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Jocelyn Ireson-Paine (Jan 07 2025 at 15:05):Thanks for the confidence boost! I think comparatively few category theorists relate to these dreams. Certainly, I don't see many applying it to visual art.
I searched about Bernstein's work on the topics of his lectures. I found a paper you might enjoy called "Rahner's Primordial Words and Bernstein's Metaphorical Leaps: The Affinity of Art with Religion and Theology" by R Masson (2006), Marquette University, https://epublications.marquette.edu/cgi/viewcontent.cgi?article=1211&context=theo_fac . Rahner was a theologian who tried to explain why some words, when used in a certain way, have the power to convey mystical or transcendent concepts . Perhaps "numinous" is the best adjective. I think pages 280 and 283 will give an idea of what such words might be.
Masson is not satisfied with Rahner's explanation of how this works, and tries to clarify it by analogising it with Bernstein's ideas. To quote his abstract:
Karl Rahner's notion of primordial words and Leonard Bernstein's conception of music as intrinsically metaphorical are engaged to suggest that there is a fundamental affinity between artistic and religious imagination. The affinity is grounded, in part at least, in metaphoric process—an elemental cognitive act in which the human spirit is stretched so that its expressions can address what lies beyond them.
I think the diagram on page 291 is the highest level "metaphoric process" that Masson sees: there are others at lower levels. However, I've not worked through this paper.
The paper did make me think of Joseph Goguen's use of colimits to explain how metaphors get their meaning. There's a concise explanation of this in Graham Joncas's blog page https://gjoncas.github.io/posts/2020-12-26-algebraic-semiotics.html , "Algebraic Semiotics: Joseph Goguen’s Semiotic Morphisms". This is an example of the formalisms that literary researchers such as Masson ought to be using, in order to more precisely convey their ideas. But as Goguen says on page 5 of https://cseweb.ucsd.edu/~goguen/pps/as.pdf , "An Introduction to Algebraic Semiotics, with Application to User Interface Design" (1998),
Semiotics has escaped this particular revolution, probably in part due to its increasing alienation from formalization during the relevant period. But I claim there is much to be gained from this unlikely marriage of semiotics and category theory (with cognitive linguistics as bridesmaid), not the least of which is a theory of representation that can be applied to topics of current interest, like user interface design, metaphor theory, and natural language understanding.
The EU thought so too. There was an EU project called COINVENT, whose aim was "to develop a computationally feasible, formal model of conceptual blending". Googling "eu conceptual blending project coinvent" finds several papers written under it, including one on varieties of blending in music. This is the paper "Creative Extensions to Conceptual Blending in Music" by D. Stefanou, https://www.coinvent.uni-osnabrueck.de/fileadmin/publications/2_enriching_the_Blend.pdf .
Alex Kreitzberg (Jan 07 2025 at 19:13):Thank you for sharing those links, I'll check them out. Glad my endorsment of Bernstein was relatable/matched a thread you'd already been pulling.
Before reading the blog you linked to I wanted to share an undeveloped thought I had regarding the relationship between metaphors and colimits.
The idea in my head is we have sentences and various implications between them,
So "sun" implies beauty, radiance, warmth, power, etc.
And "Juliet" to Romeo implies, grace, beauty, charm, radiance, etc.
So if "Juliet is the Sun" is somehow a colimit, then it should imply beauty, radiance, etc.
So we can "name the nameless" by "matching against" named stuff.
This very rough intuition was what I needed to convince myself a metaphor and an analogy were not the same thing :sweat_smile:.
I have to say I generally don't present ideas like this to people as "done math" - Because I have a lot more room to grow in category theory. Until I'm stronger at this I'm sure I have a high risk of confusing myself.
But yeah I did want to at least mention I can see the thread you're pulling. When I try to explain similar ideas to close friends, if my explanation is poor they look at me like I'm crazy...and I get worried myself :laughter_tears:.
Jocelyn Ireson-Paine (Jan 07 2025 at 20:36):Could you explain more about why a metaphor and analogy are not the same thing? To start with (and this is not a criticism of you), I'm not sure that there's a single standard definition of one versus the other.
To test this, I asked ChatGPT "Various sources distinguish or don't distinguish between metaphor and analogy. List the various distinctions (or lacks thereof) that have been made." It came up with seven different pairs of meanings. One of these, which I'd not seen before, is that metaphors are intended to evoke emotions, whereas analogies are usually intended to clarify, illustrate, or persuade. Although it might have been hallucinating, I did then find similar differences mentioned elsewhere.
I think that ❝we can "name the nameless" by "matching against" named stuff❞ is a really nice point. Something that could stand on its own as the subject of a paper. You should write it. Even if you can't make it rigorous, it would be good for humanists just to know that there is this formalism, that it can describe certain kinds of semiotic situation concisely, and that it can inspire new ideas. It would be like explaining the advantages of the notation "" over the sentences below (from al-Khwārizmī's Algebra, c. 830 AD):
What must be the amount of a square, which when twenty-one dirhems are added to it, becomes equal to the equivalent of ten roots of that square?
Halve the number of roots; the moiety is five. Multiply this by itself; the product is twenty-five. Subtract from this the twenty-one which are connected with the square; the remainder is four. Extract its root; it is two. Subtract this from the moiety of the roots, which is five; the remainder is three. This is the root of the square which you required and the square is nine …
One paper that does this is: Paul Vickers, Joe Faith and Nick Rossiter (2013), "Understanding Visualization: A Formal Approach using Category Theory and Semiotics", in IEEE Transactions on Visualization and Computer Graphics 19 (2013), 1048–1061, https://arxiv.org/abs/1311.4376 . They start with the Piercian semiotic triangle, whose vertices represent thing, symbol and concept, and whose edges the processes of going from one to the other. The semiotic triangle predates category theory, but they treat it as a categorical diagram, so that thing, symbol and concept become objects, and the edges become morphisms. Semiotic morphisms, though of a different kind from the Goguen ones that Graham Joncas writes of.
They then specialise the objects a bit, so that the things are data and the symbols are charts. And they then extend the range of objects and morphisms, and use this extended framework to formalise familiar concepts in information visualisation, such as ambiguity and chart junk.
By the way, Max Ernst wrote some things that are strongly reminiscent of colimits. I don't think he was naming the nameless, but he was certainly trying to generate it. See the quote in http://www.j-paine.org/lds/rules/forcibly_blending_II.html :
A ready-made reality, whose naive destination has the air of having been fixed, once and for all (a canoe), finding itself in the presence of another and hardly less absurd reality (a vacuum cleaner), in a place where both of them must feel displaced (a forest), will, by this very fact, escape to its naive destination and to its identity; it will pass from its false absolute, through a series of relative values, into a new absolute value, true and poetic: canoe and vacuum cleaner will make love. [Etc.]
On another topic, I get the looked-at-as-if-crazy thing too. I've tried to explain the ideas in http://www.j-paine.org/lds/minhyong/catsemart.html#13 and http://www.j-paine.org/lds/book/pl_power_of_line.html , and few people get them, unless artists. You might find them interesting, by the way: they're about the emotion and mood conveyed by line and shape, so ought to be related to how music does the same.
Both links are to works that I'd not describe as "done math" either. In my case, lack of funding is one problem, as with the category-theory demonstrator. I've not had the chance to do more than list some ideas superficially, and mark them for later exploration in depth, should the means ever arise.
(I am, though, pleased that I at least got the sheaves animations to a point where proper maths can start. I think I should put "Will build mathematical animations for money" in my profile.)
Alex Kreitzberg (Jan 07 2025 at 21:21):I hope to be a bit braver someday. But for now I'm too much of a beginner to write much on this I think.
But I'll try my best to answer your question on why analogy and metaphors are different given my understanding of the terms.
An analogy is almost scientific, and in many definite cases can be made exact.
Two things are analogous when the relationships between their respective parts are the same. If they're analogous in more than one way usually it's useful to specify the analogy you want.
I wouldn't be surprised if in essentially all scientific/mathematical examples, you could express an exact analogy as a functor.
Metaphors are first a specific writing device in the form of "A is a B".
Not "A is like a B" and especially not "A is like a B with respect to C". The phrasing "that man is a bear" is critical.
You're not supposed to clarify the parts that you believe A and B have in common. Because you want to suggest all the commonalities. Contending with these large clusters of meanings adds to the metaphor's power.
Bernstein likes to use the word "ambiguity" to describe this, but that phrasing sells metaphors short I think. If you're trying to understand an object with its set of arrows, you need to specify all the arrows somehow. That might superficially look like an "ambiguity" to a person who thinks you must have only one arrow in mind.
Morgan Rogers (he/him) (Jan 07 2025 at 23:01):Jocelyn Ireson-Paine said:
One paper that does this is: Paul Vickers, Joe Faith and Nick Rossiter (2013), "Understanding Visualization: A Formal Approach using Category Theory and Semiotics", in IEEE Transactions on Visualization and Computer Graphics 19 (2013), 1048–1061, https://arxiv.org/abs/1311.4376 . They start with the Piercian semiotic triangle, whose vertices represent thing, symbol and concept, and whose edges the processes of going from one to the other. [...]
They then specialise the objects a bit, so that the things are data and the symbols are charts. And they then extend the range of objects and morphisms, and use this extended framework to formalise familiar concepts in information visualisation, such as ambiguity and chart junk.
There was someone on this Zulip a few years ago who was always talking about semiotics and claiming that categorical ideas were relevant. I understand that the idea is appealing, because the diagrammatic language of category theory really does enable one to write down compelling-looking diagrams. Unfortunately it's easy to fall into a kind of trap with this language: in this example, once they arrive at their pleasing, symmetric diagram with all the concepts they could think to include, and they've managed to extract some plausible explanations from it, they are incentivized not to scrutinize it any further from a mathematical point of view because they are trying to present their formalism as robust. For the benefit of anyone reading, here is the diagram they take a while constructing:
image.png
There are a number of aspects of this paper which make it hard for a category theorist to take seriously:
I am not pointing this out to discourage you; I know from first-hand experience that mathematical formalization of concepts is Hard, and you have to grasp at all the straws you can. Rather, I would encourage you to try to see through the authoritative tone that any academic paper taking itself seriously ends up with. If there is some genuine connection here that you have the patience to sift out, that could be exciting.
Notification Bot (Jan 07 2025 at 23:04):6 messages were moved here from #practice: communication > Using Interactive Web Pages to Explain Sheaves by Morgan Rogers (he/him).
Alex Kreitzberg (Jan 07 2025 at 23:48):I feel somehow my comments tend to invite moving threads, sorry Rogers I don't mean to trigger your moderator hat :sweat_smile:. I hope you're threading because you find the topics interesting, rather than off topic.
I really did just want to note a common interest, not derail a conversation. I'm not sure how comfortable I am talking about these ideas at length in a public forum.
I want to solidify my understanding of category theory a lot more because I'm sure I have a lot more weak or dangerous intuitions to cull.
Alex Kreitzberg (Jan 08 2025 at 00:37):I guess I'll add I've never heard of the field "semiotics" but thumbing through it quickly, I get the impression that I've been stressed by a lot of related questions because of my interest in music and art.
Here's a drawing of mine:
Six_of_Pentacles.png
So I'm trying to figure out in the above how to communicate "six", "balance", "donations", blah blah blah. And probably the most powerful device I'm using is that the hand is "white" against "grey", though it's certainly not the only one.
It'd be great if category theory could help me answer these questions, but I know that's asking a lot. So instead I've been trying to focus my formal energy on making sense of what my tools can physically do.
So I think hard about bezier curves and whether there is a useful connection between them and [[Moore path categories]], for example.
I'm convinced I could make a vector style image file type inspired by Moore path categories.
But it's been very difficult to talk about this. It's like I ask "Does everyone see the same color?" And someone says "that's the great problem of 'what is quality?'" And I go "well crap how do I know people interpret my work the same way?"
Then after a bit of time later, I learn our vision yellows with age because of a layer in our eyes. So I go "AHA! I need to punch up the blues if my audience is old people!"
Clearly I'm communicating in a confusing way with respect to these issues, I'll try to do better but it's very hard.
Morgan Rogers (he/him) (Jan 08 2025 at 09:12):Alex Kreitzberg said:
I feel somehow my comments tend to invite moving threads, sorry Rogers I don't mean to trigger your moderator hat :sweat_smile:. I hope you're threading because you find the topics interesting, rather than off topic.
The topic shifted to some more philosophical stuff, not so related to "illustrating sheaves with interactive websites", that's all :)
Jocelyn Ireson-Paine (Jan 08 2025 at 09:18):Morgan, thanks very much for that. I do appreciate your criticisms.
It's why I joined this forum, to be able to discuss with people who
know better than I do.
That said, I can't see your reply on Zulip. I'm replying by email, and
I don't know where my reply will end up. I also can't see my post that
you're replying to any more, or Alex Kreitzberg's. I don't know
whether I'm misnavigating the thread, or a moderator has deleted or
moved the posts?
Morgan Rogers (he/him) (Jan 08 2025 at 09:19):I sent the notification of moving the messages to the new topic rather than the old one @Jocelyn Ireson-Paine, my apologies.
Jocelyn Ireson-Paine (Jan 08 2025 at 19:22):Thanks. Rather slowly, I realised that I could find the thread by searching for "analogy". And here I am.
Jocelyn Ireson-Paine (Jan 20 2025 at 12:10):Have you read Douglas Hofstadter's Metamagical Themas and Robert French's The Subtlety of Sameness? If you can get it and have the time, the chapter on "Analogies and Roles in Human and Machine Thinking" is where Hofstadter explains his ideas in most detail. He devises "analogy microworlds" to isolate the essential features of an analogy problem without cluttering it with real-world information. French does something similar, for a slightly different kind (though analogous...) kind of analogy.
There's a brief introduction to some of Hofstadter's work in "The Copycat Project: A Model of Mental Fluidity and Analogy-Making", https://pcl.sitehost.iu.edu/rgoldsto/courses/concepts/copycat.pdf , by Hofstadter and Melanie Mitchell. I think the idea of "transporting a rule" on page 206 relates to your remarks about ambiguity, not clarifying the parts in common, and specifying arrows.
It would be really good if someone would do a literature survey and write up all the categorical models of analogy, and all the other computational and mathematical models which are described clearly enough that this can be done. As far as categorical models go, I can think of five.
First, there's John Baez's "[...] every sufficiently good analogy is yearning to become a functor." Quoted in e.g. Tai-Danae Bradley's blog post "What is a Functor? Definition and Examples, Part 1" (January 31, 2017), https://www.math3ma.com/blog/what-is-a-functor-part-1?s=03 . I suspect this was an off-the-cuff remark, not a serious attack on how analogy works.
Correction: I'm wrong about off-the-cuff. In a reply, John says:
It was definitely not off-the-cuff, but it wasn't supposed to be an explanation of analogies. It was pointing out that in mathematics, we are always happy when we can solidify an analogy between categories to the point where we can treat it as an actual map from one to the other, because then we can do rigorous work with it.
Second, there are Joseph Goguen's use of colimits to explain metaphor. And various pieces of related and derived work, such as ontology merging, database merging, and conceptual blending. Some hefty theoretical support has been developed for these. As just one example which I happened to come across, Răzvan Diaconescu (2017), "3/2-Institutions: an institution theory for conceptual blending", https://arxiv.org/abs/1708.09675 .
Third is Jairo Navarrete and Pablo Dartnell (2017), "Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge", PLoS Comput Biology 13(8), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5589272/ . The abstract says they're using coequalisers: "as a formal model of re-representation that explains a property of computational models of analogy called 'flexibility' whereby non-similar representational elements are considered matches and placed in structural correspondence."
Fourth is Steven Phillips (2014), "Analogy, cognitive architecture and universal construction: a tale of two systematicities, "PLoS ONE 9:e89152. doi: 10.1371/journal.pone.0089152 . And, I think in more detail, "What is category theory to cognitive science? Compositional representation and comparison" (2022), Frontiers in Psychology, https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2022.1048975/full . Figure 3 in the latter shows the proportional analogy problem "Mare is to foal as cow is to calf" represented as a natural transformation.
I think the structure here is identical to that of the first drawing in "Natural Transformations" (April 7th 2015), a blog post by Bartosz Milewski, https://bartoszmilewski.com/2015/04/07/natural-transformations/ . This shows a left-hand "blueprint" category containing objects and arrows in the shape of a stick-man, and functors F and G from it to regions of a right-hand category which are in the shape of a pug and a pig.
And fifth, I think I once came across a use of spans or cospans to model analogy in an n-Category Café discussion. I can't find it now.
Maybe it's worth saying that some of this work evolved from, or was inspired by, Gilles Fauconnier and Mark Turner's work on conceptual blending. I have the impression that they describe several aspects of blending which I've not seen make it into the categorical models. Someone ought to check that and see what else from their work could be fruitfully categorified.
Here's an example of what I mean, in "E pluribus unum: Formalisation, Use-Cases, and Computational Support for Conceptual Blending" (2014) by Oliver Kutz, John Bateman, Fabian Neuhaus, Till Mossakowski and Mehul Bhatt, https://www.academia.edu/82298254/E_pluribus_unum_Formalisation_Use_Cases_and_Computational_Support_for_Conceptual_Blending . The authors remark that
This [simpler kinds of matching source and target spaces] relates also to Fauconnier's suggestion that it is actually what is done with the result of blending, termed elaboration (or 'running the blend'), that is the most significant stage of the entire blending process. Elaboration "consists in cognitive work performed within the blend, according to its own emergent logic" (Fauconnier 1997: 151). This makes it evident that something more is required in the formalisation than a straightforward recording or noting of a structural alignment: a new blended theory should also be 'logically productive', with new and surprising entailments which may well be quite specific to the blend.
I have not seen any work which addresses the question of how to test or formalise that a blend is logically productive.
Note that most researchers seem to think only of textual analogies, but not visual. That's reflective of a bias in applied category theory as a whole, which would much rather not get its suit dirty with paint, pencil, pastel, or anything else that can't be neatly typeset in Unicode. Here are examples of visual conceptual blending:
j_w_taylor_beer.jpg
above_the_clouds_midnight_passes.jpg
assembled_collage_25p.jpg
They are by: Punch cartoonist J. W. Taylor, reprinted in Mike Lynch's blog post "J.W. Taylor" (May 18th 2008) , http://mikelynchcartoons.blogspot.com/2008/05/jw-taylor.html ; Max Ernst, Above the Clouds Midnight Passes; and Monty Python animator Terry Gilliam, reprinted in his book Animations of Mortality.
Finally, see this cartoon. beau_peep_touch_nose.png . From Roger Kettle and Andrew Christine (1990), Beau Peep Book 11, from the Daily Star, Express Newspapers. You need to perceive sensible samenesses.
John Baez (Jan 20 2025 at 18:36):I suspect this was an off-the-cuff remark, not a serious attack on how analogy works.
It was definitely not off-the-cuff, but it wasn't supposed to be an explanation of analogies. It was pointing out that in mathematics, we are always happy when we can solidify an analogy between categories to the point where we can treat it as an actual map from one to the other, because then we can do rigorous work with it.
Alex Kreitzberg (Jan 20 2025 at 20:25):That was how I interpreted your comment Baez
One can form an analogy between triangles and tetrahedrons, by noting that in both cases a line from their verticies to the centroid of their opposing lines/faces all coincide at a centroid of their face/volume.
But this isn't really a proof of anything. And I guess if you believe Hofstadter, the mechanism of the mind behind these comparisons happens all the time and is generally meaningless. So what do you do?
Maybe if you make a certain parallel projection from the tetrahedron to a triangle, then maybe you could show the intersection defining the centroid of the volume, maps to the centroid of the triangle (you'll definitely get lines that originate from its vertices intersecting somewhere).
At least with how mathematicians typically use the word "analogy", there's hope for some map. (Though I'll also see mathematicians use the word when they suspect the compared things are common examples of a theorem, or definition, we haven't figured out yet. Still, you can only directly use the analogy after you have the theorem or definition.)
There is no such hope for metaphors. They are a tool in creative writing. The "is" in a metaphor is supposed to be between irreconcilable things. Or at least you're always permitted to make a metaphor between anything you want. It depends on what you want to communicate. This is very different from how a mathematician typically uses the word "analogy". Though I know broadly people like to conflate the terms in colloquial speech.
Aside: After reading Baez's comment on Functors. It did get me to wonder if whenever one had a structure preserving map in math, whether it's extremely likely to be a Functor. That has often provoked me to try and find categorical structures for the mapped things.
But I've really struggled with this. I'm not sure how a parallel projection should, or could, be made into a functor. This has made me wonder if I'm being silly. Still the temptation to think of segments as arrows, and points as objects is very strong. But the composition needs to make sense, and line segments don't compose into line segments (right?)
John Onstead (Jan 20 2025 at 21:14):I don't know if this will be helpful, but I think of analogies as being spans (structures of form ) rather than direct maps (of form ). In this way, you can think of a span as "picking out" some common feature in both A and B. I think this way since an analogy is generally a 4-ary relation : " is to as is to ". A span can therefore be read as " is to as is to ". But take this with a grain of salt as I'm no expert in any of this!
Alex Kreitzberg (Jan 20 2025 at 22:10):My understanding is the etymology of analogy is "proportionate", I think the term literally came from in the context of similar triangles (so there is an implicit transformation from one triangle to the other).
In which case a group homomorphism is a very definite generalization of this,
And then a functor is a definite generalization of that.
That's a big part of why I'm very drawn to thinking of an analogy as being about maps "first". Of course that doesn't perfectly match how I've seen people use the term.
"When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean — neither more nor less"
John Baez (Jan 20 2025 at 22:22):Alex Kreitzberg said:
But I've really struggled with this. I'm not sure how a parallel projection should, or could, be made into a functor. This has made me wonder if I'm being silly.
It's possible you are trying to do something hard before doing something easier.
When you have an analogy between categories, you can try to formalize it as a functor (or a span of functors, or...). But when you have an analogy between objects in a category, you can try to formalize it as a morphism in that category (or a span of morphisms, or...).
So, have you thought about what categories the triangle and tetrahedron are objects of, such that the projection map is a morphism of categories? There are at least 3 good options.
John Baez (Jan 20 2025 at 22:24):Once you understand this pretty well, you can try to 'categorify' the situation by figuring out systematic ways to treat those objects as categories in their own right.
I think going one step at a time like this would make your job easier.
Morgan Rogers (he/him) (Jan 21 2025 at 11:05):Jocelyn Ireson-Paine said:
Have you read Douglas Hofstadter's Metamagical Themas
It's nice to have an excuse to read more Hofstadter, his GEB was very influential on my younger self, although the fact that this was published 40 years later will lead me to examine carefully how far down his own rabbit hole he may have descended by the time he wrote it..! I'll check it out if I come across it.
There's a brief introduction to some of Hofstadter's work in "The Copycat Project: A Model of Mental Fluidity and Analogy-Making",
My reaction to this was frustration. An analogy is a semantic thing, trying to see what arbitrary rules people infer for syntactic 'analogies' is at best of limited psychological interest in my opinion, but I suppose it is a sensible starting point if one is trying to produce a system simulating human reasoning (which is the goal of that work as I understand it from the first several pages). But I agree that the process Hofstadter describes does feel like an inference of a context/category in which the transformations are taking place, with the objects inferred by identifying structure in the strings of characters and the morphisms inferred by which transformations are allowed.
Jocelyn Ireson-Paine said:
Third is Jairo Navarrete and Pablo Dartnell (2017), "Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge"
Did you read this? To me this is an example where category theory is introduced as a convenient "language" without having a direct bearing on the content. The model they actually use is one from logic, with a formal syntax and constrained semantics, although they are quite informal with its set-up. The 'domains' they use to model things are selected ahead of categorical considerations, and the category theory invoked consists of: in a first instance drawing a commutative square in the category of such domains; in a second instance using a presentation of an arbitrary domain as a quotient of a free one (which is where the coequalizers show up). Going to the trouble of talking about category theory has made things clearer, since it has made them consider the mappings between the formal objects they model interpretation with -- this is always nice to see. Yet this doesn't add anything much to the fundamental content of the models they use.
Addressing all of these references would take some time. My main cautionary note is: employing the language of category theory is not the same as using category theory, in the same way as using numbers is not the same as using number theory.
Alex Kreitzberg (Jan 21 2025 at 17:10):John Onstead said:
I don't know if this will be helpful, but I think of analogies as being spans (structures of form ) rather than direct maps (of form ). In this way, you can think of a span as "picking out" some common feature in both A and B. I think this way since an analogy is generally a 4-ary relation : " is to as is to ". A span can therefore be read as " is to as is to ". But take this with a grain of salt as I'm no expert in any of this!
I thought about this more, I think I borked my generalization, but I'm not sure XD.
So I'll add my working thoughts are more like this, I accept this may not fully capture the intuition for all people.
There's a list of relationships, say
And sometimes you have an isomorphism that preserves the structure, label it ϕ. Then the relationships that involved the preserved structure don't change.
So you could say the relationships are the same because of , and broadly speaking having lets you do more math.
But if you noticed that two different objects all have the same relationships with respect to the structure you care about. But you haven't found an isomorphism to prove this yet. Suspecting the existence of such an isomorphism is inductive reasoning by analogy, in my view.
Using isomorphisms is just doing ordinary deductive math. (Incidentally, that's why I really like Baez's phrasing that good categorical analogies yearn to be a functor. They aren't a functor yet!)
Sometimes this inductive reasoning can be a real theorem.
For triangles, you can prove the existence of the isomorphism, in this case a similarity, by observing that enough relationships are the same, in this case proportionalities.
So instead of writing
I'll say - therefore the triangles are similar.
When it becomes a theorem people stop using the word analogy.
Noting two vector spaces are isomorphic if they have the same dimension is another example of a "deductive analogy" in this view.
After spelling all that out, I'm inclined to say the yoneda lemma is how I've thought about analogies in the past. Because it makes it a theorem that if the arrows of two objects are "the same" then the objects are isomorphic.
This definitely isn't the end of the story for analogies, for example they don't have to imply an isomorphism, but they can still be useful.
But I think it's true that the yoneda lemma models how I made analogical thinking into deductive thinking.
Edit:
Yeah I think this is true of myself, I remember calling group homomorphisms one way analogies when I first learned them. I think it bothered me the maps didn't have to be invertible. Most people don't expect so much from their analogies XD.
Alex Kreitzberg (Jan 21 2025 at 17:39):It's good we have category theory. It gives us exact names for so many important distinctions between technical transformations.
David Egolf (Jan 21 2025 at 18:02):As another idea: Let be a category, and view the categories and as containing a lot of information about and . Specifically, our philosophy is that each morphism to tells us something about , and how the morphisms to relate also tells us something about . This information is contained in .
Then a functor takes information about and produces information about . Such a functor also facilitates other styles of reasoning that relate information about to information about :
So one might be tempted to call a functor an "analogy between and " because it provides ways of transporting information about one of these objects to information about the other one of these objects.
John Baez (Jan 21 2025 at 18:23):By the way, the usual ways we get a functor are by 'pushing forward' along a morphism or 'pulling back' along a morphism (in the case where has pullbacks). People discuss these cases endlessly because of their importance in logic and type theory.
John Baez (Jan 21 2025 at 20:06):Also btw, is a classic example of category of 'things over ', so it's often used in the descent question I recently explained. In good cases pushing forward along a morphism is left adjoint to pulling back along that morphism and this adjunction is monadic, so we can use the idea of 'monadic descent', which I also explained.
David Egolf (Jan 21 2025 at 20:18):On a possibly related note, I seem to recall we get a functor by pulling back along in , and that this functor sometimes(?) has a left adjoint. If I'm remembering this correctly, then the fixed points of the resulting adjunction might be considered as candidates for "concepts" present in the analogy between and .
John Baez (Jan 21 2025 at 20:23):Btw, most of the technology I described applies only when you start with a morphism from to , while you are not assuming that.
John Baez (Jan 21 2025 at 20:29):David Egolf said:
On a possibly related note, I seem to recall we get a functor by pulling back along in ...
I don't know what you mean by that. What's pulling back along ?
Given a morphism , in I know how to push forward along : just postcompose with . This always gives a functor . If that functor has a left adjoint we call that 'pulling back along '.
But that's apparently not what you're talking about.
David Egolf (Jan 21 2025 at 20:38):I am imagining a functor that is getting pulled back along . The idea is to view the functor as describing certain information about , and then pulling back along gives us some corresponding information about .
In this case, the pullback square involved lives in .
John Baez (Jan 21 2025 at 21:09):Okay, that makes sense to me now. Since has pullbacks this pullback will always exist.
David Corfield (Jan 21 2025 at 21:41):John Onstead said:
I don't know if this will be helpful, but I think of analogies as being spans
And Jim Dolan would have agreed.
John Onstead (Jan 21 2025 at 22:15):David Corfield said:
And Jim Dolan would have agreed.
That's interesting! I like the connection with how the object at the center of the span acts as a sort of abstraction. It seems to align with those philosophies of analogies that state analogies are built from finding common abstractions. Even after just looking at the wikipedia page for "analogy" I've come to see those certainly aren't the only philosophies of analogies, and that entire books can and are written about the philosophy behind analogies!
I also just though of another reason why I like spans over maps based on what I've been reading, and it's due to symmetry. Spans are symmetric (which is why categories of spans are dagger categories) but maps are not unless they are isomorphisms. In addition, analogies are also symmetric: the analogy "A is to B as C is to D" is generally "isomorphic" to the analogy "C is to D as A is to B". I think there was a discussion on here a while ago about whether all analogies were like this but I'm not sure if it came to a conclusion one way or the other!
Alex Kreitzberg (Jan 21 2025 at 22:36):I've heard someone explain that if two predicates are consequences of the same theorem, or examples of the same definition, then they're analogous. I think that's an example of a span in that sense.
I'm open to the possibility of there being multiple common concepts that are all called "Analogy". Or maybe "Analogy" is too big to be safely defined in just one way. Still, this conversation has warmed me up to "Analogies want to be arrows", despite how vague that is :joy:.
Alex Kreitzberg (Jan 21 2025 at 22:40):I'll think about spans more though, maybe I don't understand them well enough.
(I'm also a bit paranoid of how easy it is to define categorical concepts in terms of each other, it's easy to go around in circles)
John Onstead (Jan 21 2025 at 23:15):Alex Kreitzberg said:
(I'm also a bit paranoid of how easy it is to define categorical concepts in terms of each other, it's easy to go around in circles)
Especially if you consider that a map can come from a span! Given a span , if we set to be the identity morphism then you just get and thus .
John Baez (Jan 22 2025 at 00:00):Every sufficiently good analogy yearns to be a functor, but most have to settle for being a span.
Mike Shulman (Jan 22 2025 at 00:17):Or at least a profunctor.
Evan Patterson (Jan 22 2025 at 00:46):Nice slogan, I'll have to remember that one!
Morgan Rogers (he/him) (Jan 22 2025 at 11:01):John Baez said:
Every sufficiently good analogy yearns to be a functor, but most have to settle for being a span.
This is going to become one of those quotes that gets gradually forgotten over time until it's just "Every sufficiently good analogy..." accompanied with a shrug or sly wink and conveying something completely different, like "jack of all trades".
John Baez (Jan 22 2025 at 16:49):My original slogan was more memorable: "Every sufficiently good analogy yearns to be a functor". Frankly I think slogans shouldn't have clarifying footnotes. But I couldn't resist issuing version 2.0 with a bug fix.
Jocelyn Ireson-Paine (Jan 22 2025 at 19:18):(deleted)
Jocelyn Ireson-Paine (Jan 22 2025 at 19:20):Alex Kreitzberg said:
I'm open to the possibility of there being multiple common concepts that are all called "Analogy". Or maybe "Analogy" is too big to be safely defined in just one way. Still, this conversation has warmed me up to "Analogies want to be arrows", despite how vague that is :joy:.
Analogy is a biological phenomenon. It's therefore bound to be impossible to define succinctly, and to have exceptions, and exceptions upon those exceptions, and ... . The first thing we need is for an exhaustive, and probably exhausting, fieldwork session, where investigators go out and collect as many examples as they can find. This will be difficult. If users are explicitly calling something an "analogy", we can be non-prescriptive, say that it is one, and drop it into our killing bottle. If they don't say that it is one, things are more tricky. How do we decide whether to treat it as an object of study? Biologists must have an answer to this, I suppose, when deciding where to draw the boundary of a species, and which samples not to study.
Then, collection done, we can have the fun of studying and classifying our subjects. The SF writer and ex- marine biologist Peter Watts has a nice quote about such work, in his blog No Moods, Ads or Cutesy Fucking Icons (Watts 2021) :
And the virtues of reducing model complexity were hammered home to me on Day One by my old doctoral mentor Carl Walters, who never let us forget that any model as complex as Nature is going to be as difficult to understand as Nature. In a lot of cases, hi-def realism is the last thing you want in your models; they have be caricatures if they're going to generate useful insights.
Jocelyn Ireson-Paine (Jan 22 2025 at 19:23):
John Baez said:
My original slogan was more memorable: "Every sufficiently good analogy yearns to be a functor". Frankly I think slogans shouldn't have clarifying footnotes. But I couldn't resist issuing version 2.0 with a bug fix.
My apologies if this was already answered (I just didn't see it) but I have to ask: what are some concrete examples of functors that are analogies? I can't seem to think of any. For instance, a forgetful functor doesn't seem much of an analogy: "an underlying set is to a topological space as a set is to a set"? That doesn't seem like a very interesting analogy. I guess the fact this particular functor preserves products might give the analogy "the product topology is to topological spaces as the cartesian product is to sets", which is a much better analogy. However, this fails in general since not every mathematical object's underlying set functor preserves products, so it's more useful to think of that analogy in terms of the fact the universal property of the product exists in both categories. A free functor isn't any better: "a set is to a set as a discrete space is to a topological space". Another example could be the fundamental group(oid) functor since this is a clear example of a "bridge" functor between two mathematical disciplines (topology and algebra), but I don't see this implies some analogy between these fields, just that a connection exists between them. If we have some good examples of analogies that can be encapsulated by a functor and analogies that can't, I think it would help in better understanding when in general this slogan might or might not apply!
John Baez (Jan 22 2025 at 19:55):I was thinking about analogies like "a covering space is to topological spaces as a subgroup is to groups".
(More precisely, pointed topological spaces.)
Here I'm imagining someone who noticed that pointed topological spaces are like groups without yet knowing about the "fundamental group" functor. Perhaps that's an unlikely scenario. Maybe more plausible is someone who noticed that a lot of things about groups have analogues for rings, without yet noticing the "group ring" functor.
John Baez (Jan 22 2025 at 20:03):However, I'll be happy if my slogan inspires a more detailed and accurate treatment of the role of analogy in mathematics.
(A mathematical theory of analogies as used outside mathematics is too ambitious for me to think about - I'll let philosophers and people working on AI tackle that.)
Jocelyn Ireson-Paine (Jan 22 2025 at 20:54):John Onstead said:
My apologies if this was already answered (I just didn't see it) but I have to ask: what are some concrete examples of functors that are analogies?
If you don't mind me changing your question to "... of functors that are parts of analogies", how about
image.png
or (depending on whether you prefer span-style setups)
image.png
The fellow looking down on the left is Ray Cummings, author of the much-loved vintage SF novel The Girl in the Golden Atom. An image search on book covers thereof is revealing.
It would be interesting to expose deep-learning programs to suitable instances of the two situations (Bohr atom and solar system) and then see how their learnt representations for each differ. I don't know enough to justify this remark, but I'd not be entirely surprised if we found that the difference betwe them could be modelled, approximately, as a functor.
Kevin Carlson (Jan 22 2025 at 20:57):What functor is this supposed to represent? It looks like the identity functor between two isomorphic copies of the walking arrow category, which is not really much of a functor.
John Baez (Jan 22 2025 at 21:18):That's what the functor is mathematically, but I believe Jocelyn is mainly interested in examples outside math (precisely what I swore to avoid talking about, but oh well), and they are using this functor to map the Earth to an electron and the Sun to a proton.
Still, this sort of analogy as a functor is an isomorphism of categories, which is an amazingly strong sort of analogy - achieved here by omitting all the ways in which the Earth going around the Sun is not like an electron going around a proton.
Jocelyn Ireson-Paine (Jan 22 2025 at 21:26):It's not much of a functor, but it is (I am fairly sure) an answer to John Onstead's question. It is also an analogy which has been studied in the cognitive science literature, and which has also been very productive in literature, spawning a number of science-fiction stories. Most of these didn't extend the analogy much beyond star/nucleus and planet/electron, because, I suppose, it breaks when you try. As John Baez has replied while I type this,
John Baez said:
Still, this sort of analogy as a functor is an isomorphism of categories, which is an amazingly strong sort of analogy - achieved here by omitting all the ways in which the Earth going around the Sun is not like an electron going around a proton.
I suspect we can learn a lot about how human analogy-making works by studying all the ways in which one side of an analogy is not like the other side of the analogy. And perhaps we can do that for examples inside math as well as outside.
P.S. It just struck me that John Baez saying "outside math" is an analogy too. After all, math is not a space. Is it?
Jocelyn Ireson-Paine (Jan 22 2025 at 21:44):Where do the phrases 'pushing forward' and 'pulling back' originate? Were they around before category theory?
John Baez (Jan 22 2025 at 22:00):I don't know. I bet they were used in math before category theory formalized them, but outside math people have been pushing things forward and pulling things back for millennia.
John Onstead (Jan 22 2025 at 23:27):Jocelyn Ireson-Paine said:
It's not much of a functor, but it is (I am fairly sure) an answer to John Onstead's question. It is also an analogy which has been studied in the cognitive science literature, and which has also been very productive in literature
Yes, it's a great example of how a structure preserving map can act as an analogy! It shows that if a mathematical object contains a structure that determines how its elements relate to each other, then a map between such objects will preserve this structure and so map one relation between things into an analogous relation between other things. In this case, the relation is "orbits" and the structure preserving map preserves this "orbit" relation between elements. I also like how you included an example of how this analogy can be thought of as a span involving a "generic orbit relation"!
John Onstead (Jan 22 2025 at 23:34):John Baez said:
Still, this sort of analogy as a functor is an isomorphism of categories, which is an amazingly strong sort of analogy - achieved here by omitting all the ways in which the Earth going around the Sun is not like an electron going around a proton.
I guess this is sort of like how, in the category Set, you can't distinguish between isomorphic sets even if you want to interpret them differently. For instance, if you have a set of three balls and another set of three galaxies, these two sets are indistinguishable in Set even though the objects are meant to be vastly different things. But I think the analogy makes sense if you're willing to work in material set theory instead of structuralism, so long as you have a good way of representing the concept of an electron, proton, planet, and star as material sets unto themselves.
Alex Kreitzberg (Feb 12 2025 at 05:45):@Morgan Rogers (he/him) Why didn't you refer to your own work on this question:laughing:
https://youtu.be/QURq2a1ezns?si=ok5lgGQzmx4R_OYg