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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 7th 2014
• (edited Jan 7th 2014)

created an entry category of being, for completeness.

A general remark: people often write that, unfortunately or not, Eilenberg-MacLane’s term “category” is not that of, say, Kant. But in fact if read this way here, following Lawvere, then the former is a good formalization of the latter, after all.

• CommentRowNumber2.
• CommentAuthorDean
• CommentTimeMar 1st 2020
• (edited Mar 1st 2020)
Hi, I am a bit new to this, but @David Corfield can probably answer this. Can someone explain to me:

1) Emptyset is interpreted as nothing and singleton as pure being. What are the rest of the objects interpreted as in Hegel's philosophy? Is it that objects are entities and a : y means "a is a y"? If so, perhaps "a : x alows us to judge a : y" means "an x is a y"?

2) What are the morphisms in this category supposed to be interpreted as? There is a table here, but I don't see morphisms on it: https://ncatlab.org/nlab/show/Science+of+Logic#FormalizationDictionary

3) How is the idea that pure being and nothing are the same to be interpreted in categorical language?

For (3), recall Hegel's Science of Logic, 132, 133, 134:

"Being, pure being, without any further determination. In its indeterminate immediacy it is equal only to itself. It is also not unequal relatively to an other; it has no diversity within itself nor any with a reference outwards. It would not be held fast in its purity if it contained any determination or content which could be distinguished in it or by which it could be distinguished from an other. It is pure indeterminateness and emptiness. There is nothing to be intuited in it, if one can speak here of intuiting; or, it is only this pure intuiting itself. Just as little is anything to be thought in it, or it is equally only this empty thinking. Being, the indeterminate immediate, is in fact nothing, and neither more nor less than nothing.

Nothing, pure nothing: it is simply equality with itself, complete emptiness, absence of all determination and content — undifferentiatedness in itself. In so far as intuiting or thinking can be mentioned here, it counts as a distinction whether something or nothing is intuited or thought. To intuit or think nothing has, therefore, a meaning; both are distinguished and thus nothing is (exists) in our intuiting or thinking; or rather it is empty intuition and thought itself, and the same empty intuition or thought as pure being. Nothing is, therefore, the same determination, or rather absence of determination, and thus altogether the same as, pure being.®

What is the truth is neither being nor nothing, but that being — does not pass over but has passed over — into nothing, and nothing into being. But it is equally true that they are not undistinguished from each other, that, on the contrary, they are not the same, that they are absolutely distinct, and yet that they are unseparated and inseparable and that each immediately vanishes in its opposite. Their truth is therefore, this movement of the immediate vanishing of the one into the other: becoming, a movement in which both are distinguished, but by a difference which has equally immediately resolved itself."
• CommentRowNumber3.
• CommentAuthorDean
• CommentTimeMar 1st 2020
I tried to ask this on MO and was told that "The quote reads like a troll pretending to be a philosopher."; sometimes I think the people here are more open minded than elsewhere.

I am also told that there are many modern interpretations of Hegel, though for the time being I am interested in the one here, if only I can figure out some of the things which do not appear on the "formalization dictionary" here: https://ncatlab.org/nlab/show/Science+of+Logic#FormalizationDictionary. What's on my mind right now is, "what do morphisms of types correspond to", which is (2) above.
• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeMar 1st 2020

Care to provide a link to the MO question? Did it get deleted?

• CommentRowNumber5.
• CommentAuthorDean
• CommentTimeMar 1st 2020
@David Roberts. I deleted it since I figured it was being recieved poorly.
• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeMar 2nd 2020

Fair enough. Not sure how much traction you’ll get here, either, since the number of people who can pass Hegelian thoughts through hopefully the clarifying lens of Lawvere’s proposed interpretation is rather small. And those few people might be busy with something else.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMar 2nd 2020

The answer to the third question is expanded on at Aufhebung of Becoming

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMar 2nd 2020
• (edited Mar 2nd 2020)

Regarding the first question: All the other objects is what appears in between as the two opposite extremes of the initial opposition are pulled apart:

$\varnothing \longrightarrow X \longrightarrow \ast$.

This being a special case of the general way in which adjoint modalities exhibit all objects as equipped with two opposite properties, see here.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeMar 2nd 2020
• (edited Mar 2nd 2020)

Hi Dean,

(1) The general rule is that types are objects and terms are morphisms. In intrinsic type theories, terms are uniquely typed, so one cannot have $a: A$ then also $a:B$ in a non-definitionally equivalent type. This section of Science of Logic addresses Aristotelian syllogistic logic. In the case of $f: A \to B$, glossed as ’All $A$ are $B$’, as it says latter there, often one works with a context, $C$. If then $A = \sum_{x:C} P(x)$ and $B = \sum_{x: C} Q(x)$, for predicates $P(x)$ and $Q(x)$, then $f$ might arise from an implication $P(x) \to Q(x)$. We’re perhaps inclined here to speak of a $c: C$ such that $P(c)$ and $Q(c)$ is true, and then have $c: A$ and $c:B$, but this isn’t strictly correct.

There’s a question of how the translation works from Hegel to Lawvere and to nLab, whether and how one requires some kind of agreement with the spirit of the writing of someone who lacked any formal means to say things we now can. But it may be easier to approach from the other side and take what’s in Lawvere and nLab as rather interesting, and reasonably inspired by Hegel. The portion on Being and Nothing begins here. As Urs writes, the adjoint modalities $\emptyset \dashv \ast$ is at the heart of this passage, expressing the unity of two moments. You can think of it generated by the adjoint triple, formed of left and right adjoints to the terminal functor $\mathbf{H} \to \ast$, so that $0$ and $1$ are distinct images of $\ast$ (full subcategories of $\mathbf{H}$).