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This is an old question on MO: “Why is a topology made up of ‘open’ sets?”.
To me, it is category theory that answers why?-questions in mathematics.
So far nobody there menioned locale and topos theory, which I think any “right” answer here would have to mention. :-)
The currently top-voted reply almost manages to make the connection to the internal logic of toposes, but doesn’t quite get around to. Same for the comments there.
The next two replies, as far as number of votes go, effectively say that there is no good reason to wonder about the nature of axioms.
Maybe that’s a difference between categorical and non-categorical thinking. There are plenty of examples where progress is made by deeply thinking about the nature of the axioms. If one does it for topological spaces, one ends up in topos theory.
So, Urs, how would the best answer to the question go?
I guess one reason that a connection to locale theory etc. wasn’t made was because the OP (Minhyong Kim) wanted an explanation which could be pitched to an undergraduate class: students that one assumes haven’t been exposed to high-level categorical thinking.
As Jim says, in classical mathematics there are lots of ways to define a topology. But with hindsight (including some awareness of constructive mathematics), it is lucky that the open-set formulation was chosen. (Recently I have been partial to the idea that a topology can be defined as a left exact comonad on the power object as an internal meet-semilattice, but this is the same thing in a general topos.) I think Dan Piponi’s answer, the first answer that Urs referred to, fits well with a constructive line of thinking; e.g., while it is sometimes difficult to affirm that two computable Cauchy sequences of rationals converge to the same real, one can affirm that they don’t by seeing they disagree after finitely many decimal places, and this latter condition is an open-set condition.
There’s a hint of Popper’s asymmetry of falsification to it: it takes just one observation to falsify a general hypothesis, but no amount of observations will verify it.
For me, the most attractive definition involves open neighbourhoods - it is historically earlier (abstracted from metric topologies), and connects to things like filters, and ’knowing numbers accurate to n decimal places’ and so on (and also is good for motivating sober spaces). There is more information contained in such data than in the definition of topology as a collection of open sets, but it can be useful at times to know this.
Perhaps in some sense talking about open neighbourhoods is partway between open sets viewed as a frame, and open sets from a point-set POV, and we can recover both from the neighbourhoods.
So, Urs, how would the best answer to the question go?
First, motivate that whatever you want to call a "space" is characterized by its presheaf of probes. as is done here.
Then point out that among all such presheaves, those that arise as probes of any space will remember how probes glue in the space, hence will respect a coverage. The notion of coverage has a single axiom that is entirely evident.
These are the two simple things you have to accept intuitively as characteristics of "spaces". The rest now follows.
Specifically, from this you have the notion of Grothendieck topos ("subcollections of probes of a space that glue").
Then from these you find the locales via one or the other discussion of aspects of localic reflection and thereby then topological spaces. In particular the mysterious fact (as witnessed by the MO discussion) that the hallmark of topological spaces is "arbitrary unions but finite intersections" follows (and much more does, since this is really the hallmark of geometric morphisms…).
Category theory really allows one to go to the root of phenomena.
Are you going to post that?
Are you going to post that?
Maybe it would be useful to write an exposition along such lines into some Lab entry.
A basis for a system of open neighbourhoods is a good example of a coverage which is not a pretopology, and Urs’ favourite example, Euclidean spaces (or even contractible spaces, in the case of locally contratible spaces) is exactly what you need to give a basis for the topology that manifolds carry. A lot of nice local topological properties are stated in terms of (open) neighbourhood bases, which makes translating them into site/sheaf-theoretic language easy.
Anyway, I’m just rambling on. I don’t have a particular barrow to push. :)
@Harry - no. ionads are to topoi as space are to locales.
Firstly, I think that this should definitely not be posted at the MO question in question. As Todd points out, the motivation for that question was pedagogical. Can you really imagine standing up in front of a bunch of physics, engineering, comp-sci students and saying that? I certainly can’t. I’m all for teaching categorically, but I think that the pedagogical path has to go from the real world to the categorical one, and you don’t get much closer to the real world than talking of measurements, approximations, distances, and neighbourhoods.
Secondly, I actually think that that last sentence applies more than just pedagogically. I’m a topologist. I read Urs’ explanation above and don’t understand a word of it. But I feel that I know what a topological space is and what it is for: it is for studying approximations. I don’t need a categorical explanation on top of that. Now, the categorical explanation might make me feel easier in my mind about dealing with the more general stuff because I can think of it as being “almost, but not entirely, unlike topological spaces”. But thinking of topological spaces as a specialisation of whatever-it-was is not going to make me think “Wow! I’ll study whatever-it-wases instead” but rather “those category theorists! There’s nothing but they won’t make it more complicated.”.
Yes, I’m playing anti-category-theorist’s advocate here a little. But it really does feel like a load of high-falutin nonsense to explain something that is, at its heart, very simple and very concrete.
As for the “mysterious fact”, maybe I’ve just lived with it so long that it is no longer mysterious. It’s like saying that is “mysterious” because it is that way around and not . It’s that way around because that’s how it works in the real world and mathematics models reality, not the other way around.
Trying to follow Urs #8:
First, motivate that whatever you want to call a "space" is characterized by its presheaf of probes. as is done here.
Then point out that among all such presheaves, those that arise as probes of any space will remember how probes glue in the space, hence will respect a coverage. The notion of coverage has a single axiom that is entirely evident.
These would seem to indicate that any particular space should be thought of as a particular (coverage-respecting) presheaf on some fixed category of probing spaces. And, accordingly, a category of (coverage-respecting) presheaves would not represent any particular space, but, rather, would represent a category of spaces.
That is, a Grothendieck topos is now motivated as a category of spaces, rather than as representing some one particular space. Ok, but…
How does this lead to the idea of “open subsets” of spaces, and from there to the idea of categorizing spaces by their “open subsets”, as organized specifically by locale theory? Yes, there is a subobject classifier around with frame structure. But why concentrate on frame structure specifically, when the subobject classifier in fact furthermore has complete Heyting algebra structure (there is a binary operator for implication around, which does not exist as a continuous function on Sierpinski space, and indeed throws into confusion my thoughts on what topological space I am supposed to think of the subobject classifier as representing).
Then from these you find the locales via one or the other discussion of aspects of localic reflection and thereby then topological spaces. In particular the mysterious fact (as witnessed by the MO discussion) that the hallmark of topological spaces is "arbitrary unions but finite intersections" follows (and much more does, since this is really the hallmark of geometric morphisms…).
How would I find myself thinking about localic reflection if I hadn’t already had reason to consider the concept of a locale?
I suppose thinking about geometric morphisms could make me understand why to concentrate on frame structure alone and not complete Heyting algebra structure in general, but I would still need to know how to motivate “geometric morphisms” in the first place (the only “geometric” motivation I know for them depends on already having motivated frames/locales…).
Finally, the mentions of “localic reflection” and “geometric morphisms” here suggest I am supposed to now think of each topos as representing a particular space, and geometric morphisms between topoi as representing continuous functions between those spaces. Which is fine, but I don’t see how to reconcile it with the discussion of probes above, which suggested thinking of a topos as representing not a particular space, but a category of spaces.
Category theory really allows one to go to the root of phenomena.
I believe this may well happen here, so I am eager to understand the perspective you are putting forth, but I’m afraid I need a little more help following you…
That is, a Grothendieck topos is now motivated as a category of spaces, rather than as representing some one particular space.
Yes, so starting from this perspective topological spaces are less fundamental than Grothendieck toposes. Exactly as you say, we start here with having categories of spaces. So next we want to identify actual topological spaces in there. So, as you do, we want to ask
How would I find myself thinking about localic reflection if I hadn’t already had reason to consider the concept of a locale?
Like this: locales are the (0,1)-toposes among the Grothendieck toposes. So we are asking here for spaces whose probes are unique if they exist.
I am supposed to now think of each topos as representing a particular space, and geometric morphisms between topoi as representing continuous functions between those spaces. Which is fine, but I don’t see how to reconcile it with the discussion of probes above, which suggested thinking of a topos as representing not a particular space, but a category of spaces.
I would say the duality between spaces and categories of spaces locally modeled on these spaces is something we get for free: every topos is equivalent to the étale geometric morphisms over itself. So in addition to learning about spaces from the probe-perspective, we learn much more. The probe-perspective informs us not just about the nature of the axioms of topology, but also about how these sit inside a much larger story.
For me, there are three basic intuitions behind the various things that fall under topology, which directly motivate, respectively, the concepts of convergence space, locale, and simplicial set. These three intuitions are sufficiently different that I usually think of them as giving three different fields. Of course, like everything else in mathematics, they’re related, and the standard notion of topological space gives us one way of doing this.
From convergence spaces, we get the standard notion by requiring an extra property of niceness; from locales, we get this same notion by asking for an extra structure of concreteness. Thus, there is clearly something important about them in that they show up in both ways, even though I don’t have an inutition that motivates them directly. (Although a topological space is also a simplicial set with extra stuff, I don’t know a good motivation for the concept in this way.)
From convergence spaces, the concept of open set does not seem very basic, although it can be defined, and then it’s clear that an arbitrary union of open sets must be open, while there’s no particular reason why this must hold for intersections (and indeed it’s easy to find counterexamples). From locales, of course, the concept of open set seems quite fundamental, and the idea that opens can be joined arbitrarily but meeted only finitarily is in the basic intuition behind locales. However, there is a duality between open and closed that makes it somewhat arbitrary, when we represent the opens of a locale concretely in a topological space, that we do so in such a way that joins become unions and meets become intersections (rather than conversely, in which case the opens of the locale would become the closed subsets of the topological space). But the standard interpretation is clearly better than the dual when you do things constructively.
Toby, thanks for sharing this.
I felt that once we have locales, then (sober) topological spaces are but an afterthought and that at this point we may as well conclude that locales is what we really care about.
So the question that remains is then: Why locales?
What would be your take on that?
I think that Steve Vickers’ book gives a good perspective both on Why locales and the relationship with ‘reality’. It is based on Geometric logic and is worth glancing at.
For what it’s worth, I believe Piponi’s answer in the mathoverflow thread is meant to spell out the same perspective as in the Vickers book. That it is the top-voted answer by a huge margin suggests that many (myself included) agree this is an extraordinarily illuminating perspective.
This logical description is just the formal frame dual of the locale picture. While definitely enlightning (this is the enlightning geometry/logic duality of topos theory), to me it still begs the question: why locales? This is the same question as: why frames? so passing there does not intrinsically buy something as far as “Why”-questions are concerned, it seems to me.
To me, the answer “because these are the (0,1)-Grothendieck toposes” goes further and provides an actual conceptual answer.
Well, noncommutative geometry needs going beyond locales, beyond Grothendieck topology and beyond topoi. Noncommutative Zariski corresponds to the exact localizations of the abelian categories of qcoh sheaves and such localizations do not commute. So the order matters – and one does not have the pullback axiom for Grothendieck topology. And in many examples we know it is clear that a good class of localizations is a good notion of a topology!
To make my last statement, #20, more vivid:
Dan Piponi’s post an MO is effectively the reverse of Tarski, 1938. People who say that this “explains why” topological spaces are what they are secretly (without knowing, maybe) saying that they take for granted the statement “The logic of space/locus is first order intuitionistic logic.”
I believe it is clear that if you asked these voters explicitly if this is what indeed they take for granted about the nature of space, a tiny fraction would agree. Therefore it seems illusional to me to accept this as an explanation of anything about topological spaces. It is a very useful equivalent formulation.
I haven’t really been following this thread, but I feel like I have to ask: what is it about topological spaces (or locales, or convergence spaces, or whatever) that invites so much philosophical verbiage? We don’t spend a long time discussing the question of “why groups?” for instance, because the answer is obvious: groups arise naturally in mathematics, so we study them. Can’t we say the same about “spaces” and be done with it? (The original pedagogical question is still interesting and important, of course.)
Re: Urs #20:
But Piponi does attempt to provide an answer to “Why frames?”. That’s the whole point of the ruler discussion!
(I’d still like to understand your perspective as well, but I shall have to return to ask more questions later)
Mike - I think you can observe a difference in how people treat the definitions of group and topological space. At research level, people do complain about or look down on the definition of topological space. E.g. think of your esteemed ex-supervisor’s attitude towards non-Hausdorff spaces. I’ve never heard anyone say that the definition of group is excessively general, but I’ve heard plenty of people say or imply that about topological spaces.
I think discomfort with the definition of topological space is common enough that an external observer - a philosopher of mathematics, maybe - could pick up on it and ask what’s going on.
so much philosophical verbiage
Grief, has pragmatism such a strong hold?
We don’t spend a long time discussing the question of “why groups?”
Maybe not here and now, but it’s been a profitable question for mathematicians to discuss since the 1870s, as it has for physicists since 1930. And it’s still an interesting question if you take it in the sense of “As a means to capture symmetry, why groups?” with the possible answer “Groups only take us so far, look to groupoids, quantum groups, 2-groups, …-groupoids.”
Similarly, we can take “why topological spaces?” to suggest the further question “What concept of space meets the needs of current mathematics and physics?”, and we can read beautiful answers by Connes, Cartier and our very own Urs.
That’s the whole point of the ruler discussion!
I appreciate that at a vague level, but I do think there is a bit of over-simplification at work here. Those undergraduates may be happy about it, but I think at the cost of having been tricked. (People are often happy when tricked. This accounts for many up-votes on internet forums ;-)
Consider this: much more common, as you may know, have been discussions that work exactly the opposite. Namely like this:
Some person would advocate intuitionistic logic. Dicussion partners would jump at him or her, claiming that this is a figment of the imagination that a main-stream mathematician would not need nor want to care about. The proponent will then make the point that intuitionistic logic is not that weird, by saying something like the following:
“Okay, intuitionistic logic may seem weird, but consider this. You all know and love topological spaces. Now, consider the logic where every open subset is the proposition ”a point is in this subset’. Inclusions of open subsets are impliciations, and so on. Then – low and behold – this logic is intuitionistic. This was a major insight of Tarski in the 1930s. “
His opponents would then nod and go “Hm, hm, all right, okay. If something as mundane as a topological space behaves this way, it cannot be too weird.”
In other words, usually people feel that they understand what a topological space is and that this helps them understand the point of dealing with intuitionistic logic.
Therefore, it here seems like a cheat to me to now suddenly turn this around and claim that intuitionistic logic is what we all happily take for granted, and the notion of topological space is to be derived from this.
@Urs: First of all, I think it’s not really intuitionistic logic that’s getting invoked to explain open sets, but geometric logic.
Secondly, professional mathematicians who are used to the concept of topological space have generally found their own ways to understand it, or made their peace with it pragmatically. Thus, it makes sense to explain something unfamiliar to them, such as intuitionistic logic, by showing that something they think they understand, such as topological spaces, gives an example of it. But this doesn’t contradict an observation that actually, the thing that they think they understand (topological spaces) is the way it is because of the thing they don’t understand (geometric logic) — and that moreover, most of us have an intuitive understanding of geometric logic coming from the behavior of real-world measurements (as long as we don’t confuse ourselves by using highfalutin’ words like “geometric logic”).
In other words, if A gives rise to B, then we can use that relationship to explain B to someone who understands A, but also to explain A to someone who understands B.
@Tom and David C: I think I expressed myself poorly. I wasn’t so much complaining about the particular discussion going on here and now (although I do think it’s important not to let philosophy overshadow pragmatism — or vice versa, for that matter).
What I really wanted to ask is, why is there a difference in how people treat the definitions of group and topological space? People who use abelian groups don’t refer to arbitrary groups as being excessively general; they just tack on an adjective for the ones that they want to study. Why are topological spaces different?
Perhaps the difference is that topology is so often stated as being about notions like space, geometry, shape, etc. I think most people will readily accept that frames (or, traditionally, subframes of powersets) are an object worthy of study, but are reluctant to conflate that particular study with the study of space, geometry, shape, etc.
It’s not that the formal concept “topologcal space”, as defined by open sets, is excessively general per se; it’s that (to those who feel this way) it is excessively general as a formalization of the intuitive concept of a shape-up-to-stretching-and-squashing-and-rearranging-and-such.
Groups, on the other hand, are less often described as about any particular intuitive concept (except perhaps symmetry*); they are presented as just some abstract formal concept from the start, and so no one has any call to quibble. Even their name contains no flavorful connotations.
[*: For which I actually would quibble. I always find it odd that groups are described as formalizing the general concept of symmetry, when I think monoids (or perhaps categories, or, if one insists, left-cancellative monoids or categories of monomorphisms) are more appropriate to this purpose. Surely the successor operation on the naturals might just as well be regarded a symmetry as any bijection? Anyway, that’s all irrelevant to this thread…]
By the way, this isn’t all that relevant, but regarding my “esteemed ex-supervisor”, he does generally prefer to use (weak) Hausdorff spaces when possible, but I think that’s largely for pragmatic reasons; for parametrized homotopy theory they had to drop the separation condition. And he enjoys working with REU students on finite topological spaces, which are of course quite rarely Hausdorff.
More generally, one might present Alexandroff spaces as an argument for studying non-Hausdorff, non-sober topological spaces: the category of posets embeds fully-faithfully into topological spaces, but the resulting spaces are not always sober. There is also domain theory, e.g. the Scott topology is also not always sober (as Peter Johnstone proved in a memorably titled and cleverly written note).
Hi Mike,
two responsens:
Concerning the question about abelian groups:
I think for abelian groups people DID ask the kind of “philosophical” question that we are asking here about topological spaces. And have answered it long ago. Going to the heart of the axioms here allows one to find spectra, -algebras, stable homotopy theory. etc.
To me a good answer to these philosophical question is of big practical relevance. It has changed entire fields of mathematics in the past. And is as we speak.
Concerning the space/logic duality: of course, if it is true that we take the relevant kind of logic as granted, then the definition of topological space follows from that. It’s just the formal dual. I am just not sure if that’s really to be taken for granted. How many people who upvoted the measurement example will readily agree that the logic of measurement will violate excluded middle?
I think what the measurement example really only motivates is “subsets that may overlap”. Nothing about closed and open and their special properties (indeed, the comments to the reply seem to witness this). To me, “subsets that may overlap” motivates directly the notion of a category with a coverage. From that I can deduce topological spaces and internal topos logic, but there is a bit of non-trivial work involved there.
But the idea behind Piponi’s description was not just “measurements of some sort, of which more than one may hold”; it was something like “affirmable” or “semidecidable” measurements, with a view to justifying that these are closed under finite conjunction and arbitrary disjunction.
In the comments, closed sets (representing “rejectable” or “co-semidecidable” measurements) were brought up, to some confusion. That’s fine, too. The point isn’t that affirmable measurements are the only kind of measurement one could imagine; just that one might fruitfully study the theory of affirmable measurements, and spaces as represented by these, and this becomes frames/locales/topology.
I think for abelian groups people DID ask the kind of “philosophical” question that we are asking here about topological spaces. And have answered it long ago. Going to the heart of the axioms here allows one to find spectra, E ∞-algebras, stable homotopy theory. etc.
I am dubious about the historical veracity of that statement; it sounds more like the “historical fiction” mathematicians tell ourselves to motivate concepts after the fact. But I’m happy to be proven wrong.
“affirmable” or “semidecidable” measurements
Exactly.
Okay, I think we have exchanged the arguments, and I agree with what you are all saying: If we assume we take this peculiar logic of “affirmable measurements” for granted, then, yes, we get topological spaces for free.
I think the point I am trying to make is: you can start with something even simpler and derive this peculiar logic.
Maybe now it is at me to invoke those proverbial undergraduates:
shall we have a bet who has more success with motivating topological spaces to kids: you starting with the geometric intuitionistic logic of affirmable measurements, or I with collections of test spaces and a notion of gluing them?
Well, I’m not interested in viewing this as a competition; I also want to understand your method of motivation. The more motivation the merrier. And if in understanding your method of motivation, I come to see it as more “fundamental”, in some sense or another, that would be great too.
(I do suspect undergraduates would have an easier time motivating standard topology with the logic of affirmable measurements than via your “First Grothendieck topoi as categories of spaces, then (0, 1)-Grothendieck topoi as particular spaces” proposal, but I’m not really equipped to judge, because I still don’t really understand the roadmap of your proposal.)
So, anyway, instead of the competition of making a bet, let’s engage in the cooperation of helping me reach enlightenment. (This is a selfish kind of cooperation, mind you…). But I’d like to do it this way: pretend I’m just an undergraduate math major neophyte with no existing knowledge of category theory, locales, etc. One day, in one of my classes, I’m shown the open set definition of topology, but I have difficulty understanding what this has to do with the intuitive conception of topology I’ve heard so often (“donuts and coffee mugs”). So I come to you for help. Give me your spiel (and I’ll ask questions as I feel necessary).
I am dubious about the historical veracity of that statement;
I don’t think the historical order matters for this. Even today, we still get insights into spectra by thinking of them as something following the same form of axioms as abelian groups. All derived algebraic geoemtry is deduced this way: find the fundamental axioms of abelian groups,then rings, then their formal duals, then their gluing, then notice that all these fundamental axioms have also more general interpretations and there you go.
It is a pity that people have such a reluctance to this game. Would people feel more free to play this game, much of the things that, say, Lurie is writing down could have been written down 50 years back. In the bulk of the DAG series, he is just playing this game seriously. The game of thinking deeply about what the fundamental axioms of “space” are.
More generally, this is – to me – what all of category theory is about: find the most general abstract formulation of anything, then endulge in how that teaches you thousand and one thing in practice.
(I do suspect undergraduates would have an easier time motivating standard topology with the logic of affirmable measurements than via your “First Grothendieck topoi as categories of spaces, then (0, 1)-Grothendieck topoi as particular spaces” proposal,
No, this is not the way you would say it to undergrads. In teaching, there are two steps:
understand yourself the topic deeply;
then find a simple way to explain it.
The order matters. In the first step all tools are allowed.
And I think it is clear that this was happening in the MO thread, too. I didn’t get the impression that the original question was “I know very well why the axioms of topology are what they are, I just don’t know how to break it to them undergrads.” No, what happened was that somebody set out to teach topology, started at point 1 above and noticed that he was unsatisfied with his conclusion already at that step.
I’ll say how I would proceed in the next comment.
Here is how I would explain spaces to undergrads.
a)
Start with “spaces as things defined by how stuff sits in them” “how probe spaces sit in them”, that is: presheveas explained by a story along the lines of motivation for sheaves, cohomology and higher stacks. (In fact I did write this as notes for a course that I once taught ).
I could also say: I’d essentially follow Lawvere in this step, who in his categories taking seriously is motivated from teaching, right in the first sentence of the abstract
The relation between teaching and research is partly embodied in simple general concepts which can guide the elaboration of examples in both.
That sounds good to me.
Lawvere talks about presheaves as the most simple general concept of spaces there around pages 16 and 17. Again, while the wording there is not designed for undergrads, the simple concepts that he talks about indeed are easily phrased in such a way. You can teach this to high school students.
b)
Observe that, in the above story, the presheaves that we think of as probes of a space will satisfy a special property: they will remember what it means for two probes to overlap.
So there will be collections of maps between probes that our presheaf sees as “the -probes in the space all overlap to form the -probe”.
Depending on taste, we can also phrase this in terms of measurements. “Is a value in the probe ?” Is it in the either of the probes ? Etc. But I think making this precise is the same as deriving the axioms of a topos (because geometric logic is essentially equivalent to topos theory, so I think if we assume we don’t know yet what a topological space is then we don’t know yet geometric logic. But okay, we had this discussion before.)
c)
We need to motivate one single axiom for these overlap relations. To help get that across, I would now secretly prepare the way for the (0,1)-truncation and say something like:
while the above is already very simple, let’s make it a bit simpler even. Let’s concentrate on the simple case where the probes can sit inside each other at most in a unique way. In other words, suppose we have a notion of sub-probes: we know that some probes are sub-probes of other probes.
Then for this case the following axiom for overlaps we take as self-evident: if if a bunch of s covers , then we find sub-probes of the that cover any sub-probe of .
This is the (single) axiom on a coverage.
At this point, I have already fully motivated/defined Grothendieck toposes, of course without saying so.
So at this point now in a better world we’d already be done. This is the “true concept of space in topology”. Grothendieck toposes. And it was most easy, I’d say, and nicely fundamental.
If my undergrads had heard an introductory course on category theory before (or along this one), I could make it even simpler: what we have done is take a collection of test objects, freely joined them together (presheaves = free comcompletion) and then divided out the relation that some cover others (sheaves = localization).
In this fashion Dan Dugger teaches spaces and higher spaces in the beginning of his notes Sheaves and homotopy theory. This also nicely confirms the quote by Lawvere above: Dan Dugger evidently thought of this way to teach it when he was working on his theorem. His theorem (i.e. “Dugger’s theorem”) says: my above way of motivating spaces is the precise blueprint for getting also all higher spaces. Lurie’s Higher Topos Theory is to maybe 60% a fleshing out of this observation of Dugger’s. (If this sounds like a weird statement let me know, and I’ll be happy to explain).
d)
Okay, so we were already very happy. To make us more happy, we again simplify the simple theory and agree to look at the case again where probes sit in at most one way inside each other. In this case, it is clear that we also want to ask what the spaces are such that they have at most one probing by each such probe.
(These are, of course, secretly the (0,1)-sheaves, but you wouldn’t say that to an undergradute. Or maybe, why not? Can’t hurt to hear the term. Is just a word for the simple idea just explained).
At this point my motivation ends. At this point we start doing math. At this point we can now with elementary means derive that these spaces with “univalent probes” are locales.
I’d have to sit down now and think about how to arrange an exercise sheet that guides students through that derivation. If you force me to, I can do that for you.
If time permits, it would also be great at this point to unwind the wealth of nice facts that now all follow from elementary reasoning. For instance the fact that our “univalent probes” now behave precisely according to the Vickery-Piponi-style measurement follows. Can be derived. We didn’t need to know beforehand what it means to say that a probing is “affirmable”. Or that we needed to know beforehand that reasoning about these spaces violates the excluded middle. All this now follows. We just had to know that spaces are things that are probed by probes that may overlap.
I am getting a bit tired now. I’ll stop here and wait for your reaction. If this looks unsatisfactory, let me know, and I can try to expand or otherwise react.
No, this is not the way you would say it to undergrads. In teaching, there are two steps:
understand yourself the topic deeply;
then find a simple way to explain it.
The order matters. In the first step all tools are allowed.
Quite right. My wording there may have been unduly flippant, for which I apologize.
At a brief look at your presentation, I think I’m still getting hung up on the transition from “a (pre)sheaf represents a space” to “a category of (pre)sheaves represents a space”. I’ll formulate some more specific questions about this later today.
My wording there may have been unduly flippant, for which I apologize.
No worries, no need to apologize. I had wanted to say that anyway in reaction to other participant’s comments. This finally made me formulate my reply. :-)
I am only now looking back at motivation for sheaves, cohomology and higher stacks and notice that this does start out talking about topological spaces. That is not necessary. I have now added the following sentence to the first paragraph there:
Actually, the reader need not even know anything about topological spaces. The only assumption we need is that a space is something that has probes by test spaces (points, say). The reader can jump right to The basic idea of sheaves.
Maybe I should re-work the whole thing one day.
If you could get me to understand this stuff, you could get anybody to understand it.
So me, as an inquisitive undergrad would ask, “What do you mean by ’probe’?”
Edit: Can a ’probe’ be defined arrow theoretically?
Edit: Would it help to illustrate with a finite example where you can list all ways to map to explicitly?
“What do you mean by ’probe’?
I think I say that in that text that I keep pointing to.
So probe means simply this: I think of a space , without showing it to you. You think of a probe space . So you are throwing balls into a dark room, photons onto a surface, electrons into a spacetime, strings into a spacetime, to see what the ambient space is like.
So for each probe that you hand me, I give you back a set, called and tell you: this is the set of ways that can sit in .
The only other rule of the game is: if you have two probes and (say a string and a membrane) and you probe by , in that you throw into like this
then I hand you back a map of sets like this
what maps the -probes of that I have told you before to the corresponding -probes given by the way you specified how sits in (and both in ).
To have a finite example:
think of a Tetris game. We have some square lattice, that is . And we have a bunch of tiles, glued from squares. Only that we don’t have them drop to the ground, we assume we can just put the tiles into the square lattice and they stay there.
So then let be the -shaped tile of length 5 + 1, say. You know what I mean, the thing that consists of 5 squares vertically on top of each other and one to the side.
And let be the tile consisting of just two squares stuck together.
Then you can start counting yourself in how many ways the -shaped tile may sit in the lattice. Depends on your choice of lattice of course.
But the -tile sits inside the -tile in exactly, let’s see, 10 ways, right? so we have that is the set consisting of 10 elements.
Write hence for the set of ways of putting the -pice into the ambient lattice. Then fix one of the ways
in the set that the 2-square-piece sits in the -piece. Now if you put the -piece into the lattice, also the -piece lands there. So we have a map (a “map of probes”)
that sends each way to put the -piece into the lattice to a way to put the -piece there.
Even today, we still get insights into spectra by thinking of them as something following the same form of axioms as abelian groups.
Of course. The point I was making is that people didn’t start out by complaining that the notion of “group” was overly general; they just added an adjective when they needed to talk about abelian groups. Then people noticed that both notions could be homotopified, giving rise to loop spaces (grouplike -spaces) and (perhaps connective) spectra (infinite loop spaces, or -spaces). Whether this came about by deep contemplation of the fundamental enigma of abelianness, or by pragmatic consideration of what structure is possessed by an infinite loop space, is beside the point.
Similarly, it seems to me that we should just talk about topological spaces, Hausdorff topological spaces, convergence spaces, locales, sites, and so on, as needed. Many of these notions can be homotopified/categorified, yielding results that are perhaps more different from each other than the corresponding unhomotopified versions. We may arrive at these categorifications by deep contemplation of the fundamental enigma of “space-ness”, or by pragmatic considerations of what we need in order to make some calculation or prove a theorem. My question is, why don’t we get on with it and use whichever structure is appropriate to our needs, rather than complaining about some definition being too general?
Would people feel more free to play this game, much of the things that, say, Lurie is writing down could have been written down 50 years back.
This is irrelevant to the main discussion, but I think that’s a ridiculous thing to say. Obviously any simple modern definition “could have been written down” many years ago in the sense that it was physically possible for someone to have done so. But simple modern definitions have only come about after lots of work by many very smart people isolated the fundamental ideas, pruned away the chaff, and made clear what aspects are essential and what are not. It’s easy to look back at people in the past with the benefit of hindsight and say “gee, it’s all so obvious, why did they take so long to notice it?” But the chances are very good that people in the not-too-distant future will look back at us and say the same thing.
I would venture to guess that if Jacob Lurie had lived 50 years ago, he would still have been a brilliant mathematician, but he would not have written Higher Topos Theory or Higher Algebra. Even imagining that those books could exist depends on a great deal of work on categories, higher categories, homotopy theory, “brave new algebra”, and 1-toposes (to name a few subjects) that was done by many people over the past 50 years. I don’t think even Jacob Lurie could have leapfrogged over all of that work in a flash of insight. Nor, if he had done so, would anyone else have been able to understand what he was doing or why.
“Man proceeds in the fog. But when he looks back to judge people of the past, he sees no fog on their path. From his present, which was their faraway future, their path looks perfectly clear to him, good visibility all the way. Looking back, he sees the people proceeding, he sees their mistakes, but not the fog. And yet all of them… were walking in fog, and one might wonder: who is more blind?” – Milan Kundera
@Urs #44: I think this is a very pretty idea and I really enjoy reading that page whenever I can. It could even be reworked to be appropriate for high school students if someone were inclined.
Back to being an inquisitive undergrad, “But you are using “space” to define “space”.”
Hi Mike,
I don’t mean the higher categorical concepts. Take Structured Spaces. Apart from the fact that there is an infinity-sign in front of everything, the first half is simply a careful abstraction of what the category-theoretic ingredients are when speaking of geometry modeled on sites of formal duals.
Of course it is no good to speculate who would have written what, but I think the point remains: simply writing down the abstract formulations of what one does anyway – for instance passing from “formal duals of rings” to “formal duals of algebras over some algebraic theory” – gives you much of what today is heralded as the “new geometry”.
@Urs 45: Thanks. I like that example, but I’m still stuck because I don’t understand how you can describe Tetris to me without me first understanding what a space is.
“But you are using “space” to define “space”.”
No, I am using “sets of probes” to define space. The space does not exist before or apart from the probes. The space is the collection of the probes.That’s how I define it. Only in giving you examples do I make use of the fact that you already can envision some spaces.
Eric, think of it from the point of view of you in physical space. You don’t really know that there is a space “out there”. What you really know is that when you point your flashlight at something, you get back a set of visual impressions, when you throw a radar wave a something, you get back a set of diffraction patterns analyzed by your radar computer. When you put your hand through a hole in the wall, you’ll get back a sensual impression of how it fits in there. You never have the space itself. You always only know its sets of probes.
Mike,
concerning the fog: I think what I am getting at is that with fog all around us, in maths, when it comes to the theory of concepts, there is a map, and that map is category theory. It’s not that we are completely clueless in the world of mathematical concepts anymore. That was true last century, maybe. But meanwhile we have seen that they are organized by category theory. (It seems to me that you and me in the last months had some fun together producing more examples of this fact.)
This all reminds me much of our last discussion concerning the POV, which was the last time that I was called “ridiculous” for highlighting the usefulness of category theory.
Sometimes I am wondering if with all the category hate mail that you and Todd have apparently seen or received in your life, you have finally forced yourself to forget how great it actually is as a tool for organizing concepts. :-)
@Urs #32:
How many people who upvoted the measurement example will readily agree that the logic of measurement will violate excluded middle?
Actually, geometric logic doesn’t violate excluded middle; the notion simply doesn’t apply! Geometric logic has no negation (or other implication). We’re used to working in accessible truncated models (like locales and Grothendieck toposes) where these (and infinitary conjunctions) can be defined (using an impredicative metalogic), but geometric logic itself has only finitary conjunctions and infinitary disjunctions (at the level of propositions).
That the logic of observable properties has no negation is, in my opinion, behind the intuition that makes people so readily (and falsely) say ‘You can’t prove a negative.’. What they should say is ‘You can’t observe the negative of a typical observable property.’. So if people complain that the logic that you introduce has no negation (and ipso facto no excluded middle), you can trot out this proverb.
I just remembered another reason why we take arbitrary unions but only finitary injections (of opens, and dually for closeds), which I learnt from Paul Taylor. See, the real rule is that we can take unions of a family continuously indexed by an overt space and intersections of a family continuously indexed by a compact space. Only when we restrict to the free spaces on a set (the discrete spaces) do we get, on the one hand, that all discrete spaces are overt and, on the other hand, that only the finite discrete spaces are compact.
(But what is it that makes arbitrary discrete spaces overt and only finite discrete spaces compact?)
@Toby #54: If you allow infinite intersections, the point is both open and closed and every topological space is discrete. I think. I spent some time trying to motivate the definitions in my dissertation.
If you demand infinite intersections of opens be open, you get Alexandrov spaces, which aren’t necessarily discrete in the sense of arising from sets; rather, they are essentially the same thing as preorders.
geometric logic doesn’t violate excluded middle; the notion simply doesn’t apply!
Okay, sure. This is the “affirmable” bit: if a proposition is affirmable (open subset), then its negation (complement) is not (closed subset) and hence negation “simply doesn’t apply” for affirmable propositions. But negation is usually defined in this context as the affirmable bit inside the negation (the interior of the complement) and that together with the original proposition violates excluded middle.
But in either case, this seems to be a state of affairs which I doubt many people would readily accept as obvious if they realized it explicitly: “A topological space is something in which you can affirm if a point is within some region, but you cannot affirm if the point is not in that region”.
I know this would have thrown me as an undergrad and much later still. (Of course I was a physics undergrad, maybe that makes all the difference.)
(I’m “bookmarking” this thread to read more carefully at a later date, but wanted to make an initial remark)
Urs, this is much better. It resolves the issue I had with your original description which was that I couldn’t imagine how to present it even in baby talk to undergraduates. Now, that is clearly because I don’t understand the concepts involved well, but if you had wanted to post an answer on MO then you should not have assumed that any readers understood the concepts any better than I do. So I was reading your original post as “This is what I would say on MO” rather than “This is the hidden message behind what I would say on MO”.
On my initial scan through your main explanation, I’d want to add one more thing which you would call “co-probes”. In the course I taught last semester, I introduced metric spaces via the idea of doing experiments. Imagine that we have a complicated experiment set up which we would like to study. To make the study simple, we start by varying one parameter, and taking one measurement. This means that we can view the experiment as a function , (and so gets us to the study of ). Thinking of the possible configurations of our experiment as the topological space, , we therefore have a path representing the parameter (a probe) and a function representing the measurement (a co-probe).
Not sure how this fits in with your general set-up. But I doubt anyone is surprised that I like the dual nature of this!
Many years ago when the world was young, people (i.e. me plus one or two others) contrasted shape theory (based on coprobes) with singular homotopy (based on probes and hence a singular complex). I even floated a version which looked at the composites of probes and coprobes… and that begins to look like a topological version of the abstract differential space structures that various nLab contributors use.
32, Urs – maybe it is good to know that the concept of abelian group is essentially earlier than of a group, and is due Gauss who used it effectively in Disquisitiones Arithmeticae.
But negation is usually defined in this context as the affirmable bit inside the negation (the interior of the complement)
No, negation is not defined in this context.
If by “this context” you mean the internal logic of the category of open subspaces of a locale (or of a topological space), then yes, this is how you define negation (if you want it). But the context that I was using is geometric logic, in which there simply is no negation. So you start with geometric logic as the logic of affirmable propositions, use this to motivate locales (or topological spaces), then notice (a derived fact!) that you can define negation in these models but (another derived fact) only in a way that (in general) violates excluded middle.
“A topological space is something in which you can affirm if a point is within some region, but you cannot affirm if the point is not in that region”. I know this would have thrown me as an undergrad and much later still. (Of course I was a physics undergrad, maybe that makes all the difference.)
This ought to make perfect sense to a physicist! One point that maybe I should make: it’s not that you can’t ever affirm that the point is not in the region; even if you don’t have the negation operation (taking the exterior), you still may know that two regions are disjoint, so affirming that the point is in one will affirm that it’s not in the other. However, you can’t necessary affirm that the point is not in the region. And this is true: if the point is on the boundary of the region (and hence not in it, since these are open regions), then no measurement, however precise, will ever affirm that it is not in the region.
So to clarify: an affirmable proposition is not simply a proposition that can sometimes be affirmed (as we can also tell by looking at the empty case: the always false proposition is affirmable, even though it can never be affirmed). It is a proposition that can always be affirmed if true (and also only if true, since otherwise one would not use the word “affirm”). If people know something about topological spaces (perhaps just a special case like metric spaces, manifolds, Cartesian spaces, or simply the real line), then you can use this example (like I did at the end of the last paragraph) and point out how open subspaces are like this but non-open subspaces are not. But you don’t need even that example. You just argue under what circumstances you would necessarily be able to affirm a conjunction or disjunction of such propositions (and also note that you usually could not necessarily affirm their negations). Then you get geometric logic. Then you can motivate locales (or Grothendieck toposes, or even Grothendieck-Lurie infinity-toposes) as models of such logic (or topological spaces as concrete models).
@Urs 52: I guess I can repeat my response from last time then. This is not based on receiving “category hate mail” but on my personal experience of mathematics.
Toby,
yes, I mean topological spaces, the topic under discussion. I was being peer-pressured in #28 to say “geometric logic” for this. I should not have given in! ;-)
Since Urs in 52 described a reaction I once had (to something formerly on the page nPOV) as ridiculing someone for highlighting the usefulness of category theory, which is a preposterous characterization of my reaction, I’ll join Mike 63 and concur with his repeated response.
I don’t see how such provocative language (proposing we take bets on who has the better explanation for undergraduates, or describing a point made in 28 as “peer-pressure”, etc.) is helping the discussion. I am however following the discussion on probes and coverages with interest.
@ Urs #64:
Ultimately, I’m responding to your #27
Therefore, it here seems like a cheat to me to now suddenly turn this around and claim that intuitionistic logic is what we all happily take for granted, and the notion of topological space is to be derived from this.
as corrected by Mike in #28. The attempted motivation (which you are calling a cheat) is logic of affirmable (necessarily affirmable with a finite amount of effort iff true) propositions to (by formalising this intuition) geometric logic to (by making accessible truncated models) locales to (by looking at every finitary logical operation that can be defined in these models) intuitionistic logic. Not the circular cheat from intuitionistic logic to locales to intuitionistic logic.
If you want to stand by your claim that doesn’t work on very many people to motivate topology through intuitionistic logic, well, I’ll agree. I just don’t think that anybody (on MO, in Vickers’s book, in this thread) is advocating that.
@ Sridhar #55:
(But what is it that makes arbitrary discrete spaces overt and only finite discrete spaces compact?)
Good question. But at least the main principle is manifestly self-dual, it’s only the discrete functor from sets to spaces that doesn’t respect this.
Mike, Todd,
that repeated response emphasizes the difference between “category theory explains” and “category theory helps to explain”. The first is “ridiculous” and “provocative”, the latter is okay? Is that the idea?
I happen to still think that my analogy between category theory in maths and the theory of evolution in biology is rather apt (including the “explains” versus “helps to explain”, issue, actually), and from the spirit of that analogy I still understand my statements about category theory here and they still look quite sensible to me. I might be convinced of the opposite by rational arguments, but repeatedly ridiculing me for this attitude only makes me think that previous bad experience you two had with related discussion made you lose your temper.
So let’s come back to the issue at hand: when seeing that MO thread title, I had expected some topos theorist would have chimed in and explained that topos theory sheds a lot of light on the “why” in the axioms on topological spaces. Your (Todd’s) comment #4 suggests that the mentioning of undergrads might have prevented you from doing so.
So let’s leave the undergrads out of the way. Suppose I asked you, and I do so hereby, “Why do you think do we define topological spaces the way we do?” and encouraged you to use in your reply as much abstract high-brow and high powered notions as you like (and as far as you feel necessary). What would be your reply?
Toby,
I’ll try to say again what I meant by “cheat”.
It seems to me that the subtlety in the definition of the right “logic of measurement” is precisely as subject to asking “why?” as is that in the definition of topological space. I felt that some people may have been “cheated” because the subtlety was more hidden on the logic side, so that they may not have realize that their original “why?”-question was not answered, but just formally dualized.
Of course, as I said before, if you say that for you there is no subtlety and that the intuitionistiuc logic of affirmable measurement is obvious then, I agree, also topological spaces are obvious.
The first is “ridiculous” and “provocative”, the latter is okay? Is that the idea?
Do I detect a note of sarcasm?
No, it wasn’t the idea AFAI am concerned. “Provocative” was identified quite clearly in 65 and refers to stuff said in this thread, and particularly the aggressive tone which I find off-putting. “Ridiculous” when I used it in the other thread on nPOV referred to the over-the-top sentence “nothing in mathematics makes sense except in the light of higher category theory”, which was eventually toned down to some statement which you are condensing here into the (IMO somewhat weaselly) shorthand “explains”. But yes, I’m with Mike in preferring to tone that down still further. (Edit: By the way, isn’t “repeatedly ridiculing me”, as if you are being attacked personally, just a little strong? I think it’s just certain utterances that seem far too strongly worded that are coming under fire.)
As for the other part: I may or may not take you up on your challenge. In #4 I was really just interpreting Dan Piponi’s answer, not a hypothetical answer of my own, and otherwise I’ve only been watching the discussion. As I say, I find your own reaction to the challenge in Minhyong’s MO question interesting and worth considering.
Do I detect a note of sarcasm?
You seem to, but there wasn’t. You say you also detect aggressive tone and provocation. I don’t recognize my contributions in this, but if you say it comes across this way, I’ll believe you.
That’s how it probably goes with online discussion where one can’t see each other, each side perceives the other as being too unfriendly.
To add my point of view, I feel that I am being subjected here to a quite unusual amount of aggressively-toned responses (as s.p.r moderators we would have cancelled any message calling any participant ridiculous, for instance) to what I think of as a matter-of-fact discussion. From my perspective it is strange that after being openly told “Don’t post this reply!” and I then dare to say that I bet my answer is better after all, it is that bet that is being regarded as a provocation.
I’d rather we all stick to discussing maths and philosophy of maths.
If a statement of mine about that strikes you as provocative, please try to regard it as an intellectual provocation, not as a personal one.
Urs: Please read my edit in #70. Let’s all take a deep breath, and turn the heat way down. Please know that everyone respects you far too much to think of “repeatedly ridiculing you” in any sort of personal way.
if you say that for you there is no subtlety and that the intuitionistic logic of affirmable measurement is obvious
I do not say this. I say that for me there is no subtlety and that the geometric logic of affirmable/verifiable/observable measurement is obvious. And I’d like it to be obvious to you, too. (Not the excluded-middle-defying intuitionistic negation; that’s not obvious and relies on size conditions to even exist.)
Suppose that and are verifiable propositions; that is, if either of them true, then sufficiently careful observation will necessarily allow you to verify it. If is true, then and are true and so can be verified, so to verify you simply verify and then verify . (We need a sort of classical measurement assumption here; treating this in a quantum fashion presumably should give, in the end, a notion of noncommutative space.) Nothing is needed to verify . So by induction, verifiable propositions are closed under finitary conjunction. Now suppose that we have a set of verifiable propositions. If is true, then (for some ) is true and so can be verified, so to verify you simply verify . (This needs no classical assumption.) Therefore, verifable propositions are closed under arbitrary disjunction. (I see no argument that verifiable propositions are closed under implication or even negation, nor under infinitary conjunction.)
For what it’s worth, I agree that higher-categorial thinking explains things even more deeply. I’m just confining myself to elementary explanations of the propositional case, as in Dan Piponi’s answer, which is sufficient for the original MO question.
And I’d like it to be obvious to you, too.
Hey Toby, it’s not that this isn’t clear to me. I’d dare say that more complicated things than this are clear to me. But my point is: I deny that it is clear as a first principle that this is related to the axioms of space.
My claim is that this is clear only in the way that for any mathematician the derivation of any fact from evident first principle axioms can be clear. But it’s not an evident first principle itself. Instead of deriving the nature of topological spaces, this statement is instead deeply influenced by experience with topological spaces.
The problem is with your very first sentence:
Suppose that and are verifiable propositions; that is, if either of them true, then sufficiently careful observation will necessarily allow you to verify it.
To my mind, this presupposes exactly what is to be derived, namely that in the notion of measuring the location of points in space, there is a subtlety with whether a proposition about them is “verifiable” or not.
I think this is a profoundly subtle point. Not that it is hard to understand in itself, but that it is a priori unclear why this is something characteristic exactly of statements about space.
Try it on young kids. Ask them if they think they could not verify the statement “This point on the table is within 5 centimeters from the edge?” and if they think they could only verify the statement “This point on the table is within less than 5 centimeter from the edge.”
The idea that something may not be verifiable because it requires passing to a limit of many verifiable observations is presupposing a notion of limits and convergence which is precisely the subtle point to be derived when asking “Why the axioms of topological spaces?”
So when I say: “If you take it for granted, then the notion of topological space follows.” I mean “for granted” in the sense of “if you claim that this is a self-evident truth to you about the concept of ’space’ that does not warrent asking ’Why’ about it”. But I think in that case you could also simply say that the notion of topological space is self-evident, and does not warrant asking ’why’ about it. And so then you are answering the question the way other proposed to answer it. By claiming that it should not be asked, because it’s “obvious”.
The quoted line makes no supposition about whether “The point is 5 centimeters from the edge” is verifiable (in the sense of being verifiable whenever true) or not. It simply talks about properties one would naturally expect auto-verifiable propositions to have, whatever they should happen to be*.
One could, if one liked, imagine “The point is 5 centimeters from the edge” and such things to be auto-verifiable as well. The frame axioms do not prevent one from doing so; they just highlight the consequence that the space this describes is discrete. And that a different account of what is auto-verifiable would lead to a different kind of space with different properties, such as connectedness and so on.
(*: And in sussing these properties out, it doesn’t use any reasoning about limits or convergence, as such; at least, I didn’t see Toby invoke such things.)
But why is there, a priori, an issue with “verifiable” at all?
Why can’t any statement about the position of any point in any space be verified? By measuring the position exactly and then checking whether it satisfies the given statement.
I am surprised to be the only one arguing that this is a priori subtle. Go ask a random person on the street: “Did you know that there are statements about the location of points in space that can’t be verified?”. I think you will get strange looks.
For thinking about the question “Why the axioms of topological spaces?” one has to try to remember how it was long ago before one learned about the notion of topological spaces and continuity.
(I do think it might be fair to say “Logic of auto-verifiable propositions is all good and great and easy to motivate as formalized by frames. But what makes this topology, in a donuts and coffee mugs way?”. I am currently happy to say “It has turned out to be very fruitful to view a lot of such abstract geometry under this particular natural lens, of starting with the connecting idea that membership in a region of space is auto-verifiable just in case there is wiggle room, and then proceeding to discuss geometry in auto-verifiability logical terms. This isn’t the only fruitful way to look at geometry; one might be interested in the more abstract view of spaces as infinity-groupoids or the less abstract view of spaces as subsets of R^n or such things, but it is one somewhat natural, very fruitful view.” (Well, this wiggle-room = auto-verifiable business isn’t the only thing I say. I have other things I say to myself as well, but this is one thing I am happy to say, sometimes, as needed)
But I may be happier to have something entirely else to say instead, once I come to understand the topos approach to motivation.)
On edit: I wrote this before I saw the above post
Maybe to strengthen the point about the “random person” from #76 further.
There are even professional scientists, among both mathematicians and physicists, who publicly deny the idea that the physical space that we inhabit can be anything but discrete. Who claim that with a microscope of good enough but finite precision, every point in physical space has a verifiable position. These people typically argue precisely that “non-verifiable” concepts such as real numbers cannot be physical.
To these people, the general notion of “topological space” is not an abstraction of the a priori evident notion of “space”, but of something different.
The challenge that we are talking about, the one in the title of this thread, then is this: suppose you talk to somebody who denies that “space” should be a notion where not every statement about its points is verifiable. Try, then, to make that person agree on a set of axioms about space the he or she does find self-evident, such then you can prove to that person that from these axioms the non-verifiability of some statements about measurement follows.
Here is the more detailed story I usually say to myself, should anyone care:
What we really want to say about continuity is this: a function is continuous at p if whenever x is infinitesimally close to p, then f(x) is infinitesimally close to f(p).
Only, what does it mean to be infinitesimally close to p?
Well, one idea* is that a point is infinitesimally close to p if you can’t tell it apart from p; i.e., if it satisfies all the same verifiable properties. And this idea can be made to work gangbusters, but now we’ve got to start thinking about what we want verifiable properties to act like**. That’s where they enter in.
(*Specifically, we end up saying that a function is continuous at p just in case the conjunction of r(x), for every verifiable property r satisfied by p, entails the conjunction of s(f(x)), for every verifiable property s satisfied by f(p) [where these infinite conjunctions are interpreted in a filter completion so as to prevent “infinitesimally close to x” from being identified with “equal to x”]. In other words, a function is continuous at p if the preimage of any open set containing f(p) contains a finite intersection of open sets containing p. With the appropriate assumptions about verifiable properties/open sets, this becomes equivalent to any other standard definition of continuity.)
(**: In particular, we will derive, if we use this framework, that if we are to avoid considering absolutely everything continuous, then we must avoid considering “is equal to p” always verifiable)
Try it on young kids. Ask them if they think they could not verify the statement “This point on the table is within 5 centimeters from the edge?” and if they think they could only verify the statement “This point on the table is within less than 5 centimeter from the edge.”
I don’t know if the kids would get the right answer. And I don’t know how old they’d have to be to understand the right answer when I explain it. But I would hope that a student with experience with measurement would understand that we can never be sure that two points are exactly 5 cm apart, even in a classical world where two points could be exactly 5 cm apart, and even if they really are 5 cm apart. And similarly, if they are exactly 5 cm apart, then they will be at most 5 cm apart, but we will never be sure of that. And similarly if one is a point and the other is a line. I think that this would have been clear to me, if somebody had explained it, when I was a frosh physics major as yet ignorant of the definition of topological space. It does depend on a certain classical point of view to think that it applies to the real world, but classical physics is still fruitful, so that’s enough to think that frames (hence locales, hence topological spaces) might be worth looking into to describe actual space. (Or rather observation in general, because space isn’t the only topological space in real life; it’s just the one that made us put ‘space’ in the name.)
The idea that something may not be verifiable because it requires passing to a limit of many verifiable observations is presupposing a notion of limits and convergence which is precisely the subtle point to be derived when asking “Why the axioms of topological spaces?”
On one hand, it’s perfectly legitimate to use notions of limits and convergence to motivate the definition of topological space to people familiar with those concepts from elementary real analysis (as taught in their calculus class), which will be most if not all of the students in one’s undergraduate topology course. On the other hand, I see no need to use these notions to talk about this business with 5 cm on the table; I’d rely on experience with actual measurements. Do you think that I might have to get out some rulers and microscopes and have them try it out? Or do you think that they will be able to verify that two points are exactly 5 cm apart?
Thanks, Toby. Good that we understand each other now.
I could now reply what I said before, that if you insist that this is obvious and does not warrant asking “why”, then, yes, there is no remaining problem.
I still feel though that this is a bit of cheating. Or maybe not cheating, but the failure to succeed in trying to forget all the things one already knows, and trying to remember what the very basic assumptions are.
Maybe to make that point more strongly, I should allude not to hypthetical kids, but to our actual grand-grand-parents of who we know their thoughts: great minds like Laplace in 1814 were not troubled by inconsistencies in the thought of
[a]n intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, […] #.
The idea that there are non-verifiable statements about space certainly was not a major one at these high times of classical mechanics just a few generations back. Maybe with hindsight we can say that it could have been obvious back then.
But I think the point of the exercise we are talking about is to try to drop any hindsight.
Sridhar,
interesting that you bring up infinitesimals in this discussion, in #79:
As soon as we want to try to understand a notion of “space” more general than that of “topological space”, for instance notions of infinitesimals, we need more topos-theoretic arguments anyway, for instance to motivate synthetic differential geometry.
I don’t think that this causes a problem for Laplace’s Demon. On one hand, Laplace hypothesises that the demon knows certain things, not that it’s something that can verify them by observation. But even so, as long as time evolution is continuous, then it’s enough that the demon can predict any future fact to any desired precision by observing the past to sufficient precision.
I see that the discussion has thankfully returned to the mathematico-philosophical question, but I wanted to finish smoothing things over and apologize for the word “ridiculous”. Along the same lines as Todd’s edit to #70, I would emphasize the distinction between calling a person ridiculous and calling a statement ridiculous — but perhaps even the latter is worded too strongly. We have to agree to disagree, I guess (and hopefully avoid re-having this argument again in the future). To me it is obvious (and supported by numerous examples from my own experience and others’) that while category theory can often tell us the right thing to do in a new situation (I think none of us around here need convincing of that), at other times the right categorical framework only arises out of lengthy consideration of examples at a more concrete level. To you (Urs) that is clearly not obvious, and it looks like none of us is going to convince the other.
As far as the question of verifiable observations go, I’m basically with Toby. My intuition fully justifies the idea that there is a notion of “verifiable statement” that is closed under finite meets and arbitrary joins. It might happen to be the case that in the real world, all statements are verifiable, but we can still imagine other worlds in which that is not the case. And in daily life as a practical matter, I think it is clear that not all statements are verifiable, e.g. the example of the ruler.
On the other hand, “intuition” is a notoriously individual thing, so if this doesn’t help you, then so be it. But it clearly helps many of us.
We have to agree to disagree, I guess
We could do that. We could also discuss the issue, since we haven’t yet. So far we have me making a statement and you and Todd rejecting it.
It started with the list of examples at applications of (higher) category theory that used to be at nPOV. Lists of examples such as this – which I would think to be further extended by examples as I tried to point out in this thread here (largely unsuccessfully, maybe) – made me make the kind of statement you were objecting to.
If I could hear some counterexamples that you are thinking of, either the wrongness of my statement would become clear to me, or I had a chance to react to the rejection.
Of course we don’t have to do this. But it would seem to be the obvious alternative to just disagreeing.
If I could hear some counterexamples that you are thinking of
Well, to start with, how about the invention of categories? Eilenberg and Mac Lane invented categories (and, more importantly, functors and natural transformations) only after they and many other people had studied algebraic homotopy invariants such as homology and cohomology for a long time and were looking for a good way to describe their structure.
By the way, the fact that something can now be considered an “application of category theory” doesn’t say anything about how it arose historically: whether the abstraction or the examples came first.
How much more can be added to applications of (higher) category theory? Will we see ’number theory’ and a link here in the future? Will random graph theory ever join the list?
Thanks Mike, thanks David for further reactions.
I’ll get back to this discussion here a little later, when I have more leisure time.
Concerning David’s question, for the moment just a quick remark and quick observation.
Modern number theory is usually fruitfully studied as a part of algebraic geometry over .This embeds it right away into topos theory. The Langlands program, for instance, witnesses that this is the useful point of view, I think. Also much of the activity concerning mathematics over the “field with one element”.
A noteworthy aside:
you may know the MO question Elementary Number Theory Text from a Categorical Perspective, whose sociology I find illuminating for our discussion here. Somebody asks a conceptual question and is immediately faced with criticism and told that he shouldn’t ask that question.
Todd there being the voice of reason.
Notice the irony: at the same time, as we speak, it is considered to be an exciting deep idea (and rightly so) followed by major mathematicians to ask exactly this question: what happens to number theory as we abstract away the properties of to arrive, say, at the “field with one element”?
Clearly, ordinary category theory has a lot to say to number theory. I was wondering about further up the ladder. I know stacks get used to deal with moduli problems in, say, elliptic curves:
it turns out that most of the problems in dealing with these moduli problems have arisen from our shoehorning the problem from the world of 2-categories into the world of categories. (Barnet-Lamb, Minor Thesis)
I can try to give more detailed comments later, but for the moment just this brief remark: David Ben-Zvi has been very much emphasizing – in research articles and expositions – the fundamental role of -category theory and -topos theory in the Langlands program and many things that go with it. If you just google around, you’ll find a lot.
And yes, it would be good to eventually add something along these lines to the list at applications of (higher) category theory.
Is there a line of argument, if we take higher category theory and physics seriously, which says that whenever we see dynamics about, there’s a good chance that there’s also a cohesive -topos around.
See a bunch of attempts to develop a -adic physics, expect there to be a -adic kind of cohesiveness.
See a quaternionic physics and perhaps there’s a quaternionic cohesiveness?
Looking back to #85, part of my worry about “Nothing in mathematics makes sense except in the light of (higher) category theory” was indeed sociological. Put in print, it looks almost guaranteed to alienate those who do not hold this view. I am trying to imagine someone (maybe you, Urs) getting up in front of a big heterogeneous crowd of mathematicians and uttering this without your tongue at least partly in cheek. (And yet, the nLab plays out before a big heterogeneous crowd, and this statement was presented with a straight face, as it were.)
But let me offer a class of examples. Erdos (1) lived his entire illustrious life with probably zero idea of, and no use for, category theory. Thousands of people have made careers by following up on his ideas, such as random graphs, probabilistic proofs, and many others. Nothing, or hardly anything there, is in the light of category theory. And so none of it makes sense? Or are you telling these fine people they could do it better with category theory? And would you be able to defend that thesis, in detail, within their own areas of expertise?
So by looking through the work of Erdos, you should be able to find many examples where the burden of proof would be on you (Urs) to defend the statement as applied to those examples. I mean, you could be right, but still the burden of proof would be on you, and I’m just not prepared to link arms with you on such a bold claim.
(1) I’m skipping the diacritical mark over the o, which I’d have to look up how to do.
Good that we are finally discussing this!
Todd, you are not quoting the full statement. The full statement was an analogy. It went like this:
Just as one says that
- ’In biology nothing makes sense except in the light of evolution’,
so
- in mathematics, nothing makes sense except in the light of (higher) category theory.
Please think about what this analogy means for a moment. What does “make sense in biology” mean here? I didn’t image that this was not clear, but if it is not, let’s discuss it!
That statement about biology does not mean that every single random effect in biology has an explanation by the theory of evolution. There are thousands of biologists, and famous ones, who never in their life appealed or will appeal to anything in the theory of evolution. The vast majority, I guess. There are probably professional biologisists who don’t even trust in the theory of evolution. (All reasons why I think the analogy is great!)
Nevertheless, when you look at what all these thousands and thousands of biologists are doing from a bird’s eye perspective, you may wonder if there is some global structural explanation for the huge random collection of phenomena that they see. And eventually it dawned on mankind: there is some global structural mechanism at work.
For instance, evolution explains why humans have problems with guessing each other’s intentions when discussing in online forums, while they are superb intention-guessers as soon as the see each other. What evolution does of course not explain is more detailed, more random phenomena, such as why human A happens to fill online forums with discussion of category theory, while human B does not.
But in the statement “Nothing in biology makes sense unless in the light of evolution theory” it is (or so I thought! :-) understood that “makes sense” is in a global structural way. From the bird’s eye perspective.
Urs, I don’t understand the analogy very well because, unfortunately, I don’t understand, and certainly am not competent to explain, the theory of evolution on anything more than a pretty vague level. (This is not to say that I have any sort of disbelief.) Nor could I affirm to what degree evolution plays an explanatory role in biology, generally. Thus, the strength of that statement of Dobzhansky (who was a theologian? and admirer of de Chardin who held similar beliefs), beyond very general philosophizing or theologizing, is not at all apparent to me. Any idea how working biologists would respond to the statement, at a scientific level? Would they openly endorse the statement as something important for biologists to believe and bear in mind? (I can see that biologists might want to pay some attention to evolution insofar as one wants theories in biology to be mutually consistent, but that’s not what we’re talking about, is it?) Or in psychology, a much more tenuous science about something much more complicated, would it have any more significance than just a very general rule of thumb, as in the example you gave of human intentions in online forums?
Not only do I have trouble understanding what scientific significance Dobzhansky’s statement has, I don’t know how far to take the analogy, or if you really mean something stronger. Dobzhansky’s statement, as I understand it, was really in a context of discussing creationism, not biology. It seems that you on the other hand intend the statement about mathematics to be a pretty strong statement about the explanatory power of category directly in the science, something far beyond general philosophizing, something that would apply directly at more detailed levels.
Taking this statement
Nevertheless, when you look at what all these thousands and thousands of biologists are doing from a bird’s eye perspective, you may wonder if there is some global structural explanation for the huge random collection of phenomena that they see. And eventually it dawned on mankind: there is some global structural mechanism at work.
are you claiming that category theory provides a global structural mechanism, giving a bird’s-eye view on or general explanation of what hundreds of workers in (say) Ramsey theory are up to?
(I’m not trying to nitpick here; I’m honestly confused by what you’re saying.)
Edit: sorry, my bad, Dobzhansky was an evolutionary biologist according to Wikipedia. I may have to revise some statements if I learn more about the paper in which the statement appears, which was published in a zoology journal, but with implications for Christian theology.
To continue Todd’s point, why can’t Terence Tao say to you “No, the general overview is provided by notions of structure and pseudorandomness. You higher category types are working out an elaborate part of the structure portion. Ramsey theorists are working out a part of the pseudorandom portion. I can see the whole as the juxtapostion of the two.”
But maybe there’s a glimmer of hope from his own lips:
It does seem that categorification and similar theoretical frameworks are currently better for manipulating the type of exact mathematical properties (identities, exact symmetries, etc.) that show up in structured objects than the fuzzier type of properties (correlation, approximation, independence, etc.) that show up in pseudorandom objects, but this may well be just a reflection of the state of the art than of some fundamental restriction. For instance, in my work on the Gowers uniformity norms, there are hints of some sort of “noisy additive cohomology” beginning to emerge – for instance, one may have some sort of function which is “approximately linear” in the sense that its second “derivative” is “mostly negligible”, and one wants to show that it in fact differs from a genuinely linear function by some “small” error; this strongly feels like a cohomological question, but we do not yet have the abstract theoretical machinery to place it in the classical cohomology framework (except perhaps in the ergodic theory limit of these problems, where there does seem to be a reasonable interpretation of these informal concepts). Similarly, when considering inverse theorems in additive combinatorics, a lot of what we do has the feel of “noisy group theory”, and we can already develop noisy analogues of some primitive group theory concepts (e.g. quotient groups, group extensions, the homomorphism theorems, etc.), but we are nowhere near the level of sophistication (and categorification, etc.) with noisy algebra that exact algebra enjoys right now. But perhaps that will change in the future.
Put his noisy cohomology in a (noisy) -topos and I’ll be convinced.
Urs, I don’t understand the analogy very well
I see. Good to know. At least now we know that we are back to square 1 of the discussion.
Looking around, I find this article here,
which summarizes what is also my understanding of the phrase as follows (p.2, 3):
It is taken to mean that the structure of living organisms only makes sense when viewed as a set of evolutionary adaptations to specific selection pressures. Nothing in biology makes sense except in the light of adaptation. For example, the dictum is cited in support of the view that psychology will make better progress once it learns to see the human mind as a set of adaptations to the human ‘environment of evolutionary adaptedness’ (e.g. Gintis, 2007). The view that nothing makes sense except in the light of adaptation has also been endorsed by some philosophers of science, […]. Philosopher Alexander Rosenberg has argued that the structure of the genome only makes sense when viewed as a set of adaptations and that individual genes are, or at least should be, defined by the purposes for which they are adapted (Rosenberg, 2001). In these discussions it is clear that what is meant by ‘makes sense’ is not only that biological phenomena are as we would expect them to be if they had been produced by these selection histories, but also that classifying biological parts and processes as the products of particular selection histories is a productive way to frame investigations into the details of their form and function.
Let me take that last sentence and make two substitutions for it to become
classifying mathematical parts and processes as the products of general abstract constructions is a productive way to frame investigations into the details of their form and function.
That’s the kind of statement that I am thinking of concerning category theory.
On the other hand, the article points out that this present understanding of the phrase is different from Dobzhansky’s original one. And I admit that I had not been aware of Dobzhansky or of his views before. I had thought this sentence was general folk lore. Well, effectively the article says that it has become folk lore.
But this is my view. I stand in front of the huge edifice of mathematics, and am overwhelmed by the sheer mass of detail. Then I see that there are patterns along which whole mathematical fields and their concepts flow from each other and from ever more fundamental concepts. I see formal categorical duality map whole fields to each other, such as number theory to algebraic geometry, see both sitting as branches in a huge tree where also analysis and differential geometry sit somewhere, all these growing out (not to say evolving ) from a common root concept in topos theory. And suddenly it all makes sense.
And so I can start to write a wiki and organize it all nicely.
Hi David,
the “pseudorandom” aspect certainly exists, but it just does not fall under “makes sense”. There is nothing to be made sense of. It’s just, as the name says, a random collection of phenomena.
(By the way: while the phenomena may be random, our tools for thinking can hardly ever be. Probability theory may be about random phenomena, but there is nothing that prevents that theory to have a good category-theoretic foundations. )
To come back to that analogy we keep discussing, biology is full of random phenomena. Much, much, more than math, of course. Still, I want to make sense of the non-random patterns, and none of these make sense without… (and so on).
By the way, do you know if Tao would list “exceptional” structures in his dichotomy under “pseudorandom”? I like to think of the dichotomy
general abstract structure exceptional structure
instead, and that explains a lot about the world to me.
For instance the notion of “group” is very fundamentally general abstract, very close to the very “root” of that large tree mentioned before. And yet, when we study the examples / models of this concept, suddenly the Monster group comes up. Who ordered that? This is exceptional structure that suddenly appears like a blossom on the tree of general abstraction. A miracle.
Add differential geometric structure (internalize to a suitable topos) and the story repeats. Who ordered E8? A very fundamental general abstract construction gives rise to an exceptional series of models, and so lots of particular (as opposed to general abstract) structure arises.
Continuing this way, with just a tiny further shift in the ambient topos, suddenly 11-dimensional supergravity appears as an exceptional structure out of general abstract cohesion (and as such turns out to contain both as well as the monster!)
Similarly some of the things that Tao lists as “pseudorandom” in his entry feel like “exceptional structure” to me. So, we want to classify it to the best that we can, and I suppose that’s what he is spending a good bit of time on. But by the very nature of randomness and exceptional structure, there is no meaning to be understood here. The list of simple groups, finite or Lie, just is what it is. We just have to determine it and then accept it.
That’s life. (And maybe in a deeper sense so than may appear on first sight…)
Urs, I think that going into vague analogies is a distraction for most of us. The disagreement Todd and Mike have with you is about the general impression on mathematics and the discussion should continue in that direction – into describing the impressions and examples of mathematics totality which are behind our personal experience, and not diverting into biology and even I daresay into non-mathematical philosophy. Todd and Mike well know that one can find an example of categorical thinking behind some parts of structure in every major field of mathematics. They also know that when it applies, the category and higher category theory simplify the things and they often teach us that.
But it is still careful picking needed to choose the math for which the category is essentially (non-artifically usuful), if at all. I see that you are excited when seeing that some new area for you, like rigid analytic geometry also fits. Yes, a concept of a space is having a categorical aspect, so hardly a good concept of a space would not be enlightened by a category theory. Similarly a definition of some algebraic structure. But somewhere else it is artificial. Say in analytic number theory, vast majority of results and reasonings can hardly be helped with introducing categories. You can pick up geometric Langlands which is a flavour of Langlands for function fields, what is geometry again and say this is having deep origin in something categorical, yes, but most people in analytic number theory really have no use from geometric Langlands. In so called hard analysis and in statistics and so on the islands of insightful use of category theory will be still much smaller and more artificial. Even if you can define the object of study categorically this does not say much about the insight to the problems and deep methods which exist there. Here is a list of problems of one harmonic analyst, harmonic analysis being among most systematic (hence closer to physics, geometry and category theory) parts of analysis. And I am quite convinced that trying to solve or elucidate those problems with any help of category theory is almost certainly a waste of time.
P.S. the list of problems given above (link to the harmonic analyst) is a list of problems which ARE meaningful to analysists and group with many similar problems, but not in a way where category theory gives a deep structure but in ways which correspond to other aspects of mathematical intuition and knowledge. It is not true that it is meaningless what is not prone to essentially categorical way of organizing (I consider it trivial that one can phrase things in some formal logic etc. and reduce, but those formalizations are not essential and can vary with the same mathematical content).
P.S. II what one can see as details and not as structure is for another mathematician the meat and subject of his work. We can look for Giry monad in statistics as a categorical gadget, while most probabilists and statisticians won’t care. They have their own problems to solve which will make you and me yawn at their seminar, and this is most of the contemporary mathematics. Of course for the basic structures in fundamental physics, geometry and general algebra it is most useful to take the Gorthendieck and MacLane points of view but there are so many more problems of specific nature. Indirectly of course they can profit from other knowledge in foundations and elsewhere, but if we look at the particular problem and not the cunmulative and revolutionary moves (which belog to systematization and more oftn are akin to category theory). Most combinatorics is really combinatorics (and math is mostly combinatorics) and develiping a system for a particular problem is usually also an overkill and also seeks for a more regular problem, not hte dirty ones.
Let me quote your paraphrase, which I agree with, but shift the emphasis:
classifying mathematical parts and processes as the products of general abstract constructions is a productive way to frame investigations into the details of their form and function.
Now let me perform a similar substitution on the first sentence of Griffiths’ explanation to obtain a statement that I do not agree with:
the structure of mathematics only makes sense when viewed as the product of general categorical abstract constructions
Can’t you see the difference between saying that category theory is a productive way to frame investigations and saying that it is the only way to see the structure of mathematics?