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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?

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    If you are interested in philosophy of mathemtics that goes off the beaten path of foundational issues try Corfield's Towards a Philosophy of Real Mathematics. It is not focused on group theory exclusively (Monster group is featured), but touches on contemporary fields like algebraic geometry and topology that most philosophers do not much discuss (or know).
    – Conifold
    Mar 21 at 3:40
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    One of the cautions for philosophy of latest ultra-abstract maths/physics (group is the stepping stone to that journey as its interest is not really in objects but transformations) was once pointed by Richard Feynman, our mundane experience of jiggling balls to imagine atoms are useless and sometimes outright wrong. So unless you studied the math/science part as an expert first, otherwise any philosophical "formulations" may be totally inappropriate or can gain very little to be applied in classical use cases... Mar 21 at 16:50
  • Stewart Shapiro Philosophy of Mathematics: Structure and Ontology. (Oxf.UP 1997), perhaps: the important word here is "structure" .
    – sand1
    Mar 23 at 22:01
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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.

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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).

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  • Dedekind's "cut" and his definition of the real numbers described by you above was the beginning of modern understanding of real numbers. From here, we gradually understand more abstract transformations/actions control more concrete object/number. All concrete objects may be dropped and replaced up to isomorphism... Structualism sounds like to claim the precedence of transformational structure over objects. What's its essential difference vs Functionalism then? Mar 21 at 21:41
  • @DoubleKnot As far as I understand, Dedekind's cuts do not belong to structuralism since they define a particular set of objects rather than a structure. Dedekind's structural definition of the reals was that of any system satisfying the axioms of a complete ordered field. I have never heard the term functionalism used in the context of philosophy of mathematics so I cannot compare it to structuralism.
    – NWR
    Mar 22 at 2:04
  • Maybe Structualism has more relation to Tagmark's MUH (a radical Structualism or Platonism?)... Functionalism is mainly used in mind related pschophysical realms. For me sounds like structualism and functionalism are of opposing views, one only cares internal while the other only cares about external input/output result. Since they're applied in different areas, maybe it's nonsense to compare them. Mar 22 at 2:18
  • @DoubleKnot I would guess that Tegmark is a structuralist but I am not overly familiar with his ideas, mostly since it doesn't make much sense to me to suggest that reality "is" a mathematical structure. Regarding Platonism, one obvious difference with structuralism is that a Platonist will insist that mathematical objects are intrinsically "something".
    – NWR
    Mar 22 at 2:26
  • Wyle once said: ”if phenomenal insight is referred to as knowledge, then the theoretical one is based on belief... But where is that transcendent world carried by belief, at which its symbols are directed? I do not find it, unless I completely fuse mathematics with physics and assume that the mathematical concepts of number, function, etc...partake in the theoretical construction of reality in the same way as the concepts of energy, gravitation, electron, etc.” This sounds a modern basis for platonism, if so MUH sounds more like radical platonism. The belief in "dx" can explain Zeno paradox Mar 22 at 2:40
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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.

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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.

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