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Hello all! I thought it would be a good idea to write an exposition of the current state of the formalization of Hegel, hiding as much mathematics as possible, so I did it! I mainly oriented myself after the dictionary you’ve set up here:
Since this is supposed to be an exposition of work on the nlab I thought I would give you a chance to take a look at it before I try to get it published somewhere (preferably in a philosophy magazine, to get them to pay attention), in case you have any comments.
I wasn’t sure how to best share the article, so I put it on Github:
https://github.com/nameiwillforget/hegel-in-mathematics
I also have some questions to completely finish the article:
In general categories, if we understand unities of opposites as adjoint modalities and define an Aufhebung of a unity $U$ as a unity $V$ such that one of its opposites contains the entire unity of the first Aufhebung, is it possible that the other opposite does not contain any of the opposites of $U$?
Is the shape of codiscrete types necessarily trivial: $\Pi\sharp\simeq *$? The example that is always given is that of codiscrete spaces, and I’ve written it from that perspective. Additionally, I feel like I don’t completely get the interplay between $\sharp=loc_{\neg\neg}$ and the logical understanding of types as propositions. I guess localizing the double negation is not the same operation as double-negating every type? What exactly does it do to the constituents of a type, if they are understood as the truths of a proposition? Or what exactly does the localization to Euclidean-topological $\infty$-gloupoids as a concrete example? I understand that it is the composite of a global section and an embedding, but how exactly does the embedding look?
Is the fermionic modality $\rightrightarrows$ or $\overline{\rightsquigarrow}$? On the Science of Logic page, it defines it as $\overline{\rightsquigarrow}$ at the beginning, but later on, and also in other nlab-articles and I think in dcct, it defines it as $\rightrightarrows$.
Who else should I mention as having worked on the formalization? As it stands, I’ve only mentioned Lawvere and Urs Schreiber.
Thanks for all the insights you’re keeping publicly available!
P.S. There clearly is some potential interest: https://www.reddit.com/r/askphilosophy/comments/q2w6ua/lawvere_and_hegel_use_or_abuse/
Thanks for sharing. Interesting.
I can’t promise to dive into this and provide much feedback, since I am too busy with other urgent tasks (not even counting the overdue referee reports to be sent…).
Just quickly on the name of the modalities:
I kept changing my mind about this, since $\rightrightarrows$ is, strictly speaking, bi-fermionic, while $\overline{\rightsquigarrow}$ is, of course, the negation of bosonic. Both of these match aspects of”fermionic” as used colloquially. But for labelling the progression of opposites, it seems more elegant to refer to $\rightrightarrows$ as “fermionic”.
Some random comments, as I read through your test:
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on p. 2:
I wouldn’t say that the adjoint triples that Lawvere considers are “based on” Hegelian dialectics. How about changing to “capture” or “give precise meaning to” or similar?
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on p. 3:
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on p. 4
I find the emphasis on “abstraction” is missing an opportunity: The point of category theory is rather that it captures qualitative aspects, where arithmetic knows only quantity. Highlighting this would serve your main point. In fact the first sentence on p. 3 almost says this already, with its emphasis on structures and relationships:
Over the past century, mathematics went through a rapid increase in abstraction, as the focus shifted first from concrete entities, like numbers, to the struc- tures they inhabit, like groups, to the relationships between those structures and the realms they inhabit, which are categories.
How about slightly adjusting this along the following lines:
Over the past century, mathematics went through a rapid increase in expressiveness, as it subsumed not just quantities like numbers, but also qualitative meaning, such as the structures these inhabit, like groups, to the relationship between those structures…
?
Next, the nice Hegel quote you have there:
These many different things stand in essential reciprocal action via their properties; the property is this reciprocal relation itself and apart from it the thing is nothing;
would be an excellent opportunity to point out that this is exactly how category theory works: Namely the nature of the objects in a category is all encoded in the morphisms which relate them to other objects! See also the nLab entry: Structuralism – Formalization of structuralism in category theory and type theory.
Next your text says:
Using category theory, we can explain Lawvere’s translation of Hegelian dialectics. For this, what Hegel calls a “Moment” is formalized as a (co)modality on a category
Please note that Lawvere never says this (and we don’t know if he even approves of it). He doesn’t refer to modalities or moments. This “modal homotopy theory” is my formulation of Lawvere’s formulation of Hegel’s formulation.
This first appears in (pre-)print in:
U. S.: Sec. 2.2 of: Quantization via Linear Homotopy Type Theory $[$arXiv:1402.7041$]$
Hisham Sati, U.S.: p. 6 of: Proper Orbifold Cohomology $[$arXiv:2008.01101$]$
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Allow me to point out two further relevant references:
A slightly more citable version of what I have written at Science of Logic is
U.S.: Modern Physics formalized in Modal Homotopy Type Theory,
expanded notes for a talk at:
What are Suitable Criteria for a Foundation of Mathematics?
FOMUS – Foundations of Mathematics, July 2016
An exposition of part of the modalities, but with more emphasis on their incarnation in homotopy type theory is:
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So much for the moment. Maybe I’ll send more comments later, if I find the time.
Another relevant reference:
especially Section 5.5 there.
As you’re already using github, we don’t you push the textfile as a PR? This way you’d not need to get infos about typos on nLab and github issues.
Even better: the .tex file as well as the pdf.
on p. 5:
footnote 6 says:
in spite of the analogy to time that Hegel had undoubtedly in mind when choosing the term “moment”,
This may have been lost in translation: No, Hegel certainly did not refer to temporal moments. In German the temporal moment is “der Moment” but Hegel is speaking of “das Moment”, which is (a somewhat archaic expression for) something like “aspect” or “quality”.
This is clear to readers of the German original, but let me try to find a reference which expands on this…
How about this:
deals with this issue on p. 44-45:
only Grégoire and Seeberger offer anything like a definition of ’[das] Moment’-as distinguished from ’der Moment’, which Hegel uses in the normal temporal sense. The others, including Findlay, seem to assume that it will be understood as ’$moment_H$’, namely, “dialectical or speculative phase,” though Findlay some times puts it in inverted commas. Grégoire defines ’moments’ as “des aspects ou des éléments d’un tout essentiellement corrélatifs” (Gr 108nl). Seeberger defines ’moment’ as “a distinguishable, but not separable, component of a more comprehensive whole or process” (S 102), offering as a, perhaps overly specific, example the organs of a living organism. An organic whole, Seeberger adds, has its being in the living totality of its moments${}_H$ and each moment${}_H$ has its living actuality only in its unity with the whole (S 55).
Or here is an example of a more contemporay use of “das Moment”:
(Here something like: “The aspect of recognition in the philosophy of law.”)
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What follows ends up being a little of a complaint regarding attribution. Please give it a thought:
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also on p. 5:
footnote 5 says
This, and definitions building on it, are original terminology.
This seems to be referring to the terminology “successive moment”. It may be hard for the reader to see what the footnote is really saying: The “successive” is your invention, but the way of speaking of moments and modalities is taken from my writing. As far as I can see.
You say
Following (Lawvere’s formalization of) Hegel, we say a
but the symbols you give are entirely taken from my writing (eg. in dcct v2, this is pp. 287). I don’t think you find this in Lawvere’s writing.
To sort this out, here and elsewhere, I suggest to follow a simple principle: Whenever you attribute anything to anyone: include a specific citation – a citation which points the reader to a concrete page or paragraph in that person’s writing which supports the claim that the notion originates there. This helps the reader to find your sources, and it helps you to verify that your attributing is accurate.
For example, further down on p. 7 you write:
Following the work of Martin-Löf [13], the part of Hegelian logic which he calls “subjektive Logik” and which has been understood as classical logic, is in this formalization identified with the deductive part of type theory, and extended to type theory in general. A “Begriff” is here understood as a type.
The reader of this line is probably bound to get away with the impression that it was Martin-Löf who suggested that his notion of “type” may be understood as a formalization of Hegel’s notion of “Begriff”. This is not true, and it is probably not what you want to say here. What is true is that one can glimpse that ML understood that “type” refers to “concept”, and from there I suggested that we should understand this as Hegel’s “Begriff”. I have referenced this in some detail here.
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One way to motivate accurate attribution in this matter is to realize that referencing/attributing is not just a matter of distributing praise, but also of distributing blame: There is no indication that Lawvere (or Martin-Löf or anyone) approve of (or even have taken note of) anything that I wrote on Science of Logic, and they might be displeased to find a text which gives a different impression.
Thanks so much for the comments! I’ll implement them today, and also put up the texfile. I’m glad about the clarification of the attribution, I tried to straighten it out, but I first learned the theory from the nlab (and dcct), then from (discussions of) Hegel and then from Lawvere, so there is bound to be some confusion there (which is also why I put it up here). But is it right to attribute identifying $\sharp$ and $\overline{\sharp}$ with intensive and extensive qualities to Lawvere? I know he wrote about them and their properties, but I wasn’t sure if he already identified them with the functors.
I generally intended to mean what you thought I meant, so I will make those passages more clear.
FWIW, I can say based on the one discussion I’ve ever had with Lawvere that he is absolutely not inclined towards $(\infty,1)$-categories, and he was in fact very dismissive. I wanted to raise the topic of Urs’ work on cohesion, to either see what he thought of it, or otherwise let him know how it extends and fulfills the premise of his vision, but we did not get anywhere near it, since the discussion died on the starting line. The further development of his interpretation of Hegel by Urs thus seems like something Lawvere would not particularly appreciate.
Lawvere’s dislike for $\infty$-category theory is known (the careful reader of “Axiomatic cohesion” will see that his ambition was to get the classical homotopy category using the 1-categorical cohesion of $sSet$ – of course it fails to account for fibrancy)…
…but what tends to confuse people – and rightly so – is that he did not connect cohesion to modalities (as far as I am aware).
@Alexander fyi if you write to open a PR on a bulk of latex text, extra line breaks after sentence ends probably help. Otherwise one can’t really comment on individual sentences all that well. (Although no idea how much feedback you’d get in the PR, so unless you’re quick with scripting, changing the existing text might not be worth the effort.)
Alright, I’ve implemented the corrections, clarified the ambiguities, expanded the sources, re-attributed the attributions and put up the texfile along with the corrected pdf. Later I’ll go through any remaining attributions and see if they check out!
“…is known” well, to insiders perhaps. The point is that we are having a discussion precisely for, ultimately, those who are not insiders.
You mean in “is known as homotopy theory”? I only meant it as a more elegant way to say “called”. I’ll replace it and add a reference to a homotoy theory introduction and the hott-book.
Thanks.
Some further random comments as I keep reading through the text:
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p. 7:
“from abstract nonsense” – this doesn’t seem to work well with the whole context of your text. (It’s a phrase that is used in texts that do need to mention category theory but only tangentially and with hesitance, signalling to their readers that there is no risk that the author will get carried away on that tangent. This is all opposite to the intention of your text.)
I’d suggest to simply state the actual reason: “from essential uniqueness of adjoints”.
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“we will now discuss homotopy type theory” – best to give a reference. (Now I see you give one at the end of the section. I’d suggest to move this to the beginning of the section. Also for the lay audience that you are addressing, there are probably better first introductions than the HoTT book: for instance Shulman 2017)
“we will now discuss homotopy type theory even though” – why “even though”? The following words instead suggest a “because”. (?) Generally, this first sentence of section 5 is hard to read. Maybe it could be broken up into two sentences.
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“This is identified with Hegel’s first law of thought” –
Such statements could be more precise if qualified as: “This state of affairs may neatly be matched/compared to…”. (At least I found this matching was really neat when I first suggested it.)
Maybe you want to capitalize and/or italicize “first law of thought”: the First Law of Thought.
Notice that this is not really Hegel’s law, but (at least) Fichte’s (1794, see at first law of thought). It is a great example of how Hegel was trying to connect to whatever he found worthwhile in other people’s “systems”.
“The self-equality … can then be understood …” – This sentence I find confusing. What you should say here is that with identities understood as paths,the canonical identity (the “reflector” in technical terms) is the constant path, which clearly always exists.
I find this simple but important point could be brought out more clearly on p. 9. I would start by beginning a new paragraph before “Furthermore…”.
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On the bottom of the page you drop the all-important univalence law without really saying much about it.
The claim that this is to be related to Hegels bold statement that “Alle Dinge sind verschieden” seems not to be brought out at all, at face value it seems surprising. To do justice to this point, I think you need to expand more here.
The way to go, I think, would be to tap into the type-theorist’s re-phrasing of univalence as “isomorphism is equality”. Then you could say that this can be read as saying “Whenever two things seem alike, they are already equal”; or conversely: “Whenever it looks like we have two things (instead of one in two guises), then they are (substantially) different.” Which, at last, is really close to what Hegel was saying.
This issue of univalence and what Fichte/Hegel sensed about equality could easily make a whole subsection in itself, if one were ambitious about it. If you don’t want to write such a subsection, maybe at least make sure that the reader understands that they are just being shown the scratching of the surface of something deep.
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just trivia, but the empy set comes out nicer, I find, when called not as \emptyset
but as \varnothing
. One might even pause to comment on that in view of Hegel’s process. ;-)
“where we derive moments from the theory of types” – I wonder how the reader is meant to understand such a claim. I think you need some lead-in explanation here which connects back to the yoga of moments that you had recalled earlier:
You might recall that Hegel was trying to feel the inner dynamics of a progression of oppositions and sublation, starting with nothing.
You would then highlight that it’s hard to “feel” this for most people, but that, luckily, you now have a mathematical calculus that formalizes it: the calculus of adjoint idempotent monads/moments and their resolutions.
Therefore, you would say, you can now try to see how far the calculus of progressions of adjoint monads can be worked out, starting from the base opposition in a topos, and how the result can be understood as matching Hegel’s ontological poetry.
Incidentally, it is this mathematical formalization of the objective logic that Lawvere is referring to in the quote which is reproduced on your p. 2.
Without explaining this, I am afraid the generic reader is bound to get away with the impression that section 6 is all weird.
I’ll write something to smoothen the edges, but I also have another question: I’m reworking the section on infinitesimal cohesion, but I’ve confused myself a bit. In particular, is an infinitesimally thickened point an $\Re$- or an $\Im$-anti-modal type? In anti-reduced type it says $\Im$, in infinitesimally thickened point and Science of Logic it says $\Re$. I’m assuming it’s $\Im$ because then they correspond to formal moduli problems? Also somewhere it says that the reduction $\Re$ doesn’t strip away all infinitesimal structure which we can see because its modal types still receive non-trivial morphism from infinitesimally thickened points, whereas $\Im$ doesn’t.
So as I understand it, based on the entries, $\Re$ trivializes infinitesimal structure by reducing it to what is implied by the real structure, whereas $\Im$ trivializes it by identifying everything connected by infinitesimal paths, which creates small rifts in the spatial structure, so that an $\Im$-modal type is a much more constrained kind of structure (one that is formally etale over the terminal object). Then a map $m:X\rightarrow \Im Y$ maps all infinitesimal neighborhoods in $X$ to infinitesimal neighborhoods in $Y$ where now all points are identified, so corresponds to the map $\overline{m}:\Re X\rightarrow Y$ that maps the real point $p$ in an infinitesimal neighborhood of $X$ necessarily to the real point in the neighborhood that $m(p)$ will be mapped to. Is that about right?
Also, I rewrote the section on the double negation modality, I think it is a lot better now (just in case anyone reads through that section).
Here a PR in the spirit I mentioned, with targeted comments possible on the changes page.
The plaintext without the diff is also here on the view file page.
(I naively do line breaks after 100 or more characters here. Can also change the PR to any other such number or give you the script.)
Thanks! I’ve merged it with the main branch. Can you add the script to the repository?
I’ve also reworked the homotopy type theory section. I did not write a subsection, mainly because I’m trying to keep the article short, but also because I don’t think I have enough expertise about Hegel (or Fichte). Hopefully philosophers will fill that hole. In any case, it should be a lot easier to read. I’ve also written an introduction to the next section which should set it up better, and added paragraphs to the subsections on elastic and solid substance that compare the three unities of unities to each other.
is an infinitesimally thickened point an $\Re$- or an $\Im$-anti-modal type?
Both conditions are equivalent:
Decomposing under the definition here we have
$\Re \,\simeq\, i_! \circ i^\ast \;\;\;\;\;\;\;\; \Im \,\simeq\, i_* \circ i^\ast$where $i_!$ and $i_\ast$ are both fully faithful and both preserve the terminal object. So
$\Re X \simeq \ast \;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\; i^\ast X \simeq \ast \;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\; \Im X \simeq \ast \,.$In natural language it seems more natural to refer to $\Re$, but in working with these structures it is $\Im$ that is doing most of the work.
Of course, thanks! I should have seen that. In any case, I think, now my picture of the infinitesimal modality is fairly clear: $\Re$ reduces the ideal part to (what is implied by) the real one, whereas $\Im$ identifies all points in an infinitesimal neighbood, thereby shrinking the space to an infintesimally flat shape, the “crystalline” shape in crystalline cohomology.
Yes, that’s right!
Or in slightly different words:
$\Re$ removes all “infinitesimal thickening” and does nothing else.
$\Im$ identifies infinitesimally close points. This in particular implies the removal of all infinitesimal thickening, but is also non-trivial on ordinary spaces.
More discussion of $\Im$ that might be useful:
from a neat synthetic differential geometry point of view in the work of Felix Cherubini (née Wellen) (here), a good exposition is in his talk at CMU 2019 (here);
in the context of partial differential equations is in my article with Igor Khavkine: Synthetic geometry of differential equations
Thanks!
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