Dear All, I hope all is well. If I may, going by Newton, Grassmann, late Professor Ronnie Brown, Professor F. William Lawvere, and Professor Andree Ehresmann, analysis and synthesis are not free-floating; they are ordered: 'synthesis after analysis' or 'composition after resolution', in the terminology of Newton: "As Mathematicians have two Methods of doing things wch they call Composition & Resolution & in all difficulties have recourse to their method of resolution before they compound so in explaining the Phaemoena of nature the like methods are to be used & he that expects success must resolve before he compounds" (please scroll to the bottom of the webpage <https://www.scienceopen.com/document/review?review=2be8ae94-4521-43b7-b9ea-a14bdef0e213&vid=c6d3ab2d-f848-4465-a1d5-b716d98e3834>). Along these lines, we have: "Neither the synthetic nor the analytic method constitutes the essence of the matter, which is the discovery of the dependency of forms on each other and the manner in which their properties are continued from the simpler figures to the more complex" (please see pp. 3-4 in <https://lawverearchives.com/wp-content/uploads/2024/12/2005-book-reviews.pdf>). The study of spaces, in the work of late Professor Ronnie Brown, involved two sequential processes: (i) subdivision of a space into small comprehensible pieces (cf. analysis) followed by (ii) algebraic inverse to subdivision (synthesis / composition / putting together those that fit together as in composable pair of maps; oftentimes composition called for novel constructs such as corners). Professor Tim Porter: please correct me if I'm mistaken. Professor Andree Ehresmann's solution to the binding problem in terms of colimits within the broad framework of Memory Evolutive Systems can also be understood as putting together / synthesis of various sensory features into which the ambient energy is analyzed by our sensory apparatus. Professor Andree Ehresmann: please correct me if I'm mistaken. It is also interesting to note that relation between synthetic and analytic operations is one of adjointness, as Professor F. William Lawvere illustrates in terms of an elementary exercise we (me, for sure) engage in all the time, albeit without pausing to reflect on that which we are doing in analyzing (please see pp. 4-6 in <https://lawverearchives.com/wp-content/uploads/2025/07/1989.cambridgetalks.pdf>). I just thought some of you might find the above of some interest. Thanking you, Yours respectfully, posina On Mon, Jun 29, 2026 at 8:58 AM José Manuel Rodríguez Caballero via Categories <categories-list@categories.org.au> wrote:
Thank you very much for the clarification regarding the evolution of the distinction between analytic and synthetic, especially how the term "analytic" was conflated with "algebraic" by Viète.
Since we previously touched on cognition in this discussion (Abdiel Mars's example), I was motivated to ask four Large Language Models (Gemini, ChatGPT, Copilot, and Claude) the following question:
How to apply bouba–kiki effect to analytic vs synthetic mathematics: https://ncatlab.org/nlab/show/analytic+versus+synthetic
In every case, the answer was the same: Kiki = Analytic; Bouba = Synthetic. Reference to bouba–kiki effect:
Ćwiek A et al. 2021 The bouba/kiki effect is robust across cultures and writing systems. Phil. Trans. R. Soc. B 377: 20200390. https://doi.org/10.1098/rstb.2020.0390
Perhaps there is a correlation between an individual's cognitive preferences regarding the bouba–kiki effect and their preference for analytic versus synthetic mathematics. I am not sure, but I find this experiment interesting because four large language models reached a consensus on what appears, in principle, to be a nonsensical question.
Kind regards,
JMRC
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