Dear José, You may find the preface to Kock's "Synthetic Differential Geometry" (https://users-math.au.dk/kock/sdg99.pdf) useful. He refers to a statement by Sophus Lie that also deals with the distinction between synthetic and analytic methods (without exactly explaining the difference). Kock then goes on to explain what *he* means by it. Best, Jon
On Jun 27, 2026, at 11:18 AM, José Manuel Rodríguez Caballero via Categories <categories-list@categories.org.au> wrote:
Dear all,
I would like to include the following distinction in a manuscript:
There are two styles of reasoning in mathematics:
analytic: structures built from a “point-set” substrate; 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?
Many thanks in advance.
Best regards, JMRC
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