Added a reference to

- Michael Roy,
*The topos of ball complexes*, TAC Reprints no.28 (2021) pp.1-62. (abstract)

Added reference to

]]>Added a reference.

]]>Added a reference to the recent article by Marmolejo-Menni on “level $\epsilon$”.

]]>Thomas, thanks for the reference to the Redding paper, it looks great!

As for Kant/Hegel, I appreciate that what you're doing is provocation. It is of course incredibly important to get people to consider Hegel as an actual logician, and indeed, thereby to broaden their understanding of what logic is. I simply think it is worth pushing the envelope further back and giving Kant the same treatment, and indeed, thereby allowing us to draw more interesting *logical* lines from Kant to Hegel. Here are a couple further reasons to reconsider Kant's status as a logician, responding to your points:

1. One of van Lambalgen's students, Riccardo Pinosio, wrote an excellent MSc thesis reconstructing Kant's philosophy of mathematics based on the idea that transcendental logic is geometric logic (https://www.illc.uva.nl/Research/Publications/Reports/MoL-2012-13.text.pdf). It's really good, and it does an excellent job of dismissing Friedman's claim that Kant's constructivism was motivated by the inability of his logic to formulate ∀∃ judgements. There's more work to be done along these lines, as the role of temporality in Kant's account of construction is still not completely explicated, but I think it's a great start.

2. I think focusing on Kant's comments on general logic is to sell short the *genuinely logical* character of his transcendental logic. By analogy, one might say that 'Well, classical first-order predicate calculus and set theory is perfectly adequate, as far as it goes...' and then go on to talk about what it *isn't* adequate for, such as reasoning about things that aren't either unstructured points or set-theoretic constructions upon them, but it would be a terrible mistake for someone to focus on the first comment at the expense of the rest of the explanation. For a long time people have simply dismissed the idea that Kant's transcendental logic is logic at all, much as they have dismissed Hegel's logic, and it is important to see that there are good reasons for counteracting the former dismissal as well as the latter. Furthermore, I would argue that combining these argumentative strategies bolsters them, both rhetorically (it is often easier to get people to take Hegel seriously by exhibiting his advances on Kant), and conceptually (it is often easier to actually understand Hegel by exhibiting his advances on Kant). However, to make good on this promise I have to justify Kant's idea that there is something *inherently logical* about relation to an object (objective validity), and show that this is encapsulated in a significant way by geometric logic. That might take some time, and is probably better to do on its own page/thread. For now, you might want to think about the issue in the following way: for Kant, general logic is fine for reasoning about both mathematical and empirical matters only insofar as it is entirely indifferent to the difference between them, and in this indifference it is also perfectly good for reasoning about Harry Potter, unicorns, or even colourless green ideas, and engaging in a panoply of other language games that might be perfectly internally consistent yet have nothing resembling purchase on the world. It is not too far a jump to interpret him as circling around the same issues that Hegel (per Lawvere) is talking about in terms of the difference between subjective and objective logic. ]]>

@Peter,

there is a syntax for query boxes, we don’t use it much these days, with discussion being held here instead, but you could still use it. That would help separate your questions/comments from what David C wrote.

]]>@#69 Peter, regarding editing pages, the general rule is that anything on nLab can be edited, with announcement here if substantial. For others’ private webs, I just correct typos.

As for my own, where that Brandom note is, I just collect together some sketchy thoughts there. I’d be happy to read your thoughts there, if you could designate them as yours.

]]>@#65: Peter, concerning the passage you criticize though I would admit the sin of ’rhetoric’ I would deny the charge of ’*dark age* rhetoric’. The intention there is to provoke the reader with the idea that the ’logical lightweight’ Hegel outdoes Kant when it comes to having a critical attitude to *traditional* logic (note the implied suggestion to view Hegel as an expansion of the Kantian project to logic) and more generally that the postKantian philosophers of the 1790 were quick to dismiss practically all preceding ’dogmatic’ metaphysics but often took the traditional laws of formal reasoning for granted. I am probably willing now to exempt at least some of the postKantians from this charge since some of them felt indeed that the critical philosophy demanded a revision of *traditional* logic e.g. Salomon Maimon published a ’Neue Theorie des Denkens’ in 1794, Jacob Siegismund Beck, a mathematician from Kant’s inner circle published a ’Lehrbuch der Logik’ in 1820 introducing transcendental concepts into traditional logic, and Fichte in 1808 lectured on ’transcendental logic’ producing a large posthumously published text. The point is that Kant did not feel this need, the Jaesche-Logik contains the famous quote that general and pure logic is dull and short and basically a closed chapter since antiquity (or something like this), a quote that made it into the 1928 textbook of Hilbert and Ackermann who obviously did not think that chapter quite as closed neither did Leibniz before them.

This does not mean that the Jaesche-Logik is unimportant for the philosophy of logic nor that Kant’s transcendental logic cannot not fruitfully confronted with geometric logic, though calling the later ’Kant’s logic’ runs into the problem that Kant admitted traditional logic as a valid form of reasoning regardless of the objective content of the concepts employed i.e. to the extent that Kant ’had’ a logic traditional logic is a better candidate for it, in my view.

That Kant’s reasoning is inherently constructive is due to his attempt to model philosophy on the reasoning with constructions in Euclid’s geometry and the later is also an albeit remote source of *geometric* logic. Anyway, I am the last person to belittle Kant who is in fact one of the brightest stars on my philosophical firmament. In the later passages of the nLab article the continuity between Kant and the postkantian systems and Hegel is stressed. I generally find it useful to view thinkers like Kant, Fichte, Schelling and Hegel to be involved in a common project of transcendental philosophy which in my view is highly relevant to contemporary philosophy or cognitive science and deserves to be formalized by methods of modern mathematics.

Concerning Ploucquet, there is a German-Latin edition of his Logic by Michael Franz available as well as an article by Redding exploring the connection between Hegel and Ploucquet called THE ROLE OF LOGIC “COMMONLY SO CALLED” IN HEGEL’S SCIENCE OF LOGIC presumably available from his homepage as a preprint. In the context of cognitive underpinning for sheaf theory the link to the Petitot paper at Aufhebung might be interesting as well.

In any case, feel free to edit or expand Aufhebung when you cannot stomach certain passages. Additional insights or views are always appreciated and generally encouraged by the nLab!

]]>This is probably a silly question, but would the right etiquette be to edit your Brandom note if I wanted to respond to your thoughts on the topic? ]]>

Well you’d have been very welcome to the workshop anyway even if just to attend.

Interesting you mention Brandom. I jotted down a note which sounds like it may be in the same direction as your criticism. (By the way, I overlapped here with Ken Westphal for a couple of years.)

When you have something to read on what you describe in the 3rd paragraph, I’d be very interested.

Unfortunately, the reading group is just a bunch of us in a room thrashing things out.

]]>I haven't written anything up on the Kant-Vickers connection yet. It's part of a cache of insights I've been slowly tripping over since discovering the A&L paper on transcendental logic and getting serious about understanding sheaves and Grothendieck topoi. Strangely, I got into all this by trying to figure out what was so unsatisfactory about Brandom's formal incompatibility semantics, which has some Hegelian philosophical inspiration, but ends up being horribly gerrymandered into classical rubbish. One of the inconsistencies between Brandom's philosophical inspirations and his formalism (as pointed out to me by Ken Westphal) is precisely that it ignores Kant's distinction between negative and infinite judgments, collapsing everything back into (classical) propositional negation, and thereby being completely unable to account for the sorts of concrete incompatibilities between predicates he starts from (e.g., between blue and the various predicates - green, red, yellow, etc. - that fall under non-blue). There are more complaints I could make on this front, as learning what's wrong in Brandom's project has been quite enlightening, but I'll stop there.

The overarching project that these ideas belong to is the development of what I'm calling 'computational Kantianism', reading Kant's transcendental psychology as essentially already the project of AGI, using contemporary work in logic/maths/compsci to make sense of Kant and using Kant to provide some overarching structure connecting this same work. I gave a talk in Dublin recently that went over some of the overarching methodological ideas of this approach, but it's not written up. I'm due to give a seminar at a Summer school in NYC directly on computational Kantianism later this month, but I'm still working from notes, trying to condense things down into something tractable. I think one can draw a useful line between Kant's insistence on the primacy of judgment as a starting point for transcendental psychology and Harper's idea of computational trinitarianism, and that this (along with certain stories one can tell about subterranean connections between Kant and constructivism in the history of mathematics: i.e., Brouwer-Heyting, topos theory, Curry-Howard, HoTT) opens up the possibilities for reflecting back and forth between Kant and contemporary work I'm proposing. The really novel thing I think can be imported back from Kant is his conception of the relationship between mathematical and empirical judgment/cognition, which I think can be roughly understood through the duality between intuitionistic and co-intuitionistic logic (judgment) and computational data and co-data (cognition). I think this line of thinking inevitably leads you from Kant to Hegel, as the relationship between imagination and understanding needs to be supplemented with that between understanding and reason, but it's nice to start with Kant's emphasis on our (computational) finitude and build up to Hegel from there. It also has the advantage of suggesting how to extend computational trinitarianism beyond HoTT, insofar as one can project something like a co-intuitionistic dual of HoTT for empirical cognition. There's a few more things I could say about this, but I'll stop myself before I get further out of my depth!

Is this reading group online? If so, I'd very be interested in tagging along. One of the things reading the work being done here has taught me is how much was lost in the transition from the Aristotelian to the Fregean logical paradigm. ]]>

Welcome Peter, from another philosopher.

I’m very interested in your “potted example”. The reading group I belong to will soon be reading Paul Redding on the lost subtleties of negation possible in term logic in his ’Analytic Philosophy and the Return of Hegelian Thought’.

Have you written on the Kant-Vickers connection?

]]>I wanted to offer a reason to reconsider the following quote from the entry:

"However critical these idealist systems had been to the claims of traditional metaphysics and epistemology they all left the traditional logic untouched and in this respect fell behind Leibniz. It is at this point where Hegel starts: he sets out to extend the critical examination of the foundations of knowledge to logic itself."

I can't speak for Fichte, but there's some good evidence that this is not the case for Kant. Dorothea Achourioti and Michiel van Lambalgen have an excellent paper titled 'A Formalization of Kant's Transcendental Logic' (http://philpapers.org/rec/ACHAFO) that makes the case that Kant's logic is actually geometric logic. The paper is aimed at philosophers, and so goes out of its way to not use category theory (it uses inverse limits of a system of inverse sets), but I have confirmed with van Lambalgen that this was a very deliberate de-categorification for philosophical consumption. I actually think that the semantics they provide is probably way too simple, but the exegetical case for interpreting Kant's logic as geometric is very good in my view, and the connection to Grothendieck topoi provides various interesting connections between Kant's transcendental psychology and mathematics/compsci (e.g., the idea that object synthesis is about tracking local invariants in varying heterogeneous data (intuition) attached to/organised by some base topology (forms thereof)).

If you want a potted example, Kant's distinction between negative judgments and infinite judgements makes perfect sense from the perspective of Steve Vickers's ideas about geometric type theory. For Kant, the crucial difference between the two (corresponding to propositional vs. predicate negation) is that the latter has existential import (relation to an object) and the former does not. Vickers's idea is that the infinitary disjunctions of geometric logic can be re-interpreted using typed existential quantification, and this is essentially what provides the existential import/objective validity of the infinite judgment from the Kantian perspective. Using a standard and overly simplistic example, judging that an extended object is non-blue excludes a determinate range of possibilities for that object, because the type of extended objects includes a colour attribute with a strictly delimited but potentially infinite range of possible variations. There's more that could be said here, but I think this indicates that Kant is more interesting for the nLab project than the above quote suggests.

I'll close with one more observation about the history of logic. I'm no expert, but the history of logic between Leibniz and Boole is really not very well articulated, and it seems that there are a lot of lines of influence that really aren't properly understood. It's all too easy to represent it as a sort of logical dark age where nothing took place. I honestly don't know how well Hegel would have understood Kant's perspective on logic, given that much of what we now understand comes from the lectures he gave on the topic, which wouldn't have been widely available at the time. However, it seems that Hegel and Schelling were significantly influenced by Ploucquet, from whom they seem to have contracted the idea of the reversibility of subject and predicate. Ploucquet is one of the significant figures recognised in the tradition post-Leibniz, but his work doesn't seem to be widely studied (it's all in latin, with no English translation from what I can gather) either on its own terms or as an influence on Hegel's logical views. It's worth avoiding 'logical dark age' rhetoric, whether it is between Aristotle and Frege or Leibniz and Hegel, as there are hidden lines of research and influence that were still in the process of uncovering and reconstructing. ]]>

From a quick glance at the section you link, to the disjointness property is meant only for the particular example. In the general case, the two subcategories are usually far from disjoint, in fact, one can think of the process of Aufhebung as a gradual level-to-level augmentation of the objects in the intersection, that contains the ’true thoughts’ where content (left inclusion) coincides with notion (right inclusion), starting from $0\cap 1=\emptyset$ up to $id_\mathcal{E}\cap id_\mathcal{E}=\mathcal{E}$.

]]>under this section, we read:

the functors L and R must actually correspond to inclusions of disjoint subcategories

I take it this is not necessary in general, and is only true here since the composites $T \circ L, T \circ R$ are *equal* to the identity. I think that in general, when the composites are merely isomorphic to the identity, the subcategories have intersection given by the equalizer of the subcategory inclusions. Is this reasoning sound?

The quote from Hegel is 113 from here:

]]>Cancelling, superseding, brings out and lays bare its true twofold meaning which we found contained in the negative: to supersede (aufheben) is at once to negate and to preserve.

At MPI Bonn this Aufhebungs-announcement is flying around (full pdf by In Situ Art Society).

I have take the liberty to add it to the entry *Aufhebung*.

added to the discussion here of Aufhebung $\sharp \empty \simeq \empty$ over cohesive sites pointer to lemma 4.1 in

- William Lawvere, Matías Menni,
*Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness*, Theory and Applications of Categories, Vol. 30, 2015, No. 26, pp 909-932. (TAC)

which obverseves this more generally when pieces-have-points.

]]>I wrote out a more detailed proof of the statement here that the bosonic modality $\rightsquigarrow$ preserves local diffeomorphisms.

It seems the proof needs not just Aufhebung in that

$\rightsquigarrow \Im \simeq \Im$but needs also that this is compatible with the $\Im$-unit in that $\rightsquigarrow$ sends the $\Im$-unit of an object $\stackrel{\rightsquigarrow}{X}$ to itself, up to equivalent

$\rightsquigarrow( \stackrel{\rightsquigarrow}{X} \stackrel{\eta_{\stackrel{\rightsquigarrow}{X}}}{\longrightarrow} \Im \stackrel{\rightsquigarrow}{X} ) \;\;\; \simeq \;\;\; ( \stackrel{\rightsquigarrow}{X} \stackrel{\eta_{\stackrel{\rightsquigarrow}{X}}}{\longrightarrow} \Im \stackrel{\rightsquigarrow}{X} )$This is true in the model of super formal smooth $\infty$-stacks, so I am just adding this condition now to the axioms. But it makes me wonder if one should add this generally to the concept of Aufhebung, or, better, if I am missing something and this condition actually follows from the weaker one.

]]>That makes me want to experiment with re-thinking about a possibly neater way of defining differentially cohesive toposes.

Something like this:

A *differential cohesive topos* is (…of course…) a topos $\mathbf{H}$ equipped with two idempotent monads $\sharp,\Re : \mathbf{H} \to\mathbf{H}$ such that there are adjoints $\int \dashv \flat \dashv \sharp$ and $\Re \dashv ʃ_{inf} \dashv \flat_{inf}$ (…but now:) and such that

(clear:) $ʃ {}_{\ast}\simeq {}_{\ast}$ and $\sharp \emptyset \simeq \emptyset$

(maybe:) $\flat_{inf} \Pi \simeq \Pi$ and $\Re \flat \simeq \flat$.

I need to go through what I have to see what the minimum needed here is. I certainly need $\flat_{inf} \flat \simeq \flat$ for the relative infinitesimal cohesion to come out right. Also $\Re \ast \simeq \ast$, which would follow from the above.

In any case, I feel now one should think of these axioms as describing a picture of the following form (the Proceß)

$\array{ &\stackrel{}{}&& id &\stackrel{}{\dashv}& id \\ &\stackrel{}{}&& \vee && \vee \\ && & \Re &\dashv & ʃ_{inf} & \\ &&& \bot && \bot \\ &&& ʃ_{inf} &\dashv& \flat_{inf} \\ &&& \vee && \vee \\ &&& ʃ &\dashv& \flat & \\ &&& \bot && \bot \\ &&& \flat &\dashv& \sharp & \\ &&& \vee && \vee \\ &&& \emptyset &\dashv& \ast & \\ }$and the question is what an elegant minimal condition is to encode the $\vee$-s.

]]>coming back to #16:

I used to think and say that in the axioms of cohesion the extra exactness condtions on the shape modality seem to break a little the ultra-elegant nicety of the rest of the axioms. There is an adjoint triple of (co-)monads, fine… and in addtition the leftmost preserves the terminal object – what kind of axiomatics is that?!

But *Aufhebung* now shows the pattern: that extra condition on the shape modality

is just a dual to the “Aufhebung of becoming”

$\sharp \emptyset \simeq \emptyset\,.$Maybe a co-Aufhebung, or something.

]]>coming back to #50:

on the other hand, of course $\flat_{inf}$ *does* provide Aufhebung of cohesion in the sense that $\flat_{inf} \int \simeq \int$.

Of course this follows trivially here, since we are one step to the left and both of $\int$ and $\flat$ correspond to the same subcategory.

]]>Ah, I should be saying this more properly (and this maybe highlights a subtlety in language that we may have not properly taken account of somewhere else in the discussion):

in the topos over the site of formal smooth manifolds, the sub-topos of $\flat^{rel}$-modal types is “infinitesimally cohesive” in that restricted to it the map $\flat \to \int$ is an equivalence.

]]>Yes, right, in the given model the $(\flat^{rel} \dashv \sharp^{rel})$-level exhibits what “we” here had decided to call “infinitesimal cohesion”, which is essentially another word for what Lawvere had called a “quality type”.

And yes, I’d agree that it would make much sense to regard $(\flat^{rel} \dashv \sharp^{rel})$ as being the “next” level after $(\flat \dashv \sharp)$. After all, the sequence of inclusions of levels

$\array{ \flat^{rel} &\dashv& \sharp^{rel} \\ \vee && \vee \\ \flat &\dashv & \sharp \\ \vee && \vee \\ \emptyset && \ast }$reads in words “a) the single point, b) collections of points, c) collections of points with infinitesimal thickening”.

And it seems clear in the model (though I’d have to think about how to prove it) that $\flat^{rel} \dashv \sharp^{rel}$ (when given by first-order infinitesimals) should be the smallest nontrivial level above flat $\flat \dashv \sharp$.

]]>