Math Talk! Dr. Emily Riehl, to infinity categories and beyond.

[@MrSamwise25]

Hi Dr. Riehl, I was recently reading your book and it helped me understand universal properties. So thanks!

[@k-theory8604]

I'm not sure Dr. Riehl spends much time checking the comments on my KZbin video, but she is very active on twitter. I'm sure if you leave this message for her there she'd appreciate it: twitter.com/emilyriehl/status/1623790539559804929

[@MrSamwise25]

@@k-theory8604 I suspected as much, but I figured it was worth a shot since I don't have Twitter 😅

[@alexandersanchez9138]

The music theory topos book is by Guerino Mazzola. I only remember that because the book is sitting on my shelf in front of my right now!

[@reubenconducts5792]

Have you been able to read/digest Mazzola's book? I am a professional musician and a homotopy theorist familiar with topos theory and I couldn't get a dozen pages into that book.... Let me also just add that I took Emily's Homotopy Type Theory course at JHU and thoroughly enjoyed it :)

[@alexandersanchez9138]

@@reubenconducts5792 Nope. It's still pretty incomprehensible to me. I'm just an undergrad, though (music composition turned math major).

[@MusicTheoryTree]

This is super exciting, Kris! I listened to Emily on Sean Carroll's Mindscape! I'm curious how this interview came to be.

[@k-theory8604]

I listened to that episode too, I love Mindscape! Well Dr. Riehl was invited to come give a talk at my university. When I heard she was coming I reached out to ask if she had time for an interview, and thankfully she did.

[@MusicTheoryTree]

That's awesome! I love that things like this can happen. If you haven't already, I highly recommend "The Biggest Ideas in the Universe" as well. I will also be taking a bit of a look at the music theory book you mentioned in this chat. I didn't expect that to come up. Delightful surprise.

[@jcreedcmu]

Totally loved those Marilyn Burns books that Emily mentioned when I was a kid (Math for Smarty Pants and The I Hate Mathematics! Book)

[@jcreedcmu]

a fun interview generally!

[@k-theory8604]

I'll have to keep that in mind for the young people in my life!

[@tylerbakeman]

You’re the one that discovered Category theory?? Wild. I’ve been programming the concepts of abstract algebra hierarchically - I should mention that I haven’t found names for everything yet. A mathematical structure has a signature of relations - similarly, a category has a signature of arrows, with the added properties of identity and commutativity, which would mean a category is a type of structure (possibly a system if there is an axiomatization, but I don’t think I have a clear understanding of what constitutes an axiom - I know the definition, it’s just not easy to draw a boundary). Anyway, category arrows are a type of relation (a mapping) that is a superclass to functions in Set. However, functions aren’t exclusive to Set, as a category, because a structure of functions could exist without the added properties of identity and commutativity. So, what does one call a structure with a signature of functions? Perhaps a formal calculus, unless a morphism also exists in a formal calculus (which sometimes it does). What does one call a structure with a signature of operations? This one is an algebraic system (an algebra) because operations have the property of closure. Outside of trying to name structures of relations, I’ve had issues with the mathematical model of structures: A signature is a collection of relations (of a structure), which relate the instances of the structure- so, it’s difficult to say: are structure and signature synonymous? I think you can say no, if you look at the signature as an abstract property of a structure (I say abstract property to avoid russels paradox: relations belong to a structure, properties are relations, structures have a property of the signature). This question can be extended for any collection of relations that belongs to a structure (signature, axiomatization, perhaps formal calculus once I find a formal definition, etc…) I can’t wait to see useful applications come out of the topic of categories (outside of the Yoneda Lemma) - which I’m happy exists. There are so many patterns in math that we can finally formally put together, to make observations. In all honesty, I don’t think categories themselves will be able to do very much outside of Yonedas Lemma, just because their too generalized. They serve better in the mathematical hierarchy, to help us to understand the model of mathematics as a whole - or to make new mathematics models - and to potential make arrows between those models. Categories will probably also be used in linguistics too. Since we have morphology. Formal language. Formal proofs… etc

[@k-theory8604]

I'm not sure if it was a joke, but just to be clear, no one in this video discovered category theory, that was done well before either of us were born. The language of category theory has already done a lot for many fields of mathematics, precisely because of how general they are. They've also already been applied to linguistics, though I'm not sure if it actually led to new insights in linguistics yet.

[@pmcate2]

Interesting she says she is a Platonist but her research sounds like it is constructive.

[@alexandersanchez9138]

I’m not sure the two philosophical positions are directly at odds with each other; working with constructive constraints leads to strictly richer structure (think about the Gödel-Gentzen translation). I don’t think Platonists actually claim to hold any special knowledge about the Platonic realm. So, you can easily imagine being a Platonist metaphysically while being a constructivist epistemically.

[@elidrissii]

Since when is Platonism opposed to constructivism/intuitionism?

[@k-theory8604]

​@@elidrissii Not being a historian, I could be way off, but i recall that intuitionism started as a philosophical stance that was opposed to platonism. Of course nowadays it's just synonymous with dropping LEM, but I think it used to be more philosophically "loaded".

[@callanmcgill]

This is conflating contructivism in the sense of Brouwer's philosophical views about the ontological and epistemological status of mathematics with constructivism qua logical system (or collection of interrelated foundational systems as in Martin-lof type theory). I have seen it said that Brouwer would have been against the axiomatic approach to constructive logic that was developed by Heyting etc. as being constitutive of his philosophical positions (and iirc he had views against the axiomization of foundations). Needless to say there is no conflict between being interested mathematically in constructive logic (the logic that is valid in any topos for example) or higher variants (HoTT) and your beliefs about the nature of mathematical entities and how we come to know about them.

[@aaAa-vq1bd]

Platonism is really popular in pure math for obvious reasons. We sort of need a source-dual theory of knowledge. However, it’s a problem that mathematical objects never actually materialize. Think of how, for example, a paper Möbius strip is not even 2D embedded in 3D but is a normal object. And we do not see the hypothetical objects of pure math. But lo and behold, morphisms can commute and two Aristotelian mathematical objects can be connected by a Platonic topos-bridge. But again, what is the actual situation? I am actively constructing structure-preserving maps “between real objects”. But what I am actually doing first is representing those real objects. And I need to do this progressively across time. That’s how humans work. However, for example, a finished definition or proof is still only a meager mind-dependent object saying things about some mind independent material stuff. At this point you have to decide what truth is. You can use a mathematical/logical framework to represent your decision.. but you will be choosing from a few possible sensible theories of truth. Let’s say we are friends so you decide that the good old fashioned reflection theory is the best. The easiest way to understand if mathematical objects is to see how they reflect the reality they correspond to. This is a corollary of the real statement which is: truth is a mapping from mind-independent reality to conception. truth itself does not contain the objects which it concerns, it merely indicates them. This is obvious anyways because we have to write them out.

i really like your channel but could you please use some audio processing or use better equipment or whatever is necessary to make the audio better? it is really stressful to listen to.

[@k-theory8604]

Audio processing is definitely something I need to teach myself at some point. Perhaps the fall reading week or winter holidays. I have invested some more money into equipment since this video, but I guess I bought a crappy brand and/or there's some issues with the cables I have because it seems the audio has turned out worse in several instances...

[@roberthuff3122]

I support all mathematicians.

[@SkaiKuutamo-dg9tq]

Deliberately misconstruing what “support” means in the context of an organization that supports a disadvantaged community.

[@roberthuff3122]

@@SkaiKuutamo-dg9tq Put your mores and logic in the queue of disadvantaged.

[@markusklyver6277]

Holy fuck

[@jonathanvalentine1325]

*Promo SM*