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A lightning quick stubby note:
abstract general, concrete general and concrete particular
Is the last one right?
I’m pretty sure that the last one is right. The middle one, I don’t know. As for the first, I clarified that the algebraic theory appears in its guise as a Lawvere theory, since I think that this is important for Lawvere: that the abstract general be a category with a concrete particular example in it.
It seems to match with John’s description. Was an ’abstract particular’ out of the question?
Was an ’abstract particular’ out of the question?
That’s what I was after, in a light-hearted chat with a colleague. We were wondering if we could break up a research paper into parts following Lawvere’s pattern. ;-)
Have to go offline now…
If, say, a particular group in the category of groups is a concrete particular in a concrete general, is an object in the theory of groups an abstract particular in an abstract general? So that $G^2$, say, or the free group on 2 generators, is an abstract particular.
I would think that $G$, at least is a (the?) abstract particular.
Is it essential to this terminology that the theory in question be a Lawvere theory, that is, a finite-product theory? Would the free finitely-complete category containing an internal category, for instance, be an “abstract general” in a different doctrine?
I think that it’s essential that it be a category (like a Lawvere theory is) rather than a more traditional logicians’ description of a theory. But the precise doctrine is surely not important. I’ve edited the page to reflect this.
I have added
an intro-sentence
the case of “abstract particulars”;
an Examples-section with something about groups.
The case of “abstract particulars” is clearest for algebraic theories with more than one “color”: the different colors are the different abstract particulars.
Would it be worth a remark to the effect that ’abstract particular’ is not Lawvere’s term, hence it’s non-inclusion in the title of the page?
Well, if you (Urs) are going to say ‘syntactic category of’, then I guess that we don’t need to specify ‘Lawvere theory’. A Lawvere theory is a syntactic category.
However, I don’t like your concept of the abstract particular, especially for theories with more than one colour (by which I assume you mean base sort). For example, if we take the theory $T$ of a vector space (over an unspecified base field), this has two sorts: one ($F$) for the field, and one ($V$) for the vector space itself. (Actually, maybe we should do modules over commutative rings so that this exists in more familiar doctrines.) Then a model of $T$ in (say) $Set$, that is a concrete particular, is not a field or a vector space but a pair of such. Similarly, an abstract particular (as I understood when writing #6) is not either $F$ or $V$ but the pair of them (together with their structure morphisms). But your definition of the abstract particular as a single generating object in $T$ suggests otherwise.
an abstract particular (as I understood when writing #6) is not either F or V but the pair of them
Yeah, I agree, I guess that makes more sense.
Somebody points out by email this comment by Lawvere on “abstract general, concrete general and concrete particular”.
I believe the implication of the email is that the entry might be in need of adjustment (it certainly is in need of further expanding it!). I don’t have the leisure to look into this right this moment. Maybe a little later. But I thought possibly somebody else here might.
I am not sure what “K” is supposed to be in the third paragraph.
Anyway, it seems that our entry currently gives a definition of “concrete particular” which does not coincide with what Lawvere actually suggested.
This reminds me: does anyone have a pointer to an electronic version of a writeup of a lecture note where Lawvere lays out his intended meaning of these terms? Just so that we get the account straight.
‘In case there is an initial theory, every fiber has a canonical “underlying” functor to the thus-distinguished fiber K (which is typically taken to be the category of abstract sets).’
Oh, thanks, true. It was a bit too late for me yesterday.
So a particular is a functor $P \to K = Set$. From this we are to built (a syntactic category of) a theory $P^*$. So what is $P^*$? The free theory on $P$? But I gather $P$ could already be a theory and the particular could already be a model. In this case the notion of “particular” here coincides with that of “concrete particular” in the $n$Lab entry.
Why is it a “category mistake” – end of paragraph 3 – to say “concrete particular” instead of “particular”?
So the “thus-distinguished fiber” is the fiber of models over an initial theory, e.g., a trivial theory placing no conditions on a set.
I had taken categorical model theory to be largely talking about dualities between categories of syntactic theories and categories of models, and reconstruction results (Gabriel-Ulmer, Makkai, Awodey & Forssell). But Lawvere’s talking of a fibered category with models of a theory sitting above that theory.
Hmm, what’s the relationship between using $Set$ with two different structures to dualize between syntactical categories and semantic categories, and using $Cat$ as a codomain for a functor from $Theories$, so that it generates a fibration of models over theories?
What’s ’further’ about what he says in the paragraph starting “A major methodological advance…”? Isn’t this just saying that models are functors from a theory to, say, $Sets$?
Can anyone explain what’s going on in the following paragraph?
By contrast both aspects abstract and concrete of a general concept are purely mental (i.e., belong to collective thinking). To cite a notorious example the category of featherless bipeds as a class is equally mental with the theory of fbps.
Is he just using the amusing device as a synonym for human beings, and then making the point that not only the construction of the concept of a human being, but also the act of grouping them as a set, are mental acts? Perhaps not, maybe he’s giving an example of
P->P*#
The particular P is a group of humans. We extract a theory, P*, which picks out the properties bipedal and featherless, then we look for all models. Hence Humans map into Featherless bipeds.
A particular by contrast is a category P equipped with a functor to K regarded as a measurement. This category itself may come from reality as a collection of experiments which are behaviourly comparable; that is it need not be construed conceptually as consisting of structures and structure-preserving maps. However such a conceptual theory P* may be derived (depending on the doctrine of generals) by considering the diagrams of natural transformations on K and close relatives (like AK where A is for example an exponential functor on K, or “arity”). That abstract general P* has over it the fiber P# which of course is a concrete general. Typically there is a functor P->P# into this double dual giving concrete models for the objects of the particular. However this functor is surely not surjective and typically not faithful. Thus there is no natural place in this account for any “concrete particular”; that would be a “category mistake”.
Here’s a thought. The Awodey-Forssell reconstruction theory goes: if you give me a category of models, $M$, for some first order theory, but don’t tell me the theory. Then every functor of a certain kind (definable) from $M$ to $Sets$ corresponds to a formula of the theory, hence reconstruction is possible. But what if you started with a collection of particulars, or maybe particular collection, with no thought of it being a complete set of instances. Then applying reconstruction, we emerge with a theory, but when we find all models of this, the map from the original particular collection to the collection of all models of the theory is not surjective. And maybe some relations between the original instances are lost in the theorising. E.g., a take a group of people, extract what it is to be a person, but I’ve lost the fact that some couples are married.
So there’s ’abstract general’ = a theory, ’concrete general’ = all its models, ’particular’ = some collectivity drawn from the world, thought to be united in some way. Then what could ’concrete’ particular mean? There are just particulars.
But Lawvere’s talking of a fibered category with models of a theory sitting above that theory.
Yes, but that’s just the Grothendieck construction of the functor that takes a theory to its category of modules, so it’s not different to what you are used to.
Thanks for all the other comments. I get back to that a little later when I have a spare second.
it’s not different to what you are used to.
Yes, but I’d still like to know whether there’s a gain in thinking in a fibered way. It’s the same thing with hyperdoctrines and syntactic categories. It seems that the latter approach has come to dominate. but maybe the two approaches are interchangeable.
What’s ’further’ about what he says in the paragraph starting “A major methodological advance…”? Isn’t this just saying that models are functors from a theory to, say, Sets?
One can consider notions of models of theories that depend functorially on the theories and hence form a fibered category over theories even if the models themselves are not subcategories of functor categories. In non-categorical model theory, say. But if they are, this is a “further” appearance of functors.
Can anyone explain what’s going on in the following paragraph?
I think he is meaning to generalize as follows: he says any functor to $Set$, from some domain $P$ is a “particular”. It need not be a products/limit-preserving functor out of the syntactic category of an algebraic theory/essentially algebraic theory.
Then he consideres some theory $P^*$ freely generated from such a “particular” and talks about whether $P \to Set$ can be related to an actual model for $P^*$.
That’s the crux that the new discussion started in #15 is about, I think: the $n$Lab says that a “concrete particular” in Lawvere’s sense is a model of a theory. But in this message to Posina Venkata Rayudu he makes the point that he does not want to speak of “concrete particulars” at all, but just of “particulars” and does not want to require them to be models, but allow them to be any old functors.
I am mildly disappointed by this in several way. I dearly wish to get hold of an authorative document – probably the best I can hope for is a lecture transcript from almost 10 years back – where Lawvere’s original idea of general/abstract/concrete/particular is recorded.
Is he just using the amusing device
I can’t parse a sentence of the form
the category of $X$s as a class is equally mental with the theory of $Y$s.
Then what could ’concrete’ particular mean? There are just particulars.
I guess that’s the whole punchline here, that the $n$Lab entry shouldn’t be attributing “concrete particular”, to Lawvere, but just “particular”.
On the other hand, the $n$Lab entry contains what I find to be a noteworthy suggestion (did I make that originally?, I forget) that it can make good sense to have the distinction
an abstract particular is a generating object in the syntactic category;
a concrete particular is a model.
I have slightly re-edited the entry now to make this clearer.
I can’t parse a sentence of the form
the category of $X$s as a class is equally mental with the theory of $Y$s.
Why isn’t it as I suggest? We can look at a portion of the world (a particular), extract a theory of it, and then look for models of that theory. But don’t expect the particular and the resulting collection of models to be identical. That’s the mistake of objective idealism.
Think of the world “as one great blooming, buzzing confusion” (William James). You take a portion of this world, and try to understand the principles that govern it. When you look at the models of this theory, don’t mistake that for reality, since models and theory are equally mental.
I see. Thanks.
Re 26, I don’t think even with your edit that it sounds like Lawvere. I don’t think he wants a concrete particular to be a model.
A particular by contrast is a category P equipped with a functor to K regarded as a measurement. This category itself may come from reality as a collection of experiments which are behaviourly comparable; that is it need not be construed conceptually as consisting of structures and structure-preserving maps.
Down at the propositional level, this might be a discrete set of objects each with a set of properties. In devising a theory, taking the particular within the mental we must transform it. Seeking the theory’s models doesn’t take us back out in the world. It will only be mental idealisation.
I don’t think even with your edit that it sounds like Lawvere. I don’t think he wants a concrete particular to be a model.
But I do. :-) At that point the entry speaks about other things one can do.
Ok, but then you probably shouldn’t say
That seems to be roughly what is suggested in Lawvere.
What about a syntactic category of a first-order theory? Are abstract particulars to be its objects, i.e., formulas?
The idea is that the entry currently first reviews what Lawvere say, then asserts that “That seems to be roughly what is suggested by Lawvere”, then says “But one can play with this” and then gives the other version.
Isn’t that what the entry says since I announced that I edited it? I’d think it is. But I really need to rest, maybe I can’s see straight anymore… :-=
What about a syntactic category of a first-order theory? Are abstract particulars to be its objects, i.e., formulas?
So the abstract particulars are the sorts.
Hi David,
sorry for having been so brief. I was really tired and exhausted when we had the last exchange yesterday, and needed to quit.
Now this morning I was about to change at geometry of physics the “concrete particular” to just “particular”.
But maybe you could help me again and further for a sec with your always very insightful comments:
I am wondering: is that really necessary or even sensible?
Even if we admit what Lawvere says in your exegesis of #27: if the collection of all models is “general concrete”, why should we not call a particular such model a “particular concrete”?
Because: we could talk about all other kinds of particulars. A particular theory, say! Or something entirely different. A particular thought in some entirely different context. But no, here we have a notion of concreteness for some given theory, so we are in a very particular context (!) and in that specific context we speak of “particulars”, not in any other one. Namely of “particular concrete realizations of some fixed theory”, or just “particular concrete” for short.
Do you see what I mean? Please let me know what you think.
(For other readers: of course we are just playing with words here. But since these words serve as technical terms for some mathematical discussion that we care about, maybe we are excused of spending a bit more time on this than might seem reasonable. )
Well, I suppose Lawvere would point us to the map he designates
P->P*#
That P is some part of the world on which we have taken some measurements. We are warned not to take it necessarily
as consisting of structures and structure-preserving maps,
but it is at least a category.
Two things can happened as we transform P into the category of models of things behaving according to the theory we extracted from P.
1) It need not be surjective. So there may be models of the theory which we never see in the world. I think this wouldn’t be so bad if P->P# were at least faithful, as then we could say of P# that it contains members of P along with some other things of a similar kind, and we might be happy to talk of these unworldly models as possible particulars. No, but even there it wasn’t that P contained particulars; P is the particular. Then the models are a similar kind of thing to ’elements’ of the particular.
2) But it gets worse because P->P# need not be faithful. So I can look at two members of P# which may have come from P, but the way I consider them to be related may be an impoverished rendition of the complexity of reality. Imagine from a group of people I extract their properties, then consider all possible things with those properties. I happen to list a couple of these possible people who in fact are the images of a real couple, but now I have to leave out that they are married. I shouldn’t then pass off the image of the real wife as a part of the original particular if my models don’t bear the possibility of being married.
Bit rambling, but I have to stand in now for someone to give a seminar on the original EPR paper, so shall need some luck.
Thanks, David.
You are talking about the EPR paradox?
Concerning your comments: I feel that that what you discuss is a second layer of subtlety which I would like to ignore for the time being. I’d be happy to come back to the question of how to “relate models to measurements” later. For the moment I just want to find agreement about what we say for “model” if we say “general abstract” for “theory”!
In your reply you are following Lawvere in that you take for granted that “a particular” is necessarily “a particular collection of measurements”. But that’s a massive restriction of the general word “particular”. This is a very particular kind of particulars, if you see what I mean.
So I would suggest we let ourselves not be distracted for the moment by the separate discussion about “measurement and models”. I want to be talking just about models.
So to repeat, I am saying:
if we call the collection of all models of a given theory a “concrete general”, then what do we call a single model in that collection? By all means, is it not the most natural to call that a “concrete particular”? Why not?
Then you could speak of “measured particulars” or “particular measurements” later, if you like :-)
Yes, the EPR paradox. I’ve jotted down what we came to while still fresh in my memory.
I’ll answer your comments when I have a moment, but must dash.
The particular P is a group of humans. We extract a theory, P*, which picks out the properties bipedal and featherless, then we look for all models. Hence Humans map into Featherless bipeds.
What makes this example notorious is that $P \to P^{{*}{#}}$ fails to be surjective; the latter also includes plucked chickens. And even before Diogenes plucked his actual chicken (if this ever actually happened), the mental category $P^{{*}{#}}$ contained mental plucked chickens that aren't the image of any actual human. Even if we add broad nails to $P^{*}$, there will still at least be mental counterexamples.
Urs, I rather like your classification of abstract/concrete general/particular. I am still not sure if (even after removing abstract particulars) it agrees with Lawvere (who is hard to understand). But if it does, then your invention of abstract particular is right and good.
Still not enough time, but let us not forget that Lawvere hasn’t hit on these terms from nowhere. “Das konkrete Allgemeine”, translated as concrete general or concrete universal, is a term used by Hegel.
“The concrete is the universal in all of its determinations, and thus contains its other in itself”
There’s the idea that the concrete universal is the universal realizing itself in all its manifestations with all of their relations. Hence, I guess, why Lawvere takes it as the category of all models. No doubt Lawvere is aware also of how Marx will take up Hegel’s ideas.
But, of course, we’re free to leave all this history behind. I see David Bohm also used such language here.
That’s a good point in #38, but still the possible failure of the mapping $P \to P^{* #}$ to be faithful needs to be thought about.
Another thought: Hegel was never one to devise terms which dichotomise. If you read of two terms, it’s never the case that they exclusively and exhaustively cover a domain. You find that the two terms turn out to be the same in a sense, while retaining their difference, and this tension is overcome (sublated, aufgehoben) in an identity of opposites. You can see how duality theorems between theories and spaces of models might fit the bill.
There’s some interesting writing by Lawvere here, which makes his thinking a lot clearer.
There’s some interesting writing by Lawvere here,
Thanks! Finally I see a link to published material on this issue. I have moved that reference to the $n$Lab entry.
Looking at it, I must say it makes the question even more pressing: if we think of the general abstract as $\mathcal{A} = \mathcal{B}^*$ and the general concrete as $\mathcal{C} = \mathcal{A}^* = (\mathcal{B}^{*})^\ast$, and if we have a word for the objects in the double dual, why on earth would it be a “mistake” to have a similar word for the objects in the single dual $\mathcal{A}$?
Maybe it’s only that in Hegel’s writing we find no word to borrow for this case? So then, that’s Hegel’s problem, not ours. :-)
still the possible failure of the mapping $P \to P^{* #}$ to be faithful needs to be thought about
To stick to classical examples, is there something about how ‘Socrates’ and ‘the greatest philosopher’ mean different things but refer to the same actual person? I can’t make this fit a failure of faithfulness (or even fullness); it’s more a failure of functoriality (which we wouldn’t want).
Actually, that particular example is more about intensional vs extensional equality.
Your Socrates example reminds me most of the idea of how terms refer. E.g., if I use ’Aristotle’ to designate the student of Plato, author of the Nicomachian Ethics, etc., then it is necessarily the case that Aristotle is a philosopher. On the other hand, if ’Aristotle’ designate a particular person, maybe I watch him running about as a boy, then it seems I should say that it is not necessarily the case that Aristotle is a philosopher, i.e., he might have become a politician.
There’s also the idea of necessary a posteriori truths, such as the morning star = the evening star, and contingent a priori truths, the metre bar standard in Paris is a metre long. As you say, more to do with intensional vs extensional.
Back to Hegel, he writes Sec 227 of The Shorter Logic:
Because it presupposes what is differentiated as a being that is found to be already on hand, standing opposite it (the manifold facts of external nature or of consciousness), finite knowing has (I) the formal identity or the abstraction of universality as the form of its activity at the outset. This activity thus consists in dissolving the given concrete dimension, individuating its differences, and giving them the form of abstract universality; or in leaving the concrete dimension as the ground and, through abstraction from the particularities that seem inessential, extracting a concrete universal, the genus or the force and the law. Such is the analytic method.
Then you’ll truly see this concrete universal through all its instances, and their interrelations.
Interesting that there’s talk of ’dissolving’ the concrete. ’Concrete’ is from Con-crescere, to grow together, to thicken, condense, curdle, stiffen, congeal. It would have been a good word to have used for a certain kind of cohesion. The Romans used concrete as a building material, but it wasn’t rediscovered until much later, so I don’t think that plays much role in the philosophical sense.
Hegel is working out of a tradition which understands:
concrete: 4. a. Applied by the early logicians and grammarians to a quality viewed (as it is actually found) concreted or adherent to a substance, and so to the word expressing a quality so considered, viz. the adjective, in contradistinction to the quality as mentally abstracted or withdrawn from substance and expressed by an abstract noun: thus white (paper, hat, horse) is the concrete quality or quality in the concrete, whiteness, the abstract quality or quality in the abstract; seven (men, days, etc.) is a concrete number, as opposed to the number 7 in the abstract.
Hmm, so why not see that second dualization as the return of the concrete, so that original observed instance and model are both concrete particulars? Another translation of the Hegel passage has
Its activity therefore consists in analyzing the given concrete object, isolating its differences,
’Concrete object’ seems close to ’concrete particular’.
So when we talk of a concrete category with functor to, say, Sets, that’s pretty close to the idea of properties/structures adhering to something.
Thanks. David, for digging out these quotes!
Its activity therefore consists in analyzing the given concrete object, isolating its differences,
’Concrete object’ seems close to ’concrete particular’.
And notably “given concrete object”, does: “the given object” = “this particular object”.
Since the topic of concrete universals came up in a reading group yesterday, I was taking a look at Stern’s article here, and extracted a couple of quotations here:
Bernard Bosanquet
A world or cosmos is a system of members, such that every member, being ex hypothesi distinct, nevertheless contributes to the unity of the whole in virtue of the peculiarities which constitute its distinctness. And the important point for us at present is the difference of principle between a world and a class. It takes all sorts to make a world; a class is essentially of one sort only. In a word, the difference is that the ultimate principle of unity and community is fully exemplified in the former, but only superficially in the latter. The ultimate principle, we may say, is sameness in the other; generality is sameness in spite of the other; universality is sameness by means of the other. (Bosanquet, The Principle of Individuality and Value, 3)
and T. M. Knox
An abstract universal has no organic connexion with its particulars. Mind, or reason, as a concrete universal, particularizes itself into differences which are interconnected by its universality in the same way in which parts of the organism are held together by the single life which all things share. The parts depend on the whole for their life, but on the other hand the persistence of life necessitates the differentiation of the part. (Translator’s notes to Hegel’s Philosophy of Right (Oxford: Oxford University Press, 1952))
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