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September 3, 2019

The Narratives Category Theorists Tell Themselves

Posted by David Corfield

Years ago on this blog, I was exploring the way narrative may be used to give direction to a tradition of intellectual enquiry. This eventually led to a book chapter, Narrative and the Rationality of Mathematical Practice in B. Mazur and A. Doxiades (eds), Circles Disturbed, Princeton, 2012.

Now, someone recently reading this piece has invited to me to speak at a workshop, Narrative and mathematical argument, listed here. Reflecting on what I might discuss there, I settled on the following:

The narratives category theorists tell themselves

Category theory is an attempt to provide general tools for all of mathematics. Its history, dating back to the 1940s, is characterised by ambitious attempts to reformulate branches of mathematics and even mathematics as a whole. It has since moved on to influence theoretical computer science and mathematical physics. Resistance to this movement over the years has taken the form of accusations of engaging in abstraction for abstraction’s sake. Here we explore the role of narrative in forming the self-identity of category theorists.

Years of hanging out around this place have given me plenty to talk about, but perhaps people have some particular insights they’d care to share.

I know some would rather avoid direct identification as a category theorist, instead describing themselves indirectly as a mathematician/mathematical physicist/computer scientist, etc. who does research in/looks to use the tools of category theory. But as the kind of person who shows up to CT2019, Applied Category Theory 2019 or SYCO 4, do you have story-like ways of thinking about your longer term research path? This might be in relation to achievements of historical figures of the tradition (Mac Lane, Kan, Grothendieck, Lawvere, etc.), things your supervisor told you, things you tell your students, or perhaps in relation to alternative ways of doing research in your area?

I imagine common themes will include: isolating the essence of an idea manifested across different situations; providing a common language; offering guidance for theory construction. These can be read from opening motivational paragraphs of books such as Tom’s Basic Category Theory:

Category theory takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level. How is the lowest common multiple of two numbers like the direct sum of two vector spaces? What do discrete topological spaces, free groups, and fields of fractions have in common? We will discover answers to these and many similar questions, seeing patterns in mathematics that you may never have seen before.

Or Emily’s Category theory in context:

Atiyah described mathematics as the “science of analogy.” In this vein, the purview of category theory is mathematical analogy. Category theory provides a cross-disciplinary language for mathematics designed to delineate general phenomena, which enables the transfer of ideas from one area of study to another. The category-theoretic perspective can function as a simplifying abstraction, isolating propositions that hold for formal reasons from those whose proofs require techniques particular to a given mathematical discipline.

What other narrative themes are operating out there?

Posted at September 3, 2019 10:51 AM UTC

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Re: The Narratives Category Theorists Tell Themselves

Another reference for the same themes you mention could be Tom’s Perspective on Higher Category Theory.

Posted by: Mike Shulman on September 3, 2019 12:18 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Thanks for reminding me! That was part of a series, including Urs on Intrinsic Naturalness (just 143 comments to sift through) and your own Confessions of a Higher Category Theorist.

Posted by: David Corfield on September 3, 2019 1:30 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

I see Tom remarked

We all feel that we know what we mean by this ‘generalized the’, but handwaving is what it is.

Does he know it’s since been resolved by type theory, generalized the?

Posted by: David Corfield on September 3, 2019 1:41 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Why not ask him?

Seriously: sure, I’ve certainly heard people say that it’s been resolved, and I know the rough idea. But ultimately, as for any claim in any area I haven’t studied intensively, it mostly comes down to trusting the expertise of experts.

There is another factor, though. Saying that the usage of “the” has been resolved isn’t like saying that the Riemann hypothesis has been resolved. The “the” issue isn’t a crisp, clear, unambiguous mathematical question or conjecture. Indeed, it’s part sociological, in that it’s partly about how practising mathematicians use language. Saying “does Tom know X” presupposes that X is true, and since X isn’t a black-and-white mathematical statement, there’s space for differing opinions. And as you know better than me, that’s the standard situation in foundational matters.

Posted by: Tom Leinster on September 13, 2019 9:14 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Sorry, I was being rather tongue-in-cheek.

But it’s not a bad idea to think in terms of contractible types to represent the universality of a construction. E.g., given a pair of types, AA and BB, we have that the type of types that behave as the product of AA and BB is contractible, hence we’re allowed to say ‘the product’. Of course, you first need to buy into type theory.

I tried to explain this in a paper of mine.

Posted by: David Corfield on September 14, 2019 10:08 AM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

I don’t think you even have to buy into type theory to appreciate the idea, which has been around a lot quite a long while, that a category of universal constructions of a given sort forms a contractible groupoid, i.e., a groupoid equivalent to a (the) terminal category.

That’s not to say I reject what Tom is thinking, especially since I don’t pretend to know what he’s thinking.

Posted by: Todd Trimble on September 14, 2019 1:15 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Thanks for forwarding the link to your paper, David. I’m looking forward to it.

I should clarify that I’m not taking a sceptical position here, just a cautious one. And despite the generous last sentence of Todd’s comment that suggests I might be having deep thoughts, I am not having deep thoughts :-)

All I have in mind is something like the following. Someone might observe (as many have) that the set theory most mathematicians pay lip service to and the set theory most mathematicians actually use in day-to-day life are rather different. And they might then pose the challenge of reconciling that difference.

Someone else might respond that ETCS answers that challenge.

But much as I’ve been an advocate for ETCS, and much as there are strong arguments behind the view that ETCS answers the challenge, I do think some caution is warranted. Maybe we can only be really sure that the original, partly sociological, problem has been solved after a new generation of mathematicians have grown up using ETCS as both their day-to-day set-theoretic language and their axiomatic framework for set theory. (And I can think of some specific obstacles that might prevent that happening.)

That’s all I mean!

Posted by: Tom Leinster on September 14, 2019 2:13 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

I didn’t say explicitly, but should have done, that the stuff about ETCS was just an analogy. It wasn’t intended to have any direct connection to the issue under discussion.

Posted by: Tom Leinster on September 16, 2019 11:12 AM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

I think there’s more to what David is saying than the long-known idea that universal constructions are unique up to a contractible groupoid of choices. The point is rather that in HoTT, the notion of “contractible groupoid” and the corresponding meaning of “the” appears directly and automatically in the foundations of mathematics and logic, in a way which is not the case even in ETCS.

In first-order logic, we can give direct meaning to “the” by postulating a definite description operator, ι 0x.P(x)\iota_0 x. P(x) meaning “the xx such that P(x)P(x)”, with an axiom saying that if there is a unique xx such that P(x)P(x) then P(ι 0x.P(x))P(\iota_0 x.P(x)) (and hence, by uniqueness, any yy such that P(y)P(y) is equal to ι 0x.P(x)\iota_0 x. P(x)).

We could try to do something similar (perhaps in ETCS) with contractible groupoids, with say an ι 1x.P(x)\iota_1 x. P(x) where if there is a contractible groupoid of xx’s such that P(x)P(x) then P(ι 1x.P(x))P(\iota_1 x.P(x)). It’s not quite that simple, because we have to have some way to specify the morphisms in the contractible groupoid too, but I think that can probably be dealt with (e.g. along the lines of the treatment of replacement in section 7 of my paper on structural set theory).

But in HoTT, once we’ve defined the type XX to which xx belongs, and the predicate PP, to prove that the type of xx’s such that P(x)P(x) is a contractible groupoid we simply prove that the type x:XP(x)\sum_{x:X} P(x) is contractible. The correct notion of morphism in this “groupoid” is automatically produced by the definitions of XX and PP. Moreover, since the proposition IsContr(A)IsContr(A) is itself a type, defined as a:A b:A(a=b)\sum_{a:A} \prod_{b:A} (a=b), once we’ve proven IsContr( x:XP(x))IsContr(\sum_{x:X} P(x)) we can simply project out its first component and obtain our ιx.P(x)\iota x. P(x) together with a witness that it satisfies PP. That is, we get a direct interpretation of “the” that applies uniformly to ordinary uniqueness and to uniqueness up to a contractible groupoid, as well as uniqueness up to a contractible \infty-groupoid, without having to add any new ι\iota operators to the syntax.

Posted by: Mike Shulman on September 15, 2019 2:01 AM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Right. I also find it exciting that we have an advance on the path initiated by Russell in 1905 on ‘definite description’, which you might say was one of the founding acts of analytic philosophy.

Looking back to Tom’s original post, he raised the issue of ‘generalized the’ in the context of a wish for a more satisfying definition of higher categories, with such a wide range of algebraic and non-algebraic notions of \infty-groupoid having been proposed.

Perhaps we could say that again we are dealing with the idea of a concept, XX, with multiple realisations, which though they look different are really ‘the same’, and so any of them is ‘the XX’.

I remember as a child being troubled by the doctrine of the Trinity. How could one thing be equal to each of three things, no two of which are the same: “three persons, one being”?

That got me thinking about the computational trinitarianism thesis. In which kind of framework could one think about how the parts are the same? Or is it that there will be no such single way, but a category-theoretic way, a type-theoretic way, and so on.

One step further and we find that all such ways are really ‘the same’.

Posted by: David Corfield on September 15, 2019 9:00 AM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

You’re right, Mike: there’s more to it than that. Perhaps it would have been better to say “you don’t have to buy into type theory to appreciate a simpler version of the idea, which is that the category of universal constructions of a given sort forms a contractible groupoid”, and that’s all that I meant. But it’s unquestionably the case that homotopy type theory has something new to say here.

Posted by: Todd Trimble on September 15, 2019 8:03 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Like many (most?) who identify as an “applied category theorist”, the main story I tell myself is that category theory is primarily a way to get compositionality (which itself is necessary for scalable reasoning in any scientific domain). It is not even unique in that regard: linguists have been thinking about compositionality for over a century, and concurrency theorists and programming language designers also have deep ideas about it. In my opinion CT distinguishes itself from other approaches to compositionality by its other, secondary benefits: mathematical cleanliness, and flexibility as a “universal language”.

Obviously “traditional” category theorists are vastly outnumbered by programmers who use monads to structure their code, with a big grey area in between the extremes. The lower bound has recently been raised by the sudden importance of profunctor lenses, which are really quite complicated. (It is rumoured, for example, that the source code of Minecraft contains an implementation of profunctor lenses in Java.) I suspect that by now, the majority of people who actually identify as category theorists approach the subject in a very different way to someone who once took a course in algebraic geometry or topology.

(Incidentally, part of the story of my research over the last few years has been discovering that “classical” CT is not totally useless. This will not be very shocking to nCafé readers.)

Besides having different motivation, applied category theorists emphasise different tools. While traditional category theory asks “what are the morphisms?”, applied category theory more often asks “what are the objects?”. Nearly all categories of interest are monoidal, which makes categories with structure relatively more important vs universal properties. The link between free monoidal categories and geometry is extremely important because it yields practical geometric methods for doing calculations in the categories of interest (ie. string diagrams) - but only up to dimension 3 (and more often 2), after which it stops being practical.

Posted by: Jules Hedges on September 3, 2019 1:33 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Thanks, Jules.

If category theory deals well with compositionality is this primarily via its relationship to operads, multicategories, generalized multicategories and their many relatives? (Whatever happen to Baas’s work on hyperstructures)?

While traditional category theory asks “what are the morphisms?”, applied category theory more often asks “what are the objects?”. Nearly all categories of interest are monoidal…

I guess someone might say that one person’s object of a monoidal category is another person’s morphism in an one-object bicategory, the latter perhaps then inclined to categorify horizontally for a multi-object version.

Posted by: David Corfield on September 3, 2019 2:12 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Currently, almost everyone (including myself) uses monoidal categories with extra structure, the usual suspects being symmetry, compact closure, bialgebras, Hopf algebras, Frobenius algebras, and daggers. There are good reasons to believe that (coloured) operads are better - several of the above structures can be viewed as hacks to squash something “really” operadic into the shape of a monoidal category. But as things are, David Spivak is the only person I can think of who’s really pushing operads for applications.

Viewing objects of a monoidal category as 1-cells of a bicategory wasn’t what I had in mind. I use the term “openification” for the process of, starting from widgets, asking what are the appropriate interfaces (ie. objects) through which “open widgets” can interact? For example, starting from graphs we pass to a category of cospans whose legs are discrete graphs, resulting in a category of open graphs. A standard sledgehammer that people use for openification is Fong’s decorated cospans.

Posted by: Jules Hedges on September 3, 2019 3:26 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

While traditional category theory asks “what are the morphisms?”, applied category theory more often asks “what are the objects?”.

It took me a few minutes to understand what you meant by this. At first I thought you were referring to the classical quip that the morphisms of a category are more important than the objects, and claiming that the opposite is true in the kind of categorical network theory that has recently appropriated the name “applied category theory”. But now I think I see that you’re referring instead to the dichotomy that around here we’ve gotten used to calling vertical versus horizontal categorification: whether you start from the objects and then add the morphisms, or whether you start from the morphisms and then add the objects.

Of course, horizontal categorification also has a long pedigree in both pure and applied-in-the-old-sense category theory. Viewing objects of a particular monoidal category as the morphisms in a 1-object bicategory — or the elements of a particular monoid as the morphisms in a 1-object category — is only the beginning. Once you’ve done that, the standard thing to do then is to consider what multi-object (bi)category that 1-object one sits naturally inside. For instance, the monoidal category (Set,×)(Set,\times) naturally sits as the hom-category from 11 to 11 in the bicategories SpanSpan and ProfProf, while (Set,+)(Set,+) sits inside CospanCospan as the hom-category from 00 to 00. In many cases, like these, the resulting (bi)category is indeed monoidal, and the one object of the 1-object version is its unit object. And we can also ask, of course, for how a particular kind of structure on a monoid or monoidal category can be generalized to a structure on categories or bicategories that induces it in the 1-object case. So I think your “openification” really is an instance of horizontal categorification.

Posted by: Mike Shulman on September 3, 2019 6:05 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

For instance, my paper Constructing symmetric monoidal bicategories, which some people doing “openification” like to cite, is part of a line of research that was originally motivated by applications to classical algebraic topology, specifically the bicategory of parametrized spectra constructed by May and Sigurdsson.

Posted by: Mike Shulman on September 3, 2019 6:34 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Thanks - I’d convinced myself at some point that it wasn’t an instance of horizontal categorification in general, but I clearly need to come back and give this some more thought

Posted by: Jules Hedges on September 4, 2019 11:49 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

I identify as a “category theorist” in the attitude, if not in the outcome of research, since before the beinning of my PhD, and I had not the occasion to spend it in a place where people actually do category theory, instead only using it (and not always well; but maybe that’s a story for a different talk of yours :-)).

Building a consistent narrative has been crucial helping me cope with the involuntary loneliness I had to face for years (and I betcha you will see why I felt so lonely after reading my comment?), so I feel I can say a lot on the topic. (It goes without saying that this is 100% personal opinion, even if strongly advocated.)

TL,DR: Category theory is all about telescopes, and the kind of category theorist you are is determined by what you’d do with a telescope. I am the kind of guy who doubts the telescope ever existed and asks questions like “yeah, but what are eyes?”.

When I was depressed, lonely and I felt misunderstood because “I looked for abstraction for abstraction’s sake” I’ve been gifted with the following analogy. It has been a precious piece of advice through the years. What follows is the transcript of a very old email:

To me, category theory is a point of view, or rather a point of observation allowing me to look better at Mathematics. Like a telescope. Something that allows you to look at stars far away that you wouldn’t see with with naked eye. If someone gave me a telescope that’s what I’d use it for. Someone else would instead unmount it, to look inside it, and when asked why he’d respond that to look even further stars you need to build more powerful telescopes; and to build stongest lenses you study telescopes, not stars.

At this point there’s a tension between these two opinions. There are no stars outside our reach, only scattered rocks, dust, and gaseous masses. There is nothing to see “there”. Well, I’d say, from a certain point of view rocks and stardust are interesting in their own right. But how can you be sure there are not other stars outside the reach of your telescope? Ok then: show me.

In short, the point is: such a tension sometimes generates friction and misunderstanding, but it also has a great dialectic potential. Try to find a balance between the two points of view, it will make your life easier.

Even if it had a completely different result than the one expected, this message had a great impact on me. It had a major in defining my narrative and my approach to Mathematics. It still does, secretly, because the passing of time is making its job in turning me into a more moderate kind of guy, but I’ll probably never be “that” kind of moderate guy. My fault, probably.

Anyway, I answered as follows.

It’s good to build telescopes. On the other hand, why do we study only the visible universe? This is a stifling limitation of our descriptive potential: in the opposite category of our universe, the following statement is true: “there are more things in philosophy, Horatio than are dreamt of in Heaven and Earth”. How can you be sure that we inhabit this or that?

Why don’t we try to formulate a general definition of “universe” in order to understand what a universe can or can’t be, and what it is forced to contain? Can a universe exist without stars? Can there be one without beings capable of observing them? And if yes, in which sense the universe “exists”? Given a universe, there’s a universe that is pointwise “dual” to the first. Right? Wrong? Is this dualization process involutory? Given a universe different from mine, how can I be sure that they are actually two? Maybe the one I’m looking at is equal to mine, just a little deformed? Put all universes in a box: what is this box, and more important, where is this box? How does the box react if I shake it? Put the universes in a pot, and boil for a while: how does the network of relations between universes change?

Your (plural) questions are “how can I study the stars I can see?” or maybe even “how can I study stars I don’t see?”. Mine is: what is a telescope and why it is what it is, and not something else? What is the act of vision throug which we employ the telescope?

Stars are merely a figment, the pretext for asking The Question, the only question that exists and lies beyond words, in a place forbidden to language.

I like Mathematics when it’s substantiated with the holistic claim to slit open the throat of the faceless Gods worshipped by the gods we created. Mathematics, and category theory, is what presupposes language. Structural mathematics is the viaticum I need to cross language unharmed to its dangers, to reach beyond, beyond beyond, to the other shore, to nirvāṇa.

Even when I try to externalize my views, this attitude doesn’t seem so strange. Observation influences the observed; language describes reality at least as much as it creates it; the world might be a mental construct (for a suitable meaning of “mind” and “world”). So, it is as important to look at the lens as it is to understand what presupposes observation, or to address the taxonomic classification of all star-like entities, in search of a general theory of “all” stars, in their own turn elements of the collection of all universes, in their own turn elements of the collection of all the meta-universes… There’s no point where this ladder stops. Truncation is (a functor and) a failure. Inside Mathematics, category theory is the cure extinguishing thirst.

I grew up reading Borges, Gnosticism, Zen buddhism, and H.P. Lovecraft. I found category theory resonating with what has always been my instinctive intuition of reality.

Emanuele Severino once wrote “dietro ogni astrazione si nasconde l’intuizione di una Totalità” (“beneath every abstraction lies the intuition of a Totality”). This defines well the purpose of category theory to my eye. In the end, I’m a mathematician because I find it the only honest way to be a philosopher, and I’m a category theorist because I find it the only honest way to be a mathematician. “Mathematics” and “honesty” have of course here the same plastic meaning of “bread” and “pyramid” for the Babel’s librarian.

PS: I value the opportunity to share my thoughts on this topic, that I care about and that I always felt is disregarded in public or private discussion. I hope I’m not being too dissonant from the statistically dominant opinion. Thank you, I hope this helped you.

Posted by: Fosco on September 3, 2019 3:41 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

I love this analogy! Currently we are benefiting from decades of people obsessing over building better and better telescopes.

Just a minor historical comment: This appears to be a reference to a famous quote by Dijkstra, “Computer science is no more about computers than astronomy is about telescopes”. Under this analogy it appears to be comparing pure category theorists to mere computer engineers.

Posted by: Jules Hedges on September 3, 2019 4:13 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Thanks! Your talk of telescopes reminded me of Jean-Pierre Marquis on Abstract Mathematical Tools and Machines for Mathematics. There’s then a question of when the machinery becomes the object of study (TPRM, pp. 19-20).

Posted by: David Corfield on September 3, 2019 5:30 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Even if meant in jest, I feel this tweet captures some of the philosophy of the Australian school of category theory, in the sense that working with an arbitrary 2-category rather than CatCat is a fruitful approach.

Posted by: David Roberts on September 3, 2019 7:47 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

And yet, arguably a central point of categorical logic is that by passing to the internal language of a category or 2-category, you can work with “elements” or “objects” while retaining the desired generality.

Posted by: Mike Shulman on September 4, 2019 12:27 AM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Now that I work for a theoretical physics institute, I sometimes say that I am “employed as a physicist but trained as a category theorist”.

Posted by: Theo Johnson-Freyd on September 3, 2019 10:03 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Surely category theory has become so vast in its ramifications that any one narrative would be hopelessly limiting. Nevertheless, some discussion of those narratives could be useful, since it seems to be a persistent mystery how category theorists think.

As far as “traditional” category theory goes, a formulation I’ve been playing around with in my mind is that category theory is an exploration of that vast swath of mathematics that is explicable in terms of that which is universal (e.g., universal properties in the technical sense). It takes training to get a sense of what an enormous terrain that is.

Of course this is much too brusque, and gives no clue about the subtlety and difficulty of the conceptual work of ferreting out that which is universal. But a key simplification in the study of universality is the algebraic yoga of adjoints and higher adjoints.

Posted by: Todd Trimble on September 3, 2019 11:40 PM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

I remember the thrill of seeing basic category-theoretic ideas in terms of lines on paper. After you’ve drawn two lines next to each other, and joined one to another, what are going to do but bend lines into caps and cups, and have three lines meeting at a point?

Then you can thicken up such a meeting and see one edge as a cup, and suddenly something of the relationship between adjoints and monads appears.

Posted by: David Corfield on September 4, 2019 9:20 AM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

I think one narrative of category theory is that we can make progress on difficult questions by thinking harder about issues that many consider settled. Mathematicians spend a bit too much time trying to climb to the top of the tree, dealing with the most complex problems they can handle. It pays to go down to the roots now and then, where small improvements can make a big difference further up.

Consider how addition and multiplication were revealed to be left and right adjoints of the diagonal! Consider how existential and universal quantifiers were revealed to be left and right adjoints of substitution! And so on.

Back when I felt nn-categories needed help, I spent a lot of time telling everyone how they spring out of thinking harder about what’s apparently the simplest thing in the world: equality.

Nowadays I’m busy trying to convince everyone that electrical circuits, signal flow diagrams and all the other networks on which our technology is based are actually morphisms in symmetric monoidal (double) categories with good universal properties. Apparently this is why these networks can do so many things. So, I’m trying to get people to take a look at humble things like circuit diagrams and think about them harder.

Posted by: John Baez on September 4, 2019 3:47 AM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

Yes, who would have thought that linked propositions (E.g., It’s raining, and doing so hard) share a common idea with fibre bundles, as dependent sums.

Alexander Borovik calls this kind of situation vertical unity.

The same ideas and patterns of thinking can be found in elementary school arithmetic and in the cutting edge mathematical theories.

Posted by: David Corfield on September 4, 2019 9:11 AM | Permalink | Reply to this

Re: The Narratives Category Theorists Tell Themselves

I’m glad this topic has been brought up!

Although I still consider myself new to the categorical way of thinking, I can safely say I find some the ways of thinking associated with it divine, subtle and meditative. I will present a significantly different viewpoint than those I’ve found so far. I’m sure these thoughts aren’t new, but I’ve been unable to find anything about CT from this perspective. I also don’t think these four points fully express the range and implications of what I think about CT and I appreciate any insights on how to elaborate it even further.

1) CT gives us a systematic way to check for coherence of thoughts.

2) CT ensures that no abstraction is leaky, ever.

3) CT allows us to compress thoughts more efficiently.

4) CT specifies exactly what is in my scope when thinking.

1) CT gives us a systematic way to check for coherence of thoughts.

Systematic in the sense that we can program a machine to do it (proof-assistants, for example). If I tell you an abstract idea in CT terms, it is really straightforward for you to check if this idea is valid. This isn’t true for natural language, which is ad-hoc.

2) CT ensures that no abstraction is leaky, ever.

Leaky abstraction is the one which doesn’t hide its internal structure properly; it breaks in some specific cases.

So many software problems seem to happen only because abstractions break in some specific cases. Then people make fixes for these abstractions by adding more abstractions, rather than removing the incorrect ones. An even bigger issue is that then these added abstractions have the same problem of being leaky, and so on. Taking this to the extreme, you end up with Slack, a chat program that eats up 6GB of your RAM. CT is different - it doesn’t even let me talk about leaky abstractions: if I define a category where associativity breaks down in one specific case, then that really isn’t a category!

3) CT allows us to compress thoughts more efficiently.

If I know enough CT, I can state a very abstract thought in a precise manner and transmit it somewhere. Emphasis on thought. Whatever is happening in my brain when I’m doing category theory, I can ensure other people can get that same process. It is literally the thought that starts out in my mind, gets compressed and transmitted over a low-bandwith communication channel and recreated in somebody else’s mind. This, of course, holds true for language as well, but category theory seems to be orders of magnitude more efficient at compression and error-correction (see bullet-point 1) ). I’m very interested in understanding how this thought compression can shed more light on the question “How do we think” and “What makes us intelligent”. Machine learning systems which we’re creating right now are effectively untyped and don’t seem to be exploiting any of the categorical structure - which I think is a shame!

4) CT specifies exactly what is in my scope when thinking.

Just like I have to worry about my scope and imports when programming, CT does the same for thinking. Perhaps more importantly, I know what is not in my scope: I can clear my mind of other thoughts that aren’t relevant. It feels like a meditative experience.

Let me elaborate by giving an example of a non-CT concept and about the needed context we need to keep in our mind. Take for instance an object in the real world, such as a light switch. When I tell you “consider a light switch”, there are so many things you could potentially have to consider! A light switch has a position, shape, material it’s made of, utility, set of ways to use it, its age, color, it has a specific function, etc.. The issue here is that it is impossible to enumerate all the notions here! Furthermore, the actual list of notions you will need is context-dependent.

Constrast this with the notion of a “category”. There are just 7 things there: objects, morphisms, identities, composition, associativity, and left and right identities. That’s it. It’s not context dependent and there nothing else hidden. In order to reason about categories, I know I can clear my mind from everything else and know that any proof will have to be based on just those 7 things. Of course, I usually need to add more structure, but then when I do add it, I always know exactly what structure is there. This is incredibly important: I have a limited computational power in my brain and I need to use it wisely. While contemplating all these different abstractions and moving throughout the thought space, I don’t want to carry too much things I don’t need. Category theory allows me go on this road trip with less luggage. Furthermore, this trip strangely feels like meditation, as both require a deep contemplative state of mind. Both category theory and meditation seem to be beginnings of thought reflecting on itself - the only difference being category theory is more structured and coherent.

Anyway, enough of my rambling :)

I hope this was at least somewhat coherent. I’ve seen people here dabble in abstractions that I’m incredibly far away from comprehending. I know my view on category theory wasn’t this elaborated (if you can call it such) when I just started doing CT. This tells me that these viewpoints might very different from ones I’ll have in 5 years and that this perhaps might be naive. Since people here are well beyond this “5 year” gap, I’d very much like to hear opinions on what I’ve said!

I’ll end with a quote that I found on reddit that perfectly sums up my category theory experience:

I barely even understand what they’re supposed to represent. So much of pure math is like that. You look at the definition of like a topological space or something and you think “what the fuck is this and who the fuck cares?” but then suddenly as you learn more and dig deeper suddenly you find an order there, a structure that is both alien and familiar. It’s bizarre beyond anything you’ve ever encountered before, yet so, so beautiful. Math is the art that stands apart from art itself, that is not only beautiful but the progenitor of all beauty. It is the structure behind structure itself, an eternal landscape that existed before we ever did, and will continue to exist after we all cease to be. Math is the ultimate, incontrovertible study of Truth itself: cold, strange, infinitely subtle, infinitely deep. And thinking about it while high is really, really fucking fun.

Posted by: Bruno Gavranovic on September 4, 2019 10:19 PM | Permalink | Reply to this

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