Abstract Unity in Material Diversity: An Introduction to Category Theory and a Defense of Mathematical Realism

April 3, 2021   /  

Name: Micah Phillips-Gary
Majors: Philosophy, Mathematics
Minor: German Studies
Advisors: Rob Kelvey and Garrett Thomson

This project is concerned with looking for abstract structural analogies between materially distinct areas of mathematics and regions of being. In mathematics, this is accomplished through category theory. By abstracting away from the internal structure of mathematical objects, the analogous external structural relations they have as objects of a category become visible. The various kinds of external structural relations (“morphisms”) between objects are discussed and material examples of these are given. We also discuss important theorems in category theory such as the Duality Principle, the Five Lemma for abelian categories and the Yoneda Lemma. The latter two in particular we see to result from the further abstraction of theorems in group theory (the FiveLemma for groups and Cayley’s Theorem, respectively). In philosophy, this search for structural analogies is accomplished through phenomenology. Using the methodology of Husserlian phenomenology, we are able to show how the correlative modes of being and possibilities of experiencing mathematical and empirical objects differ materially while instantiating the same abstract structures of being and experience, respectively. By not straightforwardly identifying the mode of being of mathematical and empirical objects (as traditional Platonism does), we are able to avoid epistemological problems while maintaining a genuine realism that acknowledges the existence of mathematical objects independent of consciousness in opposition to constructivist positions like those of the later Wittgenstein and the later Husserl.

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Micah will be online to field comments on April 16:
noon-2pm EDT (PST 9-11am, Africa/Europe: early evening) and 4-6 pm EDT (PST 1-3pm, Africa/Europe: late evening)

35 thoughts on “Abstract Unity in Material Diversity: An Introduction to Category Theory and a Defense of Mathematical Realism”

  1. Micah- Wow. What a cool, “meta”-project. Category theory is no simple matter. Congratulations on diving into some really deep stuff in both disciplines. Your professors are so proud of you.

  2. Congratulations, Micah! What an amazing combination of philosophy and mathematics. So, are we causally linked to mathematical objects, by way of understanding them? Or does being exposed to them count as such a ‘mental link’? And do mathematical objects exist without smart mathematicians such as yourself to observe/describe them? These are really fascinating ideas. Thanks for sharing your work!

    1. Thank you very much! On my view, there is no causal link between mathematical objects and human beings. Rather, there is something analogous to a causal link. What is necessary for knowledge is that what is known serves as an explanatory ground for the act of knowing. In the case of everyday perceptual knowledge, this is accomplished by way of causation. In the case of mathematical knowledge, this is accomplished by way of the mathematical object in question structuring a diagramamtic instantiation. In both cases, the object known enters into the explanation of why my act of knowing intends the object that it does rather than some other object. The idea that a causal link is required for knowledge in all cases takes the conditions for knowledge that are valid in the case of empirical objects and absolutizes them, taking them as valid for knowledge in general without recognizing the material differences in the ways that mathematical and empirical objects exist.

      As to the existence of mathematical objects independent of mathematicians, on my view they do exist, albeit only in an ideal sense. That is, they exist insofar as this existence is identical with the ideal possibility of their being particulars which instantiate these objects. Certain mathematical objects also have an existence as instantiated (namely, when there actually are such particulars). Of these objects, some (such as “2”) exist as instantiated independent of mathematicians (there still would be (at least) 2 things in the world even if there were not persons). Others are only instantiated in cultural products (diagrams constructed by mathematicians) and so independent of mathematicians they only exist ideally. Further, even the ideal existence of mathematical objects is identical with the ideal possibility of their being mathematicians who would cognize them. But, this is simply to say that it must be logically possible for a mathematician to cognize a mathematical object for this object to exist.

  3. This was so cool! I have always enjoyed the abstract/logical nature of philosophy but I love that you made an overarching connection between this and mathematics. I followed what you said quite well and I the amount of thought that it took to understand and explain all of this astounds me! Great job!

  4. This is truly fascinating stuff, Micah. I particularly enjoyed how you avoided the inherent skepticism in ontological monism. I also think arguing that the examination of real or possible objects is all that’s required to have knowledge of mathematical objects is an elegant addition to the claim that mathematically structured things exist independently of human cognizing them.

    1. Thank you! Apologies I missed your question time. I credit your Roundtable for helping me to recognize that propositions are not non-derivatively intentional.

  5. Micah, this is excellent work and something I know you’ve been interested in for a long time. Congratulations on both this presentation and completing the I.S. project. Well done!

  6. Amazing topic. Your construction of the existence of mathematical objects avoids all of the problems associated with the usual conservative Platonic understanding, while not falling into a skepticism! You’ve positioned yourself perfectly in a debate which often very quickly devolves into reductio ad absurdums about “finding numbers in the wild” and created an elegant explanation for how we can, in fact, have objective numericals.

  7. Hello and congratulations, Micah! A great example of interdisciplinary work. Thank you for sharing your project today!

    1. Thank you! I credit you for freeing me to explore my mathematical interests by letting me write what would’ve been my IS topic had I been an English double as my FYS project.

  8. Excelletn work, Micah! I’m curious how you might elaborate on your ontological pluralism. Since different “regions of being” need not be causally connected, I’m wondering if there’s any specific schema of regions and/or connection between them. I personally get nervous around equivocal accounts of being, but I very much appreciate your non-Platonist realism.

    1. Thanks, Scotty! “Regions of being” is a bit of misnomer in that it refers to objects as they exist in different ways and not necessarily numerically distinct objects. To say that different regions of beings are causally connected is a mistake on my view because causation is a materially determinate concept that takes on a different significance with regard to different regions (physical events as such are causally related, while psychical events as such are motivationally related). Different regions of being are related primarily by instantiation of one kind of object in the other, being “one over many” (e.g., 2 is instantiated in any collection of two things insofar as the existence of 2 consists in the ideal possibility of there being two things, but also an everyday empirical object is an identity across a multiplicity of potential acts of perceiving it) and by identity (e.g., the living body exists in a different way than the physical body, but they are identical). Naturally, I haven’t worked out the latter kind of connection (which I think is the most relevant to your concerns) because I think the only numerical identity that mathematical objects have with another region of being is that between mathematical objects as ideal possibilities of instantiation and mathematical objects as actually instantiated. The picture I have in mind is a kind of realist perspectivist version of Davidsonian anomalous monism. There are relations of identity beween objects as described in different ways, but there is no such thing as an object not existing under some (potential) description. Such a view can be modelled well by category theory.

      I should also note that my account of being is not entirely equivocal. There are certain formal and transcendental logical truths that hold for any kind of object whatsoever and so say what being in general is. These concern the possible abstract structures of objects and the requirement of any object whatsoever that it can be thematized in intersubjective discourse that respects basic normative conditions for discourse. It is simply with regard to the filling out of these abstract structures that I deny that there is any univocal sense of being. Thus, a universal is not an exemplar (as on a traditional Platonist view), nor is it even the kind of thing that can enter into causal relations.

  9. Congratulations, Micah! This is a fantastic project. I also wanted to commend you on publishing some related work in the journal Stance! You have much to be proud of.

    1. Thank you very much! I was happy to see Ryan’s work in the same issue.

  10. The concluding sentence of your abstract is beautifully dense. I enjoyed the thoughts it invoked about my own understanding and view on philosophy & mathematics. Thanks for sharing!

  11. Hi Micah, it has been a great time living in the German suite with you for the past four years!

    I’m very impressed with your research, I can tell you put a lot of time and effort into it! What part of your research are you most proud of?

    1. Thanks, Sydney. It has been nice getting to know over the past four years. I think I’m probably most proud of chapter 3 (the part summarized in the poster) and chapter 6 (the core of my philosophy of mathematics, summarized in the video). Chapter 3 was the biggest challenge to get through this year (requiring the most mathematical proofs), whle chapter 6 drew the most from my background knowledge on Husserlian phenomenology and so felt like the the culmination of years of reading and research I’ve done in my spare time.

  12. Quite interesting! Though, you should make explicit your notion of creativity. It seems that any system with some form of permutative/combinatoric capacity of inputs/stimuli can be said to be creative. But can the climate be seen as creative for it can “create” complex structures such as snowflakes? Are Humans a complex of such creative machines? Do not randomisers have a causal background?

  13. What an unbelievable presentation about an interesting topic! You’ve been a philosophical and mathematical inspiration for me Micah in every class I’ve had the pleasure of taking with you. The way you have connected these ideas makes me want to learn more about both category theory and phenomenology. I wish I had more substantive questions, but I wish the best in the future!

  14. Micah,

    Fantastic! I’m not much of a philosopher, so perhaps I am not fully on the same page as some of the other commenters here who have studied philosophy, but my first question would be to ask if mathematics is incomplete, then how does this change the way we view it phenomenologically, if at all?

    My next question is to ask what you’re plans are for next year? Very excited to hear about them.

    1. That should be (your) not (you’re). Type too fast and you will make silly mistakes!

      I guess I also have a follow up question, as well. Do other mathematical philosophers agree with your view of mathematics through a phenomenological lens? Or do other researchers feel that perhaps there is a more appropriate way to view the connection of abstract objects to the world around us? Perhaps there is no consensus at all.

      1. Hi Isaac! Thank you very much for your comments. The incompleteness of mathematics fits very well with a phenomenological picture (Gödel was very sympathetic to a realist phenomenological view). Formal mathematical systems on my view are analogous to particular angles from which an empirical object is seen. There’s always more to my coffee cup than what I see from one particular angle; there’s always more to a mathematical object than can be proved in any formal system. The analogy is of course more complicated than that, because establishing that you’re dealing with the same mathematical object (the same abstract structure) as defined in a different formal system is rather tricky. This is one of the places that category theory comes into play, showing us how different lower-level mathematical objects instantiate the same abstract structure (the same higher-level mathematical object).

        As to any consensus in the philosophy of mathematics, you can’t get philosophers to agree about whether or not chairs exist or what “is” means. My view is fairly unusual in that most philosophies of mathematics either start with mathematical objects floating in the ether (Platonism) and then struggle to explain how we can know them or start with formal system described simply as the blind manipulation of symbols (formalism) and then struggle to explain how these manipulations have any relation to the world. Naturally my position is that you can’t start from these extremes and then try to explain the connection, you have to start with the connection (as it is given in experience) and then you can make sense of the objects and the formal systems that realize them.

        As for next year, I am still applying to programs in the UK and Germany. Hopefully I will be able to attend a program in logic and the philosophy of mathematics in order to continue my interdisciplinary research, but I am also applying to general philosophy and mathematics programs. In short, nothing set in stone, but hoping for the best. I am particularly interested in learning more about the category of all categories (which I discuss in the conclusion of my IS project) as well as making sure I have a firm basis in axiomatic set theory.

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