Philosophy of Math that talks about group theory, (or other stuff that's math but not numbers or geometry)?

Question

I have just realized today that anytime I've read something about the philosophy of mathematics, the focus is on numbers, figuring out what numbers are, whether they're real, the relationship of arithmetic to logic, things like this. Geometry is sometimes mentioned, generally by way of reference to Euclid.

(Heck, as I type I'm realizing I am not sure I've seen discussion of imaginary numbers, only real ones! This I'm sure exists but it does seem lilke it doesn't pop up as a common or typical topic in the field.)

I'm wondering if there's much or any work out there in which philosophers of mathematics instead engage with things like group theory (the study of symmetries, whether of mathematical objects, geometric objects, or any other kinds of objects).

The interest of this question for me lies in the fact that contemporary practicing mathematicians will sometimes say that the number line and complex plane and geometric spaces can all be construed as examples of a more abstract thing called a group, and that a lot of the properties of the former can be seen as consequences of facts about the latter. This tempts me to try to think more about groups as archetypical mathematical objects than numbers or figures.

Is this an ongoing research project that exists? Any good authors to look for?

Answer 1

Zalamea’s research may be interesting — I’m thinking of Synthetic Philosophy of Contemporary Mathematics as there’s a fair bit of analysis of the work of Grothendieck and other recent abstract/pure mathematicians, and also a pretty broad-ranging review of lots of other works that could be starting-points for investigating the philosophy of contemporary maths. He reviews for instance a book of Tymoczko’s, New Directions in the Philosophy of Mathematics which has some affinities with Z’s project but I can also recommend it as interesting and maybe relevant to your question in its own right.

Answer 2

You will interesting commentary here:

https://mathoverflow.net/questions/352298/could-groups-be-used-instead-of-sets-as-a-foundation-of-mathematics?r=SearchResults

The "received views" generally ignore such possibilities because of history and folklore. Category theory and its child homotopy type theory are changing that situation. But, they do not do so by addressing the classical criticisms with respect to "second order" or how "verificationalism" does not properly distinguish between truth and provability.

Among the traditional foundational authors, Russell acknowledged how group theory and projective geometry could impact the significance of logicistic sentiment embodied by the analysts arithmetizing the study of real numbers. In fact, prior to his logicist period, Russell wrote a book on geometry proposing how projective geometry might correspond with Kant's "pure geometry" associated with external spatial intuition. (The common disparagement of Kant in which this is assumed to refer to Euclidean geometry has been shown to be misinformed by a translation by Ewald in which Kant actually calls for the development of alternative geometries long before such alternatives had been realized in the nineteenth century.).

Laughably, as logicians began to "represent" truth values with 0 and 1, other avenues of mathematical research began representing proper mathematical objects (such as finite geometries obtained from finite groups) using the two-element Galois field. The sixteen basic Boolean functions that ground classical propositional semantics correspond with the 4-dimensional vector space over GF(2). This vector space has the form of a finite affine plane. If you perform negations and de Morgan conjugations and their composition on some fixed representation of truth tables, you will be performing collineations in this plane.

Additionally, this vector space corresponds with a group of order 16 which carries a finite configuration called a Kummer configuration. String theorists are interested in the continuous correlate to this configuration, and, Melmendier has specifically correlated the theta functions of interest to these 16 vector space elements.

I guarantee that you can use devise a compositional system over 16 constants in which truth tables are recovered and with which you can implement propositional logic.

Arithmetization? Logicism? Formalism?

No one can can prove the "truth" of their presuppositions. And, as you learn about the famous distinction between "syntax" and "semantics" look up "pragmatics" as it had been associated with Rudolph Carnap's work. People trying to sit on a three-legged stool with only two legs look ridiculous.

Answer 3

It sounds to me like you are talking about structuralism.

Structuralism is the philosophical view that mathematics is the study of abstract structures (i.e., patterns) as opposed to the objects and relations that instantiate these structures. Accordingly, mathematics studies the abstract structure of the objects and relations that realise a given structure rather than the natures of those objects and relations.

From a structuralist point of view, the objects of a given structure such as a group have no nonstructural properties - i.e., no internal nature or composition. So, for example, "cross-structural" statements involving mathematical objects such as equating zero with the empty set have no objective truth-value.

I believe that structuralism has its roots in the work of Dedekind in the late 19th century. For example, Dedekind wrote:

If in consideration of a simply finite system X, set in order by a transformation f, we entirely neglect the special nature of the elements; simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order-setting transformation f, then are these elements called natural numbers.

(Dedekind, Was Sind und Was Sollen die Zahlen?, 1888.)

Another example of Dedekind is his definition of the real numbers as a complete ordered field (unique up to isomorphism).

Answer 4

Whether imaginary numbers are 'real' is a topic as old as they are. They have been found to causally implicated in physics though.

Group theory is abstract algebra, capable of expressing types of geometry as special cases, just as algebra allows us to use different types of number lines. It's like a mapping that can let us link different types of mathematics, so I would see it as mistake to see it as 'more fundamental' in the way of particles to atoms say, rather than recognise it as versatile. Many physics cases do not involve the most reductionist strata of explanation, because it would not be tractable. What is important is the whole structure of explanations, and checking their consistency.

You might be interested in https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_Everything because continuous symmetries & their relation to conservation laws were understood through group theory.

Linked this to the Mathematical Universe Hypothesis, and the kind of thinking here https://phys.org/news/2014-12-universe-dimensions.html we could understand E8 as the metamathematical space of all physics, that our universe
& it's initial conditions, can be understood as 'within'. It pretty much ends up being a physics topic, but philosophers should be considering this area more.

I suggest mathematics is not 'found' in the world, but us a language that compounds knowledge about experiences into salience landscapes that foreground useful behaviour, and make understanding dynamics tractable. In this view, the best maths depends on what we are doing, just like the best language on phrasing on what we want to say. And new areas of focus and discussion will of course develop and refine what can be done and said.

You might like this concise video that links the old questions about numbers, to the challenge of non-Euclidean geometry, and where we are now.

Answer 5

This's a field I'm quite interested in, but I found it difficult to drum up any supervisor interest in research exploring the philosophical function of the concept of Representation in mathematics. Basically, I think the issue is that the contemporary study of Group Theory has a surprisingly high bar to entry, in a way that Set theory seemingly does not. By the time you're at the graduate level and prepared to properly dive into the mathematics, you're probably not contemplating studying a PhD in Philosophy.

Representation Theory might be usefully described as the study of the concretization of abstract mathematics, and one of the proported virtues of Category Theory is that it is a generalization of Representation theory. We show how mathematical methods used in more concrete settings also apply to our abstract objects of interest by constructing canonical representations of our abstracta in the concrete domain, and this in turn allows for the extrapolation of results from one subfield of mathematics to another.

But is representing mathematically constructive, or is it a conservative logical extension of the basic underlying concrete domain? What consequences do theorems in abstract mathematics have for different Reverse mathematics foundations, and what kinds of logical strengthenings or weakenings would affect key proofs and methods in algebra? How do standard results in abstract mathematics relate to the mathematical ontologies of structure, set and number, and what kind of reasoning is the homomorphic map? Would a mathematics grounded in Univalent foundations have any reason to be called the same thing as a Set Theoretic view of the mathematical universe? Etc. Etc.

I think there's important work to be done here, but the real challenge is that there's not a lot of demand within either academic mathematics or philosophy to do it, and without the financial backing of the academy, it's probably not getting done by a rogue unfunded PhD philosophy grad student.

Answer 6

Taking into account what Joseph Weissman and nwr answered I would like to add that there is a master's thesis written by Oscar Javier Pérez Lora (in spanish) which addresses some closely related topics. Also, I think that you might find useful the work of Albert Lautman, in his Mathematics, ideas, and the physical real, perhaps you would be specially interested in Section 1, Chapter 3, and the introduction by professor Zalamea: Albert Lautman and the Creative Dialectic of Modern Mathematics. Also, the book Philosophy of Mathematics by Stewart Shapiro discusses in several occasions about structuralism, wich may be of your interest as well. And last but not least, Israel Kleiner's book A History of Abstract Algebra could be a nice way to introduce oneself -I think-.