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The SEP article on category theory says:

Categories, functors, natural transformations, limits and colimits appeared almost out of nowhere in a paper by Eilenberg & Mac Lane (1945) entitled “General Theory of Natural Equivalences.” We say “almost,” because their earlier paper (1942) contains specific functors and natural transformations at work, limited to groups. A desire to clarify and abstract their 1942 results led Eilenberg & Mac Lane to devise category theory. The central notion at the time, as their title indicates, was that of natural transformation. In order to give a general definition of the latter, they defined functor, borrowing the term from Carnap, and in order to define functor, they borrowed the word ‘category’ from the philosophy of Aristotle, Kant, and C. S. Peirce, but redefining it mathematically. [emphasis added]

But this seems like something of a throwaway historical grounding if one considers that Kant's (or Aristotle's) categories of ontology seem far removed from modern categories of sets, setoids, groups, groupoids, etc.

However, homomorphisms are important to modern category theory. So looking through the (Meiklejohn translation of the) first Critique:

That is to say, the three categories [of quantity], in which the unity in the production of the quantum must be homogeneous throughout, are transformed solely with a view to the connection of heterogeneous parts of cognition in one act of consciousness, by means of the quality of the cognition, which is the principle of that connection.

... But this very synthetical unity remains, even when I abstract the form of space, and has its seat in the understanding, and is in fact the category of the synthesis of the homogeneous in an intuition; that is to say, the category of quantity, to which the aforesaid synthesis of apprehension, that is, the perception, must be completely conformable.

... In all subsumptions of an object under a conception, the representation of the object must be homogeneous with the conception; in other words, the conception must contain that which is represented in the object to be subsumed under it. For this is the meaning of the expression: "An object is contained under a conception." Thus the empirical conception of a plate is homogeneous with the pure geometrical conception of a circle, inasmuch as the roundness which is cogitated in the former is intuited in the latter.

And so on and on; there are 18 results for "Find: homogeneous" in the first Critique. (Emphasis added in the above.) Now, per the Greek roots of -geneous and -morphism as "species/kind" and "form/shape," are there uses of homogeneity and homomorphism that invoke roughly equivalent concepts? Not identical, but similar enough for category-talk of homogeneity in Kant to be what Eilenberg and Mac Lane had in mind regarding the significance of homomorphisms in their category-talk, so far as these authors sought to ground their metatheory historically?

For what it's worth, the nLab discussion of this topic makes no overt mention of a possible homogeneity-homomorphism overlap as such. The source material they quote involves some fairly direct claims that modern category-theory talk inherits Aristotle and Kant's (and even Hegel's!) category-talk.


Note: my question is, to be honest, very similar (you might say homomorphic to!) this earlier PhilosophtSE question. Conifold there remarks that:

However, the idea that objects are merely relational placeholders in a structure was certainly alien to both Aristotle and Kant.

I would be cautious about using a phrase as implicitly strong as "certainly alien," here, though, seeing as Kant did say in the first Critique (the quote after the ellipsis is from here):

On the other hand, the internal determinations of a substantia phaenomenon in space are nothing but relations, and it is itself nothing more than a complex of mere relations. ... That I, when abstraction is made of these relations, have nothing more to think, does not destroy the conception of a thing as phenomenon, nor the conception of an object in abstracto, but it does away with the possibility of an object that is determinable according to mere conceptions, that is, of a noumenon. It is certainly startling to hear that a thing consists solely of relations; but this thing is simply a phenomenon, and cannot be cogitated by means of the mere categories: it does itself consist in the mere relation of something in general to the senses. In the same way, we cannot cogitate relations of things in abstracto, if we commence with conceptions alone, in any other manner than that one is the cause of determinations in the other; for that is itself the conception of the understanding or category of relation. But, as in this case we make abstraction of all intuition, we lose altogether the mode in which the manifold determines to each of its parts its place, that is, the form of sensibility (space); and yet this mode antecedes all empirical causality.

Kant's relevant phrasing is even harder for me to follow in the Transcendental Aesthetic, since he seems to have the formal intuition of space/spatial form of intuition as relational in some basic capacity, but as anteceding relationality in another (also basic) one. (Since he subtly distinguishes formal intuition from forms-of-intuition, perhaps space considered as a formal intuition is more/less relational than space as a form-of-intuition?) Regardless, I'm also having trouble reading Kant as saying that categories themselves are primarily/merely relational; he seems (to me) to say, "No," on one level, but, "Not quite," on another... So, I'm not sure any of this is especially pertinent to Conifold's observation.


EDIT:: TL;DR version: are the following (roughly) equivalent?

  1. (Kant) A is homogeneous with B.
  2. (Category theory) A is homomorphic to B.

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Daniel Sutherland (in Kant's Mathematical World) suggests, like you do ("per the Greek roots of -geneous and -morphism as "species/kind" and "form/shape," are there uses of homogeneity and homomorphism that invoke roughly equivalent concepts"), that Kant's terminology here derives from Greek mathematics and relates to genus/species trees. I, however, find the interpretation proposed by David J. Hyder, linking Kant's use of the term to Kästner (teacher of Gauss), more plausible, as it is known that Kant was profoundly influenced by people like Euler and Lambert (with who he corresponded, and to whom he planned to dedicate the first Critique) - and Kästner is closely related to these remarkable individuals.

What does homogeneous mean then? Sometimes, when Kant speaks of the relation of spontaneous representations (concepts) to sensible representations, he says that they're inhomogeneous. This means that there must be sensible representations which, as it were, represent the pure logical structure of the category sensibly. For example, successivity (one moment in time necessarily following another) sensibly represents causality. Kant calls these pure schemata. This is the case in two of your quotes.

If we're, however, speaking not of epistemology, but of mathematics, then homogeneity is a property of quantities whose aggregates are representable as numbers (i.e. linearly orderable). E.g. blue and red or sounds and colours are not homogeneous, but various shades of gray are. ("In Kant’s and Kästner’s terminology, number applies to aggregates that can be linearly ordered as whole composites, but which need not have intrinsic order" - D. Hyder, Kant on Time II, p. 527)


Regarding your second question, Kant of course was aware of various "pan-relationist" ideas, since he was originally taught by metaphysicians in the Leibnizian school. Kant in fact argues that, because space and time are relational structures, they must be ideal. The quote that you cite is exactly where he says it, closely following Leibniz. During the Kant-Eberhard controversy, Kant even says that his Critique in some sense reasserts many important points made by Leibniz regarding the relational character of space, time and sensation (for that see first chapters of R. Pippin's Kant's theory of form).


I will just point out one thing: it is a common mistake to make when reading Kant to search for profound ideas where Kant simply uses terminology that he borrowed from other authors, but which nowadays isn't used commonly. The answer to most questions like "Did Kant anticipate x?" is: he did ancipate x insofar as he developed ideas from Lambert, Leibniz, Euler, Newton etc. which anticipate x. The manifestation of these influences is sometimes (but not always) terminological convergence between whoever works on x nowadays and Kant.

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  • If you can link to PDFs of the mentioned texts I’d appreciate it Commented Apr 18 at 11:50
  • @JuliusHamilton Which ones exactly? Commented Apr 18 at 12:37
  • All of them, but if you don’t want to, it’s ok, I can track some down myself. Thanks Commented Apr 18 at 13:29
  • @JuliusHamilton I think most of them are quite easy to find. Commented Apr 18 at 13:55
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I have nothing deep to say but I really like this question and I might be able to contribute a few small points.

But this seems like something of a throwaway historical grounding if one considers that Kant's (or Aristotle's) categories of ontology seem far removed from modern categories of sets, setoids, groups, groupoids, etc.

As far as I know, yes. There is no deep connection between the terminology of category theory and the original fields those terms came from. At least, there was none overtly intended. Category theory was invented by algebraic topologists who were trying to abstract very specific scenarios they were studying in I believe homological algebra. It was not initially philosophically motivated, and it was not initially intended as a “new foundations of mathematics”, as it became. I know Mac Lane explicitly says they borrowed the term “functor” from Carnap, but “category” is an everyday term and I do not think they would need to “consult with Kant” before deciding on its use. Mathematics is full of ultimately somewhat arbitrary terminology. The terminology does not always have “perfect conceptual presentation”. Similar to natural language, sometimes a term is originally used in one context, but it migrates to new contexts as the field itself evolves. There are probably good examples of names in math that are almost like historical accidents. I recently came across an argument that calling “open sets” “open” in topology is one such example.

So there is not, on the surface, a reason to associate the word “category” with Kantian philosophy. On the other hand, clearly Mac Lane was interested in philosophy, at least being aware of the work of Carnap, so you could make a more subtle argument that there is some non-obvious connection after all. You could also argue that there are parallels between how the concept of a category evolved and the Kantian idea of a category. But those are different claims than that Mac Lane and Eilenberg actually thought categories were “Kantian” when they first presented the idea of them.

However, homomorphisms are important to modern category theory.

Homomorphisms are generally a type of function between “algebraic structures” (groups, rings, etc.). An algebraic structure is a set plus some closed operations (“closed” meaning, a function from the set back to the set; assigning its own values to its own values). The operations fulfill certain axioms. One such axiom could be f(x, y) = f(y, x). Another could be there x such that for all y, f(x, y) = y. A homomorphism is a map from the elements of one algebraic structure to the elements of another, so that all of the axioms commute with the map. One of the simplest algebraic structures is a semigroup. It has a binary operation whose only axiom is associativity: operation(operation(a, b), c) = operation(a, operation(b, c)). Consider a map f from one semigroup to another. If f(op(op(a, b), c)) = op(f(op(a, b)), f(c)) = op(op(f(a), f(b)), f(c)), then what this means is we can do the semigroup operation on 2 elements and then take that element under map f, or we can take 2 elements under map f and then apply the semigroup operation. In either case, the combination of the map and the operation will still act associatively.

A homomorphism is not the same as a morphism in category theory. A morphism is actually a bit simpler.

A category is technically a directed multigraph, plus a binary operation on arrows. That means, it is a graph, where the edges between nodes have a direction, and there can be multiple distinct edges between the same pair of nodes, and there is a binary relation which associates an arrow to certain pairs of arrows. (Intuitively, it has a concept of “path equivalence”, where two specific arrows is “equivalent” to a different single specific arrow).

Morphism is just a synonym for arrow. It can best be thought of very abstractly, and simply. It doesn’t have to have any specific “meaning”. It’s literally just an arrow, like in graph theory, or any type of diagram we see in daily life, pointing from one object to another.

That said, it turns out that if you look at a collection of say, semigroups, and then you look at the possible homomorphisms between them, that meets the criteria of a category. So there are categories where the morphisms are interpreted or denote “homomorphisms between certain algebraic objects”, but that is just one of myriad things the morphisms could “represent”. That would be a semantics for the syntax, where the syntax is basically a childishly simple diagram. A category can be abstract: you can study just the properties of the diagram, without saying the morphisms “stand for something else”.

That is to say, the three categories [of quantity], in which the unity in the production of the quantum must be homogeneous throughout, are transformed solely with a view to the connection of heterogeneous parts of cognition in one act of consciousness, by means of the quality of the cognition, which is the principle of that connection.

I cannot understand this unfortunately. I do not know what the “three categories of quantity” are, “the unity in the production of the quantum”, or the “heterogeneous parts of cognition”. That said, this sounds an awful lot like a functor. A functor is kind of like a “homomorphism”, between categories. It is a mapping between the objects and arrows (nodes and edges) from one category to another, where basically the category axioms still hold whether you do the mapping before or after checking the axioms.

I’m going to be honest with you and say that I made tons of mistakes in the above that I now have to think about, but thanks for helping me realize how many further questions I have on the above.

Anyway, it sounds like it would be a cool exercise to explore if Kant was basically suggesting there was a functor (more likely than a homomorphism) from the noumena of the physical world to the phenomena of consciousness!

I haven’t read the rest of the question since I’ll need to understand Kant better in order to go on.

By the way, here’s an essay relating Kant’s thought with a modern mathematical view of fundamental physics. It is not category theory but it is abstract algebra.

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