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Here’s another thought, specifically in relation to the analogy with biology and evolution. My understanding is that whenever a biologist asks a question like “why does such-and-such organism have so-and-so behavior or properties?” the answer is always “because it evolved that way”; the interest is in figuring out how and why that evolution happened. In other words, evolution supplies the framework that specifies what an answer consists of to a very basic class of questions. The only thing I can think of which plays a similar role in mathematics (specifying what answers consist of) is the notion of formal proof.
Yes, proof is one of a few concepts playing a similar answer-format-defining role, but I don’t think it’s quite the only one. The format of answers to questions like “How can I determine…?” is specified by the notion of an algorithm, for example.
Also, while proof and theory-building of course play giant, ubiquitous roles in mathematics, and could perhaps be said to together comprise the whole subject, it seems to me that specifically formal proof and formal theory-building play rather lesser (albeit still very significant) roles in mathematical practice.
But I seem to be getting away from the point. I agree that category theory, albeit extraordinarily useful extraordinarily often to a wonderful degree in illuminating and unifying mathematics, is not nearly the end-all, be-all of mathematics, nor ought we expect it to be even in an idealized future. I would be surprised if we could not all agree to this, but perhaps we cannot.
(From my outsider’s position, I am torn, though, on whether this is actually analogous to evolution’s position within the modern study of biology or not. I can see ways to interpret Urs’ statement about the role of category theory on which it is not so outlandish; perhaps we are to understand to take it as a guiding principle at a less than literal level, which may be how the analogous statement about evolution must be understood as well, an acceptable dash of dogmatic exaggeration having been used in the wording simply for aesthetics.)
Can’t you see the difference between saying that category theory is a productive way to frame investigations […]
“Investigation into form and function”, mind you.
and saying that it is the only way to see the structure of mathematics?
Not sure if I recognize myself in the way you put this question. But I am interested in your list of ways of seeing the structure of mathematics. What’s another one?
My understanding is that whenever a biologist asks a question like “why does such-and-such organism have so-and-so behavior or properties?” the answer is always “because it evolved that way”;
No, that’s not right.
There is loads and loads of randomness in biology, of course. For instance different species in the same habitat on different continents tend to look and behave similarly to some extent. But they are still very different from each other. Not all their properties can be explained by big patterns. Just some global structural properties can.
But I am interested in your list of ways of seeing the structure of mathematics. What’s another one?
Not all meaning and content is structure. Take for example one of the central topics in analysis – regularity of solutions of PDEs. Often the proofs of regularity go that you prove weaker level of regularity, and once you know that analyse the equation one more time and gain more and than you continue. Every student of the subject learns this philosophy and the usual way how the gain is accomplished (so this does not reduce just to iteration, there is much more to say here). Is it that this is not about the meaning of mathematics (which you reduce to structure now) ? Or you think that category theory helps here in any way ?
There are many unifying ideas in combinatorics, and more or less most mathematics reduces to combinatorics. For example matroids. They do so many beuatiful things systematically and it is not categories.
Is it that this is not about the meaning of mathematics
You think that a rule of thumb for how to decompose a special problem can be about the meaning of mathematics?
Langlands correspondence is as a special phenomenon as it is regularity of solutions of PDEs. There are people who dedicate life to any of the two phenomena nowdays, the saga of the two are infinite and boundless. Reducing to a naturality square is not any different from some other decomposing. Somebody generalizes solutions to weak solution in some sense, somebody else generalizes from a manifold to a stack.
Your denial is very strange. There are obvious places in deep mathematics where categories do not help to elucidate the deep content. There are ways of thinking there, which are highly appreciated among the experts. So how can one claim that categories is the only way to understand the meaning as the minds in those areas do it differently ? It is hard to comprehend your attitude.
Langlands correspondence is as a special phenomenon as it is regularity of solutions of PDEs.
The Langlands correspondence hardly “is regularity of solutions of PDEs”!?
From #103:
But I am interested in your list of ways of seeing the structure of mathematics. What’s another one?
How about model theory?
109 English misunderstanding: One is as special as another. Langlands correspondence is as a special phenomenon as the regularity of solutions is a special phenomenon. Please reread my post 107 in this light.
@Sridhar 102: But proofs and algorithms are really the same thing! (-:O
There is loads and loads of randomness in biology, of course.
My understanding is that “evolution” encompasses that. Wikipedia says
Evolution is any change across successive generations in the heritable characteristics of biological populations.
For instance, biologists study “convergent evolution” which is the phenomenon of unrelated species still coming to look or behave similarly. Random genetic drift is also a cause of evolution.
But I am interested in your list of ways of seeing the structure of mathematics.
I don’t think I would say there is any one way of seing “the structure of mathematics”, other than tautologously by doing and understanding mathematics. Particular fields have useful organizing principles that display some large parts of their structure. Zoran has mentioned a couple. Geometries organize the study of 3-manifolds; the Yoneda lemma organizes the study of category theory. And some ideas, like category theory (but one might also include things like formal logic and model theory), can play an organizing role across subjects and disciplines. But I see no reason that there should be one “way” to see the structure of all of mathematics.
The effort to think up counterexamples to some principle we don’t quite get seems a little fruitless to me. Would it not be more interesting if we could take specific challenges to Urs’ position, and see where they led? Rather than same vague expression of category theory’s usefulness, why not take up some of the bold claims to be found throughout nLab, and put them to the test? E.g.,
Slogan.
Thousands of definitions of notions of cohomology and its variants. From the nPOV, just a single concept: an -categorical hom-space in an -topos.
Well, David, in this one some variant of the slogan will IMHO eventually succeed to kill any concrete counterexample. There are concrete and very interesting challenges to this principle, which I believe will be resolved in some sort of enriched or noncommutative variant/generalization eventually, like the question of bialgebra cocycles !
Hi,
here an intentionally late reply, since I need to take the speed out of this discussion, otherwise it is using up too much of my time.
In reaction to the claim
whenever a biologist asks a question like “why does such-and-such organism have so-and-so behavior or properties?” the answer is always “because it evolved that way”;
I disagreed and said
There is loads and loads of randomness in biology, of course.
To wich Mike replies
My understanding is that “evolution” encompasses that. Wikipedia says
Evolution is any change across successive generations in the heritable characteristics of biological populations.
But this does not answer generally “why does such-and-such organism have so-and-so behavior or properties?” in a way more insightful than “because it happened to turn out this way”.
Also the weather proceeds by successive changes, and still it makes no real sense to ask “why did this raindrop land precisely on my shoe?, while it does make good sense to ask about global structures, such as “why did the medieval little ice age occur?” (possible answer: because of cyclic lows in solar radiation).
Similary, a biologist does not ask: why does Mr X have a slightly longer neck than Mr Y, because that’s just random variation, but it does make sense to ask “Why does the giraffe have a long neck?” (and the answer is apparently not the obvious answer that they taught us in school).
All this has, I think, its obvious analogs in mathematics. It’s not for nothing that category theory is the theory of universal properties, and not of all properties.
Hi Mike,
I said:
But I am interested in your list of ways of seeing the structure of mathematics.
you replied:
I don’t think I would say there is any one way
Right, that’s what I understood. Therefore I thought I’d ask for your list.
You say:
some ideas, like category theory (but one might also include things like formal logic and model theory), can play an organizing role across subjects and disciplines.
Okay, so we have
category theory,
formal logic,
model theory
Some questions:
how does formal logic play an organizing role? Certainly it is the very substance that mathematics consists of. But is it organizing?
On the other hand, via categorical semantics there is an immense overlap where formal logic is a way to speak about categories, and vice versa.
Concerning model theory: I see people speak of two branches of model theory, one that has been entirely absorbed within category theory, one that has not (from Model-theoretic imaginaries and coherent sheaves, thanks to David Corfield here):
Model theory has evolved in two sharply different directions. One is set-based, centred around pure model theory and applications to various mathematical structures: here even the language of category theory is only beginning to be heard. In contrast is the sort of model theory which is set in rather general category-theoretic, or topos-theoretic, contexts
What is it with the first branch? Which apects is it concerned with that category theory does not inform us about?
(Not a rethorical question, I would like to understand this.)
David writes:
, why not take up some of the bold claims to be found throughout nLab, and put them to the test? E.g.,
Slogan. Thousands of definitions of notions of cohomology and its variants. From the nPOV, just a single concept: an ∞-categorical hom-space in an (∞,1)-topos.
I am thinking of that very entry (cohomology) as providing a rather good amount of argument for that slogan. There is a quite long list of classes of notions of cohomology listed there and explained to fit this slogan.
Since I don’t control the world, I can’t stop people from using the word “cohomology” in a way that does not fit this. But I don’t see many examples of notions that violate this slogan and at the same time have proven useful.
An example to keep in mind: the traditional definition of Lie group cohomology did not fit in, as discussed there.. Then it was realized by Segal and later independently by Brylinski that the traditional definition was not good. They gave a refined definition. I showed that this refined definition does fit the slogan.
I have had already long discussions with Zoran about other examples that don’t fit. To my mind these discussions were nerver resolved. I’d be happy to try to reactivate them. (But maybe we should do that in another thread.)
For instance here Zoran points to bialgebra cocycle. But the first thing that this entry mentions is that, a) there is a standard notion of bialgebra cohomology which is precisely in line with the above slogan, and b) there is a differnt notion promoted by a single person so far.
I haven’t checked yet if Majid’s definition fits the pattern. But also, after having checked that a rather long and encompassing list of accepted and tested definitions fits, I don’t quite feel compelled to check that everyone’s personal definition also does. At some point I’d say the burden of proof reverses. If your notion of cohomology does not share this basic property of a vast number of other accepted notions, then maybe you shouldn’t call it by the same name.
If we don’t do a minimum of notion book-keeping, we’ll end up in a mess of notions. See the other thread.
Re #119, there’s nothing I disagree with there. The proposal of a bold characterisation of a cluster of notions is often a very fruitful activity (think Eilenberg-Steenrod). Testing that characterisation is a means to help understand some situation, e.g., I’d like to learn what results from your discussion with Zoran about his proposed counterexamples.
Of course, it isn’t a black or white issue whether, after attempts to reformulate it to make it fit, one responds to a counterexample by excluding it, or else gives up on one’s characterisation. Nor is it black or white whether one accepts a reformulation so as to fit a proposed characterisation as doing justice to the concept. But hopefully there are reasonably unequivocal cases, perhaps your Lie group cohomology case.
The Eilenberg-Steenrod axiomatisation of homology and cohomology threw up interesting phenomena such as that a previously designated homology, Cech, was no longer a homology. And then cohomologies emerged not obeying the dimension axiom.
I’d like to learn what results from your discussion with Zoran about his proposed counterexamples.
Me, too. Zoran still has query-box complaints at cohomology, which we tried to discuss here. But I am not sure if I understand the remaining complaints. Probably my fault, would be happy to try again.
But this does not answer generally “why does such-and-such organism have so-and-so behavior or properties?” in a way more insightful than “because it happened to turn out this way”.
Yes, my choice of wording “the answer is always ’because it evolved that way’” was poor, sorry. If you read on in #101 you’ll see that I said
the interest is in figuring out how and why that evolution happened. In other words, evolution supplies the framework that specifies what an answer consists of to a very basic class of questions.
So I shouldn’t have said that evolution is the answer, but rather that it tells us what it means to be an answer.
@Urs #118: I think you’re missing the point. The point isn’t that there are other subjects which do what category theory does, or something comparable to it. The point is that category theory does not organize all of mathematics.
For instance here Zoran points to bialgebra cocycle.
I moved my answer to this (Urs 119 passage) in another thread.
But also, after having checked that a rather long and encompassing list of accepted and tested definitions fits, I don’t quite feel compelled to check that everyone’s personal definition also does.
Urs, great majority of cohomology theories are the abelian ones, hence fit already into the Grothendieck’s 1956 Tohoku unification or Cartan-Eilenberg, or more modern to the derived categories setup. Once this taken into account, the list reduces to a very short list to check on, not thousands. Then the effort becomes much more interesting. The bialgebra cocycle is one of very few examples which are not unified already 55 years ago (or defined later from the start in the same canonical way). I do not understand excitement on redoing many examples from 55 years ago and in the same time calling the effort to do examples which are really out of that old list not interesting.
I am only looking at the latest replies here now. Needed to save myself from this discussion.
I guess it’s good to stop here. Sorry for stealing all our time with this.