With the purpose to clarify my ideas about terminology I would like to ask in Mathematics Stack Exchange what is the difference of two verbal expressions: when a professional mathematician understands that a contemporary and remarkable work in mathematical research (you can to think in historic papers/articles, thesis, books, monographs or a work that corresponds to a mention of an award) is mainly a synthetic work and when, in contrast, the professional mathematician considers that the work of his/her colleague was mainly analytic.
Question. Provide a clarification about when a mathematician understands, that a great mathematical work due to a contemporary mathematician, is mainly synthetic and when the mathematician should to undestand that such work is mainly analytic. Many thanks.
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Professional mathematicians do not classify research as synthetic or analytic. The analytic/synthetic distinction comes from philosophy, and it is very rare for mathematicians to use these words in their philosophical senses. Based on my experience, I believe few mathematicians are even aware of these philosophical concepts, and ever fewer could give definitions of "analytic proposition" and "synthetic proposition". (I belong to the middle camp: I'm aware that the analytic/synthetic distinction is a thing, but I don't think I could adequately define these terms.)
I have never heard a mathematician describe research as analytic or synthetic. The exception is in the context of analysis as a mathematical discipline and fields like synthetic differential geometry or analytic number theory that have these words in their name. But here the mathematicians are using these words refer to the discipline, not the philosophical analytic/synthetic distinction - even if e.g. the use of the term "synthetic" in "synthetic differential geometry" can possibly be traced back to the philosophical concept.