nLab synthetic mathematics

Idea

In “synthetic” approaches to the formulation of theories in mathematics the emphasis is on axioms that directly capture the core aspects of the intended structures, in contrast to more traditional “analytic” approaches where axioms are used to encode some basic substrate out of which everything else is then built analytically.

analytic	synthetic
structures built out of “point-set”-substrate	axioms imposed on the available types

Often the “synthetic approach” is just referred to as “axiomatic”. For instance model categories were introduced as “axiomatic homotopy theory” and indeed they may be regarded as providing a synthetic axiomatization of homotopy theory, which is not based on (but does subsume) the traditional “point-set model” provided by topological spaces.

Examples

• synthetic geometry axiomatizes the nature of objects in a plane or hyperplane without speaking of the plane or hyperplane itself as a collection of points and without speaking of these figures as being subsets of points.

• synthetic differential geometry is refinement of this to contemporary research-level mathematics.

  Here instead of defining the concept of smooth manifold analytically by “point-set methodology” as is traditional (a set of points equipped with a topology and a smooth structure, etc.) one imposes one single axiom scheme in the ambient topos (the “Kock-Lawvere axiom” in this case) which essentially ensures that for all objects/types $X$ there exists also an object/type $T X$ which behaves as the tangent bundle of $X$ should. These axioms then turn out to have models in smooth spaces (which include smooth manifolds) but also in other, such as algebraic geometry and supergeometry.

Alternatively one may set up synthetic differential geometry via axioms for differential cohesion.

• synthetic homotopy theory – With the advent of homotopy type theory, which may be regarded to some extent as a further abstraction of axioms similar to those of model categories, it became more common to speak of this as “synthetic homotopy theory”, as, for example, in the HoTT book. For a list of work carried out in homotopy type theory, see at mathematics presented in homotopy type theory

• synthetic topology – There are many different approaches for doing topology synthetically. Some of them include:

  • synthetic Stone duality, which sets up the topos of light condensed sets via axioms

  • abstract Stone duality, which sets up the category of locally compact locales via axioms

  • real-cohesive homotopy type theory, which sets up the (infinity,1)-topos of Euclidean topological infinity-groupoids via axioms

• formal category theory, a synthetic approach to category theory

• synthetic differential topology

• synthetic domain theory

  • synthetic guarded domain theory

• synthetic algebraic geometry

• synthetic probability theory

• synthetic computability theory

• synthetic Tait computability

• synthetic $(\infty,1)$-category theory

Relation to constructivism

Synthetic approaches are naturally compatible with constructive mathematics/intuitionistic mathematics, but synthetic mathematics is about a different issue than constructivism. For instance even most constructive proof assistants in existence have in their libraries defined basic mathematics concepts, such as topological spaces, in the traditional analytic way.

Relation to computer science

There is at least some similarity between synthetic mathematics and domain specific embedded programming languages, see for instance (Hudak 98, section 3.2). In (Hudak 98, figure 2) this shows aspects of a real-world DSL for “geometric region analysis” embedded in Haskell which under the relation between type theory and category theory/computational trinitarianism one immediately recognizes as a fragment of synthetic geometry.

Philosophical arguments

Proving an arithmetic result in Peano arithmetic, or more generally a statement in an appropriate axiomatic theory, ensures axiomatic purity (a notion of purity of methods) , cf. Lehet 2021).

Moreover, such proofs are certain to be valid because of the nature of the fundamental objects considered, as captured by the axioms of the theory, and not by accident because of their particular construction in such and such a system.

Related concepts

• analytic versus synthetic

References

General

• Andrej Bauer: Synthetic Mathematics with an Excursion Into Computability Theory, talk at University of Wisconsin Logic Seminar (Feb 2021) [blog, slides:pdf, pdf video:yt]

• Wikipedia: Analytic-synthetic distinction

On the axiomatic method:

• Ellen Lehet: Impurity in Contemporary mathematics, Notre Dame J. Formal Logic 62 1 (2021) 67-82 [doi:10.1215/00294527-2021-0003]

Synthetic homotopy theory

Discussion of synthetic homotopy theory (cf. synthetic homotopy theory, typically understood as homotopy type theory):

• Ulrik Buchholtz, Sec. 3.1 of: Higher Structures in Homotopy Type Theory [arXiv:1807.02177]

• Mike Shulman, slides 37 onwards in: Homotopical trinitarianism:A perspective on homotopy type theory, 2018 (pdf)

• Egbert Rijke: Classifying Types [arXiv:1906.09435]

Via cubical type theory:

• Anders Mörtberg, Loïc Pujet, Cubical synthetic homotopy theory, CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (Jan 2020) 158–171 [doi:10.1145/3372885.3373825, pdf]

Via simplicial type theory:

• Emily Riehl: Synthetic perspectives on spaces and categories [arXiv:2510.15795]