Way back at the end of last year I put out a call to mathematicians I know: hop on Skype and chat to me for a while about the work you’re doing at the moment. The first person to answer was David Roberts, a pure mathematician from Adelaide. 

We had a fascinating talk about one thread of David’s current work, which involves all sorts of objects I know no more about than their names. I had intended to release this as a podcast, but the quality of my recording was very poor and it turns out I’m terrible at audio editing, so instead here’s a transcription. Assume all mistakes are mine, not David’s.

If you’ve ever wanted to know what it’s like to work in the far reaches of really abstract maths, this is an excellent glimpse of it.

DR: I’m David Roberts, I’m a pure mathematician, currently between jobs. I work – as far as research goes – generally on geometry and category theory, and the interplay between those two. And also a little bit of logic stuff, which I thought I’d talk about.

CLP: So is that what you’re working on at the moment?

DR: That’s one of the things – I have lots of irons in the fire, on quite a few different projects. This is something in progress.

CLP: OK. So, logic.

DR: I guess more precisely, topos theory, which is a many-headed beast, but the classic metaphor is the elephant and the blind man feeling it – it has all these different aspects, and all of them are true, but each aspect feels very different when you’re focusing on just that aspect. There’s a famous textbook called Sketches of an Elephant and that’s sort of the bible of topos theory in some sense. It has all these different things you can do with it. But my interest is specifically in the logical aspect of topos theory; even more precisely, I mean things like building new models of mathematics, like new models of set theory. That’s very broad.

CLP: That is very broad, so I’m going to try to think of a question that will help us narrow it down. Could you in a few sentences give me an idea of what the aim is? How do you intend to use topos theory to investigate logic?

DR: Do you know about the continuum hypothesis?

CLP: Yep, I know that one.

DR: We don’t really know the size of the set of real numbers. The set theory axioms tell us that we have natural numbers – countable infinity – and then the real numbers are a lot bigger, by Cantor’s work, and then we don’t really know if there are any infinite sets between the real numbers and the natural numbers. That’s like baby problem number zero. That property persists all the way through all the different sizes of infinite sets. So if you say, “here’s an infinite set, and it’s bigger than these other sets”, we don’t know how much bigger, and the fun part is that you can build new models of set theory – mathematical universes – where you can adjust the size of these sets, almost at will. This is a well-established subject in set theory, but the particular aspect which I would like to look at is, the category theory side of it is still not worked out. The key word, I think, is forcing – that was invented in the 60s by a set theorist called Cohen – he won a Fields medal for this, settling the continuum hypothesis by saying, well, you just can’t prove that it’s true: you can assume that it’s false, or you can assume that it’s true, as you like. But with his technique you can make new mathematical universes, new models of set theory, but you can only adjust up to a certain level. You can massage the sizes of your sets – the relative sizes of the infinities – up to a certain point, and after that they just do whatever they do, you can’t touch them. Then Easton, a set theorist in the 70s, came up with a much more expanded method which means you can basically change the size of almost every infinite set as you like, up to some conditions which he specified. This method is still a little bit exotic; I’d like to get a better handle on it.

CLP: So how do you intend to do that?

Alexander Grothendieck

DR: This is where topos theory comes in. Topos theory was originally invented by Grothendieck in the 60s to think about algebraic geometry. To go “well, how do we probe what geometry is when it’s not real geometry”, and you can’t picture it and it doesn’t behave like you’d think. The topos is like a collection of measuring sticks: you look at your geometrical object with a collection of measuring sticks and it tells you everything you need to know about this object. A little bit later people like Bill Lawvere and Myles Tierney came along and they realised that the properties of this topos, that Grothendieck and his collaborators had come up with, had a very simple set of axioms which could also be used to say, “this is what sets are like”. At the raw bottom of mathematics, “let’s behave like this”. If you tweak them a little bit and look at slightly different axioms, you get this interesting geometric stuff. This is why I used the elephant analogy – if you’re looking at the foundations of mathematics and think you’re feeling the leg, it might actually be the trunk, and if you’re a geometer you feel the tip of the nose and it’s this soft squishy thing with interesting geometry. So people realised that the method of forcing devised by Cohen had a really nice interpretation with a strong geometric flavour, coming from topos theory. What I’d like to do is to understand Easton’s method of forcing in this topos theory, but the naive thing you want to do will not work – it becomes meaningless if you try to do the same thing that was done for Cohen with Easton’s work. Another key word is ‘sheaves’ – the category of sheaves forms a topos and you can do forcing using this method of sheaves. In the Easton method, you say “I’ll adjust this infinite set, and then I’ll adjust the next infinite set”, and so on. You basically work through the whole universe that way. The question is, how do you make this not just an infinite process, but a process the size of literally every set that can exist? An infinite set is pretty big, but the collection of all sets is too big. You want to come up with a process, and do it as many times as there are things in the mathematical universe, and then at the end still have something left. You don’t want to end up with just a mish-mash, you want to keep some of the nice properties you started with.

CLP: That was good. I think we’ve got right up to the top of where you want to end up, and how you intend to do it. There were many things in that I didn’t understand, so we’ll try and work our way down now.

DR: Cool.

CLP: So, you want to… create a topos out of sheaves which will allow you to create a model of set theory – is that right?

DR: Yep.

CLP: And then you use this method called forcing to adjust the sizes of the infinities in such a way that you end up with something which still looks like the rest of maths that people do when they don’t think about weird infinity problems?

DR: That’s right.

CLP: OK. Cool. So, adjusting the sizes of infinities. I think that’s probably the key to it and I don’t quite get that. The continuum hypothesis is that we don’t know if there’s something in between the countables and the reals, is that right?

DR: Correct.

CLP: And so, adjusting that – would that mean putting more different sizes in between? Different sizes of infinities?

DR: You take the real numbers, and you basically force there to be more of them. 

CLP: Right, easy!

DR: You just go, “I have my set of natural numbers, and I don’t really know how big the real numbers is but what I’ll do is I’ll force a set between the natural numbers and the real numbers”, which in essence just makes more real numbers. You have to insert a strict inequality so you’re pushing the real numbers bigger, to make a space on the number line, as it were. And then you have a set which is between the natural numbers and the real numbers.

CLP: And we’re not inventing any new kinds of numbers?

DR: Not even numbers, no, it’s just pure sets. The real numbers are just a set, I don’t care about them as numbers. They’re just points.

CLP: And the problem here is that you could introduce sets in a way that the structure you end up with doesn’t look much like the real numbers we understand at the moment?

DR: Sometimes you might want to, and sometimes nothing might happen – I might not want to change what the mathematical universe looks like, except for an extra set or two, or a hundred, or pick a number – I can insert that many between the size of the natural numbers and the size of the real numbers. It’s like imaginary numbers where you don’t have a square root of -1, so I’ll posit it and note the properties it has, and then complex numbers just work. So for these infinite sets you posit that they exist, and think carefully about how to write down a system where they’re there, and then you work with it. And it works fine.

CLP: OK. So these sets in the middle don’t have an interpretation as anything in particular? They’re just created because you wanted them to work in a certain way.

DR: Right. It’s like an unknown set, I’ll call it $x$, and I’ll declare it to have these properties: it’s larger than the natural numbers, and smaller than the set of real numbers, and then everything else needs to exist for it to make sense.

CLP: OK, I think I’m happy with that now. That gets us between the natural numbers and the reals. And then you want to do that for the entire hierarchy of infinities.

DR: Yeah. There’s some you can’t touch because of the way the structure is weird –

CLP: That’s interesting! Are these like… are we talking about omegas or alephs? I can never remember which.

DR: Alephs.

CLP: Aleph nought ($\aleph_0$) is the countable set – natural numbers, $\aleph_1$ is the reals –

DR: Well, not necessarily: that’s the continuum hypothesis. The reals are the power set of the natural numbers. There are two ways, starting from the natural numbers, to get another infinite set. Which I didn’t learn when I first learned about this as an undergrad, and the continuum hypothesis didn’t make sense to me – how do you get another infinite set aside from what we’ve got? There’s two ways: one is you take the power set, the set of all subsets of the naturals – that’s the reals – or you do this omega thing. You form a set called $\omega_1$, which is essentially “how many ways can I make crazy infinite queues from natural numbers?”

CLP: Infinite queues?

DR: Yeah, like ordinal numbers, if that makes sense. You can have an infinite queue, and then after that another countable infinite queue, and after that have two people waiting. And they have to wait for – let’s say we list all the even numbers above , and then list all the odd numbers, and then list 2 and 4. And that’s just the natural numbers but in a strange order. If you look at all the ways of doing that, that gives you what’s called $\omega_1$. And that has size $\aleph_1$.

CLP: I can see that that’s a very, very, big set.

DR: And that’s kind of crazy. That really is $\aleph_1$: it’s the thing that is the next larger after $\aleph_0$. It’s quite complicated, really, you’re forming these compounded infinite queues. The real numbers are just all arbitrary subsets, so there’s no a priori reason why these two things should be the same size. Maybe they are, maybe they aren’t.

CLP: And the way you’ve described it, leads me to think maybe they’re not because one sounds bigger than the other. But that’s not how infinity works, is it?

DR: Well, the size of the set of real numbers is greater than or equal to the size of $\aleph_1$, but Kurt Gödel constructed a mathematical universe in which they are literally the same size – there’s a one-to-one map between them. But that was very very special, and somehow a bit too special. That’s like a tiny skeleton of the mathematical universe.

CLP: So in order to construct that he had to add in some axioms or something like that?

DR: No, he only considered sets that could be built using very restrictive methods, called the constructive universe. That’s essentially saying “I only want sets that I can be very explicit about”. You don’t talk about something in terms of “there are some sets that have some properties and I don’t care what they are but I know that they’re there”. That’s a very narrow view of what the mathematical universe looks like. Anyway, he did that and people were like, “Wow, the continuum hypothesis is at least consistent with the rest of mathematics”. Up until then, we had no clue: Cantor had no proof either way.

CLP: You’ve managed to say ‘Gödel’ and ‘consistent’ in the same sixty second span, which I think is the law.

DR: Right, so this isn’t completeness, but the continuum hypothesis is at least consistent, it’s not wrong. It won’t contradict anything. But then when Cohen came along and introduced forcing, we realised that the continuum hypothesis being false is also consistent. That’s when set theorists said logicians can go and they can dial up a mathematical universe with the real numbers whatever size they like. Subject to some small constraints. The trick is to realise that you can do that to more sets. I can take the real numbers, and I can consider the set of all subsets of the real numbers – which is very very big – and then I say “maybe I want to change the size of that”, maybe I want it to be very big, or maybe just as small as it can possibly be, or somewhere in between. Or you could pick your favourite infinite set, and say you want the next infinite set after that to be yea big. So you dial that in and create your mathematical universe with these specified sizes of infinity. But at some point it has to stop, you can’t go beyond.

CLP: Where does it stop?

DR: You can pick a set – an infinite set – of infinities, and say, “I will adjust these”, and then after that things just fall into place as they should. There’s sort of a ceiling. If you consider the mathematical universe as a tower of infinities rising up, at some point you’ve adjusted the natural numbers and the real numbers and a bunch of other things but then there’s a ceiling above which you’ve stopped your manhandling of the sizes of infinity, and they just do whatever they do. You haven’t touched those in the sense of, the axioms of set theory take over. And the trick is, how do you do this when you don’t give yourself a ceiling? When you’re talking about the whole mathematical universe, your argument has to somehow sit a little bit outside because you’re trying to get a handle on everything.

CLP: Is this where you’d say the logic comes in, or have we been doing logic the whole time?

DR: The whole thing is logic the whole time. It’s like, “how do you build a mathematical universe?” You can do it using what’s known as material set theory, which is traditional ZFC set theory, the set theory of Gödel, Cohen, Cantor, that’s one way of doing it, but then you have these really different methods which come out of topos theory with its geometric roots and category-theoretic language. These give a very different perspective on what is at some level the same thing, but where we’re looking it’s not obvious why they’re the same. At heart they’re the same thing, but using different techniques altogether. So we have to reinvent or rediscover the key idea – what’s the engine that runs it?

CLP: OK. Let’s get this clear then. I’m just about OK with the way the set theory works – you described all that. By the engine that drives things, do you mean that in set theory it was the way you construct these sets?

DR: The engine is from the Easton forcing, which is that you need to adjust pretty much everything in the mathematical universe.

CLP: That’s the trick you’re using.

DR: It’s like seeing a regular petrol engine and saying “that’s pretty good”, and then seeing a rotary engine and saying “that does the same thing, but in a really different way”. And if you have even the concept sketch of how that works, maybe you can do something different with it in the end. Because topos theory, and the logic of topos theory, and the topos theory approach to sets, has routes into nontraditional logics, and saying “I don’t really want the axiom of choice to hold”, or “I would like to be constructivist in my logic”.

CLP: All the stuff my lecturers warned me off?

DR: Or better, computable, right? If you want all mathematics to be computable: you prove a theorem, and the proof is a computer program which gives you the answer.

CLP: I’m quite comfortable with computability, and yeah, it’s not very satisfactory. It’s quite small, isn’t it?

DR: Topos theory sort of interfaces with that. We’ve sort of warmed up now. The big question is, “is there an analogue to Easton’s result in these nontraditional logics?” And that’s a completely open question. Can you do what classical set theorists have done, in other foundational systems? For me, that’s the exciting bit, because if I can just reprove Easton’s theorem using topos theory, people will go, “Oh that’s nice, well done, that’s been known since the 70s, you just used a funny way to prove it.” But if I can say, “here’s Easton’s theorem in all these other curious logics”, or perhaps prove that it doesn’t work, that’s kinda cool.

CLP: Yeah, that does sound good. In order for me to understand just how cool that would be, I need to understand what a topos is.

Apparently ‘topos’ is Spanish for ‘moles’, so a google image search turned up this pic of a very hungry fella eating a worm.

DR: Ahh, yes. Let’s see.. are you happy with what the word ‘category’ means?

CLP: Yes – well, you tell me if I’m happy with it. I’ll have a go and you correct me. A category is a – not just a set – it can be a class of things, and some arrows between those things. And that’s as detailed as I’m willing to go.

DR: The ‘things’ you think of as the mathematical objects, and the ‘arrows’, you want to think of as something like a generalised notion of a function. In notation, you might write a function from numbers between 0 and 1 to the reals as $(0,1) \to \mathbb{R}$. That’s a function. And we just say, “what properties do functions have?” and “what axioms capture that notion?” A topos is a category: you have objects you’re interested in – let’s say sets, it doesn’t have to be, and often they’re quite interesting things carrying analytical information, but it can be as simple as just sets. And then we ask, “what properties do sets have”, for instance compared to vector spaces? For example, I can take the disjoint union of two sets and that’s still a set, but I can’t always take the disjoint union of two vector spaces and still have a vector space. And I can take the cartesian products of sets, I can form the set of functions between sets, you have indicator functions for certain sets. You can sort of pull out a small selection of these properties and say, “whatever my category of objects is, can I have these properties which are quite set-like”. And then you say… I can’t avoid this: in the Silmarillion, Tolkien’s Valar descend into Middle Earth, into Arda, and they take on roles. They’re like, “now we’re stuck in Middle Earth, and we are constrained by the rules”. You have a topos that’s very interesting, and you go down inside it, and now it’s your entire mathematical universe and you obey the rules of your topos. It can even look very much like ordinary mathematics, or it might look like these constructivist or computable or what-have-you worlds. You can pretend you’re working with sets, except the logic is different, if you’re thinking inside the mathematical universe represented by the topos.

CLP: OK. I think I can see where your elephant analogy came from now. Someone who’s blindfolded might say the elephant’s got rough skin, or a leg will feel the same as the trunk, or something like that. So you’re describing it by the properties it has, like you said you can describe a set as something you can take a disjoint union of and get the same kind of thing.

DR: Yeah, you don’t think about what the sets are made of, all you know is the things you can do with them, from the outside.

CLP: So this is really not constructivist thinking.

DR: It can either be constructivist or not. You’re not saying things like, “I have these specific elements”, you’re saying “I have this set of elements”. It’s like a homogenous blob: you can’t decide whether things are in or out. Or it could be literally like a set, made up of discrete things that you can point to.

CLP: OK, I have an idea of what a topos is, and the kinds of… topoi(?) you work with. Then you said something about sheaves, which is a word that every time I see it I get someone to explain it to me, and then it leaves my head shortly afterwards.

DR: Think of a Riemann surface – a torus, for instance. A torus is a nice Riemann surface: if you zoom in close, it looks like the complex numbers. You can bung some complex coordinates on it and as you move about things change as they should for complex numbers. But every function on complex numbers on the torus which is complex differentiable is constant. Which is kind of bad, right? I want to understand what the geometry of this space is doing by looking at the behaviour of functions on it, and solutions to differential equations. But every function that’s defined globally is constant. That’s rubbish. I can’t tell anything apart because everything is sent to a number. But if you look at a little patch, just zoom in on a bit and take a little disc and take a complex function there, then you actually get lots of them. You’re really just looking at ordinary complex differentiable functions on the disc and you can do ordinary complex analysis, harmonic functions, all these things. And as you take a different disc, you get different functions. You want to look at all the possible functions that could be defined on that little disc. Or maybe you do something weird like the chop the torus in half and leave off the boundary circles, what functions can you have now?

I thought you might enjoy a picture of a torus, as a visual palate cleanser. This one’s by Wikimedia user Leonid_2, and used under the CC-BY-SA 3.0 licence. Take a breather, contemplate the torus for a bit, then press on.

Or think about periodic functions on the cylinder – that’s like taking a doughnut and taking away a half. That’s just a cylinder, and you’re thinking about periodic functions like $\sin$, $\cos$, trigonometric functions. And this makes sense as complex functions. But I maybe want to consider all periodic complex functions. So a sheaf is a way of collecting all these bits of local information together. I might have two little discs that overlap each other, and I have a function on one and a function on the other, and they overlap. If I have two functions that agree where they overlap, then that’s really just one function on the union of the two discs. I can always restrict functions to smaller domains. Where sheaves came from is this: I can’t just look at all functions on the entire torus, because there are only constant functions, but I could look at the system of functions when I take these little local patches together, or restrict the domain of my functions. That is the sheaf of homomorphic functions on the torus.

That’s a lot of information because I’m thinking about all possible coordinate charts, all possible functions on each one, and that somehow replaces the collection of globally defined functions. And that’s just one sheaf, so then you might have another sheaf which is vector-valued functions, or the sheaf of functions with values the unit complex numbers. You basically collect all of these together, even ones like differential forms or all sorts of weird and wonderful gizmos. Each one of those is now an object in my category. A single sheaf is a huge amount of information about all the functions of a single type and then you collect all of those into a category and that will tell you everything about a torus, or any other sort of geometric object.

But the fun part is, it’s very significant to realise you don’t need to start with just a geometric object. You can start with anything which sort of behaves like a geometric object, and that’s very broad. One of these things very like a geometric object comes out of set theory, which is where the link happens. You say, “what I’m going to do is create an algebraic structure which is enough like something geometric” – it has this order and these things are compatible or not, and you can form the union of them when they’re compatible, which is like functions compatible on the discs. You have this type of behaviour, which is purely algebraic, and it’s built out of things you want to be true about your new universe of sets, or partial information which you’d like to build into the universe of sets. You build something which is kind of geometric – it’s kind of like when you say I need $i$, the square root of -1, and then you’ve got to build all the other complex numbers out of that. You have this condition which needs to be true, so you need to adjust a bunch of things and you end up with a whole list of conditions. They look vaguely geometric, and then you go “take sheaves on it”, do topos magic. The thing you get, when you go inside it and work inside it, looks like the universe of sets again.

CLP: Wow. I’m almost there, but I don’t quite have a picture in my mind of what the geometry looks like.

DR: Oh, it’s completely nongeometric.

CLP: You mentioned how the square root of -1 gives you the complex plane. I know a very small bit about geometric group theory, which might be closer to what you want. The Cayley graph of a group has some geometry.

DR: It’s like a partially ordered set, and you can only really talk about finite approximations to the things you want to be true: I want there to be a new set in between the set of natural numbers and the real numbers in size, but you can’t literally just give it a symbol and work with it. You can talk about finite approximations to the information – facts that a particular set should obey – and you get a huge lattice of little factoids that start from “I don’t know anything” to “I know specifically how these bits and pieces interact”.

CLP: Alright, and then the way that they fit together – what you have to know to get from one fact to the other – is where your geometry comes from?

DR: Yeah, like, I can forget some information for instance by going up the tree towards knowing nothing, then I can look at another fact and say “well, they both tell me something about a little bit of the same thing”, so that’s the same as one slightly larger fact. You get a big lattice of what are called these forcing conditions. They all have to be true for there to be a new set, so I need to make all of them true simultaneously. When you build the category of sheaves on this geometric-like thing, it’s really telling you about this new set. It’s like taking all possible ways of probing the torus –

CLP: And then you know everything about the torus.

DR: – yeah, so this collection of facts about the new set, you know they’re all true while you’re inside the category of sheaves, so it’s like you’re inside the universe of set theory where the new set really exists.

CLP: OK, wow. We now have a topos, which is… a category of sheaves.

DR: Which is a partially ordered set of facts about my new set.

CLP: And your new set is…

DR: It’s like an ideal element that you find inside the topos. Another way to think about it is when you have the integers, and you form the ring of polynomials with integer coefficients. You have a slightly bigger ring with a new thing called $x$, and maybe you say “I’ll add in a relation to say $x^3+3x+1=0$”. Then $x$ is a thing, with properties, but what is it? It’s just $x$, right?

CLP: OK…

DR: The problem is you’re doing this with all of set theory.

CLP: This thing we’ve constructed, is it a model of all possible set theories, or is it one particular set theory?

DR: One particular set theory.

CLP: And you’re looking at all the different ways you can do that.

DR: Yeah, and that allows you to dial up a set theory to have these properties. So I might want the continuum hypothesis to be true for the first five infinite sets and then false after that.

CLP: What would the analogue of the forcing method look like in this world?

DR: Taking the sheaves and then working inside the category of sheaves is forcing. In the material set theory of Cohen, you write down some syntactic expressions and create new names for sets, for this partially ordered set of conditions you’d like to be true. Set theorists may disagree, but my view is – and other topos theory people too, this isn’t original with me – that Cohen’s process is a composite of “I have set theory, I pass to the category of sheaves, and then I use a theorem which is quite detailed and complicated and rebuild Zermelo-Fraenkel set theory out of my topos”. There’s an equivalence between the two approaches but it’s detailed and involved and not hugely enlightening, so in my point of view Cohen’s work is really complicated because you do the composite of three things all at once, and the easy part is the sheaves, the meat of the forcing. The other part is a complicated transitional language between one viewpoint and another, with a step in the middle so you can get back what you started with.

CLP: So have we now reached where you are? Is this as far as this idea has got?

DR: Everything I’ve said until now, that stuff is heading back to Lawvere and Tierney in the early 70s. It’s classical topos theory, in a sense. It happened right at the start when they realised this has so many facets that we didn’t build in. So where I’m picking up is, you’re trying to build one of these topos things, you start with a partially ordered set of information which you want to be true about your new set you want to insert into the universe. The problem is if you want to adjust everything in the universe, you don’t have a partially ordered set, you have as much information as the whole universe. And so you can’t do sheaves in this setting. You can’t naively build up a topos, you might have to assume new axioms and then it doesn’t really work. The whole thing is just, ugh.

There’s a book called Fine Structure and Class Forcing by Sy Friedman. Class forcing is this “let’s change everything in the universe” type approach. Oh, man! I read that chapter in class forcing so many times and I was like, I can say “OK, that’s a thing”, but what does that even mean? How I think about sets, I can see what he’s trying to get at, and then when he actually goes and applies the technique, that’s not something I can touch. It’s just crazy. Earlier on I described this process where I change the first set, and then the next one, and so on. I have a partially ordered set which has all the information about my first set. Then I move up one rung, and I have another partially ordered set of facts which describes the next set. Then you have a ladder, which is as long as the universe, with one rung for each thing you want to change. So you do that sequentially and then at the end you take the union of everything, and it still has to be a model of set theory.

CLP: So you have a tower of mathematical universes, which themselves contain this tower of infinities which we started with in set theory, and doing that in a way that makes sense is the aim of the game.

DR: You take the union of all these things and you still have what you think of as a universe of sets rather than what I think Douglas Adams called “the whole sort of general mish-mash“, all possible views of the universe at once.

CLP: Wow. I think that’s a good place to stop. Is that where you wanted to get to?

DR: That’s the punchline. How do you push past classical logic, that’s the question.

CLP: Well, I definitely have an appreciation of it now, and following you on Google+, some of your posts will make a bit more sense. Thank you very much for that!

You can follow David Roberts on Google+, where he posts not only about topos theory but also open access publishing, women in STEM, and more impossibly large Australian spiders than I care to share a world with.

If this has piqued your interest, you can watch a lecture on class forcing and topos theory David gave at IHES in 2015.