aboutlogic #12 | Urs Schreiber – Quantum (Physics, Computing), Topos & Homotopy Theory

00:00:00: What drives you?

00:00:01: Well, what drives me is I always wanted to understand how the world actually works.

00:00:05: That's what string theory is about the last couple of years.

00:00:20: What I've been doing.

00:00:21: this amazing interest currently in quantum technologies and particularly what's called topological quantum technologies, turns out even though everybody was talking about these fields as if everything has already clear cut.

00:00:32: just a matter going to lab engineering it.

00:00:36: there are actually lots of theoretical open problems that haven't really been addressed.

00:00:46: Hello Everybody This is the latest about Logic podcast.

00:00:51: And we are very happy to have Osz Schreiber here, so Osz as a senior scientist for University of New York in Abu Dhabi and he's well known for high level quantum stuff.

00:01:07: if I may summarize this like that Maybe Oz, can we start?

00:01:14: Can you give a better description of what you're doing.

00:01:19: Right!

00:01:21: Thanks first of all for having me.

00:01:22: that's great honor.

00:01:24: it's great to see.

00:01:25: do this kind of podcast so... What I'm doing research-wise?

00:01:31: yeah in the last years We've been entering.

00:01:34: The subject has now come to be called quantum which is Just Quantum, which is a bit surprising for people who have been brought up studying quantum mechanics.

00:01:42: But now there's this shield just called Quantum concerned with you know quantum phenomena, quantum technologies.

00:01:48: originally I've and still i've been working on aspects of high-energy physics related to string M theory.

00:01:56: so You know with Hisham Sati my long term collaborator we had the Long project program of understand mathematically understanding what does it what is a brain in M theory?

00:02:05: and by doing so we eventually found some structures that are actually interesting.

00:02:11: A brain

00:02:11: is B-R-A-N-E right?

00:02:17: Yes, yes!

00:02:17: Right yeah thanks for stopping me.

00:02:18: No consciousness yet

00:02:20: Like like in membrane it's higher dimension

00:02:24: of... Yeah I know i just wanted to clarify this with the audience.

00:02:28: So there's what i'm doing topological quantum physics with relation application of higher topos theory, which is I guess the reason why i'm kind of connected to some aspects of logic.

00:02:41: Yeah can you give it a more sort of mundane account?

00:02:47: what this really about?

00:02:50: You come from string theory but you're not focused on string theory in the moment... But anyway how would person who's interested.

00:03:07: What drives you?

00:03:08: I mean...

00:03:08: Right, well what drives me is that i always wanted to understand how the world actually works and whats going on here.

00:03:17: so research has been doing very much in fundamental physics trying to understand basic building blocks of reality.

00:03:25: right thats what string theory is about.

00:03:29: And yeah, so more recently it's kind of exciting for me that we're actually entering or making contact to experimental physics.

00:03:38: In our last articles we make actual predictions about phenomena like might be seen in experiment but is still driven by understanding fundamental physics.

00:03:49: So there you know if I just look at the last couple years what i've been doing this amazing interest currently and what's called quantum technologies, in particular topological quantum technologies.

00:04:06: And turns out even though everybody is talking about these in the field as if everything has already clear cut.

00:04:11: just a matter of going to the lab engineering it, there are actually lots of theoretical open problems that haven't really been addressed.

00:04:21: so we're looking at where trying Some open problems, theoretical problem like what is an onion for instance?

00:04:30: That's something that occupies me these days.

00:04:40: Yeah, that's very good.

00:04:41: you say this because the problem already starts.

00:04:43: That is how it has always been introduced in all of our textbooks and I think there was a wrong way to go about them – not quite correct!

00:04:50: Of course they have some truth to it.

00:04:51: otherwise...

00:04:52: For one moment can we first tell people what are fermions or bosons?

00:05:01: In the standard model for particle physics an assumption is made by elementary particles falling into these two classes.

00:05:09: The one cloud is called bosons.

00:05:10: That's all the particles that transmit forces, for instance... ...the photon which is a quantum of light is the archetype typical example of a boson.

00:05:21: and on the other hand there are stuff these forces act upon in matter made by universes And they consist mostly of fermions like electrons and quarks.

00:05:36: So there was a big insight back in the last century, that there's this dichotomy.

00:05:42: These bosons and fermions.

00:05:44: mathematically they fall into different representations as one says linear representation of symmetric group.

00:05:50: so there is funny effect figured out many decades ago where wave functions for several electrons changes its sign when two of these electrons exchange their places, like in theory if you were to exchange two electrons.

00:06:09: In the universe turns out that corresponding wave function describes probability amplitude for whole system.

00:06:16: be at some state which is its sign Which of course something not actually observable because only square of this way functions.

00:06:25: observer but there's a face called the sine and bosons don't have those signs.

00:06:29: so when the wave function doesn't change when you exchange per fermions pick up this sign.

00:06:37: And then some time ago, I think they started in the seventies people wondered whether there might be other possibilities that... Whether there are particles such as when we exchanged them?

00:06:48: The wave function picks another factor maybe a complex phase which is not plus one but it's maybe some root of unity right.

00:06:59: so some complex number has modulus one, but it's a different phase.

00:07:04: And that was for long time with just the theoretical curiosity.

00:07:07: so these things have then be called anions.

00:07:09: So this is a pun on any possible face That might appear Any-on right?

00:07:16: This general convention every particle gets called something On and so The Any-ons or those that Might of any Phase.

00:07:24: So people

00:07:25: Yeah theoretically looked at this in the past Just out of curiosity, really.

00:07:32: Checking what method... So

00:07:35: how is this related to the notion of spin?

00:07:37: I mean, obviously bosons are even half and fermions are odd-half spins.

00:07:47: Does that make any sense or it's just related Or Is not really relevant?

00:07:53: No, it's absolutely related.

00:07:54: It is one of the more subtle aspects with motor physics this called spin statistics theorem which says that in a consistent relativistic local quantum field theory the fermions if you first characterize just by that exchange phase they also have to have a spin-one half or odd spin Which actually not so easy to see but... First of all its observed Electrons have spin, bosons has a spin of one.

00:08:26: But that is the case for quite subtler regions actually So kind of amazingly with the rise or new fashion with quantum technologies with technology progressing to the point that various corner effects had been discussed theoretically for a long time, they suddenly became actually measurable and testable in experiment.

00:08:52: That led to the remarkable situation or has lead to those situations.

00:08:56: we're now where onions have actually been observed.

00:09:00: There were claims before that, but they were the first to maybe definitely actually see.

00:09:14: That onion phase in the laboratory and has been confirmed by a few groups since which is kind of remarkable because it's there are other claims as big companies on this planet who have been claiming to see something related uh... But he had never been reproduced for people.

00:09:29: But in so-called fractional quantum whole system sees onions are not actually seen.

00:09:36: Yeah, but I wanted to answer your question whether they are in between bosons and fermions.

00:09:40: The big difference is these onions aren't elementary particles... Compound

00:09:45: particles?

00:09:46: Yeah

00:09:46: there's a bunch of them as being solitons or so that kind of excitations of the entire system in some sense.

00:09:54: So they're rather different from bosons and fermions, there's a bit of mathematical coincidence about this.

00:10:02: exchange statistics.

00:10:03: so I think it is technically wrong to speak of onion statistics as braiding phases that are being picked up but its not quite the same thing as boson and fermion.

00:10:12: But anyways i mean thats a subtlety maybe with little concern here.

00:10:16: And these onions can be useful for building a quantum computer?

00:10:23: Or is it more theoretical interest or what we have?

00:10:28: Well,

00:10:28: both certainly.

00:10:28: I mean they're suddenly just theoretically interesting but in particular now there's a big interest in them.

00:10:33: because yes this the idea of how to actually go about building large-scale useful quantum computers to equip with what was called topological protection and onions are THE way.

00:10:47: This big hope currently is very small, tiny quantum computers have been built in the lab that run for a few microseconds.

00:10:53: So-called NISC machines noisy machines so quantum computers that in principle behave like one expects quantum computer to expect but just on very tiny scales.

00:11:02: they're not actually useful for actual computations.

00:11:07: and there's one big scheme most of community are pursuing how you know, enlarging the existing quantum machines to useful size and then it's called quantum error correction.

00:11:17: So there is yes if that hardware sucks but we just make lots of it too and insert a lot of redundancy Then we can keep fighting the errors that keeps appearing in these systems so that We can push the boundary off how long this machine run without decohering becoming useless?

00:11:36: This kind.

00:11:39: You have a leaking bucket Right?

00:11:41: And you want to transport your water somewhere.

00:11:43: Your bucket is leaking, so what do we just say?

00:11:45: okay let's run many, many leaking buckets then maybe can still have some hope of bringing our water.

00:11:52: The idea for topological quantum computing was that it should fix the bucket and try a non-leaking bucket.

00:11:58: Non-leak cubit, topologically cued!

00:12:01: So there are certain fundamental quantum effects in the sense insensitive to local small perturbations, so they only depend on some global phenomena which means that if you have a little bit of noise shaking your system somewhere it will not actually disturb the quantum state unless you completely break.

00:12:23: So this is an idea where with topological quantum effects you might be able to stabilize corner computers at a fundamental hardware level and Anjans is one or big idea for how to go about this.

00:12:35: The point being that the system picks up when two onions are exchanged is, at least in theory it only depends on the onion actually being exchanged or changing positions.

00:12:46: It doesn't depend upon HOW they do so.

00:12:49: They can wiggle around each other and do all kinds of funny stuff.

00:12:53: As long as they have changed position the system will pick a phase.

00:12:56: So that means it only depends on topology And if that could be used to actually construct quantum gates, which would then be called topologically protected.

00:13:09: That is in theory at least one way of having robust quantum logic gates and hence quantum computers.

00:13:18: It doesn't exist at the moment but it's an idea for how you go about it.

00:13:22: Very good!

00:13:23: Very useful!

00:13:24: Very interesting.

00:13:26: Yeah, maybe let's move a bit on.

00:13:28: So far it is more about physics which also I mean adopting in physics are related i think but okay.

00:13:37: so you use quite high-level mathematics.

00:13:41: uh...I would say?

00:13:42: I mean..i think that if u do like higher categories and your interested in homotopy type theory can explain what the connection between you're interested in physics and your sort of interest in these mathematical theories which are quite advanced, let's say.

00:14:05: I mean...

00:14:08: Right!

00:14:09: Yeah so um i was always motivated by understanding physics.

00:14:13: So i started looking at what is known about when i'd say physics- i mean Mathematical Physics ,I guess?

00:14:22: And then it's a well-kept secret.

00:14:24: Everybody who has actually looked into these things knows this, that large swaths of even mathematical physics are using I would say inadequate mathematics for the kind problems one wants to describe here is very simple example which illustrates where this problem already starts and with the Topos theory comes in.

00:14:45: so you know its folklore.

00:14:48: everybody difficult with quantum field theory, right?

00:14:52: Everybody knows oh there's these infinities that they run into and the need to renominate in something is very difficult mathematically.

00:14:58: So one might ask well where do these difficulties actually start?

00:15:02: And there was a very simple original fact that I sort of know but not widely appreciated which is this Of course The configuration space off physical fields Which you know permits all of spacetime is a mapping space.

00:15:16: So if, for instance you have the scalar field like the Higgs field or something one configuration of that field think thing about magnetic fields that currently inhabits all this space we're living in.

00:15:28: so it's essentially okay with a magnetic field.

00:15:30: I'm cutting some corners but If we speak about the Hicks field for instance its literally true That a configuration of these fields are mapped from spacetime to some target space like the complex numbers, which to each point in our space that we inhabit assigns a number or something more complicated than a number but assigned something and some target space.

00:15:51: So the whole configuration is a map from spacetime points to whatever values that field takes right?

00:15:58: Be it be at a complex numbers are some spin representation or group or something.

00:16:05: And now so this... The space of all these maps mathematically, the mapping space.

00:16:13: The topological space of all these maps.

00:16:16: that is a configuration space on this system.

00:16:18: in the same way if you look at just single particle its configuration space it's just ordinary space.

00:16:24: so R three right?

00:16:25: It's just number.

00:16:26: positions or set-of position and particles can inhabit now for field not just R Three but its map from R Three into some other space.

00:16:36: What you want to do is, you wanna do differential geometry on this configuration space.

00:16:39: You're gonna say okay there's a symplectic form in the space.

00:16:42: we can produce some prequantum line bundle with the symplective form being its curvature so that it can quantize and then analyze some Hamiltonian vector fields.

00:16:51: So your going to do differential geometries of this space.

00:16:53: This what physics tells us after you've sorted out your configuration

00:16:57: Can you quickly say something about differential geometry?

00:17:01: Oh I see.

00:17:04: So differential geometry is the study of geometry that's sensitive to infinitesimal, very small tiny variations.

00:17:20: As opposed to topology where we care only about a broad global structure over space and differential geometry actually cares how exactly it looks locally what is, for instance if I move a particle at just the tiny bit what is the distance that has been moved?

00:17:36: as opposed to just knowing i have moved it across the universe.

00:17:40: did it encircle some hole in the universe or not?

00:17:43: this is what topology knows and differential geometry we know what happens locally.

00:17:47: so most of physics is concerned with differential geometry.

00:17:49: uh for instance the momentum of a particle its velocity vectorist an object in differential geometry.

00:17:56: If you wish are not visible in topology.

00:18:00: what controls physical systems, the velocity at a certain instance of time determines how this system evolves.

00:18:08: So differential geometry... Yeah so that's why we need to do differential geometry

00:18:12: on these... Can you say it is an abstract way in looking at some smooth geometry?

00:18:17: I mean if they have naive idea about geometry and sort of internal surfaces then what can be done with them?

00:18:26: And differential geometry?

00:18:28: to look at the geometry in this sense, or is it a fair description?

00:18:38: Right.

00:18:38: Of course that's the technical term where we speak of a smooth space.

00:18:41: if has been equipped with an ocean of how to differentiate on and hard-to-do this local analysis for it.

00:18:47: yeah That's right.

00:18:49: Yeah!

00:18:49: Of course If this helps The audience then differential geometry about smooth structure as opposed just topological structure.

00:18:57: And now Now, the funny thing is that now we're looking at a mapping space from one smooth space or spacetime manifold.

00:19:08: Or just spatial manifold into this target manifold and it turns out only under very constraining conditions.

00:19:15: Is there as smooth manifold structure an ordinary smooth structure?

00:19:19: That you may find in the textbooks on this mapping space.

00:19:24: So technically if the domain space has to be a compact manifold functions, which is exactly what you don't want to assume in physics.

00:19:31: You wanna assume there's a compactness as a mathematical condition.

00:19:34: that kind of means your spaces is bounded it ends somewhere and then certainly not what we wanna assume for physical space.

00:19:41: yes

00:19:42: And so so strikingly The most on the very first step in doing field theory not even quantum field theory, just classical field theory.

00:19:51: You want to say okay on which space are we doing differential geometry now?

00:19:55: So that's what you're doing in this mapping space... ...which doesn't have a textbook smooth structure.

00:19:59: so unless your restricts artificial situations.... ...you cannot actually open the text book and do differential geometry on this space.

00:20:08: it is kind of striking!

00:20:10: This is known for people who work with this.

00:20:12: and yet And this is part of the reason why these infinities and stuff crop up in physics because what those people do, they put some tricks instead doing their actual differential geometry.

00:20:24: They just kind work with formal Taylor series which things that look like could come from a differential geometry without actually being such.

00:20:33: so you wave your hands... ...and then on to something different than Newton would have told us had he known about field theory.

00:20:43: And maybe to cut the story short, because I'm changing this out too long-windedly here.

00:20:49: Topos theory...

00:20:55: So isn't that often mathematicians there is a conflict?

00:20:59: The mathematicians say that physicists are cheating and do things which actually not really mathematically justified!

00:21:07: This basically such case where they do you structures, which are really not applicable because of what users say?

00:21:17: Because these bases aren't compact and just pretend to do it.

00:21:20: And somehow things work

00:21:22: out.".

00:21:22: You will know saying actually we can fix this... We make both mathematicians and physicists happy sort-of.

00:21:33: Yeah absolutely!

00:21:35: That's exactly the summer.

00:21:37: yes Exactly, there's all these steps in physics famously.

00:21:41: So where the mass is lagging behind physicists just do stuff they kind of know what they want right so that They have this application on mind.

00:21:50: and so if their math it not quite available You you do something?

00:21:53: That looks like its going to write direction Because because your not a pure mathematician you haven't an application in mind.

00:21:58: that happens Not only in physics It happens another fields but of course in physics as most.

00:22:02: yeah noticeable Yeah.

00:22:03: And then at some point things go wrong.

00:22:06: You know, this is where these words renormalization come from right?

00:22:09: So there's a famous step in quantum field theory.

00:22:11: you start quantizing your field theory and then at some point we say oh!

00:22:14: We need to renormalize.

00:22:16: I mean the word literally says...you make needs normal again

00:22:20: Right?!

00:22:20: You just pretend everything should work And you only stop when you run into an actual contradiction When things actually don't work.

00:22:27: Then you'll say okay let's step back and try to fix it.

00:22:34: Yeah, right.

00:22:34: So you get something that is nonsense.

00:22:37: so they say oh well okay all right yeah we knew we cheated a bit.

00:22:40: so let's backtrack and fix things.

00:22:44: And then once you've renormalized Well by I mean eventually like over long periods of history some things actually are sorted out.

00:22:54: so renormalization was eventually understood.

00:22:57: Actually renormalizations now mathematically will find thing but next cancel anomalies.

00:23:04: That's another thing that goes wrong, and then so things keep going wrong.

00:23:06: you just kept going exactly.

00:23:08: And the question is actually if right I mean it's kind of urgent in these days.

00:23:12: So when i started studying physics Right there was already at the time where everybody said okay we know everything about this standard model.

00:23:18: now We need to understand deeper physics or we need go through the foundations?

00:23:21: Then evident questions.

00:23:22: well If already the physics that you claim to Understand Is not on solid ground You don't really know what are they talking About?

00:23:30: how can go further to ask for more foundational physics, if you're already lost really with the non-foundational stuff.

00:23:37: So that is the motivation for saying let's check first what are these things?

00:23:42: And in some cases it's not so hard... It's a big mystery!

00:23:45: You just have to look at and know some modern math ...and then the answers are there

00:23:50: right?!

00:23:50: This gets us on top of the theory we asked about.

00:23:53: Turns out its very easy.

00:23:55: actually give a smooth structure to mapping space or Topos theory does.

00:24:02: It produces for your Cartesian closed category where mapping spaces, the internal homes are on exactly same footing as all other objects.

00:24:10: so there is a larger physicist.

00:24:14: since what?

00:24:16: Since beginning of last century they work in this category of smooth manifolds which it's just not large enough and doesn't contain these mapping spaces but the categories of smooth manifold that fully faithfully Inside the topos that are called, we call it the Topos of Smooth Sets.

00:24:32: The topos on all smooth manifolds.

00:24:38: Can you for a moment explain what's the relevance of a topos here?

00:24:43: It is like... You basically replace the theory working in talk about sets or types and also constructions And you move them into a different sort of word.

00:25:01: You put yourself in to the world, into different topples which looks very much like what we are used too but is somehow different?

00:25:11: Is this also good way for us say it?

00:25:14: Yes absolutely!

00:25:15: And of course, topos have these two aspects.

00:25:16: There's always the logical and geometric aspect to it.

00:25:19: but yeah what drives... What drives the story from physics here?

00:25:23: is this geometric aspect?

00:25:24: Yes so from this original Grotendieg style aspect a topos is kind of just a better universe in the mathematical sense A better category or on the logical terms even a better category of spaces Of whatever notion of geometry you're interested.

00:25:38: for every notion of Geometry there is a Topos that contains spaces of that geometric kind and all their limits, co-limits in mapping space.

00:25:50: So it's a context which constructions you want to be doing on your objects do exist in the natural canonical way.

00:26:00: That is how this begins...

00:26:03: What happens are these manifolds didn't exist or they did not exist.

00:26:15: If you go into a topos which is given by some particular construction, then inside the topos what was achieved before and now becomes... completely reasonable construction.

00:26:35: Yeah exactly, so for instance these mapping space the spaces of field configurations.

00:26:39: they just exist now.

00:26:40: They are formed by what's technically called the internal home Formed in this topos internal from your spacetime manifold which is not an object Of their topos to say the complex numbers Which was also know?

00:26:52: An object of that topos.

00:26:53: one can form This mapping space or internal home and continue operating on it, or what can I ask for?

00:27:00: differential forms?

00:27:02: So do all the things that a differential geometry wants to do and if they just exist yes.

00:27:09: That's kind of synthetic mathematics side.

00:27:14: you work inside this topos in certain principles holes there which are not developed analytically in your meta-serial to say.

00:27:29: Yeah,

00:27:30: so I would say that's for people entering this here.

00:27:34: there is kind of the next step.

00:27:35: So first of all you can just embed your manifolds into the topos without talking hybrosynthetic mathematics.

00:27:41: It's a new definition That it more flexible.

00:27:45: but then once you are in these more general context You want ask okay?

00:27:50: What does such and such stomach construction which you're used to from the textbooks, where it's explained in terms of sigma algebras or some local constructions.

00:27:58: What does that mean now?

00:28:00: To do this same construction on a general object in this top post?

00:28:03: right so we need to lift the stung definitions into new objects and say okay I just mentioned what is a differential form for mapping space.

00:28:13: You can't find any text books but only find out what is the difference between an ordinary manifold.

00:28:18: So now you have to lift their definition.

00:28:22: one can just do this by trial and error.

00:28:25: But of course, the more satisfactory in most sustainable ways that you ask well what is the characteristic property actually off that construction?

00:28:33: What I want their construction to be like?

00:28:35: what properties should it actually satisfy?

00:28:38: so they're not just defined.

00:28:40: i can't go define something new now for my new spaces but wanna be assured there's a right generalization.

00:28:46: so first going say any given analytic definition had before let's analyze Why did we make this definition in such a way that we didn't?

00:28:55: What are its abstract properties?

00:28:57: and then just try to say whatever has the same as abstract property is.

00:29:02: now in more generality, it's correct generalization.

00:29:05: That is the synthetic approach where you say instead of building something concretely analytically let us check if it quarks like a duck or walks like a dog all these things than probably a duck.

00:29:21: Yeah, exactly.

00:29:22: And so this is how the physicist it was interest as I wasn't at that time right?

00:29:26: I didn't actually had no background in logic and in foundations of math nothing.

00:29:31: It's just interested physics.

00:29:32: Well This Is How One Gets Drawn Into The Story Now.

00:29:35: Then Next You Ask Okay!

00:29:37: How Can i now Formalize The Kind Of Physics That Actually Want to be Doing?

00:29:40: What Does A Jet Bundle In The Topos So That I can say what's a differential Equation or an Euler Lagrange?

00:29:46: So All These Things That People Usually Do.

00:29:49: you want To do it in the topos And that leads you down this rabbit hole, and then before long.

00:29:54: You find yourself looking at modal operators and stuff... ...and people will tell you there's something to do with logic!

00:30:01: How did it

00:30:03: happen?

00:30:05: You come through logic from physics trying to understand the basic building blocks of the world in a systematic way.

00:30:16: Yeah

00:30:18: Maybe very nice question.

00:30:21: Is this all guided by theoretical consideration, so the math should look nice?

00:30:26: or is there their hope to decide these things experimentally?

00:30:31: in which top us?

00:30:32: We live.

00:30:33: So because we can In theory decided with manifold will if right.

00:30:38: and then you have this older picture of physics.

00:30:42: Is there a chance for that or it's purely For them?

00:30:47: No, I would say it's certainly not for the math.

00:30:50: Not just for the maths right?

00:30:51: It is about trying to understand physics.

00:30:53: Of course within the physics you can still make a distinction.

00:30:55: Are we going after formal physics or theoretical physics?

00:31:00: Or are they even trying to connect with what experimenters actually measure in laboratory?

00:31:06: of course large huge fraction.

00:31:09: The physical community has decoupled from experiments Just asking abstract questions in between math and experiment.

00:31:19: really that are formal physics or theoretical physics.

00:31:23: But of course these things thing together even this formal physics, Even though it has been going on now for decades without contact to experiment The idea is still that someday in the future, we will be able to also connect back-to-experiment.

00:31:39: That's of course always... ...the ultimate goal.

00:31:41: but then there are also concrete questions.

00:31:44: No!

00:31:44: Nothing like this is motivated just by the math.

00:31:47: But I would say if you're an end user for math and want actually use stuff Then they have some motivation In keeping the maths elegant because it becomes very complicated Because as a physicist Your career will not end with studying this definition.

00:32:08: This is just a zero step, you know?

00:32:10: You want to use these definitions.

00:32:11: so if the definition already goes all over that place and becomes... ...just needs your whole career study in itself then it's going be useful for

00:32:19: you.".

00:32:21: So what's for me at least big motivation of actually going for elegant maths because its' just the zero-step.

00:32:31: Isn't there a lot of historical examples for this?

00:32:36: I mean, think about Faraday who had an intuitive understanding of electromagnetism.

00:32:47: But then there was Maxel and others... ...who developed the math to get really concise description on it.

00:32:56: And i think that's happening continues to happen with Einstein and generativity, he had a naive understanding was being more naive but they have an understanding what's going on.

00:33:08: But then to really make it precise you had to develop some new mathematics yeah?

00:33:13: And going onto quantum mechanics as you can see I mean Bohr, Copenhagen There is a naive understanding, even quantum field theory that we have fields which are wave function of fields or whatever.

00:33:34: So there's the naive understanding.

00:33:36: what going on?

00:33:37: but then if you need to develop new maths it actually makes us precise.

00:33:42: And as all this is saying, there's unavoidable need to develop the language.

00:33:53: To talk about the word reflecting our thought.

00:33:56: maybe intuitive understanding of the words?

00:34:00: Yeah absolutely!

00:34:01: This a back and forth that has been going on for eons For ages.

00:34:04: it starts with Newton really Newton wants just his intuition I think that's fair to say about his laws, and then develops calculus in order to put it on mathematical grounds.

00:34:15: It starts right there!

00:34:16: And yeah...it keeps going for Neumann invents the foundation of quantum mechanics people invent differential geometry for matrix making like this has been going on a long time but this progress back-and-forth between math and physics got somehow stuck.

00:34:36: sometime in the sixties of the last century with an invention of quantum field theory, there was a kind of disconnect suddenly where physicists just kept going and mathematics or the mathematical formulation has been falling behind.

00:34:50: And I don't know exactly why... That's maybe an interesting discussion one could have, but it just happened.

00:34:57: And I don't think necessarily happens to all... Some people thinks like oh the mass doesn't exist!

00:35:01: But that is not a case.

00:35:02: you can go and look for it.

00:35:04: You need to understand both.

00:35:07: of course It gets harder The further time progresses.

00:35:10: Back in days it was easier For Scholar To be expert actually In both mathematics and physics.

00:35:16: These days very hard.

00:35:17: Even most physicists only understand a tiny fraction of the field physics.

00:35:21: And, of course most mathematicians are still in a tiny sector of mathematics.

00:35:24: So it becomes harder and harder to kind of scan both worlds To see which puzzle pieces actually do need to be fit together.

00:35:31: so that probably explains In parts why things have slowed down.

00:35:35: but on the other hand.

00:35:36: at this point we really need to mention Levere.

00:35:38: William Levere The you know widely known as widely known is maybe the father of At least elementary topos theory right enough After a category theory, he was all motivated.

00:35:48: If you read his interviews... He keeps saying it again and again!

00:35:52: ...he was all-motivated by understanding physics.

00:35:54: He actually the one early on who said that we need toposes in order to describe what's called continuum mechanics.

00:36:01: So as far I understand or recall this story from his interview is that As a student started at the physics department.. ..he was handed the task of solving some problems in fluid dynamics, continuum dynamics And so that involves vector fields Assignments of directions on a manifold in uh, In a smooth way.

00:36:19: and he at that point.

00:36:20: He said well What is the vector field which?

00:36:22: Of course was a silly question for everybody else around him because there was meant to be long known since Since the days off of Einstein now than nineteen twenties.

00:36:31: but he asked it again.

00:36:32: It says what-what is it actually?

00:36:34: And I'm not sure how well this is knowin' That Lavire's development of Topos theory as an elementary topo series so As context for a universe of mathematics was motivated, he keeps saying this.

00:36:48: Was motivated by his desire to find the foundation for physics or at least continuum mechanics.

00:36:54: so that's why it develops what is called synthetic differential geometry.

00:36:57: because he thinks... So this way of synthetically doing differential geometry in topos is because they think.

00:37:02: I thought That Is The Way To Go.

00:37:06: and And That Is A Way To go yeah?

00:37:08: So turned out be right even though i don't ever see.

00:37:16: I don't think he recognized that people had started applying it to real-world physics problems, but yeah.

00:37:22: He was certainly the one who envisioned this first.

00:37:27: Maybe can i try change angle a little bit as somebody interested in foundations of mathematics?

00:37:33: I might would say this back and forth is clear.

00:37:36: I strongly believe in that, but i would see it as like.

00:37:40: there are very different mathematical theories And the moment you get interest from physicists This theory will get a boost and it will get dominant.

00:37:52: Every student needs to learn about vector spaces because The physical application

00:37:57: so-to

00:37:58: speak.

00:37:59: So I was wondering if any thoughts category theory, topos theories and their links to physics.

00:38:06: And what this might have to do with the foundations of maths.

00:38:10: or would you say set theory

00:38:12: entirely?

00:38:15: So did... Would you think that a reason one might argue why Category Theory might become a foundation is it gets more interest in contrast to Set Theory.

00:38:27: Or Is That Nothing You Would Think About?

00:38:34: Oh, I see.

00:38:34: You're asking maybe... So on the one hand category theory has slowly becoming more prominent among some physicists as opposed to set theory.

00:38:48: and you are asking is this intrinsically about a fashion?

00:38:53: Is it just that the physicist ended up liking category theory for some superficial reasons or is it for some more profound systematic reason?

00:39:03: Is that the question.

00:39:06: Yeah, this and what might have as long-term results of a mathematical community?

00:39:12: if you want to conjecture something in that direction

00:39:17: Right!

00:39:18: I can say something about this but maybe you could tell me your thoughts are because i'm certainly not a set theorist The reason, or whenever category theory has actual success in physics.

00:39:39: I mean these days as if i may say this let's put some polite words.

00:39:42: there is you know what called applied category theory now that this attempt to kind of push category theory on on applications, but I think the real role of let's at least say the natural original role category theory is a theory building.

00:40:00: That came into being.

00:40:01: it was meant to be an underlying framework in order come to grips with concepts in algebraic topology.

00:40:08: that's why Eilendberg and McLean introduced it... And that what its best for theory-building right?

00:40:15: Whenever you have question Solve solve this equation in a well-defined context or solve.

00:40:22: This problem Which has already been completely defined and you just need to work it out.

00:40:27: But when the question is a meta question like

00:40:29: what?

00:40:29: Is there actual theory we're looking for for instance in math?

00:40:32: A good example, as What is the field with one element?

00:40:35: I don't know if you know The story right.

00:40:36: this was.

00:40:37: this was a big question that occupied people some sometime ago.

00:40:40: It was not not a theorem you could prove, but it was a conceptual question.

00:40:44: People were after good definition.

00:40:46: or in physics... You want to know new theory of physics and what is M-theory?

00:40:54: So that you cannot solve an equation somewhere.

00:40:57: It's different kind of problem And that kind of problems.

00:41:00: theory building That is what category theory is good for.

00:41:03: gives your structure What theories are like which limits to any, you know and stuff like this.

00:41:12: And I think that is certainly one reason why category theory where it's successful has been successful in physics?

00:41:18: And maybe there... I feel

00:41:23: as if your answer just in a way when you said for example of differential geometry That's the basic idea of construction, which I thought was in what we call a universal property.

00:41:43: Instead like those are differences and set theory... We do concrete constructions for assembly language or mathematics but to build these theories We really need to have more high-level ways of talking about things.

00:42:01: and these are universal properties.

00:42:03: And this is given by category theory, and type theory... ...and top theory and so on.

00:42:10: I would say structural mathematics right?

00:42:13: Yes!

00:42:15: Yeah i guess that's what it meant when you said it was about theory building.

00:42:18: You want something with some properties then all the structural theories tell how go about finding that thing where.

00:42:29: set theory certainly doesn't answer these kinds of questions, right?

00:42:33: But maybe they need to... Maybe you could say what set theory is from your perspective.

00:42:38: What is that drive-set theory like this days may be?

00:42:41: can you save it?

00:42:42: I would be interested in hearing.

00:42:45: Yeah and i mean i would claim And um This might be empirically wrong That these foundational questions aren't In the focus at all anymore.

00:42:52: So no more working mathematician really cares about that because we're under a crisis.

00:42:57: Right, we have Martha's sufficiently rigorous.

00:43:01: So said theory is a branch like every other branch interested in either higher infinities or definability and has very concrete questions And relations to other fields.

00:43:19: I mean yes the question of consistency Is maybe it'll be misguided focus.

00:43:27: Also important and ignored is expressibility.

00:43:32: I mean this the same as programming languages, all programming languages you have are too incomplete so there's no difference.

00:43:38: You can compute or write your programs with two-in-machines but two in machines aren't a very good way to program right?

00:43:47: And the question for me was what would be better than structuring your constructions?

00:43:52: And this is completely, as far I can see.

00:43:55: it's ignored in sets here.

00:43:59: There are just built things in assembly English...

00:44:05: But you mean that you can abstract there too right?

00:44:07: You can

00:44:08: add syntactic sugar...?

00:44:10: Yeah but thats not what its about!

00:44:12: At least that my impression of infinities

00:44:18: and...

00:44:19: We need to talk more set theoreticians, I guess.

00:44:22: We will

00:44:23: invite some of them as well

00:44:25: with M... But can i move also a little bit on because one thing that we want you hear from us is how does this lead us to homotopy type theory and these things?

00:44:38: Because I know your are interested in the stuff too.

00:44:43: maybe you could say something about

00:44:47: it.

00:44:49: So we've talked about how one way that you can see the toposes, ordinary toposes show up in physics by providing your context for discussing more general spaces.

00:45:00: Differential geometric space or I mean there's also super-geometric space and so forth.

00:45:06: anyway it is a way to give home to those kind of spaces which appear in discussions on physical systems but then they keep going.

00:45:15: We're talking about mapping spaces.

00:45:18: Next, you might want to say okay we want infinitesimals in our space.

00:45:24: that leads us into these toposhes that model synthetic differential geometry.

00:45:29: but then you keep going and saying well next a very big aspect of physics is the presence of gauge fields.

00:45:38: so is the quantum of light.

00:45:44: And a big insight that goes back to Maxwell, which we started with.

00:45:47: I think it was illuminated later.

00:45:54: this electromagnetic field in which light is the wave and an instance what's called gauge fields are very interesting.

00:46:08: this kind of issue with the notion of equality that drives research into homotopy type theories.

00:46:14: So, a big insight from what is it?

00:46:17: The forties and fifties in the last century are that the actual configuration of magnetic electromagnetic field was not this faraday tensor but something different called gauge potential which the faradays tensor derived.

00:46:34: This gauge potential whatever it is, so differential one form locally.

00:46:39: Is not defined up to equality.

00:46:42: So turns out that if you even though the mathematical description involves this object If you physically ask for the actual nature of the electromagnetic field You have to regard some Of these gauge potentials as being isomorphic.

00:46:56: The physicist says gauge equivalent Even thought they're not equal.

00:47:01: So it's a very big deal in physics that gauge fields have this notion of gauge equivalence between them, which replaces the notion of equality.

00:47:11: And if you formulate this in mathematical form It means that for instance the configuration space of the electromagnetic field on some space is not actually a set... ...it's also not a smooth-set so its'nt even an object of your top walls or sheaths and smithmanical etc.. Its actually a groupoid.

00:47:29: So it's a set with gauge transformations between its elements, the set of notions of identifications.

00:47:36: And that is kind of striking because that just arrives from in nature.

00:47:40: you observe and I think physicists have been discussing this much longer even though more informally or improperly maybe but they've discussed as much longer than mathematicians actually like decades before mathematician starting becoming very serious about these notion.

00:47:59: That leads you further.

00:48:00: You know, we're still this physicist trying to understand what is actually the fundamental description of The universe and so he realized no We just discovered toposis as being a good context for describing physical space but it's not sufficient with these.

00:48:13: toposes need To be promoted to something whose objects look like group points And that off course Is an notion of higher topos?

00:48:21: Of which or multiple type theory provides the corresponding syntax.

00:48:26: So this is how how this homotopy aspect enters the story.

00:48:30: It's in the notion of gauge fields, kind of remarked... And then if I may just add a sentence to this and it doesn't end there at least In these more explorative theoretical part of physics that is not always entirely connected in physics, you want to go further.

00:48:56: Next there's what is called higher gauge fields that have high-gauge potentials where even the gauge transformations between different field configurations don't have a notion of equality when we have to keep going and say they may be two different gauge transformations relating pair or physical field configurations which themselves are related by what physicists call a gauge of gauge transformation.

00:49:20: And it keeps growing this way.

00:49:21: There're ever higher gauge exactly matches under the relevant translations, the most of higher homotopy types in mathematics.

00:49:35: Which is really remarkable.

00:49:37: actually

00:49:41: just because we promised this an earlier episode when we had Graham Priest here as a guest who looked into Hegel for his contradictory logics so the electric method and things like that.

00:49:56: And back then we said, but will have another guest who also said something on Hegel.

00:50:02: so since we are already overtime at would promised it which you'd be willing to say a few words in this aspect than we came from physics or computer science mathematics and also philosophy.

00:50:14: So he touched quite alot of disciplines

00:50:17: right?

00:50:17: Yeah maybe I can tell your story how i arrived because just to continue what we've already said, so one realizes now that physics needs to take place in some higher topos.

00:50:32: What's not called an infinity comma-one topos?

00:50:35: So the next question is as you already mentioned... Now you want to formalize these universal properties!

00:50:39: You wanna say well maybe it's sort of a random infinity comma on top but there should be such things, you want one that inside which physics can take place.

00:50:50: As we said there something like differential form should exist in their the standard infinity comma topo.

00:50:57: so... So he asked now okay what extra structure is it?

00:51:00: and then as we said You wanna formulate this in an elegant universal synthetic way not by declaring lots of tiny little tidbits but saying very broadly What do you need?

00:51:09: And I looked at these stuff idempotent monads and co-monads acting on this topos being an adjunction to each other.

00:51:21: And the funny story actually goes like so I had no knowledge when i did of modalities, didn't know anything about Hegel or all these things... So I was giving a talk in Oxford.

00:51:35: you can still find it on YouTube.

00:51:36: Higher Gauge Theory In A Cohesive Infinity Topos where we were presenting these experiments saying okay We need higher topos a sequence of adjoint monads on it such and such.

00:51:47: And then I said, look if we have this abstractly is my synthetic context that i can construct gauge fields into their quantization and so forth in the same evening by coincidence.

00:51:56: So there was some conference or I don't know which topic Some higher geometry conference in Oxford.

00:52:02: The same Evening There Was A Math Colloquium In Oxford Independent Where Peter Johnston Was Speaking Unrelated.

00:52:08: He Didn'T Attend The Conference But I Saw On Some Schedule I Said Oh Look Here's A Math Coloquium Just This Afternoon by Peter Jones, I just went there because Peter Jones of course is a famous toposterous that was just generally interested.

00:52:20: So i sit there and this was really striking experience because it goes to the board.

00:52:24: he says oh i want to tell you about an idea my colleague William LaVeer at their time who came out with his article on axiomatic cohesion and he said look laveria studying toposis in which they exist as sequence operators.

00:52:41: I was like, this is really crazy because it was exactly the structure that i had just been writing to the board in a few hours earlier on that same day.

00:52:48: so after this thing went home and said okay i need to clearly read what Levere has said about this.

00:52:54: Clearly there's some kind of synthesis going on here.

00:52:57: then i realized that what leviere calls axiomatic cohesion or on a one-top was actually proclaimed on these infinitive toposes.

00:53:09: So then in the wake of this I started digging into this, so i started talking about Le Vierre on the N forum it's on the n-lab discussion side and tried to put his pieces together.

00:53:20: you know bring myself up to speed with what LeVierre had already done.

00:53:24: Then David Caulfield, you know one of the regulars of the N-Forum who's a philosopher of mathematics so he has philosophy background.

00:53:31: He knew what I had never heard off!

00:53:33: He knew that LaVierre had this interest in Hegel and he kept telling me...he kept bringing up the name Hegel The first dozen times or so.

00:53:43: i just completely ignored him..I thought come on ,i want to focus your answer math let's not get distracted too much, but he kept saying it at some point.

00:53:52: I said all right okay.

00:53:53: Let me look into to see what is actually going on.

00:53:55: so then i started reading what Levere actually said about Hegel and then uh... It clicked!

00:54:01: Well yeah that just tried.

00:54:02: and then I start reading Hegel And now went through a bit of friends event for couple months because they seemed bizarre.

00:54:09: things would match nicely.

00:54:11: So Levere who haven't seen this had realized or at least declared, but realized for himself that the funny kind of logic of opposites and duals it is going on in Hegel's science of logic.

00:54:28: Livia claimed this was actually well formalized by a notion of adjoins specifically by iterated adjoints by sequences, not just of a pair or the joint fungus but triples.

00:54:43: And if you have a triple of a joint fungus that gives you a pair over joined modalities by forming the corresponding monads and co-monads.

00:54:50: so that's really.

00:54:51: Levine never quite said it that way.

00:54:52: But what is going on?

00:54:53: His triples of adjoins give your adjoint modality And modality is a mathematical incarnation of equality, right?

00:55:03: It gives math an notion of quality.

00:55:05: You cannot just say how much and true or false but you can say how it's something!

00:55:10: Is that necessarily

00:55:11: true?!

00:55:12: Is possibly

00:55:13: true??

00:55:13: Is it discreetly

00:55:14: true???

00:55:20: So he keeps talking about the the qualitative negation.

00:55:25: So he develops a kind of logic of quality, it's not true and false.

00:55:30: And this is I think why so many people who try to apply ordinary logic To Hegel get frustrated or get lost.

00:55:37: Lavia observed...he calls it objective logic in his writings that what Hegel was envisioning In an intuitive way Is a model logic Of modalities.

00:55:52: And that's how I got interested into Hegel.

00:55:54: It was really just in the wake of what Lévière saw, and then I looked a bit deeper.

00:55:57: maybe I knew more physics than Lévyère knew... ...and I got very excited about seeing you can actually push this further than Lèvière had.

00:56:06: The sequence of modalities keeps going!

00:56:08: You could follow quite few stages through Hegel's writings….

00:56:12: …I thought it would be interesting.

00:56:14: if i may add one sentence?

00:56:16: What I found fascinating reading Hegel book is Says he wants to make philosophy a science again.

00:56:23: So even though the the signs of logic as the book is called strikes many people these days It's been completely out there and do it complete different from what we understand as modern Science.

00:56:31: The preface says no He wants to turn philosophy into a science.

00:56:35: And what he meant was?

00:56:36: He wanted to make it kind of a science of ontology, he wanted to have actual facts about the original notions.

00:56:43: What does that mean to be not-to-be too two to ever quality?

00:56:50: What does it mean for things to be in contradiction?

00:56:52: He wanted this kind of precise sense, even though he was completely lacking from a modern perspective the formal tools.

00:57:02: intuitively, which makes it so hard to read.

00:57:04: It's more like a poem but through the lens of what La Vier observed you can actually play this game.

00:57:11: and there is an ontology where we say well let's take a toposth and every toposst has an initial opposition.

00:57:20: in the sense of Hegel-LaVier There are always the monad on the topos then sends everything to the terminal object and the monad that sends everything into the initial object, empty object.

00:57:32: And they are adjoints to each other!

00:57:35: This is an initial opposition if you regard these monads as being qualities which exist just in every topos.

00:57:45: So if you look at this famous notion of Resolution Aufhebung in German, resolution of opposites and Levere gave a recipe for what would mean formally.

00:57:55: There is the notion that when we have an adjoined pair of modalities it means to be resolved by another modality.

00:58:00: namely both modal types become modal even though they are opposite being two each other.

00:58:06: They're both modal for next guy.

00:58:08: And then there's another adjoint opposition and then find a new opposition.

00:58:14: You can resolve again to find the newer position, this actually turns out to be you know well-defined mathematical process And it turns out that whole sequence like twelve modalities that are actually needs in physics starting with this differential geometry going To super geometry at the end.

00:58:31: so So reaching the fermion said we started the discussion but if she fits into the scheme of Opposition and Aufhebung

00:58:39: Okay, so Thorsten if you don't have a very urgent question then let us.

00:58:44: thank you for being here.

00:58:46: For giving us insight into these many different areas of interest

00:58:51: You have yes?

00:58:52: Thank you very much.

00:58:57: And for the audience we will link to this talk.

00:59:01: you mentioned an Oxford.

00:59:02: We were linked to the end lab that your short dimension which is a huge resource.

00:59:06: and if you have any other reading recommendations We will also link to them

00:59:12: below.

00:59:13: And for the audience, we can also say subscribe!

00:59:16: There's a hype function I learned about recently like the video comment and we'll see us in two weeks with our next AboutLogic podcast.

00:59:26: so once again thank you all.