://arxiv.org/abs/2002.01653 It's a lot of prose wit
by xelxebar 7y ago
Here's what looks like the original paper; it's a mere 3 pages long:
https://arxiv.org/abs/2002.01653 https://arxiv.org/abs/2002.01653
It's a lot of prose with hand-wavy analogies. My takeaway is that the author seems to have become enamored with intuitionistic logic but seems to lack much concrete experience working with it.
For anyone intrigued by constructive mathematics, there's a nice talk and paper by Andrej Bauer (nice coincidence) called "Five Stages of Accepting Constructive Mathematics." It's a nice mix of prose and rigor of varying levels:
http://math.andrej.com/2016/10/10/five-stages-of-accepting-constructive-mathematics/ http://math.andrej.com/2016/10/10/five-stages-of-accepting-c...
The metamath[0] proof verifier also has a database of theorems on intuitionistic logic:
http://us.metamath.org/ileuni/mmil.html http://us.metamath.org/ileuni/mmil.html
It can be neat to compare proofs and theorems there with their counterparts in the classical logic database.
[0]:http://us.metamath.org/ http://us.metamath.org/
- mirimir 7y agoThanks. I gave up about 20% in, thinking that whatever it is, it's either BS or over my head. Which is where much cosmology and theoretical physics leaves me. And yes, that probably says more about me than them ;) But whatever, I rather think that subjective time involves movement at the speed of light through some n-dimensional manifold. But maybe that just reflects the SF that I've read. Such as Stephenson and Egan.
- craftinator 7y agoHah, I came here to say much the same thing about the flow of time, much for the same reason (Stephenson, Egan, Asimov).
- mirimir 7y agoAh yes, Asimov. It's been decades since I read his work. What would you recommend?
- fpoling 7y agoThe article argues for intuitionistic mathematics, not constructive mathematics. This is important. Just as classical mathematics is constructive math plus some arbitrary unprovable assumption (law of excluding the middle), intuitionistic mathematics is constructive math with another unprovable assumption (the existence of the choice sequence). The article also made a claim that physics assumes classical mathematics. Which is wrong. The equations of physical models and their solution stays exactly the same in constructive mathematics as in classical one.
- xelxebar 7y agoI'm not sure the original paper has enough substance to precisely pin down the author's preferred axioms. In practice, the adjectives "intuitionistic" and "constructive" are used with enough author-specific meanings that you just need to read their definition each time. FWIW, the term "intuitionistic mathematics" sounds a bit odd to my ear. Usually you hear about "intuitionistic logic" or "constructive logic" which are both part of the larger program of "constructive mathematics." Anyway, the "Five Stages" paper I link does a good job of introducing the broader ideas of constructive mathematics and, perhaps, gives a taste of what Gisin is excited about. > The equations of physical models and their solution stays exactly the same in constructive mathematics as in classical one. I'm not a cosmologist, but AFAIU the Hilbert formalism of QM relies on Hilbert spaces always having a basis, which is famously equivalent to the Axiom of Choice. I'm not sure you can formulate the concept of self-adjoint operators without that, at least for arbitrary Hilbert spaces. My suspicion is that AoC simply broadens the class of phase spaces to include pathological ones that "don't actually matter," so your point probably stands in all practical applications. However, it would take (hard) work to carefully extricate AoC from the current formalisms.
- fpoling 7y agoI got impression that intuitionistic mathematical analysis refers to original Boyer papers. Which is in modern terminology is constructive mathematical analysis plus an axiom of the choice sequence. Intuitionistic logic on the other hand is a classical logic without the law of excluded middle. This is what the constructive mathematics uses and which was also used by Boyer. And yes, as in practice all physically measurable spaces are of finite dimensions, so using a formalism that works assuming AoC for arbitrary spaces is OK. The only danger is that one reads too much from it and assumes that the real world is like that.