A few clarifications on “synthetic” vs “analytic”. These terms are by no means original to Kant but go back at least to Plato and Pappus. Very roughly, for Pappus analytic reasoning involved reasoning backward from the desideratum to something known, whereas synthetic reasoning was just the opposite. Viéte seems to have coined the use of “analysis” for, essentially, solving algebraic equations in which the unknown, as a variable, is manipulated as if it were a known number. There is thus a somewhat accidental evolution toward “analytic” methods as meaning, essentially, algebraic ones, and synthetic, geometric, over the early modern period. “Analytic geometry” is in fact a very well-attested term, and was indeed the first college course in mathematics for Americans as late at Halmos’s undergraduate years, as you can read in his automathography, if I recall correctly. The formulation in terms of “point sets” versus “axioms on types” in the nLab would of course have been unimaginable to any mathematician before at least somewhere between Hilbert and Bourbaki, so it should be thought of as a novel, well, synthesis, not as a neutral report of how mathematicians have generally conceived of this distinction. KC P.S. The historical material above can be found in Boyer’s History of Analytic Geometry, freely available via the Internet Archive’s digital library.
On Jun 27, 2026, at 4:20 AM, Paul Taylor via Categories <categories-list@categories.org.au> wrote:
I would like to include the following distinction in a manuscript:
There are two styles of reasoning in mathematics:
1. analytic: structures built from a “point-set” substrate; 2. synthetic: axioms imposed directly on the available types.
At present, the only source I have found stating this distinction in this form is the nLab page on synthetic mathematics https://ncatlab.org/nlab/show/synthetic+mathematics
Could anyone point me to a published reference where this terminology or this formulation of the analytic vs. synthetic approaches is explicitly discussed?
In Greek "synthesis" means "putting together" and "analysis" means "taking apart".
Immanuel Kant used these two words for a philosophical distinction that I will leave someone else to explain.
By long-standing usage, "analysis" in mathematics is the study of differential equations and related matters, but I don't see what is being "taken apart".
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".
Then Bill Lawvere, Eduardo Dubuc, Anders Kock and others studied "Synthetic Differential Geometry" by treating manifolds as objects in some suitable topos and using its logic to deduce hitherto difficult geometric theorems.
After that, there were Synthetic Domain Theory, Synthetic Topology, Synthetic Recursion Theory and Synthetic Algebraic Geometry.
However, I'm not sure what those topics "put together".
So altogether there doesn't seem to be any clear meaning to the use of these words in mathematics, so far as I can see.
Paul
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