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Peva Blanchard (Aug 11 2024 at 11:05):I am reading this interview of Lawvere (thanks to @Ivan Di Liberti 's suggested readings).
Lawvere is answering a question of the journalist about the "usefulness" of CT.
Everyday human activities such as building a house on a hill by a stream, laying
a network of telephone conduits, navigating the solar system, require plans that
can work. Planning any such undertaking requires the development of thinking
about space. Each development involves many steps of thought and many related
geometrical constructions on spaces. Because of the necessary multistep nature
of thinking about space, uniquely mathematical measures must be taken to make
it reliable. Only explicit principles of thinking (logic) and explicit principles of
space (geometry) can guarantee reliability. The great advance made by the theory
invented 60 years ago by Eilenberg and Mac Lane permitted making the principles
of logic and geometry explicit; this was accomplished by discovering the common
form of logic and geometry so that the principles of the relation between the two are
also explicit. They solved a problem opened 2300 years earlier by Aristotle with his
initial inroads into making explicit the Categories of Concepts. In the 21st century,
their solution is applicable not only to plane geometry and to medieval syllogisms,
but also to infinite-dimensional spaces of transformations, to “spaces” of data, and
to other conceptual tools that are applied thousands of times a day. The form of
the principles of both logic and geometry was discovered by categorists to rest on
“naturality” of the transformations between spaces and the transformations within
thought.
I'm having trouble understanding the "problem solved by Eilenberg and MacLane", the "common form of logic and geometry", and the fact that this problem has been opened by Aristotle (I'm aware of Aristotles' Categories, but I don't see the relation).
I feel like my question is the kind of question that requires multiple books to answer, but I'll ask anyway: what did Lawvere mean in his reply?
Joshua Meyers (Aug 11 2024 at 20:58):I'd also love explication on:
John Baez (Aug 11 2024 at 21:03):My guess: they "permitted making the principles of geometry and logic explicit" by developing category theory, laying the groundwork for the unification of geometry and logic in sheaf theory and topos theory. See Mac Lane and Moerdijk's Sheaves in Geometry and Logic for details.
Morgan Rogers (he/him) (Aug 12 2024 at 07:02):Notice that Lawvere chooses his phrasing carefully: he doesn't say that Aristotle posed the problem, but that he opened it. He is suggesting that Aristotle's Categories of Concepts are an incomplete theory that CT completes to something which can actually be applied to thinking about logic and geometry. It sounds like a grand claim (great for an interview) but I don't know how impressive it really is.
I might compare it to the observation that high school algebra provides formal notation and methods facilitating solving of problems that only a handful of people were able to pose or solve a few thousand years ago.
Matt Earnshaw (Aug 12 2024 at 10:42):In the essay on space and quantity Lawvere writes "In his Lyceum, Aristotle used philosophy to lend clarity, directedness, and unity to the investigation and study of particular sciences". Then, after lamenting the bourgeois philosophy of the 20th century, he notes that mathematicians meanwhile had to act as their own ""Aristotles" and "Hegels" as they struggled with the dialectics of "general" and "particular" in their field", and he sees Eilenberg and Mac Lane in particular as struggling with the "leap from quantity to quality", which eventually results in the introduction of category theory.
I don't think many people understand category theory as originating from this problem, so it is worth clarifying. I think the central example of quality in Lawvere's sense (axiomatized in the axiomatic cohesion paper) is the homotopy category, and the central example of quantity is (co)homology. As far as I understand, E-M spaces and perhaps other aspects of their work of which I am ignorant, indeed deal with the "leap" between these.
So a narrow claim we might extract is that E-M "solved" the problem of the relation between quantity and quality (seen through the lens of Lawvere's analysis of quantity/quality types), which is certainly a central part of Aristotle's theory of categories. Though I think we can agree the phrasing is a bit rhetorical.
A more general speculative claim implied by the first part of his reply (amongst remarks here and there) is that CT "solves" the problem of lending "clarity, directedness, and unity to the investigation and study of particular sciences". I think it's clear that this is rather work in progress.