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Matteo Capucci (he/him) (Jun 16 2021 at 09:30):Cross-posting from Twitter:
https://twitter.com/mattecapu/status/1405095207058673671?s=20
Is someone aware of a resolution of Zeno's paradoxes in the framework of cohesive topos theory? I'm particularly interested in the arrow paradox https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
- hom(—, matteo) (@mattecapu) Matteo Capucci (he/him) (Jun 16 2021 at 09:31):https://twitter.com/mattecapu/status/1405095209432649729?s=20
The 'solution' I have in mind is that at a given instant, the arrow not only has a location but also a velocity. This is a primitive way to express the fact that points of a smooth manifold are 'held together' by their infinitesimal neighbourhoods -- i.e. their tangent vecs
- hom(—, matteo) (@mattecapu) Fawzi Hreiki (Jun 16 2021 at 09:40):Well the classical resolution is that you need calculus. Just cohesion is not enough.
Fawzi Hreiki (Jun 16 2021 at 09:40):You don't have a notion of motion at an instant with just the cohesion axioms - but you do with something like SDG
Matteo Capucci (he/him) (Jun 16 2021 at 10:50):David Corfield suggested looking at the jet comonad here https://arxiv.org/pdf/1701.06238.pdf
Matteo Capucci (he/him) (Jun 16 2021 at 10:52):I suspect that or some of the other modalities involved might take on the job of 'sticking points with vectors' together into a curve
Matteo Capucci (he/him) (Jun 16 2021 at 10:52):I still need to wrap my head around it though
Simon Burton (Jun 16 2021 at 11:08):I was also thinking about Zeno's paradoxes the other day. They talk about how the idea of position is incompatible with the idea of motion, which is basically the content of the Hiesenberg uncertainty relation. Maybe the Stone-von Neumann theorem is another resolution of Zeno's paradox....
Fawzi Hreiki (Jun 16 2021 at 11:59):The solution is that the one point space is not a generator in the category of differential spaces (but the walking tangent vector is)
John Baez (Jun 16 2021 at 15:11):Interesting collision of thoughts here! So someone should try to get ahold of quantum mechanics, or at least the canonical commutation relations between position and momentum, starting from synthetic differential geometry.
Fawzi Hreiki (Jun 16 2021 at 15:37):I only meant in the context of classical physics. I have no idea about quantum mechanics.
Matteo Capucci (he/him) (Jun 16 2021 at 15:45):John Baez said:
Interesting collision of thoughts here! So someone should try to get ahold of quantum mechanics, or at least the canonical commutation relations between position and momentum, starting from synthetic differential geometry.
I suspect this might involve non-commutative differential geometry. In that setting one might approach deformation quantization 'geometrically', i.e. by considering the deformed algebra of observables again dual to a 'non-commutative' space.
Matteo Capucci (he/him) (Jun 16 2021 at 15:46):Simon Burton said:
I was also thinking about Zeno's paradoxes the other day. They talk about how the idea of position is incompatible with the idea of motion, which is basically the content of the Hiesenberg uncertainty relation. Maybe the Stone-von Neumann theorem is another resolution of Zeno's paradox....
Mmh I don't see the relationship with Heisenberg's uncertainty, would you care to elaborate? Also, what's Stone-von Neumann theorem and how does it resolve Zeno's paradox?
John Baez (Jun 16 2021 at 16:29):The Stone-von Neumann theorem says, very roughly, that the only irreducible representation of the canonical commutation relation
is the usual one that we see in quantum mechanics. My advisor would kill me for summarizing it this way because what I said is not really true: there's some very important fine print. But the basic idea is that it lets you get ahold of wavefunctions, etc. starting from this relation.
John Baez (Jun 16 2021 at 16:37):Calling the Heisenberg uncertainty principle a resolution of Zeno's paradox sounds strange, but it does tend to undermine the assumptions behind that paradox, since it says you can never know both the position and velocity of an object precisely - so the idea of a moving arrow having a definite position at any time is basically wrong, it's an idealization that's not true in reality.
John Baez (Jun 16 2021 at 16:42):So, perhaps the "walking tangent vector" so important in synthetic differential geometry - the infinitesimal space such that is the tangent bundle of a space - should be quantized, e.g. using deformation quantization as Matteo suggested, yielding a noncommutative space.
John Baez (Jun 16 2021 at 16:43):I'm not sure how this would work. In general it's the cotangent bundle of a space that gets deformation-quantized to yield a noncommutative space (i.e. a noncommutative algebra, seen as a generalized "space").
Fawzi Hreiki (Jun 16 2021 at 17:04):The problem I see here is that SDG is inherently covariant wheras noncommutative geometry is inherently contravariant (spaces are defined by their non-commutative algebras of functions)
Matteo Capucci (he/him) (Jun 17 2021 at 08:09):I don't understand your remark, @Fawzi Hreiki , variant respect to what?
Fawzi Hreiki (Jun 17 2021 at 09:33):In algebraic geometry (commutative or noncommutative), you start with what you want to be the algebras of functions and you define spaces as those things which should make that work. In SDG, you start directly with the category of spaces so you don't have all that much (or any?) machinery to talk about commutative vs. non-commutative.
Fawzi Hreiki (Jun 17 2021 at 09:34):The only thing you can do perhaps is start with a not necessarily commutarive ring object in your topos of spaces, but I don't know how far you can get with just that.
Fawzi Hreiki (Jun 17 2021 at 09:39):It seems to me that non-commutative geoemtry can only work from this contravariant algebra-of-functions direction which stands directly in contradiction to the idea of 'synthetic geometry'.
Matteo Capucci (he/him) (Jun 17 2021 at 10:14):Fawzi Hreiki said:
In algebraic geometry (commutative or noncommutative), you start with what you want to be the algebras of functions and you define spaces as those things which should make that work. In SDG, you start directly with the category of spaces so you don't have all that much (or any?) machinery to talk about commutative vs. non-commutative.
The approaches to SDG I'm familiar with actually mimick algebraic geometry using -rings in place of commutative rings. That's how you get the Cahiers topos, for instance, see first section here.
Matteo Capucci (he/him) (Jun 17 2021 at 10:15):I don't know whether -rings can be made non-commutative in a natural way, since they inherit their commutativity from that of
Fawzi Hreiki (Jun 17 2021 at 10:57):No those are about building models for SDG - not the actual axiomatics.
Fawzi Hreiki (Jun 17 2021 at 11:00):Sure there will be many algebro-geometric models for SDG. An even simpler example is the topos of co-presheaves on finitely presented -algebras for a field.
John Baez (Jun 17 2021 at 16:31):Topos theory is naturally suited to commutative rings, not noncommutative ones, because a category of commutative monoids in some symmetric monoidal category is automatically cocartesian.
John Baez (Jun 17 2021 at 16:32):Thus, its opposite (e.g. the category of affine schemes) is cartesian.
John Baez (Jun 17 2021 at 16:33):This fails magnificently for noncommutative monoids, where you typically get a noncartesian tensor product. This is the origin of quantum entanglement, etc.:
John Baez (Jun 17 2021 at 16:34):So, we should expect "noncommutative algebraic geometry" to work quite differently from ordinary commutative algebraic geometry.
John Baez (Jun 17 2021 at 16:38):A not extremely clear explanation is here:
John Baez (Jun 17 2021 at 16:38):This has lots of references, but someone should explain the basic ideas better.
John Baez (Jun 17 2021 at 16:42):I guess the paper by Kontsevich and Soibelman is a more energetic attempt to explain noncommutative algebraic geometry.