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• CommentRowNumber1.
• CommentAuthoralexei
• CommentTimeSep 1st 2016
Hi guys,

I'm one of the founders of Arbital, a website for crowdsourced, intuitive math explanations. We are currently doing a collaborative project to explain the Universal Property concept in Category Theory. If you'd like to check it out, take a look here: https://arbital.com/project/
Once we are done, I'll post a link here so people can read the explanation. :)
• CommentRowNumber2.
• CommentAuthorfastlane69
• CommentTimeSep 2nd 2016
• (edited Sep 2nd 2016)

Well, thing is, so far I don’t see any intuition reflected in your site but rather a rehash of the same theorems you find on nLab or any Cat Theory book in the same one-dimensional representation of a single line proof (the sentence).

It’s only when you can draw your ideas, express them in two-dimensions, that you can say you truly understand them at an intuitive level.

So IMO if you really want to build a site that emphasizes intuition, make it a visual site.

Most mathematicians cannot or do not draw their ideas. Witness the lack of graphics on the nLab and category theory is the most visual of all the maths. Thus they are a poor source of intuition which is to be expected since intuition is a “the ability to acquire knowledge without proof, evidence, or conscious reasoning”, which is anathema to how mathematicians work

Seek out artists and musicians who are also mathematicians or physicists who know or use category theory; these are the people that aren’t using math to build more math but using math to apply it to various areas of their life. They thus have a stronger intuition about what the basic concepts mean and more importantly they can visualize that concept and share it with you in laymans terms (mathematicians have little patience for the layman)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeSep 2nd 2016

You who call yourself “fastlane69”, it’s time for you to take yourself back a bit. Given that you have not made a single useful contribution here yet, you are not in position to cast criticism on serious people’s serious work. The site that alexei advertised provides good introductory discussion suitable for people with interest in genuine mathematical thinking. If you do not find yourself among these, you are free to spend your time elsewhere.

• CommentRowNumber4.
• CommentAuthorfastlane69
• CommentTimeSep 2nd 2016
• (edited Sep 2nd 2016)

Given that you have not made a single useful contribution here yet,

Be careful mixing absolute quantifiers (“not made a single”) mixed with subjective qualifiers (“useful”); they tend to bite you in the ass: Music theory

Sorry if my personal belief that one-dimensional proofs (sentences) are inferior to two dimension proofs (diagrams) offends you.

But thanks for going out of your way to showcase how “mathematicians have little patience for the layman”.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeSep 2nd 2016
• (edited Sep 2nd 2016)

I agree with Urs’s #3, and completely repudiate the assertion that we practicing categorists shun visualization and graphics. Commutative diagrams occur all over the place in the nLab. It would be wonderful if other graphics (like string diagrams) were easy to implement here, since in practice we use such things all the time at the blackboard and on paper.

• CommentRowNumber6.
• CommentAuthorRodMcGuire
• CommentTimeSep 2nd 2016
• (edited Sep 2nd 2016)

Your site and project on Universals looks like interesting technology. I understand that you are using universals just as a test case to see if your approach can easily teach about them.

I like that short explanation pop up when you mouse over a technical term but as far as I’ve experimented there can’t be two popups active at the same time, which makes using both to understand something difficult.

How much are you using a “knowls” like approach which transcludes text into the viewed page, as discussed somewhat in this nForum thread: page-length-and-subdivision? It might be possible for learners to construct a customized view of a page (that is remembered) that inlines short explanations where needed and suppresses information they think they no longer need because they already know it, or maybe even hides information they feel is too advanced and distracting. I noticed that at the bottom of your page universal_property/ you have a button Proof that there is no initial field that seems to be a suppressed transclusion.

Something I’ve thought about a bit involves parameterized definitions. For example if the page text reads

• “the product $X_1 \times X_2$ in the category $Quiv$

then clicking to get a definition of product might inline one using those names. E.g. something involving

• $\pi_1: X_1 \times X_2 \to X_1$
• $\pi_2: X_1 \times X_2 \to X_2$

rather than giving a canned definition involving objects $A$ and $B$.

• CommentRowNumber7.
• CommentAuthorDexter Chua
• CommentTimeSep 2nd 2016
• (edited Sep 2nd 2016)

Non-maths comments: The page loads very slowly for me. It seems like a lot of javascript is running behind the scenes to make the page load, and moving my mouse around also eats up all my CPU. This makes the site very difficult to use. (nLab also loads slowly, but it is the server that is taking a long time to respond, rather than than my computer struggling to render the page)

Maths comments: I’ve always found the usual description of the universal property of a product (and limits in general) in terms of a factorization condition awkward, at least for pedagogical purposes. It has always been more natural for me to essentially describe the universal property as an adjunction (without mentioning the word adjunction), as I have recently written in universal construction. Viewing the universal property as a natural isomorphism of functors, I find that the definition-as-adjunction emphasizes the “isomorphism” part, while the factorization definition emphasizes the “natural” part instead. To me, the isomorphism is the actual content, while naturality is just a technical coherence condition, so the usual factorization approach seems rather awkward to me. Am I the only person having such a view?

• CommentRowNumber8.
• CommentAuthorfastlane69
• CommentTimeSep 2nd 2016
• (edited Sep 2nd 2016)

I’ve always found the usual description of the universal property of a product (and limits in general) in terms of a factorization condition awkward, at least for pedagogical purposes. It has always been more natural for me to essentially describe the universal property as an adjunction […] Am I the only person having such a view?

Factorizers embrace fragmentalism; adjuntizers embrace relationism.

You and I feel more comfortable in the latter while others feel more comfortable in the former.

One might even say (or has already been said) that these viewpoints form an adjunction themselves as the focus is on objects in the former and arrows in the latter. But I digress…

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeSep 2nd 2016

Am I the only person having such a view?

No; I think I feel similarly. Similarly, for adjunctions generally, there’s the hom-set formulation which appeals to higher concepts (like using $Set$ and presheaves and representables), and there’s the more elementary nuts-and-bolts formulation with units and counits and triangular equations. The former is probably more intuitive, but the latter has its obvious place.

Factorizers embrace fragmentalism; adjuntizers embrace relationism.

To be blunt, I find this opposition a bit silly, if it means that working categorists or mathematicians are being thought of as belonging to one or another camp (of “factorizers” or “adjuntizers”). Categorists speak a variety of languages or dialects fluently and move between them with ease: there is no such thing as “factorizers” or “adjuntizers” which limits them to one way of looking at things (like universal properties).

• CommentRowNumber10.
• CommentAuthorfastlane69
• CommentTimeSep 2nd 2016
• (edited Sep 2nd 2016)

To be blunt, I find this opposition a bit silly, if it means that working categorists or mathematicians are being thought of as belonging to one or another camp

Philosophically, do you consider all adjunctions to be “oppositions” given that I stated:

One might even say (or has already been said) that these viewpoints form an adjunction themselves as the focus is on objects in the former and arrows in the latter.

I would hardly consider objects and arrows to be in “opposition” to each other but rather “complementary”.

Also, my specific word was “embrace” and maybe I’m wrong (English is my third language) but excluding a viewpoint does not logically follow from embracing a viewpoint; embracing my mother is not a rejection of my father.

Categorists speak a variety of languages or dialects fluently and move between them with ease: there is no such thing as “factorizers” or “adjuntizers” which limits them to one way of looking at things

Would you also say that because there are categorists that “embrace” english and german fluently, there is no such thing as an english-speaking or german-speaking category theorist? Or rather could we say that every category theorist has a primary language in which they “feel more comfortable” expressing themselves? I propose to you that the analogy holds insofar as while a categorist may have both reductionism and relationism in their toolkit, they may “feel more comfortable” using one tool over the other.

Would you further say that in writing a paper for publication, does a categorist move between those two languages with ease? Or rather could we say that they write the paper in the language “they feel most comfortable” and then translate to other languages as need be? I propose to you that the analogy holds insofar as a categorist may start building a concept from a wholly reductionist or relational viewpoint and then translate between them as necessary.

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeSep 2nd 2016

Or rather could we say that every category theorist has a primary language in which they “feel more comfortable” expressing themselves? I propose to you that the analogy holds insofar as while a categorist may have both reductionism and relationism in their toolkit, they may “feel more comfortable” using one tool over the other.

In the specific case of universal properties being describable in various ways? No, not really. It’s too basic a concept for this to be in any way an issue for a well-practiced categorist. I repeat: there is no such thing as “factorizers” and “adjuntizers” in the categorical community. I think you made that up.

Getting back to what Urs is saying: given your present level of experience with category theory, I find that your tone in these discussions is pretending to more authority than you possess in these matters. The first, fourth, and fifth paragraphs in your #2 are expressed particularly obnoxiously in this respect. “Most mathematicians cannot or do not draw their ideas”: what a load of bull.

• CommentRowNumber12.
• CommentAuthorfastlane69
• CommentTimeSep 2nd 2016
• (edited Sep 2nd 2016)

“Most mathematicians cannot or do not draw their ideas”: what a load of bull.

Category theorists use planer proofs (diagrams) in combination with linear proofs (sentences) moreso than any other mathematical field.

Yet they are also in the minority and no mathematician in my experience shares the physicists delight of drawing on the back of the envelope.

Hence, I confidently stand by that statement.

• CommentRowNumber13.
• CommentAuthorTodd_Trimble
• CommentTimeSep 2nd 2016

The fact remains that mathematicians, most of them, make drawings to aid them in their thinking, all the time. Even if you don’t know it.

But come to think of it, it’s a time suck to be engaging with the trollery of #2, so I’ll stop here.

• CommentRowNumber14.
• CommentAuthorDexter Chua
• CommentTimeSep 2nd 2016

I’ll add that there is a difference between mathematicians not drawing at all, and mathematicians not drawing in publications. Drawing decent diagrams suitable for publication is hard and time-consuming.

• CommentRowNumber15.
• CommentAuthorDavidRoberts
• CommentTimeSep 2nd 2016
• (edited Sep 2nd 2016)

@fastlane - you forget people who work with geometry :-).

I can provide ready counterexamples of mathematicians who delight in mathematical play, pattern-finding and drawing their subject matter, and I live in a quiet corner of the world at a small university. I have a colleague who gets out scissors, tape and other making equipment when trying to understand a geometric concept, which is a fair sight more visual/tactile etc than category theorists with their pasting diagrams.

(Please take the rest of this post as diplomatic - adjust for the usual problems with internet communication lacking non-verbal context)

I politely third (or fourth?) the implict request that you consider your relative experience with category theory (and perhaps mathematicians in general? I don’t presume to guess how many you know personally and in a professional context) before making pronouncements to others that might come across as representative, if only of this small community. Note also that not all category theorists are like the ones who frequent this forum and the nLab.

Please also note that there have been in the past people that have joined in our little experiment and tried to use category theory in their own area, with not terribly great results. So we are slightly wary of protecting the relatively high signal-to-noise ratio here, while trying to be open and welcoming. Technically, the nLab is not directly intended to be a public resource - that is a happy accident.

• CommentRowNumber16.
• CommentAuthorfastlane69
• CommentTimeSep 3rd 2016

Please take the rest of this post as diplomatic

I thank you for that.

before making pronouncements to others that might come across as representative, if only of this small community.

I haven’t adequately taken the size of this community into account in my posts and thus I risk representing a global, nLab opinion instead of just a local, personal opinion.

You are absolutely correct; I see that now, thanks.

• CommentRowNumber17.
• CommentAuthoralexei
• CommentTimeSep 3rd 2016

I find these kinds of arguments fascinating, but also a bit sad. The nice thing about Arbital is that we encourage different explanations of the same topic from different angles, because different versions work for different people. And sometimes the reader needs to see something from multiple angles before they really grok it. If you would like you write up your favorite explanation of Universal Property on Arbital, we’d be very grateful. And when we post in on Reddit, Hacker News, and other websites, we can find out which one people prefer. ;)

So IMO if you really want to build a site that emphasizes intuition, make it a visual site.

@fastlane69 (and everyone else!): We have support for images and interactive, embedded iframes. If you’d like to write another explanation that you think would be better, we would love to have it. We can even pair you up with an illustrator and/or a programmer to make interactive widgets.

Seek out artists and musicians who are also mathematicians or physicists who know or use category theory

@fastlane69: Any tips on where to find them? :)

How much are you using a “knowls” like approach which transcludes text into the viewed page

@RodMcGuire: Some. You’ve seen the text hidden behind a button. We also have text that you can see by hovering over a question mark (https://arbital.com/p/Arbital_markdown/). And we even have text that conditionally appears based on what the user knows or wants.

The page loads very slowly for me.

@Dexter Chua: Sorry to hear that. We’ll look into it to see if we can fix it.

• CommentRowNumber18.
• CommentAuthorTodd_Trimble
• CommentTimeSep 3rd 2016

Alexei: sorry about that. I applaud you and your coworkers in this project.

I actually agree with the essential point from fastlane that visual logic in more than one dimension can be incredibly valuable. There is a load of public experience with Feynman/Penrose/Peirce diagrams that this is the case. (Other parts of his comments I reject as not well-founded, but let’s let that pass.)

• CommentRowNumber19.
• CommentAuthorfastlane69
• CommentTimeSep 3rd 2016
• (edited Sep 3rd 2016)

@fastlane69 (and everyone else!): We have support for images and interactive, embedded iframes. If you’d like to write another explanation that you think would be better, we would love to have it. We can even pair you up with an illustrator and/or a programmer to make interactive widgets.

Emphasizing this aspect of your project would give you a unique selling proposition that will set you apart from the competition . And small though your competition may be, you will still need a strong USP so that a person would consider creating content there more advantageous than creating it elsewhere, such as here.

Despite the unfortunate way it was read into, this was the tenor of my original post: that you are going to have to work doubly hard to make sure your site stands out before you make any requests for user generated content since, at this point, it showcases nothing technologically or conceptually new and has less content than what is already out there. Not to mention you are intrinsically serving a niche population so you are, in effect, competing with other open-source projects, like nLab, for the same “customers”: people that want to contribute to the categorical cloud.

Please understand, I want you to succeed. I want more categorical variety online. I’m sorry if I gave the impression this was not the case. But IMO and by my experience trying to drive people online for social change, physics education, and mathymusical art, you are going to have to do A LOT (A LOT) more work up front yourself before you can expect a tipping point of user generated content.

@fastlane69: Any tips on where to find them? :)

I wish I had realistic tips for you but you are honestly better served by the industry devoted to answering those types of questions.

• CommentRowNumber20.
• CommentAuthorPatrickStevens
• CommentTimeSep 4th 2016
The thing with the NLab and with Wikipedia is that in my experience, neither of them is really trying to *explain* their contents. NLab is a self-proclaimed lab book, whose explanations are broadly speaking only really useful if you already know a lot of related content. Wikipedia… well, everyone's tried to learn maths from Wikipedia. It's an encyclopaedia, not a teaching resource.

Re the 2D-drawings vs 3D-graphics thing: I don't think very visually myself, which may be why I'm struggling to imagine a helpful 3D graphic that might help with universal properties, particularly the product. Could anyone point me in a plausible direction?
• CommentRowNumber21.
• CommentAuthorfastlane69
• CommentTimeSep 4th 2016
• (edited Sep 4th 2016)

@patrick, here is some inspiration:

1999 AMS paper on visualization and math

2009 Visual Group Theory (changed my perspective on group theory and math in general)

2016 Lego Math on Pinterest <– (personal fav of mine)

• CommentRowNumber22.
• CommentAuthorfastlane69
• CommentTimeSep 4th 2016
• (edited Sep 4th 2016)

[accidental double post]

• CommentRowNumber23.
• CommentAuthorPatrickStevens
• CommentTimeSep 4th 2016
(I'm not here to talk about group theory, but this is to give an idea of my own intuitions, and why I'm asking for help.)

I looked through Visual Group Theory's first three chapters, and I'm afraid I found the diagrams almost entirely unhelpful. For instance, the pictures of reflections and rotations of rectangles: it takes me a long time to verify that the pictures are correct, because I can't really reflect or rotate objects in my head. (I can say "the symbol 2 should be at <this position>", but I can't say in which orientation or reflection the glyph should appear.) The Cayley graphs were pretty but, as far as I can see, don't serve any purpose; certainly they didn't pattern-match to any intuition I currently have, and I didn't get any new intuition from them. I'm just much happier manipulating symbols and traversing graphs than dealing in pictures.
• CommentRowNumber24.
• CommentAuthorfastlane69
• CommentTimeSep 4th 2016
• (edited Sep 4th 2016)

For instance, the pictures of reflections and rotations of rectangles: it takes me a long time to verify that the pictures are correct, because I can’t really reflect or rotate objects in my head

This is a very interesting perspective to me as someone that works the other way around and relies on visualizations to guide the symbols.

For example, given 4 objects I can tell right away if it is a C2xC2 or C4 symmetry by the way they move/rotate (for real or in my head) and I can then translate that to the necessary mathematical symbols and language.

I know you’re not here to talk about group theory, but I’m very curious if you could share a bit as to how you intuit or think about about C2xC2 vs C4 if not in terms of their symmetries (ie rotations).

• CommentRowNumber25.
• CommentAuthorDexter Chua
• CommentTimeSep 5th 2016

I think about $C_4$ as the cyclic group of order 4, and $C_2 \times C_2$ as the product of the cyclic group of order 2 with itself. I don’t think any geometric conception will help me with that.

Of course, I would not doubt that geometric visualization helps in some cases - I would love to read books on geometry or topology that are full of (relevant) pictures, or even animations! But sometimes things are inherently not geometric, and forcing yourself to give a geometric interpretation would only confuse matters. I’d say number theory is a good example of this, but some algebraic geometers might come out and correct me…

• CommentRowNumber26.
• CommentAuthorfastlane69
• CommentTimeSep 5th 2016

But sometimes things are inherently not geometric, and forcing yourself to give a geometric interpretation would only confuse matters

In my experience I have never run into a situation where a geometrical outlook was a hindrance; in fact, it has been critical to my works in fields as diverse as physics, musics, genetics, etc.

Since I’m clearly visually biased, could you give an example where a geometric interpretation would confuse matters? It is my opinion, at worst it won’t contribute any new “insight” but I don’t see how it can actually confuse.

I think about C4 C_4 as the cyclic group of order 4, and C2×C2 C_2 \times C_2 as the product of the cyclic group of order 2 with itself. I don’t think any geometric conception will help me with that.

The symmetry group (geometry) of each group is radically different: C4 has the symmetry group of a a circle while C2xC2 has the symmetry group of torus. As such, when I view a four-element system, I immediate rotate them in my head to see if I can map it to a circle or a torus; this then tells me the underlying structure and guides my investigation. I don’t know if that view is of any use to you in your work, but that is how it helps me make sense of things.

• CommentRowNumber27.
• CommentAuthorDexter Chua
• CommentTimeSep 5th 2016

Please enlighten me – what do you mean by the symmetry group of a group? The only definition I can think of is the automorphism group of group, which (I believe) should be $S_2$ and $S_3$ respectively.

• CommentRowNumber28.
• CommentAuthorDavidRoberts
• CommentTimeSep 5th 2016

C4 has the symmetry group of a a circle while C2xC2 …

No. You are probably thinking of $C_4$ as the fourth roots of unity inside the circle $U(1)$ (aka $\pm1,\pm i$), and $C_2$ as the square roots (aka $\pm 1$), hence $C_2\times C_2 = \{\pm 1\}^2 \subset U(1)^2$. But this is not how these objects are defined.

• CommentRowNumber29.
• CommentAuthorfastlane69
• CommentTimeSep 5th 2016
• (edited Sep 5th 2016)
I visualize C4 by it's [cycle graph](http://mathworld.wolfram.com/CycleGraph.html) which has C2 bilateral reflection symmetry along any axis as per it's [multiplicative group modulo n](https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n).

I then think of V4 by either its cayley graph, a 2-element by 2-element moebius strip,[like a two cogs, two teeth linkage as in this animation](http://mathworld.wolfram.com/MoebiusStrip.html)
or as a 2-by-2-element torus
[consisting of classic two interlocked rings as in this animation](https://en.wikipedia.org/wiki/Torus); they all look the same to my eye and seem to give the right V4 symmetry relationships.

If my visualizations are incorrect, thanks for letting me know since that would obviously be of great benefit to me.
• CommentRowNumber30.
• CommentAuthorDavidRoberts
• CommentTimeSep 6th 2016

You are embedding finite and discrete objects into continuous geometric objects - this is extra structure that doesn’t come for free. You also have to distinguish between $C_4$ the group and $\mathbb{Z}/4$ the ring. The former doesn’t have a multiplicative group.

cayley graph, a 2-element by 2-element moebius strip

this makes no sense, since moebius strips are objects with globally nontrivial topology, whereas you should be thinking of finite things, and in particular the Cayley graph.

Just because some continuous object has a subgroup of its symmetries that looks like a specified finite group, it doesn’t mean that’s the only way you should think of the finite group. If you want to think of finite groups as automorphisms of particular things, then find the thing of which they are all the symmetries. This is what finite group theorists like to do.

1. $C_4$ can be seen as the collection of all orientation-preserving symmetries of the vertices of an oriented square (hence no reflections)

2. $C_2 \times C_2$ is all symmetries of the vertices of a rectangle.

In particular, I don’t care about the continuous geometry of the square, the fact it looks like a little patch of the plane, just the relative arrangement of the vertices.

In fact, from a category theory point of view, you really should think of all possible objects that admit symmetries of the type of your finite group, and how these interrelate preserving the particular symmetries. This tells you everything about the group.

• CommentRowNumber31.
• CommentAuthorfastlane69
• CommentTimeSep 6th 2016
• (edited Sep 6th 2016)
>Just because some continuous object has a subgroup of its symmetries that looks like a specified finite group, it doesn’t mean that’s the only way you should think of the finite group.

Never said my parent group was continuous; I work exclusively in finite groups. Nor did I say it was the "only way" to think about it; it's just my preferred way. The perspectives you mention are canon and entirely compatible with my visualizations, heck they are my visualizations, just in words!

>C2×C2 C_2 \times C_2 is all symmetries of the vertices of a rectangle.

*Un-oriented* rectangle, very important property. C4 is composed of directed links making it oriented; C2xC2 is composed of un-directed links making it un-oriented... exactly the same way (to my eye) a circle is oriented and a moebius strip is un-oriented.

> Z/4 \mathbb{Z}/4 the ring. The former doesn’t have a multiplicative group.

No, but C2 does and thus we can still determine $V_4^x = C_2^x x C_2^x = C1 x C1$.

>In fact, from a category theory point of view, you really should think of all possible objects that admit symmetries of the type of your finite group,

This is probably the fundamental difference in our perspectives: I am not interested in "all possible objects" because I'm not a mathematician and free to do so; I'm interested in "very specific objects" because I'm a physicist and bound by phenomenology.
• CommentRowNumber32.
• CommentAuthorDavidRoberts
• CommentTimeSep 6th 2016
• (edited Sep 6th 2016)

Just to point out the obvious: a moebius strip is in fact non-orientable, not just unoriented. Also a circle may or may not be oriented (and it may not be embedded in some other space, either!), so you are making choices here on your geometry that isn’t forced.

Also note that I carefully said the vertices of a rectangle (assumed unoriented because I deliberately specified the square to be oriented) - I don’t care about the edges per se. Merely four points that stand in some relation to each other without even being embedded in some plane. A kind of modified finite incidence geometry if you will.

But it’s late here and I don’t think it’s productive to argue this on and on.

• CommentRowNumber33.
• CommentAuthorfastlane69
• CommentTimeSep 6th 2016
>But it’s late here and I don’t think it’s productive to argue this on and on.

What you consider argument I consider instruction.

Good night and thank you for taking the time to argue with me.
• CommentRowNumber34.
• CommentAuthortrent
• CommentTimeSep 7th 2016

So IMO if you really want to build a site that emphasizes intuition, make it a visual site.

not a practical comment, but a philosophical one (b/c people often seem to think that there is visual thought and abstract thought, nothing else):

visualization is great, but its important to remember that the visual is just one sensory modality - what about intuition that we can gain from eg proprioception and olfaction? olfaction seems to me an exceptionally rich way of gaining structural/mathematical intuition (there are so many basic notes existing in extremely complex relationships …. ) yet, not only do we not have anything approaching a mathematical theory of fragrance composition, we don’t even have a philsophico-conceptual theory of fragrance composition!!! just folklore & singular poetico-philosophical reflection (which someone not immersed in the world of perfume can learn about for example by reading a book like master perfumer Jean-Claude Ellena’s Diary of a Nose). there is of course much wisdom and beauty in such material* … but it would be nice to have rigorous (mathematical) theory too. i think if an alien civilization encountered ours, one of the things that would strike them as most absurd is how we have so neglected something as extremely fundamental as scent. who knows what kinds of exotic mathematics are to be discovered in that terrain.

anyways, on a (by comparison to the above) more practical note:

while we can’t formulate rigorous theories of other modalities without years upon years of work …. perhaps we can take away now the lesson that we shouldn’t privilege visual over abstract a/meta-modal thought, but rather we should strive to learn as much as we can from/through both. the visual is just one modality, and while there are certainly elements of the universal in it, we don’t know the extent to which it is transferable across domains. privileging the visual, we risk parochialism.

*and even if in the future we have a mathematical theory of fragrance composition as advanced as say mazolla’s mathematical theory of music is … i think folklore and poetic philsophy will always play an important role