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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 2nd 2013

Late last night I was reading in Science of Logic vol 1, “The objective logic”.

I see that the idea of cohesion is pretty explicit there, not in the first section of the first book (Determinateness, which has the discussion of “being and becoming” that Lawvere is alluding to in the Como preface) but in the second section of the first book, “The magnitude”.

There the discussion is all about how the continuous is made up from discrete points with “repulsion” to prevent them from collapsing to a single and with “attraction” that keeps them together nevertheless.

This “attraction” is clearly just the same idea as “cohesion”. One can play this a bit further and match Hegel’s Raunen to formal expressions involving the flat modality and the shape modality pretty well. I made some quick notes in the above entry.

On the other hand, that section 1 about being and becoming seems to be more about the underlying type system itself. Notably about the empty type and the unit type, I think

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeNov 3rd 2013

On the other hand, that section 1 about being and becoming seems to be more about the underlying type system itself. Notably about the empty type and the unit type, I think

Is this what you mentioned back here

the initial void is the empty type $\empty$, the tzimtzum-line drawn into the emptiness that draws a distinction is $\empty \to \empty$, which is the unit type.

Now a stiffer Hegelian test. What is this in type theory?

Every human being is an entire world of representations buried in the night of the ’I’. The ’I’ is thus the universal in which abstraction is made from everything particular, but in which at the same time everything lies shrouded. It is therefore not a merely abstract universality, but a universality that contains everything within itself. (Shorter Logic, Sec 24)

$I = Type$?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 3rd 2013

Yes, so I think we can set up the following rough dictionary between Hegel’s terms and terms in homotopy type theory, and I think this works pretty well (at Science of Logic I have been compiling passages which support these identifications):

Hegel’s logic type theory
moment modality
ground antecedent
entering into existence term introduction
immediacy of reflection reflector term in identity type
all things are different intensional identity
being, One unit type
nothing empty type
becoming adjoint cylinder $\emptyset \dashv \ast$
moment of repulsion flat modality $\flat$
moment of attraction cohesion, shape modality $\int$
continuum adjoint cylinder $\int \dashv \flat$

But concerning that piece from Shorter Logic I need to check. Not sure. yet.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 3rd 2013
• (edited Nov 3rd 2013)

By the way, if you have a minute I’d appreciate you looking over some other sentence that I have added to Science of Logic.

For instance right at the beginning I added a remark that “logic” here is understood in the sense of Heraclitus’ “logos”. I think that remark helps sort out the possible confusion as to why this is not a book on how to handle logical conjunctions and qualifiers.

I have googled around to find some authority agreeing with me on this. I found an old text by Heidegger on “Hegel and the Greeks” which says this pretty explicitly. But probably there are more/better references.

One curiosity I ran into while looking into this: the German online resource for Hegel, Hegel in Projekt Gutenberg, is sad compared to the English hypertext version, even if the latter sits on the marxists.org site (sigh). The German edition is just the raw text and apparently almost blindly OCR-ed (it starts out with weird typo mismatchs right in the first line). It doesn’t even have a margin when opened in a browser window. On the contrary the English version is carefully hyperlinked, paragraph-numbered and usefully decomposed into sub-pages. Funny how it goes. Der Prophet gilt nichts im eigenen Land.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 3rd 2013

I take that back. A very good German online edition is at http://hegel.logik.2.abcphil.de.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeNov 3rd 2013
• (edited Nov 3rd 2013)

Here’s a fuller version from a different translation in note 1 of section 24:

By the term ‘I’ I mean myself, a single and altogether determinate person. And yet I really utter nothing peculiar to myself, for every one else is an ‘I’ or ‘Ego’, and when I call myself ‘I’, though I indubitably mean the single person myself, I express a thorough universal. ‘I’, therefore, is mere being-for-self, in which everything peculiar or marked is renounced and buried out of sight; it is as it were the ultimate and unanalysable point of consciousness. We may say ‘I’ and thought are the same, or, more definitely, ‘I’ is thought as a thinker. What I have in my consciousness is for me. ‘I’ is the vacuum or receptacle for anything and everything: for which everything is and which stores up everything in itself. Every man is a whole world of conceptions, that lie buried in the night of the ‘Ego’. It follows that the ‘Ego’ is the universal in which we leave aside all that is particular, and in which at the same time all the particulars have a latent existence. In other words, it is not a mere universality and nothing more, but the universality which includes in it everything.

There ought to be an equivalent passage in ’The Science of Logic’.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeNov 3rd 2013

Thanks, David. So clauses like

receptacle for anything and everything … the universal in which we leave aside all that is particular

make me think of unit type/terminal object. But I claimed this is already the right interpretation for Hegel’s “Being”. I am not sure yet what the new aspect is supposed to be with “I”.

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeNov 3rd 2013

Thinking more on this, following the Husserlian line, picked up by Martin-Lof, the trace of the ’I’ in type theory would be the judgement sign $\vdash$ itself. But then I don’t see that fitting with what Hegel writes.

As I mentioned elsewhere, Weyl does seem to have been very influenced by Husserl. He speaks of any coordinate system as being a trace of the human subject. Covariance then takes on special significance for him as reasserting the universal over a particular coordinate system.

Back to Hegel, to understand what he means by ’Logic’, one would need understand this division;

Volume One: The Objective Logic. Book One: The Doctrine of Being; Book Two: The Doctrine of Essence.

Volume Two: Subjective Logic. The Doctrine of the Notion

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeNov 3rd 2013

the trace of the ’I’ in type theory would be the judgement sign ⊢ itself.

That’s good, because that corresponds to the unit type! Under slicing. Each type is also a stand-in for the slice over it, hence for the entire topos of types dependent on it.

On the one hand, there is the unit type $\ast$ which has no internal properties, etc. it “just is” along the lines of Hegel. At the same time it is the whole type system, in the sense that

$\ast \vdash ...$

and hence

$\vdash ...$

is the global context of everything.

It would be fun if we could identify these two roles of types/contexts in Hegel’s text. Not sure yet…

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeNov 3rd 2013

In §329 it says

The moments which constitute the Notion of the one as a being-for-self fall asunder in the development. They are: (1) negation in general, (2) two negations, (3) two that are therefore the same, (4) sheer opposites, (5) self-relation, identity as such, (6) relation which is negative and yet to its own self.

Now given that “moment” translates well to modality in previous examples, this is reminiscent of the double negation modality.

The one as the unit type is indeed the double-negation modal type.

I had seen more occurences of “negation of negation” elsewhere. But now I have trouble finding them again…

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeNov 3rd 2013

Ah, I found the place I had in mind, book 1, section 1, chapter 2, A. c) Etwas

This is all about the double negation modality. Somehow.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeNov 3rd 2013
• (edited Nov 3rd 2013)

Concerning the global context and the unit type:

maybe we can match the difference between a type and its context to Hegel’s distinction between “determinate” and “indeterminate” being.

I suppose “quality” should well match to the notion of “type” (of what type is this term? which quality does it have?). Then I guess “determinate” goes along with having a quality

§130 Because it is indeterminate being, it lacks all quality;

(Not that we necessarily need to stick to what he is saying, but it is fun to see to which extent we can.)

So then this reminds me a bit of the term introduction rule for function types, by which we can move stuff from the left of the turnstile to the right. But not sure yet.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeNov 3rd 2013
• (edited Nov 3rd 2013)

I should call it quits now, it’s getting too late for me. Here is my proposed dictionary Hegel$\leftrightarrow$ HoTT so far.

Hegel’s logic modal homotopy type theory
quality $\sim$ type
determinate being of quality $X$ $\vdash X \colon Type$
moment modality
ground antecedent
entering into existence term introduction
immediacy of reflection reflector term in identity type
all things are different intensional identity
being, One (context of) unit type
nothing empty type
becoming adjoint modality $\emptyset \dashv \ast$
moment of repulsion flat modality $\flat$
moment of attraction cohesion, shape modality $\int$
continuum adjoint modality $\int \dashv \flat$
moment of discreteness flat modality $\flat$
moment of continuity sharp modality $\sharp$
quantity adjoint modality $\flat \dashv \sharp$
vanishing of infinitesimals reduction modality
moment of two negations double negation modality
something (-1)-truncation modality, classically double negation modality

Just a proposal. Search the entry Science of Logic for these keywords to find paragraphs that are supposed to make this dictionary plausible.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeNov 4th 2013

I added something to explain the relation of the Science of Logic to the Shorter Logic, Part I of the Encyclopedia. There’s a triadic structure running through all his work, which needs to be thought of as a circular process. He says somewhere that we could start anywhere in the circle.

It is amazing how suggestive Hegel was in his Logic, but I’m still stumbling over the very first steps. Being is associated with $\ast$, Nothing with $\empty$, then becoming is their unity of opposites. But why would he write what he does in 134, about their sameness (and distinctness)?

Would you write it that way? Why not the way discussed before of starting with Nothing (as inductive type perhaps), then $\ast$ arising from the autoequivalence of $\empty$?

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeNov 4th 2013

Hmm, but you did see this

This is the image of the universe before creation. That black screen with nothing on it except that intensional dependent type theory is running in the background. And no input yet: in the Coq-kernel, in the register for “context” it reads:

empty

as in

Inductive : empty : Type :=

and then nothing.

We are in the empty state. I am suggesting this picture as a good formalization of the point that Hegel tries to describe in sec 86-87. Or that the Zohar assumes when it says Ayn Sof,

as though the Nothing comes first, whatever first could mean here.

Is there a coinductive definition of $\ast$?

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeNov 4th 2013
• (edited Nov 4th 2013)

Concerning the very first steps in Hegel’s text:

I think it is important not to take Hegel too much by the word, as he was clearly lacking the language for what he wanted to express. He keeps oscillating between apparently saying that “nothing and being are the same” and “nothing and being are not the same”, explicitly for instance in §152

In this being and nothng are distinct moments; becoming only is, in so far as they are distinguished.

I think he had some intuition, some feeling of some pre-physics dynamics, but hardly the means to express it faithfully. Clearly. So when checking which formalization might capture this intuition well, we need to allow for that.

Here in this case I now tend to take Lawvere by the word and assume that wherever Hegel produces paradoxical statements of the form “moment A is the same as moment B” and “moment A and moment B are distinct”, then what we are to extract from this is one adjoint modality $A \dashv B$. We are to assume that Hegel, who had no way to speak of adjoint modalities, is expressing to us how an adjoint modality may feel like without having the words to describe it accurately. I read his text like poetry, not like formal text.

And I think that makes good sense. Of course it means there is lots of room for us to make errors in trying to formalize Hegel.

But here I think it works well: while at some points he does say things like that “nothing and being are the same”, I don’t take this verbatim, but just, by Lawvere, as the indication that I am to look for an adjoint modality between modalities that formalize both “nothing” and “being”. And then with the remaining text about what “nothing” and “being” are to be like, I find it rather compelling to identify them with the modalities constant on the empty type and on the unit type, respectively. And that does form an adjoint modality, and its unity-transformation does very well express Hegel’s statement that “everything is a stage in between nothing and being”.

So that’s how I end up concluding that whatever else he says about nothing and being, it’s all about $\emptyset \dashv \ast$.

I mean, even if we grant that he was after something 200 years ahead of his time, we are not to expect that there is a precise map from his text to type theory. It’s surprising enough if there is a rough such map.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeNov 4th 2013
• (edited Nov 4th 2013)

Concerning #15: yes, I think other philosophies, such as the old cabbalistic texts, and then notably Spencer-Brown, express a creation myth which I find well formalized by saying something like “in the beginning there was the type $\emptyset$ and hence the type forming operation $\not = \to \emptyset$ and from this was obtained $\ast = \not \emptyset$”.

I have a little note on this which I kept on my private web, but now moved over to my public web, for you to see, have a look at

to see what I mean.

I don’t find this particular intuition/insight expressed in Hegel’s Science of Logic. But maybe it is in one of the many and lengthy remarks, not all of which I have read through. But I do think this could be added as a 0th book to Hegel’s text. He starts with telling us about $\ast$ and $\emptyset$ and that there is a tension $\emptyset \dashv \ast$ between them in which every other type $X$ is a stage $\emptyset \to X \to \ast$. He does not tell us about how $\ast$ itself arises out of $\emptyset$ by applying the available type constructors. But he should have.

• CommentRowNumber18.
• CommentAuthorDavid_Corfield
• CommentTimeNov 4th 2013

OK, let’s cut Hegel some slack.

By the way, I know you don’t go the Cafe now, but I wonder what you make of a discussion I’m having there in the thread The HoTT Approach to Physics.

There’s quite a difference between Husserl and Hegel. With the former it’s all about the structure of subjective intentionality. Focus on the judgement ($\vdash$) is then natural. With Hegel it’s more theological, The Idea - Geist - God, works through its internal contradictions to create universe and mind. The universe is thus rational, there’s no gap to cross, as with other philosophies, between a mind with its concepts and an independent world.

The real is the rational, the rational is the real.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeNov 4th 2013
• (edited Nov 4th 2013)

But also with type theory the distinction blurs: while we start out calling

$\Gamma \vdash A \colon Type$

a “judgement”, as if it is some act of us humans, we know that the semantics of this is just that there is an object $A$ in a slice category over $\Gamma$. Phrased this way it is not so much something I am free to judge or not, it just is.

So the difference between making the “subjective” judgement

$\vdash \emptyset \colon Type$

or

$\vdash \ast \colon Type$

and the objective fact that there is a (type-)universe in the context of $\ast$ fall togeher.

It’s the “unity of opposites” of syntax-semantix duality, if you wish.

• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeNov 4th 2013

It’s the “unity of opposites” of syntax-semantics duality, if you wish.

Is there an adjoint cyclinder in this case? Just because there’s a dual equivalence doesn’t mean there’s a adjoint cylinder.

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeNov 4th 2013

Does anything interesting happen in that string of adjunctions of length 7 mentioned in this MO answer between $Ab$ and $Ab^{\to}$? I think I’m right that all the composite endofunctors on $Ab$ are identities.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeNov 4th 2013

Right, I was wondering, too, if that adjunction is usefully embedded somehow in an adjoint cylinder. I don’t know.

But I think the point I’d like to make remains in any case. Better maybe to think of propositions-as-types. The proposition “the type X has determinate being” is identified with the (hprop-truncation of) the whole (type-) universe itself, and at least classically that is $\ast$.

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeNov 4th 2013

That discussion about strings of adjoints is interesting, yes. I meant to get back to it at some point. I don’t know yet what to make of that 7-functor example, but potentially there is something very interesting hidden there, especially when lifted from abelian groups to spectra.

• CommentRowNumber24.
• CommentAuthorDavid_Corfield
• CommentTimeNov 4th 2013

I think the order is

$coker \dashv 0 \to (-) \dashv codomain \dashv identity \dashv domain \dashv (-) \to 0 \dashv ker.$

Some part of that was reminscent of the tangent $(\infty, 1)$ construction.

If $Set$ is characterised as here by that string of 5 adjunctions, I wonder if $\infty Grpd$ has a similar characterisation.

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeNov 4th 2013

I think in #19 and #22, you’re realising the interpenetration of the Objective Logic and Subjective Logic. As Inwood explains here, a first approximation has Objective Logic as metaphysics and Subjective Logic as concerning judgement. But of course Hegel would never maintain a sharp dichotomy for long.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeNov 4th 2013

Thanks for the pointer to Inwood! That’s useful. I enjoy seeing the diagram of the section outline on p. 263.

That reminds me: I had begun wondering if we can usefully find adjoints of adjoints here in some way. But not sure yet.

I should say that I can’t be thinking about all this much more during this week now, as there are too many other things that need my attention. But I hope to come back to it next weekend.

• CommentRowNumber27.
• CommentAuthorTobyBartels
• CommentTimeNov 4th 2013

even if the latter sits on the marxists.org site (sigh)

IME, that's a pretty high-quality site with a wealth of material of all sorts, as long as it's related to Marxism. I'm not surprised that it has good Hegel sources.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeNov 4th 2013

I can believe that. But I sighed when I went to the home page at marxists.org and was greeted by Che Guevara.

• CommentRowNumber29.
• CommentAuthorDavidRoberts
• CommentTimeNov 4th 2013
• (edited Nov 5th 2013)

@DavidC - you should probably could think of the arrow category of $Ab$ as the category of 2-groups in $Ab$, namely strict symmetric 2-groups. Then things like coker and ker of an arrow are $\pi_0$ and $\pi_1$ of the 2-group. However the domain and codomain functors are not invariant under (weak) equivalences, so I’m not sure how to fruitfully think of those from the 2-group POV.

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeNov 4th 2013

Not sure about the 2-group story.

First, notice that this example extends the one at Cohesive diagrams in a cohesive infinity-topos.

• CommentRowNumber31.
• CommentAuthorDavidRoberts
• CommentTimeNov 5th 2013

OK - I made the statement less emphatic.

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeNov 6th 2013
• (edited Nov 6th 2013)

There is something wrong with the translation of “Dasein” as “determinate being” in the standard translations.

“Dasein” means “existence”, “being there”. That it has to do with “determinate being” is not common sense, but is what Hegel says: the first sentence of §188 in the Original is

Daseyn ist bestimmtes Seyn

The standard translation would have to say here

Determinate being is determinate being

which is nonsense, and so the standard translation does something even worse and completely makes up something by writing:

In considering determinate being the emphasis falls on its determinate character

This sentence is not a translation of anything that Hegel writes.

• CommentRowNumber33.
• CommentAuthorDavid_Corfield
• CommentTimeNov 6th 2013

So what would be a better translation of

Daseyn ist bestimmtes Seyn?

Existence/Being-there/Being-in-the-world is determinate being?

But your thought is that this is the type theoretic: nothing exists which is not typed.

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeNov 6th 2013
• (edited Nov 6th 2013)

I feel that “existence” is the good direct translation. The standard English translation of Hegel is maybe unnecessarily heavy. Of course the original is not exactly a newspaper article either, but it does have a certain light poetic flow to it.

For instance §178, reminds me of Michael Ende’s “Perelin der Nachtwald” in Die Unendliche Geschichte

Das Werden ist auf diese Weise in gedoppelter Bestimmung; in der einen ist das Nichts als unmittelbar, d. i. sie ist anfangend vom Nichts, das sich auf das Seyn bezieht, das heißt, in dasselbe übergeht, in der anderen ist das Seyn als unmittelbar d. i. sie ist anfangend vom Seyn, das in das Nichts übergeht — Entstehen und Vergehen.

That’s a poem “– Entstehen und Vergehen”.

The English “ceasing-to-be” seems unnecessarily heavy here. “becoming and ceasing” works, too, doesn’t it?

It’s fun that in German one has this “Sein/Dasein” rhyme, and Hegel is fond of that, but I would directly translate it as “being/existence”, which seems good enough, too.

But your thought is that this is the type theoretic: nothing exists which is not typed.

Yes, it seems to me that the notion of “type” in the original sense “of which type is this term?” is pretty much Hegel’s “quality”, “of which quality is this term?”. Maybe also his “determinateness”, as in “what determines the nature of this term?”. But I am not sure yet if I get the “determinateness”-issue correctly.

But read the entering paragraphs again with this in mind and try to play around with reading it as being about typing the empty type and the unit type etc. I think it works fairly well.

• CommentRowNumber35.
• CommentAuthorUrs
• CommentTimeDec 28th 2013
• (edited Dec 28th 2013)

I am lacking one pair of opposites:

First, by §322 it is clear that “ideality” is the unity of the opposite moments of “being for self” and “being for one”. Second, by §324 it is clear that “ideality” itself is a moment opposite to “reality”.

So $ideality \dashv reality$ is a “second order” duality (as is $quality \dashv quantity$, for which it is clear what the four sub-moments are).

Hence it must be that “reality” is itself the unity of two opposites. But which? I cannot find this in the text.

[ hm, maybe “Realität” is the same as “Wirklichkeit”. The latter is the unity of “outer” and “inner”, he says. ]

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeDec 29th 2013
• (edited Dec 29th 2013)

There is something neat going on in the section on “measure”. From just glancing over it I used to think that this is about “measure” roughly in the sense in that it appears nowadays e.g. in “measure space”. And that used to annoy me, because the formalization of such measure in homotopy type theory which I know did not fit at all into the exegesis of Hegel that we had so far.

Namely recall that we made what I think was a rather well justifiable point that with Lawvere’s adjunction interpretation of Hegel’s unity of opposites the following dictionary makes good sense

$\array{ & attraction && repulsion \\ quality : & \int &\dashv& \flat \\ & \bot && \bot \\ quantity : & \flat &\dashv& \sharp \\ & discreteness && continuity }$

But now next Hegel says that “measure” is the unity of the opposites of quantity and quality, hence that, by this dictionary, these cohesive modalities together should be thought of as encoding “measure”.

That used to annoy me, because $\int \dashv \flat \dashv \sharp$ don’t encode anything similar to measures as in “measure space” or similar. Instead, they encode differential cohomology, hence gauge fields.

But now that I actually read Hegel’s chapter on “measure” I am struck by a pleasant surprise: what he means by Maß (measure) is really Maßstab hence what here is translated as “standard” but which we may just as well translate as… gauge.

Hegel’s “measure” is really “gauge”. (!) He makes this crystal clear in (§714 ), which could almost be from Weyl’s historical discussion of gauge invariance:

A measure taken as a gauge in the usual meaning of the word is a quantum which is arbitrarily assumed as the intrinsically determinate unit relatively to an external amount. Such a unit can, it is true, also be in fact an intrinsically determinate unit, like a foot and suchlike original measures; but in so far as it is also used as a standard for other things it is in regard to them only an external measure, not their original measure. Thus the diameter of the earth or the length of a pendulum may be taken, each on its own account, as a specific quantum; but the selection of a particular fraction of the earth’s diameter or of the length of the pendulum, as well as the degree of latitude under which the latter is to be taken for use as a standard, is a matter of choice. But for other things such a standard is still more something external. These have further specified the general specific quantum in a particular way and have thereby become particular things. It is therefore foolish to speak of a natural standard of things. Moreover, a universal standard ought only to serve for external comparison; in this most superficial sense in which it is taken as a universal measure it is a matter of complete indifference what is used for this purpose. It ought not to be a fundamental measure in the sense that it forms a scale on which the natural measures of particular things could be represented and from which, by means of a rule, they could be grasped as specifications of a universal measure, i.e. of the measure of their universal body. Without this meaning, however, an absolute measure is interesting and significant only as a common element, and as such is a universal not in itself but only by agreement.

This way our Lawvere-ian exegesis of the first book of “Science of Logic” is now the following rather pleasant system:

$\array{ & & attraction && repulsion \\ & quality : & \int &\dashv& \flat \\ gauge & \bot & \bot && \bot \\ & quantity : & \flat &\dashv& \sharp \\ & & discreteness && continuity }$
• CommentRowNumber37.
• CommentAuthorDavid_Corfield
• CommentTimeDec 30th 2013
• (edited Dec 30th 2013)

Good to see you’re back with Hegel! I’ll take a look at what you’ve written in a moment, but looking back at Science of Logic, at §178

Becoming is in this way in a double determination. In one of them, nothing is immediate, that is, the determination starts from nothing which relates itself to being, or in other words changes into it; in the other, being is immediate, that is, the determination starts from being which changes into nothing: the former is coming-to-be and the latter is ceasing-to-be

it seems Hegel wants there to be a term that unites coming-to-be and ceasing-to-be, which jointly constitute the double determination of something. Unfortunately, becoming is the obvious term to use for coming-to-be, but it is also used for the doubly determined thing. Does the German use unconnected words for becoming and coming-to-be?

• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeDec 30th 2013
• (edited Dec 30th 2013)

Good point. Yes, in the German original the two words “Werden” and “Entstehen” are genuinely different, while of course related. The latter is a bit more like “developing”. (One would say “Hier entsteht ein Flughafen” for “Here an airport is being built”.) The former is not really a noun at all, or at least is not commonly used this way, the common word is “werden” meaning “will” as in “Wir werden…”/”We will…”.

One can see that the English translator had some trouble with this, for in the German original §176 starts out as

Das Werden, Entstehen und Vergehen, ist

which in the English translation §176 is simply truncated to just

Becoming is

So the unity of opposites here is

$\array{ Werden &\colon& Nichts &\dashv& Sein \\ && \emptyset &\dashv& \ast }$

and then Hegel means to in addition identify directionality in there in two ways, Entstehen and Vergehen.

I am not sure yet how this is to be formalized. What I observe is that in unity of opposites there is already a directionality of sorts, there is always a flow from the co-modality to the modality via counit followed by unit.

$\Box X \longrightarrow X \longrightarrow \bigcirc X$

In the case of “Werden” this is the natural transformation

$\emptyset \longrightarrow X \longrightarrow \ast$

which evidently sort of flows from left to right, from nothing to being. Not the other way around.

I need to think more about this. But now I have to interrupt for the moment, am not supposed to be online right now…

• CommentRowNumber39.
• CommentAuthorDavid_Corfield
• CommentTimeDec 30th 2013

The opposite directions of coming-to-be and ceasing-to-be seem to have more of the flavour of a category with duals, e.g., the birth and death singularities (e.g., p. 23 of Carter’s Categories for Knotted Curves, Surfaces and Quandles).

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeDec 30th 2013
• (edited Dec 30th 2013)

This is something I am still trying to see, how to bring some linear logic into the picture.

I keep wondering if this might be better related to book 2 “Wesen”, but that I haven’t absorbed much at all yet.

Because the cap and cup (birth and death) naturally appear in the linear logic as part of the QFT picture where they are explicitly birth and death of “things” (fields). This seems to be different in flavor from the “being” of book 1, which is more general, it seems, including the being of concepts etc (for instance of the notion of “field” in the first place).

Judging just from headlines, such “things” would be the topic of book 2, section 2, chapter 1. But I am not sure yet how to extract an interesting formal statement from the text there.

• CommentRowNumber41.
• CommentAuthorDavid_Corfield
• CommentTimeDec 30th 2013

We noticed a while ago (#17) that Hegel didn’t begin with nothing and generate the one from it, as we might do. I wonder if there’s some difficulty he has separating the ’abelian’ from the ’topos’.

• CommentRowNumber42.
• CommentAuthorUrs
• CommentTimeDec 30th 2013
• (edited Dec 30th 2013)

Since #17 refers to formation of function types (into $\emptyset$) while (the way we are reading it) Hegel says everything in terms of adjunctions, maybe one would get a more refined exegesis here if the hom-adjunction

$(-)\times \;\emptyset\; \dashv \emptyset \to (-)$

is put to the forefront. This is of course equivalent to the constant $\emptyset \dashv \ast$ but read this way the $\ast$ here is indeed realized as $\emptyset \to \emptyset$.

• CommentRowNumber43.
• CommentAuthorUrs
• CommentTimeDec 30th 2013
• (edited Dec 30th 2013)

or better, the adjunction $(\emptyset \dashv \ast)$ is also equivalently phrased as

$\underset{\emptyset}{\sum}(-) \;\dashv\; \underset{\emptyset}{\prod}(-) \,.$

(with context extension left implicit).

This adjunction manifestly says: “We are in the context of nothingness, then there are two opposite movements: either consider all the nothingness there is, that’s nothing, or else consider the negation of that nothingness, that’s the context of everything.”

Put this way, Hegel’s opposite of dualities “$Werden \colon Nichts \dashv Sein$” here is after all very similar to the tzimtzum.

A formal proof of the intrinsic equivalence of two very different looking schools of mystic ontology :-)

• CommentRowNumber44.
• CommentAuthorDavid_Corfield
• CommentTimeJan 3rd 2014
• (edited Jan 3rd 2014)

I see, Urs, you’ve added to Type-semantics for quantization.

How will the modality part of the story feed into the linear part? I keep thinking there must be something to that string of 7 adjunctions (#21 and #24). So in the case of $Spectra^{\to}$ there’s an adjunction string of six modalities beginning with a monad, the first unit being the pushout square for the cokernel in $Spectra$, the last counit being the pullback square for the kernel. $Spectra^{\to}$ is tangent to $Spectra$, isn’t it.

• CommentRowNumber45.
• CommentAuthorDavid_Corfield
• CommentTimeJan 3rd 2014

Re #36, if you have gauge (measure) as mediating quality and quantity, then you should have actuality mediating reflection and appearance, according to the structure of the book.

Then ’the idea’ mediates subjectivity and objectivity. Finally, at the highest level, ’the notion’ mediates ’being’ and ’essence’.

• CommentRowNumber46.
• CommentAuthorUrs
• CommentTimeJan 3rd 2014
• (edited Jan 3rd 2014)

How will the modality part of the story feed into the linear part?

A first order infinitesimal extension of a space $X$ is equivalently a module for the functions on $X$. That’s how the modalities know about the linear part of the story.

But I am still thinking how to best set it up. I have that the differential cohesion modalities give for each type $X$ an etale topos $Sh(X)$ which is equipped with an $\mathbf{H}$-structure and an $\mathbf{H}_{\Re}$-structure sheaf $\mathcal{O}_X$ (where $\Re$ is to denote the reduced objects), hence is an $\mathbf{H}_{\Re}$ structured (infinity,1)-topos.

In DAG8 it is observed that we can say “$\mathcal{O}_X$-module” by saying: lift of the $\mathbf{H}_{\Re}$-structure sheaf to an $\mathbf{H}$-structure sheaf.

But this only works if the reduced objects are not just coreflectively but also reflectively embedded into all objects. This is the case if the underlying site is the opposite of the tangent category of the opposite site. But it is not the case for instance in cases like the Cahier topos, hence in synthetic differential geometry.

Therefore I am still not sure which approach best to adopt here. Another axiomatic way to define modules from the adjunctions is to say that $Mod(X)$ is the category of pointed objects in $\mathbf{H}_{/X}$ which are annihilated by the $\Re$-modality. This is something I have been trying to explore a bit more in the few minutes that I currently have. But not sure yet.

if you have gauge (measure) as mediating quality and quantity, then you should have actuality mediating reflection and appearance, according to the structure of the book.

Then ’the idea’ mediates subjectivity and objectivity. Finally, at the highest level, ’the notion’ mediates ’being’ and ’essence’.

That relates to my question above in #35, where I was trying to go further with the identification. It is not always clear to me how the structure of the section outline of the book relates to the structure “explained” in the text.

Over at Science of Logic I have tried to collect some evidence that the following exegesis is fairly accurate:

$\array{ & \text{für sich sein} && \text{für eins sein} \\ ideality & \Re &\dashv& \oint \\ & \bot && \bot \\ reality & \oint &\dashv& \tilde \flat \\ & outer && inner }$

But while the exegesis in #36 works strikingly well, it seems, this identification here I am not as convinced about. May need more fine-tuning, or maybe our little translation project here just breaks down.

Recall how the above identification was obtained: Hegel’s polemic against Leibniz’s monads we found made it clear that points with no interaction with other points are “für sich” while those that have some “antennas” to feel around them are “für eins”. This is well captured by reduction/infinitesimal shape modality. Now since Hegel says that being-for-self and being-for-it are the two moments of “Ideality”, that gives the first row above.

One might be able to think about it this way: this adjunction is about infinitesimal extension, and that is a “non-real” “idealized” extension. The tangent as the “idealization” of a curve at a point. Not a very strong argument maybe, but that’s what came to mind.

Now for the second line as far as the modalities are concerned, that follows from adjointness. So if we are on the right track here we should be able to read off from Hegel what the right labels are. He says that dual to “ideality” is “reality” hence “Wirklichkeit”, and that this is the union of “inner” and “outer”. That’s how the above table comes about.

Now concerning “reality”, I am not sure, but by what I recalled at the beginning we have that the $(\oint \dashv \tilde \flat)$-modality (infinitesimal shape $\dashv$ infinitesimal flat) controls the construction of structured étale toposes for each type. Maybe one could argue that this makes the type “become a reality” in some sense.

But I am not super-convinced yet of all this. If you have further suggestions, let me know. We may have to play around a bit more here with both the text and the adjuncitons.

Finally concerning that long string of adjunctions which you mention: yes, I agree that sure looks like it is relevant for something. But I still don’t have anything deeper to say about it, alas.

• CommentRowNumber47.
• CommentAuthorDavid_Corfield
• CommentTimeJan 3rd 2014

I mentioned di Giovanni’s translation, which seems a huge improvement. E.g., where Miller translates

• Daseyn ist bestimmtes Seyn;

as

• In considering determinate being the emphasis falls on its determinate character;

Di Giovanni has

• Existence is determinate being;
• CommentRowNumber48.
• CommentAuthorUrs
• CommentTimeJan 3rd 2014
• (edited Jan 3rd 2014)

Yes, thanks, I agree, this is a much better translation.

But remind me, what aspect of our discussion do you mean this to be a comment on? On the notion of “reality”?

Concerning Daseyn (Dasein), let’s note the following: by §208 we have that Dasein is being equipped with quality. By the discussion reviewed above in #36, we argued that Hegel’s “quality” is well-expressed by (shape modality $\dashv$ flat modality). Indeed, this adjoint modality is precisely what gives the types what I would call their “geometric” quality (whereas $\Gamma$ and hence (flat $\dashv$ sharp) encodes the “cardinality” and hence the “quantity” underlying types).

So it seems that types in the bare type theory with just the $(\emptyset/nothing \stackrel{becoming}{\dashv} \ast/being)$-adjunction have an indeterminate “being”, while in the presence also of $(\int \dashv \flat)$ they also have a determinate Dasein/existence.

That seems to make some sense.

And then further, by my previous comment, with also the $(\oint \dashv \tilde \flat)$-modality given, the types even have “reality”. Does that sound reasonable?

• being of types = presence of $(\emptyset \dashv \ast)$

• existence of types = further presence of $(\int \dashv \flat)$

• reality of types = further presence of $(\oint \dashv {\tilde \flat})$

(?)

• CommentRowNumber49.
• CommentAuthorDavid_Corfield
• CommentTimeJan 3rd 2014
• (edited Jan 3rd 2014)

I was just mentioning the di Giovanni book having noted it on the entry, and remembering your criticism of the online one in #32.

Do you keep in mind the structure of the whole work when looking for possible ?

There are three books (Being, Essence, Notion), each of which has three sections, each of which has three chapters, each of which has three subsections (A, B, C), many have which have three sub-subsections (a, b, c).

Should it matter where the triads appear? I mean Gauge: quality $\dashv$ quantity corresponds to the whole of Book 1. Quality as attraction-repulsion occurs in Book 1, Sec 1, Chap. 3, Subsec. C, apparently three levels below.

But perhaps it isn’t as tree-like as that. In fact, I know Hegel conceived it as a circular chain.

Anyway, will think about what you wrote in #47 later.

• CommentRowNumber50.
• CommentAuthorUrs
• CommentTimeJan 3rd 2014
• (edited Jan 3rd 2014)

Do you keep in mind the structure of the whole work when looking for possible ?

I’d try to, but I have mostly been looking for pieces of text that serve as plausible evidence that some interpretation is sensible. By this method I have trouble seeing all the structure of the section outline accurately reflected in the text.

For instance the second book, third section “Wirklichkeit” (actuality/reality) starts out with a sentence saying that “Wirklichkeit” is the union of “Wesen” and “Existenz” (of “Essence” and “Existence”). But going instead by the section outline it would have to be the union of “Wesen” and “Schein” (“Essence” and “Appearance”). Rather “Existenz” appears as the title of the first chapter of the second section, not as that of the second section itself. Moreover, by §1190 we have instead that Wirklichkeit/Actuality is the union of “inner” and “outer”.

How to deal with this? The same kind of “problem” arises all over the place, it seems to me.

Let me suggest this: there is maybe a duality

• poetry $\dashv$ logic

Hegel’s text seems to have both of these moments.

• CommentRowNumber51.
• CommentAuthorUrs
• CommentTimeJan 3rd 2014
• (edited Jan 4th 2014)

addendum to #48, in view of p. 11 of Some Thoughts on the Future of Category Theory: in other words we have the following situation

$\array{ being && existence && reality \\ \\ && && Red \\ && && \bot \\ && \int & \subset & \int_{inf} \\ && \bot && \bot \\ \emptyset &\subset& \flat & \subset & \flat_{inf} \\ \bot & & \bot && \\ \ast & \subset & \sharp }$

where “$\subset$” denotes inclusion of modal types

• CommentRowNumber52.
• CommentAuthorUrs
• CommentTimeJan 4th 2014

concerning the second book: I have trouble getting as much out of this as out of the first. But here is one thought:

He says the “Wesen”/”essence” is, first of all, “reflection”, and I suppose it means “reflection of/on the being”. Moreover, this is “nur Schein”. (This seems to be ill-translated in the English translations that I have seen, which make “Schein” be unqualified “appearance”. But Hegel speaks of “nur Schein” and that is more like “illusion”. “Es scheint nur so” means “It only seems to be”. So instead of “appearance” it is more like “superficial appearance”).

Taken together this reminds me of the object classifier, the type universe: this is manifestly the self-reflection of a type system/infinity-topos in/into itself, and it is clearly something of the superficial appearance of the full thing, whithout actually being the full thing.

With this in mind it is interesting to see the second book dwell on the notion of identity a lot. As if a faint glimpse of univalence.

Unfortunately, beyond that I get little out of the detailed text of book two.

But this is my broad impression: book one is about the type system, book two about the type universe/the type of types, and book three about the deductive system. And the circularity of it all is the fact that for any choice of object language and metalanguage we can consider taking the metalanguage itself to be the object language of a higher order metalanguage, and so forth.

• CommentRowNumber53.
• CommentAuthorDavid_Corfield
• CommentTimeJan 4th 2014

In #51, by p. 11, which page is this in the text? Maybe your version has different pagination?

At Some Thoughts on the Future of Category Theory, you had p. 12 in

This is what Lawvere is talking about from the bottom of p. 12 on

which I’ve changed to p. 6, as printed in the book. That agrees with what you had for the next page reference:

This is what Lawvere calls the skeleton and the coskeleton on p. 7

• CommentRowNumber54.
• CommentAuthorUrs
• CommentTimeJan 4th 2014

Oh, sorry, right, I was giving inconsistent page numbers.

The page I mean is page 7 as numbered in the book, the one that starts out with the words “adjoints which are full inclusions”. It first explains adjoint moments. Then towards the end of the page it says “The basic starting example of all this” and then gives the $(\emptyset \dashv \ast)$-adjunction, calling it (non being $\dashv$ pure being). The implication is that any $(\flat \dashv \sharp)$ is another notion of being, more determinate than the tautological pure being.

• CommentRowNumber55.
• CommentAuthorUrs
• CommentTimeJan 4th 2014
• (edited Jan 4th 2014)

I had a typo in #51 (fixed now): the pattern is like this:

$\array{ reines\;Sein && Dasein && Realitaet (?) \\ \\ && && Red \\ && && \bot \\ && \int & \subset & \int_{inf} \\ && \bot && \bot \\ \emptyset &\subset& \flat & \subset & \flat_{inf} \\ \bot & & \bot && \\ \ast & \subset & \sharp }$

This is not only correct now, but suggestive of how the pattern might continue:

$\array{ && && && \bigcirc \\ && && && \bot \\ && && Red &\subset& \Box \\ && && \bot && \bot \\ && \int & \subset & \int_{inf} &\subset& \bigcirc \\ && \bot && \bot \\ \emptyset &\subset& \flat & \subset & \flat_{inf} \\ \bot & & \bot && \\ \ast & \subset & \sharp }$

(On the other hand, arguably it’s too small a sample to make much deduction from.)

• CommentRowNumber56.
• CommentAuthorDavid_Corfield
• CommentTimeJan 4th 2014

The infinitesimal spaces, which contain the base topos in its non-Becoming aspect, are a crucial step toward determinate Becoming, but fall short of having among themselves enough connected objects, i.e. they do not in themselves constitute fully a “category of cohesive unifying Being.” (p. 10)

So that’s like infinitesimal cohesion (infinitesimal cohesive (infinity,1)-topos).

And, he says, it can be thought of as of dimension $\epsilon$, although dimensions are not linearly ordered (p. 11).

The idea behind the identification of the levels in a category of Being with dimensions is that a higher level is a more determinate general Becoming, that is, it contains spaces having in them possibly-more-varied information for determining processes.

Does this relate to hlevel? But it sounds from p. 9 like a more standard notion of dimensionality.

On the other hand, the following sounds rather homotopic:

The relation between the trivial level and the base level above it is only the first case of a possible strong relation between two levels which (hoping not to do too great an injustice to Hegel) I will call Aufhebung relative to the given category of Being: this is the relation between a lower level and a higher level whereby the first level is not only included (on the left and equivalently on the right, or simply that the longer downward functor factors across the shorter one) in the higher, but moreover that the longer left adjoint inclusion factors across the shorter right adjoint inclusion; equivalently, the higher coskeleton functor fixes both the skeleta and the coskeleta in the sense of the lower level. (p. 8)

I haven’t a clue what the next part of the paragraph means:

A very simple picture (not involving toposes) involves taking the basic downward functor to be any given map from a seven-element totally ordered set onto a three-element one, which just amounts to a partition of the big set into three non-empty closed intervals;

Have we encountered the dual situation described here?

Unity-and-identity-of-opposites, the Aufhebung relation between two such within a given unity: this is a second proposed philo- sophical guide. It is not limited to distributive categories, nor is the dual case of an inclusion which has both left and right adjoint re- tractions without interest (p. 8)

Section III is largely a mystery to me.

• CommentRowNumber57.
• CommentAuthorUrs
• CommentTimeJan 4th 2014
• (edited Jan 4th 2014)

The infinitesimal spaces, which contain the base topos in its non-Becoming aspect, are a crucial step toward determinate Becoming, but fall short of having among themselves enough connected objects, i.e. they do not in themselves constitute fully a “category of cohesive unifying Being.” (p. 10)

So that’s like infinitesimal cohesion (infinitesimal cohesive (infinity,1)-topos).

Ah, thanks for reminding me of that part of the text. I had forgotten about it. Hm, indeed, he talks about a category of infinitesimal objects being included with left and right adjoint coinciding. Somehow later he started to say “quality type” instead and not have the relation to infinitesimals be so “explicit”. So thanks for pointing this out.

The idea behind the identification of the levels in a category of Being with dimensions is that a higher level is a more determinate general Becoming, that is, it contains spaces having in them possibly-more-varied information for determining processes.

Does this relate to hlevel? But it sounds from p. 9 like a more standard notion of dimensionality.

hlevel would indeed be an example if only it had a further adjoint: the inclusion of n-truncated objects is reflective, but not coreflective.

I haven’t a clue what the next part of the paragraph means:

A very simple picture (not involving toposes) involves taking the basic downward functor to be any given map from a seven-element totally ordered set onto a three-element one, which just amounts to a partition of the big set into three non-empty closed intervals;

You are supposed to think of these posets as categories. Let’s write $[n]$ for the linear order (category) on $(n+1)$ elements, as usual. then he considers a functor

$[6] \longrightarrow [2]$

and observes that this is equivalently a partitioning of the 7-element linear order into three subintervals (the fibers of the three objects of $[2]$). This has a left and a right adjoint

$[6] \stackrel{\leftarrow}{\stackrel{\longrightarrow}{\leftarrow}} [2]$

which take an element in $[2]$ to the minimum or maximum element, respectively, of the interval in $[6]$ corresponding to it.

Now an intermediate level for this is a factorization

$[6] \stackrel{\leftarrow}{\stackrel{\longrightarrow}{\leftarrow}} [k] \stackrel{\leftarrow}{\stackrel{\longrightarrow}{\leftarrow}} [2]$

with $2 \lt k \lt 6$. This is now a decomposition of the interval of length seven elements into $(k+1)$ sub-intervals, which in turn are then subsumed into three sub-intervals.

Lawvere proposes that this factorization is a formalization of Hegelian “Aufhebung” (sublation) of the notion of “being” which expressed by

$[6] \stackrel{\leftarrow}{\stackrel{\longrightarrow}{\leftarrow}} [6]$

if the composite left adjoint $[6]\leftarrow [2]$ factors also through the right adjoint $[k] \leftarrow [2]$. Because if this is the case that the duality exhibited by $[6] \stackrel{\leftarrow}{\stackrel{\longrightarrow}{\leftarrow}} [2]$ is “removed” by the duality $[k] \stackrel{\leftarrow}{\stackrel{\longrightarrow}{\leftarrow}} [2]$ (or maybe the other way round).

Hm. I suppose if there is something to be gained from this, then we have yet to think about it…

• CommentRowNumber58.
• CommentAuthorUrs
• CommentTimeJan 5th 2014
• (edited Jan 5th 2014)

?

I have only come across this by accidence just right now, but it feels like I ought to have known of its existence for a while.

Superficially, from the table of contents etc., this has considerable overlap with our discussion here, or at least with its motivation, aim or spirit. I don’t know yet how much overlap there is in content. Rodin explicitly talks about Hegel’s Logic and Lawvere’s ideas in section 4.8. (But it seems adjoint modalities are not being mentioned by Rodin.)

• CommentRowNumber59.
• CommentAuthorDavid_Corfield
• CommentTimeJan 6th 2014
• (edited Jan 6th 2014)

For some reason I didn’t notice that final diagram in #51. In view of your comment #30, if one looks to match up some of the six modalities with the cohesive ones, I think

• $A \to B \implies A \to 0$ would correspond to $\sharp$
• $A \to B \implies A \to A$ would correspond to $\flat$
• $A \to B \implies B \to B$ would correspond to $\int$

That leaves three modalities

• $A \to B \implies 0 \to B$
• $A \to B \implies 0 \to coker(f)$
• $A \to B \implies ker(f) \to 0$.

Could it be possible that an extra modality appears below $\ast$ in #51 (the diagram begging for another column of three), and then the diagram collapses horizontally to give a string of six modalities?

(One reason for horizontal collapse might be that tangent extension and diagram extension coincide in a stable setting.)

• CommentRowNumber60.
• CommentAuthorDavid_Corfield
• CommentTimeJan 6th 2014
• (edited Jan 6th 2014)

In case that appeared like guesswork, it makes sense in terms of the adjunctions at cohesive (infinity,1)-topos.

The correspondences also makes intuitive sense in terms of clumping of $A$ together, separating out elements of $A$, etc.

Naturally, then

• $A \to B \implies 0 \to B$

would correspond to $Red$, getting rid of the infinitesimal fringe.

• CommentRowNumber61.
• CommentAuthorUrs
• CommentTimeJan 6th 2014

I see, this is a very good point, yes. Yes, you are right, the story in #55 begs for restricting it to the case where some of the incusions are actually equivalences.

Let me think about this. Maybe it will be useful not to focus on just spectrum objects for the moment, not to overly reduce the options. As pointed out in that MO discussion, this string of 7 adjunctions applies in any context of pointed objects (with all limits and colimits). Such a context is, incidentally, the focus of a new section 2.3.4 at Type-semantics for quantization (schreiber). Maybe it will be useful to think of that length-7 string in that context…

• CommentRowNumber62.
• CommentAuthorDavid_Corfield
• CommentTimeJan 7th 2014

I never asked what you intend by the right hand column of the lower diagram in #55. Are they extreme forms of modalities in some sense?

• CommentRowNumber63.
• CommentAuthorUrs
• CommentTimeJan 7th 2014
• (edited Jan 7th 2014)

I don’t know what that unspecified rightmost column would mean. All I observe is that the 8-tuple of modalities of which I know the meaning happens to arrange in a pattern that has an evident extension. This however is not an answer, but is a question: is this telling us something? Why this pattern? Can we give a reason to consider systems of modalites forming duch a pattern more fundamentally than by observing that it happens to yield a rich and interesting formal system that captures a wealth of established maths.

• CommentRowNumber64.
• CommentAuthorDavid_Corfield
• CommentTimeJan 7th 2014
• (edited Jan 7th 2014)

Regarding #61, looking at pointed objects, the extreme modalities of the six are now

$f: A \to B \implies \ast \to cofiber(f)$

and

$f: A \to B \implies fiber(f) \to \ast.$
• CommentRowNumber65.
• CommentAuthortrent
• CommentTimeJan 7th 2014
Urs, have you looked into Whitehead at all? I'm only barely acquainted with his philosophy, but it looks like there are two computer scientists (who are interested in category theory) at Northumbria who are trying to do for Whitehead something like what you and Lawvere do for Hegel:
• CommentRowNumber66.
• CommentAuthorTodd_Trimble
• CommentTimeJan 7th 2014

David, should I read $\Rightarrow$ in #64 as $\mapsto$?

• CommentRowNumber67.
• CommentAuthorUrs
• CommentTimeJan 7th 2014
• (edited Jan 8th 2014)

@trent,

thanks for the pointer. Just looked at it. These slides conclude in saying that what Whitehead called, apparently, the “Category of the Ultimate” should correspond to the hyperdoctrine or maybe more generally dependent type theory of some given (or any?) locally cartesian closed category (as in the relation between category theory and type theory.)

I didn’t know that this is something that one can find suggested in Whitehead’s writing. That is interesting.

What David and myself have been discussing here, following Lawvere, does start out just like this, but then there is one more step. The setting is an ambient locally cartesian closed category, yes, usually assumed by us to be a topos, or rather it is the plain dependent type theory validated by any such, not a specific one. Then the idea is that each “determination” or “moment” in Hegel’s text corresponds to adding to the axioms of (intensional, homotopy) dependent type theory one modality in the form of an idempotent monad/comonad, and declare for each (or some) unity of opposites that Hegel discusses that the corresponding dual moments form an adjoint pair.

If we only have the locally Cartesian closed category and no further modalities added (as is apparently the situation considered in the note you point to) then, so the translation story goes, this corresponds to what Hegel called reines Sein, “pure being” completely “undeterminate” completely without “quality”. Adding these modal operators means making the “being” more “determinate” by giving “quality” to it.

Anyway. I should maybe try to get hold of that text by Whitehead. David probably knows more here.

• CommentRowNumber68.
• CommentAuthorDavid_Corfield
• CommentTimeJan 8th 2014
• (edited Jan 8th 2014)

Todd #66, Yes.

I was wondering whether there’s a connection with the appearance of fibers and cofibers in def. 2.5 of Urs’s note.

• CommentRowNumber69.
• CommentAuthorDavid_Corfield
• CommentTimeJan 8th 2014

On Whitehead, there’s a useful page at SEP, and another on process philosophy. It’s very likely there are lines of influence from Hegel to WHitehead, through Bradley and other British Idealists.

• CommentRowNumber70.
• CommentAuthorUrs
• CommentTimeJan 16th 2014
• (edited Jan 16th 2014)

By the way, changing topic a fair bit:

While I evidently feel sucked into it every now and then, it would seem clear that having the kind of discussion that we are having here is a sure means to utterly marginalize yourself among most imaginable peer groups.

In view of this it seems a striking fact worth mentioning here at some point, that, as far as I can tell without being there, a vaguely similar kind of discussion is all en vogue in one non-negligible corner of the big market of science popularization produced by professional scientists, notably in the US.

For one, at least until recently there seemed to rage among US scientists a public discussion, preferably held, apart from the author’s personal blogs, in the New York Times, I gather, about “universes from nothing”, see e.g. Carroll 12 for what seems to be the typical attitude and Albert 12 for a quick review and more thoughtful critique. A critique that asks for something going a bit deeper.

One of the participants of this discussion, if it makes sense to think of one public discussion here, who stands out as apparently taking that critique to heart and aiming for something deeper indeed, is Max Tegmark. Having a name as a solid sober cosmologist, a few years back he posted “The mathematical Universe” (arXiv:0704.0646) which is easily one of the most vague and speculative articles in the gr-qc archive already notorious for its indulgence in vague speculation bordering on the un-scientific.

Tegmark’s arguments in detail are rather different than what we discuss here, but they are clearly driven by a similar spirit. Just a few days back he turned that into a book, “Our mathematical universe” which, judging from this website, is clearly aimed to go big in the public debate, caliber of late-night-show invitations, I assume.

You are probably aware of all this, I don’t know. But somehow it seems in view of our long discussion here, this is an elephant in the room which maybe deserves to be pointed to once.

• CommentRowNumber71.
• CommentAuthorDavid_Corfield
• CommentTimeJan 16th 2014

Regarding the current view of Hegel, I added to our page Hegel’s %22Logic%22 as Modal Type Theory a comment and reference for a revival in his fortunes. Still there’s a distance between this and people taking the Science of Logic as having something to tell us in its details. That’s not to say there aren’t exceptions, e.g., Stephen Houlgate is a respected philosopher at Warwick in the middle of a three year research fellowship, during which time he is

…writing a book on Hegel’s Science of Logic (concentrating in particular on the doctrine of essence and the doctrine of the concept).

Even there, it’s noticeable that his further interests are not with Hegel’s views on natural science, but rather on his political philosophy, aesthetics and philosophy of religion.

Re Tegmark, I remember some discussions with John Baez many year ago. I may have this wrong, but I seem to recall feeling towards his position with something resembling my feeling towards set theory, as opposed to what attracted me about category theory, i.e., I wanted a more constrictive kind of framework.

Given that we (or you, at least) are still finding out about the particularities of the kind of mathematics necessary to make sense of our physics, the idea that we should work with as loose a notion as:

A mathematical structure is precisely this: abstract entities with relations between them,

strikes me as wrong.

• CommentRowNumber72.
• CommentAuthorUrs
• CommentTimeJan 16th 2014
• (edited Jan 16th 2014)

Regarding Tegmark: yes, absolutely, in detail his methods are not what we’d find appropriate. It is curious that, once he realizes that physics should be founded in mathematics, he does not go beyond a fleeting thought as to what the decent foundation, in turn, of mathematics would be. Also his emphasis of Gödel incompleteness very much reminded me of your comments on this in your “Towards a philosophy of real mathematics”. Lawvere or anything in that ballpark seems to be unknown to him.

Nevertheless, while I find his answer inappropriate, I am struck that he finds such a resonance with asking the question.

But then, you know, this might not be unrelated. Lawvere’s substantial thoughts on the foundations of physics in mathematics are now 50 years old and I suppose the number of people to take genuine notice can be counted on the fingers of a hand. That’s in part because Lawvere does not present his thoughts the way that would get him an invitation by Letterman or to the TED show, but in parts it is also because even if he did, most people could still not connect to the category and topos theory involved. On the other hand, Tegmark’s tools, while comparatively naive, are exactly what the audience, and this is the educated audience already, is prepared to digest.

Anyway, this was just a side thought. But maybe it might be worth adding a brief digression on these social phenoma to Hegel’s “Logic” as Modal Type Theory . Something like: “While analytic philosophy removed metaphysics from the body of acceptable philosophy for its lack of intellectually acceptable tools, it is noteworthy that, nonwithstanding, the need for a metaphysics seems to be as strongly felt among modern physicists as it must have been to Hegel and his predecessors, as witnessed by—” and then something as alluded to in #70.

• CommentRowNumber73.
• CommentAuthorDavid_Corfield
• CommentTimeJan 16th 2014

Regarding the last paragraph, that would have to be phrased carefully. It’s true that for the Vienna Circle ’metaphysics’ was something to be eradicated. But it came back with a vengeance in the later stages of the last century.

Now, there’s a flourishing field of analytic metaphysics - take a look at SEP on possible worlds. Although some see possible worlds as a useful fiction to help make sense of counterfactual truths in our world, others, e.g., David Lewis, were committed to the reality of possible worlds.

However, little attention is paid to the actual physics of our universe, and naturally the most inadequate formal tools are used, untyped modal logic.

• CommentRowNumber74.
• CommentAuthorUrs
• CommentTimeJan 16th 2014
• (edited Jan 16th 2014)

Okay, thanks. Good to have you available for these matters!

I had a go at the entry and started skecthing some structure indicated by some keywords, to see where we are headed. None of this is meant to be set in stone, just to record some thoughts and to have a condensations seed for further development and refinement. Please feel invited to drastically edit what you find there.

• CommentRowNumber75.
• CommentAuthorUrs
• CommentTimeJan 16th 2014

I see that you have edited, thanks.

Something broke in the sentence following the first mentioning of Carnap. I have tried to fix it now but please check if the intention of your last edit is still properly reflected.

• CommentRowNumber76.
• CommentAuthorDavid_Corfield
• CommentTimeJan 16th 2014

Coming to think about this, it’s not perhaps as easy as you might imagine to separate out physicists’ and philosophers’ approaches to metaphysics. I mentioned on the page Saul Kripke as instrumental in the rise of analytic metaphysics. Now, one could say that his metaphysics wasn’t terribly well-informed by physics. But then came David Lewis, about which it’s harder to say that, and by the time the philosophers of physics join in I’m not sure it is easy to characterise what distinguishes the likes of Tegmark from the likes of, say, David Wallace.

• CommentRowNumber77.
• CommentAuthorDavid_Corfield
• CommentTimeJan 16th 2014

SEP on nothingness is an eclectic mix, but it shows that philosophers today aren’t averse to trying to say why there is something.

• CommentRowNumber78.
• CommentAuthorUrs
• CommentTimeJan 16th 2014
• (edited Jan 16th 2014)

re #76:

While I see what you mean, let me ask: who exactly do you think of re “the likes of Tegmark”? Wallace focuses on the interpretation of quantum mechanics, which seems to me to be, while related, rather different from (an attempt at) a genuine ontology of physical reality. Maybe it’s weird, and interpretationists like Wallace talk about how exactly it might be weird, but there is a deeper question here. Weird or not: why?

By the way, concerning interpretation of quantum mechanics: we did collect some references, but I still haven’t seen the kind of discussion which I am imagining here, which would namely go like this, please let me know what you think:

suppose we formalize classical mechanics as some theory in, say, first order logic or type theory or the like. Then there are propositions we can formulate and check to be true or not from the axioms. Then we may at least imagine asking whether these axioms of classical physics allow physical universes that contain “observers”. And then we may ask what seems to be the crucial question: suppose you can show that the physics which you have axiomatized in some formal logic may contain “observers”, of sorts, then:

will these observers regard as true the true propositions in the ambient formal system? Will they even be able to see a reflection of these “ambient” propositions, “internal” to the universe that they observe?

This is clearly a quite elusive question, already for classical mechanics. Now replace in the above “classical mechanics” with “quantum mechanics” and then this question should be largely the question for the “interpretation” of quantum mechanics.

But viewed this way I think there is a surprise: a priori if you write down any set of axioms, it would seem rather surprising that if (huge if) these axioms “describe universes which contain observers” that these observers would have sensations of perceptions of anything even remotely similar to the ambient formal system by which we are describing all this.

Viewed this way, it seems unsurprising that it is hard for observers like us to fully match our sensations of perception to the formal systems with which we describe and predict these. On the contrary, viewed this way it seems surprising that in classical mechanics we can imagine that this does work to some extent.

In other words, below the superficial surface of the question it seems to me that the fact that requires extra attention is not that quantum mechanics seems weird to our sensory apparatus, but that there is an approximation by classical physics where it does not.

In any case, to come back to the original point: while Wallace and others think about aspects of this question, Tegmark and maybe we here are after something more fundamental, it seems to me, namely the question: whence these axioms to begin with? Quite independent of how we feel about “interpreting” them.

• CommentRowNumber79.
• CommentAuthorDavid_Corfield
• CommentTimeJan 16th 2014

OK, so you want to distinguish between these two issues. We could do with terms to distinguish them, because metaphysics is used to cover both.

I think with Hegel the distinction won’t be clean. The internal contradictions of “The Idea” require a physical universe to work itself out, including the occurrence of sentient and rational beings such as ourselves. “The real is rational and the rational is real”. The subjective and objective logics coincide. The universe necessarily generates self-understanding.

So that’s rather different from all possible mathematical structures exist, one of which is the structure of our universe which contains intelligent beings. Tegmark’s account seems not so dissimilar to other forms of all possible worlds exist.

Collingwood has a Hegelian-like cosmology in “The Conclusion of 1934” in The Principles of History, but allows for multiple worlds.

• CommentRowNumber80.
• CommentAuthorDavid_Corfield
• CommentTimeJan 16th 2014

For the Oxford philosopher Derek Parfit’s arguments, see Part I and Part 2.

• CommentRowNumber81.
• CommentAuthorUrs
• CommentTimeJan 16th 2014
• (edited Jan 16th 2014)

re #79:

Wait, I am with Hegel on this, and it’s one of the neat aspects of his thinking. What I am saying is that it is but one aspect of the whole of his logic. Interprertation of (quantum) mechanics is part of metaphysics, but is not more than one chapter in a bigger story.

Also, just to be sure we are not talking past each other allow me to amplify one distinction (probably this is clear, let me just say it nevertheless): we ned to distinguish well between the “multi-verses” of cosmology from the “many worlds” in interpretations of quantum mechanics. The former is a respectable topic for metaphysical debate, but the latter is what I find a plain confusion.

• CommentRowNumber82.
• CommentAuthorDavid_Corfield
• CommentTimeJan 16th 2014

I agree that’s what makes Hegel interesting, the derivation of substantial structure from the unfolding of a logic, right up to the existence of matter, life and mind. But I would say that makes him exceptional, so I’m not sure why Tegmark is being presented as someone taken up with the same kind of metaphysical quest. The idea of all possible universes has never gone away. The idea that any universe is just structure has been commonly held for the past century. Sure he’s adding some mathematical detail to the story, but it’s a questionable mathematical framework.

And sure about that distinction in the second paragraph. Many worlds has always seemed to me misguided.

• CommentRowNumber83.
• CommentAuthorUrs
• CommentTimeJan 16th 2014
• (edited Jan 16th 2014)

I don’t want to defend Tegmark’s answers, and not the depth of his questions, but I appreciate the direction of his question. Let me expand.

First, for the record I suppose in one sentence the whole content of his “Mathematical Universe Hypothesis” is simply this:

“There exists ’mathematical structures’ $[$ possibly he’d rather want to say: formal theories in the sense of formal logic $]$ some of which are such that they axiomatize physical universes, of which in turn some are such that they accomodate substructures that may be called ’observers’, which hence, by definition, have some kind of observation of a reflection of the original ’mathematical structure’ as their perceived physical universe. “

That’s, it seems, the thought that he expanded first into an article and now into a book. Compared to Hegel a crucial distinction of this thought is that it has a vast amount of arbitrariness in it, whereas Hegel goes a decisive step further in suggesting that there is a unique formal structure (the Logic) which somehow comes with a canonical set of axioms (all those opposing moments). I am not advertizing Tegmark as being on par with Hegel here, but what I do think is that while falling short of that, he at least aims for something going one more step in that direction as compared to, for instance and as far as I can tell, what Wallace does.

For Wallace, I suppose, but certainly for various contemporay particle physicists/string theorists-turned-philosophical (such as those criticized in that note by Albert), the foundation of everything, the origin from which all reasoning about the world departs, is regarded to be quantum field theory/string theory. Interpretation of quantum physics is attempted on that basis alone, without the idea that one can take a further step back.

A while back Motl on his blog had a cute explicit version of this (I’d need to search for the link again): he told his readers about the hierarchy of higher order structures in the Universe (the usual ladder containing steps like biology$\leftarrow$chemistry$\leftarrow$physics ) and advertized as the absolute bottom of this hierarchy and hence of all reality… string theory. Now let’s not worry about string theory versus plain QFT, that’s not the point here, the thing is that some physical theory is regarded as being the ontological zero. I believe that more or less explicitly, this idea is prevalent among contemporary physicists-turned-philosophical.

And it is on this background that I say: Tegmark is one of the few of these who at least has an inkling that this is in fact not yet the ontological origin. That if physics is founded on QFT (or strings) and if QFT has, as it does, a mathematical formulation, that then there is at least one step further down to be taken and a world of mathematics to be explored, and the question to be asked how that existence of physical law (and hence of observers with their Subjective Logic) might flow from mathematical law.

• CommentRowNumber84.
• CommentAuthorUrs
• CommentTimeJan 17th 2014

here is that post which I mentioned above, the one with the table that ends the hierarchy of emergence of phenomena with a physical theory.

• CommentRowNumber85.
• CommentAuthorDavid_Corfield
• CommentTimeJan 17th 2014

Ok, I was pointing out that, leaving aside philosophers of physics, we’ve seen many ’maximalists’ who believe in the existence of all possible (in some sense) universes. Generally they’re not very restrictive about what this ’possible’ means, but then neither is Tegmark.

I can’t think of anyone other than Hegel who argues for considerable restrictions. There was the approach of Leibniz, who looked to explain God’s choice as an extremal problem, out of all possibilities, ours is in some sense the best.

• CommentRowNumber86.
• CommentAuthorUrs
• CommentTimeJan 18th 2014
• (edited Jan 18th 2014)

After following some of the links re Tegmark that Peter Woit collected yesterday here I feel a bit sickened of it all. I have removed the pointer to that “MUH” from our entry now, I suppose that’s what you gently tried to make me do anyway. This is a bit too undisciplined to cite with a straight face.

In general, i feel words are in the way here. (And are too cheap, too.) What we’d need is a kind of “proof without words”. Something like: notice Hegel dreamed this and Lawvere suggested that, and now we put this together in our formal alchemy cauldron, like this.. and now watch what happens…

I hope I’ll find more time later. Currently it’s all very constrained.

• CommentRowNumber87.
• CommentAuthortrent
• CommentTimeApr 9th 2014
• (edited Apr 9th 2014)
"Tegmark is one of the few of these who at least has an inkling that this is in fact not yet the ontological origin. That if physics is founded on QFT (or strings) and if QFT has, as it does, a mathematical formulation, that then there is at least one step further down to be taken and a world of mathematics to be explored, and the question to be asked how that existence of physical law (and hence of observers with their Subjective Logic) might flow from mathematical law." - Urs
(apologies for not embedding the quote, I don't know how)

Judging by this* popular article which I just came across on aristotelian realism in philosophy of mathematics, it looks like (mathematician/philosopher) James Franklin can be added to the list of people who have an inkling that physics in not yet the ontological origin. His stance of Aristotelian Realism rejects the Platonist idea that mathematics is actually existing in a realm of forms outside of space and time, and the nominalist idea that mathematics is just a language. Aristotelian Realism sees mathematics as actually existing, but in the physical rather than ideal platonic world. I take it this means that Franklin sees math as more ontologically basic than physics (and I presume his book** which is coming out tomorrow will go into more detail on his idea of the relationship between math and physics).

**which is a formal philosophical text, not a work of popular science
• CommentRowNumber88.
• CommentAuthorDavid_Corfield
• CommentTimeApr 9th 2014

I’ve corresponded with James. We both did some work on the idea of Bayesianism applied to mathematics. E.g., it’s possible to have (rational) degrees of belief in the truth of mathematical statements.

• CommentRowNumber89.
• CommentAuthortonyjones
• CommentTimeApr 9th 2014
There is a project uniting whitehead and quantum physics through category/topos theory by Michael Epperson and Elias Zafiris:
http://www.amazon.co.uk/Foundations-Relational-Realism-Topological-Contemporary/dp/0739180320/ref=sr_1_1?s=books&ie=UTF8&qid=1397064115&sr=1-1&keywords=zafiris+quantum

http://www.csus.edu/cpns/epperson/

http://users.uoa.gr/~ezafiris/
• CommentRowNumber90.
• CommentAuthorUrs
• CommentTimeApr 9th 2014
• (edited Apr 9th 2014)

Meanwhile I want to take back that quote in #87 above. The idea that some mathematics describes physics (or “is” physics, if you insist) is not what I thought would be a matter of discussion, since as an insight this dates from 1623 which has become proverbial in the 1960s. What I think is interesting to wonder about is how the particular mathematics that actually appears in theories of fundamental physics comes about “naturally”, as it were.

But informal debate of that only works well under controled circumstances. I enjoyed a lot how well it worked between David and me, but in the end communication on these matters won’t proceed by exchange of English language, but only by exchange of something more formal, I believe.

• CommentRowNumber91.
• CommentAuthortrent
• CommentTimeJul 3rd 2014

Just thought I’d throw this paper out there b/c - while not on hegel and homotopy type theory - it is on the related topic of a heavily hegel influenced philosopher (Robert Brandom) and the dialogic approach to constructive type theory. (Apologies to Zoran for the academia.edu link, it’s the only place where this paper is hosted and I don’t have an academia.edu account so I can’t download it / host it somewhere else.)

Knowledge and its Game Theoretical Foundations: The Challenges of the Dialogical Approach to Constructive Type Theory

• CommentRowNumber92.
• CommentAuthorDavid_Corfield
• CommentTimeMar 1st 2015

If Nature is to be seen as a representation of the Idea, a model of the theory, does Hegel give any suggestion that we should consider all such representations? I think Lawvere somewhere takes the ’concrete universal’ to be something like the category of all representations, from which one might be able to reconstruct the theory, under some duality.

• CommentRowNumber93.
• CommentAuthorUrs
• CommentTimeMar 11th 2015
• (edited Mar 11th 2015)

“necessity and contingency” (PN§193)

But before one considers this ambitious question, we should think still a bit more about which axioms we have, and if there are more to add. The more axioms, the less contingency in the choice of models.

We talked before about the question whether the bouquet of exceptional extensions of products of the superpoint might be axiomatized. If so, that would characterize one model of physics which is conjectured to be nature.

I don’t know, but I notice that with the recent addition of the third level of determinations, we may use the fermionic modality $\e$ to speak of the superpoint: $\mathbb{R}^{0|1}$ seems to be the only object (besides the trivial one, the unit type itself) in the standard model which is purely fermionic in the sense that it is an $\e$-anti-modal type.

(I have no proof of this uniqueness, but it would seem so. Certainly it is the unique representable for the sites of definitions which I know of. I should think more about it.)

We could/should also add the axiom that $\mathbb{R}^{0|1}$ carries an essentially unique non-trivial group structure. Write then $\mathbb{R}^{0|q}\coloneqq (\mathbb{R}^{0|1})^{\times_q}$.

Next, we want $\mathbb{R} = \mathbb{R}^{1|0}$, characterized as the continuum line object such that $\mathbb{R}$-localization is the $\int$-modality.

Then we are looking for nontrivial cocycles in the form of morphisms

$\mathbf{B} \mathbb{R}^{0|q} \longrightarrow \mathbf{B}^2 \mathbb{R}$

Those that we are after are the bilinear pairings $\{\Gamma^a_{\alpha \beta}\}$ induced by real irreducible $spin(d-1,1)$-representations $N$. The extensions of $\mathbb{R}^{0|dim(N)}$ that these would classify are the super-Minkowski spacetimes $\mathbb{R}^{d-1,1|dim(N)}$ “the first other-being, namely spatial-being, of the point” (PN§256a)

The question is if there is anything that singles these out these cocycles in a way that could be captured axiomatically.

• CommentRowNumber94.
• CommentAuthorDavid_Corfield
• CommentTimeMar 11th 2015

How do we go from

$\mathbf{B} \mathbb{R}^{0|q} \longrightarrow \mathbf{B}^2 \mathbb{R}$

to real irreducible $spin(d-1,1)$-representations? Is this presupposing that we want super-Minkowski spacetimes, or has that signature emerged naturally?

E.g., could we not see the cover $Spin(n, 2)$ of the anti de Sitter group?

• CommentRowNumber95.
• CommentAuthorUrs
• CommentTimeMar 11th 2015
• (edited Mar 11th 2015)

I don’t know, that’s exactly the question I was raising.

What I was trying to say was that the axiomatics has now evolved to the point that we may speak abstractly of maps of this form. The next step is to try to see if there is general abstract reason to pick the specific such maps, which we know in the standard model to produce spacetime.

And, sure, it might be deSitter or anti-deSitter just as well. Good point. But in any case the question is: why? On general abstract grounds.

• CommentRowNumber96.
• CommentAuthorUrs
• CommentTimeMar 11th 2015
• (edited Mar 16th 2015)

….

• CommentRowNumber97.
• CommentAuthorUrs
• CommentTimeMar 16th 2015
• (edited Mar 16th 2015)

Regarding further the question of how to motivate super-Minkowski space from “first principles” starting from the superpoint, here is a thought.

Consider $\mathbb{R}^{0|1}$, regarded as an abelian super-Lie algebra. As such it carries no nontrivial $d \geq 2$-cocycle.

So consider next $\mathbb{R}^{0|2}$. Its nontrivial cocycles in degree $d\geq 2$ are precisely a 3-dimensional space of 2-cocycles, spanned by the three elements $\psi^1 \wedge \psi^1$, $\psi^2 \wedge \psi^2$ and $\psi^1 \wedge \psi^2$.

Now in this low dimension this happens to coincide with the span of the three elements $\bar \psi \Gamma^0 \wedge \psi$, $\bar \psi \Gamma^1 \wedge \psi$, $\bar \psi \Gamma^2 \wedge \psi$ for the Majorana representation of $Spin(2,1)$. Hence forming the super Lie algebra extension of $\mathbb{R}^{0|2}$ by these three cocycles, we get 3d-superMinkowski spacetime as a kind of universal extension of the superpoint:

$\array{ \mathbb{R}^{2,1|2} \\ \downarrow \\ \mathbb{R}^{0|2} &\stackrel{\oplus_{i = 0}^2 \bar \psi \Gamma^i\psi}{\longrightarrow}& \mathbf{B}\mathbb{R}^3 }$

This now in turn carries at least a little brane bouquet, as there is then next a string-cocycle on $\mathbb{R}^{2,1|2}$.

From this perspective, one might next hope to see $\mathbb{R}^{3,1|4}$ in a similar manner (which will carry the cocycle for the super 2-brane in 4d). Not sure yet if there is a similar no-input way to get to that from just $\mathbb{R}^{0|4}$.

• CommentRowNumber98.
• CommentAuthorDavid_Corfield
• CommentTimeApr 21st 2015
• (edited Apr 21st 2015)

Science of Logic is becoming quite unwieldy as a page. I wonder if it is time to think of splitting it into smaller parts.

By the way, am I missing something? In the The method (absolute Idea) section, in Definition 2, when $\bigcirc \dashv \Box$ this concerns two different injections and one projection. (And then $\Box \dashv \bigcirc$ for two projections and one inclusion.) But when we come to examples, example 2 has the two even/odd injections as of the form $\Box \dashv \bigcirc$, and example 3 has the two projections, floor and ceiling, as $\bigcirc \dashv \Box$.

Also, it would presumably be worth spelling out in example 2, that the ’even’ comonad is the composition on $\mathbb{Z}$ that is greatest even number less than or equal to $n$, and the ’odd’ monad is least odd number greater than or equal to $n$.

• CommentRowNumber99.
• CommentAuthorUrs
• CommentTimeApr 21st 2015

Thanks for catching that typo! Fixed now.

I’d rather not split the page, as it means a lot of work in itself and then continuous additional work for whenever one wants to cross-link across splits.

Regarding the example: this should be added at adjoint modalities – simple illustrative examples and then we may or may not copy it over.

Thanks!

• CommentRowNumber100.
• CommentAuthorUrs
• CommentTimeApr 22nd 2015

I am still looking for the “right” idealistic term to use for the situation that a moment $\bigcirc$ is equivalent to localization at some type $\mathbb{A}$, i.e. $\bigcirc \simeq loc_{\{\mathbb{A}\to \ast\}}$. If you have any suggestion, let me know.