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Stream: theory: mathematics

Topic: Synthetic vs structural mathematics?

Andrea S (Dec 04 2025 at 21:21):

Is there a meaningful distinction between "synthetic" mathematics and "structural" mathematics?

I would define a structural approach to math as something like: "Study mathematical concepts exclusively through their extrinsic properties to find characterizations of those concepts invisible from intrinsic definitions," which already feels very Yoneda Lemma to me.

I haven't been trained in the "synthetic" approach to math, so while I can point to lots of examples of papers doing synthetic math, I'm not sure if there's a different goal in mind.

Vincent R.B. Blazy (Dec 04 2025 at 21:57):

For me, synthetic maths is more about formalizing the abstract concepts or objects one wants to study, directly in a formal system designed for that purpose. And not by defining them analytically in terms of other objects, themselves synthetically formalized (such as sets). The link between both is tha any analytic definition amounts to an interpretation (model) of such a synthetic formalization, into any the theory the definition is made in.

Alex Kreitzberg (Dec 05 2025 at 01:05):

The description at the beginning of [[structuralism]] seems nice:

In philosophy, structuralism is the point of view which emphasises that the entities considered are parts of a system and that their very meaning and identity is defined according to its relation to the rest of the system. For instance, if one part of the system changes in time, all other parts change as well. The system has features which are not just the composition of features of its constituent parts.

So, I guess my understanding of those last couple of sentences would be - if you add an object to a category such that it has no arrows to it, then you would know the resulting category could no longer have either an initial or a terminal object. Because those are both characterized by universal properties, which depend on every other object in the category. So this is a "structuralism" flavored feature of category theory.

Whereas in synthetic euclidean geometry, if you carry out the classic equilateral triangle construction, you only need to attend to the axioms that were involved in the construction of that triangle.

Note, I think "synthetic" is often synonymous with "axiomatic" in modern terminology, which is arguably a more basic concept than "structuralism" depending on what one means by the terms. In the above I'm interpreting "synthetic" in the euclidean sense of axioms referring to "actual geometric figures" which are then "combined".

Amar Hadzihasanovic (Dec 05 2025 at 07:41):

I think it is useful to think in terms of what each term is contrasted with: synthetic is usually contrasted with analytic, while structural is contrasted with material.

My understanding is that the first distinction is chiefly about a proof-centred vs model-centred mathematics. In analytic mathematics, we are interested in a particular class of objects, and what statements are true of that class of objects; there is typically no "flexibility" about "what the theory is about". In synthetic mathematics, we are interested in what statements follow from other statements. We may read the latter as statements about some class of objects, and in the best case there is a soundness-and-completeness theorem which says that a family of statements characterises precisely a class of objects as its class of models, but one may do away with it and simply think of the "object of study" being "objects that satisfy these axioms".

Amar Hadzihasanovic (Dec 05 2025 at 07:54):

So this distinction, I would say, is about whether we consider objects or proofs to come "first".

The structural/material distinction, as I understand it, is about specification-centred or implementation-centred mathematics. That is, is the object I am studying anything that meets some specification, anything that behaves like ...? Or am I studying a particular, concrete, object, or particular implementation of that specification?

I would say that this overlaps with the first distinction in the particular case where the "specification" is given by "a list of axioms", and the framework in which I am working is proof-theoretic, but I think it is somewhat richer and can have instances that only partially overlap...

Vincent R.B. Blazy (Dec 05 2025 at 09:06):

@Alex Kreitzberg, @Amar Hadzihasanovic, I guess all that fits with my account above, right? With "axiomatic" precisely meaning choosing a system and it’s axioms to formalize ideas, the "model-centredness" of analytic foundations matching with the interpretations I talked about, "class of objects" being tacitly either still informal abstractions or objects in a target theory their definition can be made in, etc. :)

David Corfield (Dec 05 2025 at 10:17):

I enjoyed finding out how the [[synthetic-analytic]] distinction evolved over time. Originally, the analytic method was the exploratory one, working out what needs to be in place for a result to be true. Then the synthetic method was the after-the-event presentation of the derivation from the discovered principles, hiding the thought processes of their discovery. One might sum it up as the difference between working backwards and working forwards.

Vincent R.B. Blazy (Dec 05 2025 at 10:21):

@David Corfield When was that origin, so that it fits historically-of-concepts with our acceptions detailed hereabove, that should have started with (Euclidean or) Hilbert’s synthetic geometry vs. Descartes’ analytic geometry?

David Corfield (Dec 05 2025 at 11:09):

Classical times. Here's Pappus, an Alexandrian mathematician, c. 300 AD:

Now analysis is the way from what is sought—as if it were admitted—through its concomitants (akolouthôn) in order[,] to something admitted in synthesis. For in analysis we suppose that which is sought to be already done, and we inquire from what it results, and again what is the antecedent of the latter, until we on our backward way light upon something already known and being first in order. And we call such a method analysis, as being a solution backwards (anapalin lysin).

In synthesis, on the other hand, we suppose that which was reached last in analysis to be already done, and arranging in their natural order as consequents (epomena) the former antecedents and linking them one with another, we in the end arrive at the construction of the thing sought. And this we call synthesis.

But these ideas derive from centuries earlier in, e.g., the works of Aristotle.

David Corfield (Dec 05 2025 at 11:17):

Synthetic has changed less in a way. When the categorical probabilists synthetically set out to do probability theory from the notion of a Markov category, we probably won't hear of the reasoning that led them to require copy-delete, etc. Then one might say they came across these constructions by working out just what was needed for certain results to hold, so analytically in the old sense.

But analysis got twisted as it went through Descartes analytic geometry to the 19th century sense in geometry of coordinate based reasoning.

John Baez (Dec 05 2025 at 12:42):

Alex Kreitzberg said:

The description at the beginning of [[structuralism]] seems nice:

In philosophy, structuralism is the point of view which emphasises that the entities considered are parts of a system and that their very meaning and identity is defined according to its relation to the rest of the system. For instance, if one part of the system changes in time, all other parts change as well. The system has features which are not just the composition of features of its constituent parts.

So structuralism is the opposite of compositionality? :smiling_devil:

Alex Kreitzberg (Dec 05 2025 at 15:54):

Maybe? I like that point of contrast.

If "compositionality" is "reductionism" then yes (if that quote is right). But if "compositionality" is "does this lego fit with every other in your kit?" then no, structural ideas are critical for compositional ones.

I think I'd say "the whole is more than the sum of its parts" is a failure of "reductionism" rather than "compositionality" with how I want to use the words. Generally in music, or poetry, you want your score or poem to say more than what you wrote down, but still possible to play and read.

So you want "compositionality" to succeed (be playable or readable), but "reductionism" to fail (it conveys more than what's literally heard).

@Vincent R.B. Blazy your views seem consistent with mine, I just wanted to observe a connotation that seemed interesting or relevant.

Peva Blanchard (Dec 05 2025 at 16:36):

@Alex Kreitzberg you might be interested in this paper (obstructions to compositionality)

Andrea S (Dec 06 2025 at 23:45):

Alex Kreitzberg said:

Note, I think "synthetic" is often synonymous with "axiomatic" in modern terminology, which is arguably a more basic concept than "structuralism" depending on what one means by the terms. In the above I'm interpreting "synthetic" in the euclidean sense of axioms referring to "actual geometric figures" which are then "combined".

and

Amar Hadzihasanovic said:

So this distinction, I would say, is about whether we consider objects or proofs to come "first".
[...]
I would say that this overlaps with the first distinction in the particular case where the "specification" is given by "a list of axioms", and the framework in which I am working is proof-theoretic, but I think it is somewhat richer and can have instances that only partially overlap...

This is very informative and I enjoyed reading the discussion, thank you!

To John's comment:
John Baez said:

[...]
So structuralism is the opposite of compositionality? :smiling_devil:

and Alex's reply:

I like Alex's distinction:
Alex Kreitzberg said:

I think I'd say "the whole is more than the sum of its parts" is a failure of "reductionism" rather than "compositionality" with how I want to use the words.

but would add that:

The system has features which are not just the composition of features of its constituent parts.

seems to be in opposition to both reductionism and compositionality. To be mildly obscene with nPOV, a reductionist view to me feels like "0-compositionality"—the compound structure is simply the sum of its parts. Where reductionism fails we observe emergent properties, and maybe a goal when solving problems with compositionality could be to investigate if those emergent properties can be studied by 1-, or higher-order composition—a compound structure is the sum of its parts and the relationships between those parts (and maybe the relationships between those relationships too in the case of composing higher-order morphisms). But in principle, some emergent phenomena may not be characterized "just" by compositionality altogether and may require an entirely different framework to capture the emergent aspects.

Probably just reiterating what I initially said, but I don't think structuralism is opposite to compositionality—to me, structuralism is the philosophical perspective that the Yoneda Lemma makes precise, which is hand-in-hand with compositionality. I was asking the question more of structuralism as a philosophy of mathematics rather than as a broader philosophical notion per Awodey's answer to Hellman's question linked in the structuralism article, and maybe this distinction is where the view of structuralism as

The system has features which are not just the composition of features of its constituent parts.

comes from? Or maybe it could be one of a few different interpretations of what structuralism is about seeing that Wikipedia already lists three distinct flavors.

Alex Kreitzberg (Dec 07 2025 at 00:08):

The paper Blanchard links to seems to make a lot of these distinctions precise. Evidently in a monoidal category, the "right" notion of arrow is a "lax functor", which allows for monoidal products to have "emergent properties". For example, $F$ lax monoidal only needs $\mu : F(x) \otimes F(y) \rightarrow F(x \otimes y)$ to be a natural transformation, rather than a natural isomorphism.

If the categories involved are monoidal preorders, with the target abelian, then it reads

$$F(x) + F(y) \leq F(xy)$$

Literally what $F$ measures of the product, or composition, is more than the sum of its parts. The point of the paper is to quantify the difference.

Amar even worked on that paper!

Mike Shulman (Dec 08 2025 at 05:45):

Alex Kreitzberg said:

The description at the beginning of [[structuralism]] seems nice:

In philosophy, structuralism is the point of view which emphasises that the entities considered are parts of a system and that their very meaning and identity is defined according to its relation to the rest of the system. For instance, if one part of the system changes in time, all other parts change as well. The system has features which are not just the composition of features of its constituent parts.

Note that that quote starts with "in philosophy". The relevant passage for this question is rather the one further down the page:

Structuralism is also the term given to a position in the philosophy of mathematics which holds that the entities treated in mathematics are simply structures. It is associated also with structural realism in the philosophy of science, which maintains either that all we can know of the world is its structure (epistemological structural realism) or that indeed the world just is structure (ontological structural realism).

Alex Kreitzberg (Dec 08 2025 at 06:36):

Those points of view are understood as deeply connected though right?

In math, aren't these two senses combined via the Yoneda lemma, Lawvere's triple adjunctions, etc?

(Though, I appreciate you pointing out that section, I did miss it)

Mike Shulman (Dec 09 2025 at 07:05):

I think of mathematical structuralism as fundamentally about the structures themselves, not about the categories that they organize into. So structuralism says that it doesn't matter (for instance) what the natural number 2 "actually is", only that it's positioned in the structure of natural numbers as the sucessor of the successor of zero. Universal properties are then a convenient way to characterize particular structures, particularly because they're isomorphism-invariant. But I see structuralism as being fundamentally about the structures rather than the universal properties.

Amar Hadzihasanovic (Dec 09 2025 at 08:55):

I agree... while category theory is often associated with mathematical structuralism, I would say that the particular flavour of CT is less of a choice of structuralism over materialism, and more of a "blurring of boundaries" between the two, a "monistic" approach... Set-theoretic mathematics have a kind of built-in dualism where "structure" is at the logical level, "matter" is sets, and "implementation" or "interpretation" is a one-way channel from logic to sets; in CT, typically, "structure" and "matter" are really just the domain and codomain of a functor (or more generally a morphism) so the same objects can play a role of structure or a role of matter depending on what side of a functor they appear on. One can make an argument that this is anti-materialistic because it de-emphasises the role of any specific matter as being "the real, actual matter" underlying mathematical objects, but one can equally make the opposite argument that, instead, this "materialises all structure": in this sense, you can see an object with a universal property as being something like the "minimal implementation of a structure"---the one that has to be represented in every other implementation---and so CT prefers to reduce a structure to its "minimal materialisation".

Alex Kreitzberg (Dec 09 2025 at 14:55):

I see, your points make sense. I'll be more careful to clarify the distinction in the future.