Dear all, To add a small historical note: the Pappus/Viète sense of analysis (reasoning backward from a desideratum) is, as Kevin and Nikita have pointed out, quite different from the Lawvere/Kock sense (axioms on types rather than on a point-set substrate). Hilbert's Foundations sit between the two, axiomatizing geometry without committing to a substrate ontology — which Patrik is right to highlight as a pre-categorical step in the same direction. Two observations from a (currently underexamined) test case — synthetic foundations for cognitive systems (agent memory, operation histories, typed transports, first-class relations) — that may sharpen Paul's harder question about what synthetic theories actually "put together": (1) The application has no plausible point-set substrate. One is forced into the synthetic line, or has to give up. When taken seriously, three categorical primitives appear naturally: an operation category with initial object (operational history), a Grothendieck opfibration over layer codes (typed transport — close in spirit to recent fibred-lifting work of Hofmann-Streicher type), and a functor ρ: Tw(M) → M lifting morphisms to first-class objects (relational reification). These three appear modular-independent in our setting: none derives from the other two over the foundational equations we use. (2) If this is not an accident, it suggests a possibly-controversial rough heuristic: synthetic methods seem most substantive where the application domain admits no obvious point-set substrate — cognitive systems, identity in HoTT, intensional computability. Conversely, where a substrate is naturally available (real analysis, set theory), synthetic approaches are at best supplemental, however elegant — ASD being the most striking counterexample to my own heuristic, which is why I am not certain of it. Whether the three-primitive decomposition generalizes across Paul's list (Synthetic Differential Geometry, Synthetic Topology, Synthetic Domain Theory, Synthetic Algebraic Geometry), and whether the substrate-existence heuristic holds up, I do not know. Both feel like interesting questions for JMRC's manuscript, and I would be glad to hear pushback from anyone who has thought along either line. Best regards, Abdiel Mars peklund via Categories <categories-list@categories.org.au> 于2026年6月28日周日 16:21写道：
Hilbert's Foundations of Geometry is obviously part of this "equation" as well.
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As I may have written before, I am more and more becoming intrigued and thrilled by the thought what foundations (both for mathematics and for computing) today would look like, had category theory been created, and would have been established even in rudimentary forms, already during the times Frege and Peano debated, i.e., categorical language would also have influenced Principia. By the time we come to the 1920's, efforts to adopt the Liar and similar paradoxes in mathematical "proof" would probably have been rejected, i.e., Gödel's "Theorems" would probably have been viewed as paradoxes (which, in my view, they essentially are), and those paradoxes would probably have been solved by now.
A "categorization" of Hilbert's Foundations of Geometry might not provide all that much new insight to foundations, but a categorization of Hilbert's and Bernays' Grundlagen der Mathematik I and II probably will, obviously including a critical and categorical examination of Gödel's "technique".
Doing so, I feel that the Foundations of Mathematics would be brought up to an entire new level of investigation, and indeed with the support of category theory.
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Best,
Patrik
On 2026-06-27 16:00, John Baez via Categories wrote:
Hi -
Geometry in the style of Euclid is called "synthetic" in distinction to using coordinates à la Dedekind, but I haven't heard coordinate geometry being called "analytic" and I'm not sure what Euclid was "putting together".
I have always heard "analytic geometry" used to mean coordinate geometry. From the Wikipedia article Analytic geometry <https://en.wikipedia.org/wiki/Analytic_geometry>:
In mathematics <https://en.wikipedia.org/wiki/Mathematics>, *analytic geometry*, also known as *coordinate geometry* or *Cartesian geometry*, is the study of geometry <https://en.wikipedia.org/wiki/Geometry> using a coordinate system <https://en.wikipedia.org/wiki/Coordinate_system>. This contrasts with synthetic geometry <https://en.wikipedia.org/wiki/Synthetic_geometry> .
Best, jb
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