Reference for the analytic vs. synthetic distinction in mathematics
Dear all, 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? Many thanks in advance. Best regards, JMRC
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
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
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 http://paultaylor.eu/delivery/categories-synthetic-analytic
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
Hilbert's Foundations of Geometry is obviously part of this "equation" as well. --- 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. --- 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 [5]:
In mathematics [1], analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry [2] using a coordinate system [3]. This contrasts with synthetic geometry [4].
Best, jb
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Links: ------ [1] https://en.wikipedia.org/wiki/Mathematics [2] https://en.wikipedia.org/wiki/Geometry [3] https://en.wikipedia.org/wiki/Coordinate_system [4] https://en.wikipedia.org/wiki/Synthetic_geometry [5] https://en.wikipedia.org/wiki/Analytic_geometry
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.
---
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.
---
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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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
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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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
http://paultaylor.eu/delivery/categories-synthetic-analytic _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
The earliest 'analytic' vs. 'synthetic' distinction as applied to mathematics (geometry) is in Pappus ( https://mathshistory.st-andrews.ac.uk/Extras/Pappus_analysis_synthesis/). 'Analytic' was later reused for the Cartesian method (but not by Descartes himself), at that time 'analysis' confusingly for us meant what we now call algebra, cf. Viete's 'ars analytica'. Nikita. On Sat, 27 Jun 2026 at 07:33, 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
http://paultaylor.eu/delivery/categories-synthetic-analytic _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
participants (9)
-
John Baez -
Jon Sterling -
José Manuel Rodríguez Caballero -
Kevin Carlson -
Nikita Danilov -
Paul Taylor -
peklund＠cs.umu.se -
Posina Venkata Rayudu -
李光熙