6 ms·

I'm not sure how knowing what an adjunction or a pullback is helps serve the mission of the undergrad philosophy department.

by platz 5y ago

myWindoonn 5y ago
Your snark is unhelpful. Here's some things philosophy undergrads should know from category theory: * Size issues. All the problems in set theory are back with a vengeance. Exploring size issues makes it clear that Russell's Paradox is not a fluke. * Topoi. A topos is a generalization of the past few millennia of maths, and their existence shows that maths is a multiverse rather than a single collection of rules. Some topoi witness refutations of LEM, making them valuable counterexamples. * Yoneda's lemma. The core of pragmatic philosophy is that objects are not separable from their effects in the world. Yoneda's result takes that to its logical conclusion: An object is equivalent to all of its effects under transformation. * Equivalence of categories. Rel is equivalent to its dual (Rel is daggered) and so is Hilb; the category Set* of sets with an extra point is equivalent to the category Pfn of sets and partial functions. These show us that parts of maths are self-symmetric beyond group theory. In addition, why shouldn't adjunctions be covered at some point? Adjunctions can't be studied without category theory (look up "Galois connections" for a history) and they're hard to avoid once we're talking about enough categories of note. I bet that the adjointness of syntax and semantics is very relevant to philosophy of language.
- platz 5y ago
  It's clear why Formal Logic is studied in undergrad philosophy programs, but there's only so much space to study the core Philosophy ideas in the undergrad time allotted, and what you're suggesting sounds like they'd be getting a math degree rather than a phil degree. It would be more appropriate for graduate phil courses to consider the ideas you highlighted.
  - nbulka 5y ago
    In my undergrad experience, there was way more choice in a PHIL degree than Math/Engineering or even English. My suggestion is to jump on the CT train early because it is new, but truly enlightening. It's the best framework we have for ontological assessment, and that is one of the three major areas of undergrad philosophy.
  - myWindoonn 5y ago
    Yes; why should we give degrees in understanding the parts of philosophy which have been disproven or refuted? The word "mathematics" literally means things which humans must teach each other. Mathematics is philosophy, but with proofs. Look at ethics for an example. Mathematics is the venue for arguing against utilitarianism. Putting those together with category theory, we get a mathematical explanation for why pragmatic ethics is preferable to utilitarian ethics.
nbulka 5y ago
Thanks I should clarify. I don't mean understanding all of the details. I mean from a high level. To be clear, I mean the incredibly useful notion of realizing that cross-domain knowledge can share structure. Logic and geometry are in some sense related in a very formal way. I was never introduced to that notion in the required logic class.
- nomoreplease 5y ago
  > To be clear, I mean the incredibly useful notion of realizing that cross-domain knowledge can share structure. Is there somewhere you can suggest that I can read more about this notion?
  - Koshkin 5y ago
    By "share structure" they mean (the various degrees of) analogy - a widely-known concept that has been made rigorous within the framework of category theory (via the notion of a functor).
  - randcraw 5y ago
    Check out this section of the SEP entry on Category Theory: 3. Philosophical Significance https://plato.stanford.edu/entries/category-theory/ https://plato.stanford.edu/entries/category-theory/ I suspect a lot of the formal work on ontology involves CT.
- platz 5y ago
  Do you think there could be a risk telling the eager-eyed philosophy students that there are truths that they should grant on principle but that they don't need to understand the details? What do they think when they take the epistemology course next?
  - nbulka 5y ago
    Yes the details are important; if you are so inclined to trace the reasoning all the way back to the axioms, thankfully they are very easy to understand! And doing so would definitely prepare them for epistemology. There's a reason that physics leans heavily on category theory.

I'm not sure how knowing what an adjunction or a pullback is helps serve the mission of the undergrad philosophy department.

by platz 5y ago

I'm not sure how knowing what an adjunction or a pullback is helps serve the mission of the undergrad philosophy department.

myWindoonn 5y ago

Your snark is unhelpful. Here's some things philosophy undergrads should know from category theory: * Size issues. All the problems in set theory are back with a vengeance. Exploring size issues makes it clear that Russell's Paradox is not a fluke. * Topoi. A topos is a generalization of the past few millennia of maths, and their existence shows that maths is a multiverse rather than a single collection of rules. Some topoi witness refutations of LEM, making them valuable counterexamples. * Yoneda's lemma. The core of pragmatic philosophy is that objects are not separable from their effects in the world. Yoneda's result takes that to its logical conclusion: An object is equivalent to all of its effects under transformation. * Equivalence of categories. Rel is equivalent to its dual (Rel is daggered) and so is Hilb; the category Set* of sets with an extra point is equivalent to the category Pfn of sets and partial functions. These show us that parts of maths are self-symmetric beyond group theory. In addition, why shouldn't adjunctions be covered at some point? Adjunctions can't be studied without category theory (look up "Galois connections" for a history) and they're hard to avoid once we're talking about enough categories of note. I bet that the adjointness of syntax and semantics is very relevant to philosophy of language.

platz 5y ago
It's clear why Formal Logic is studied in undergrad philosophy programs, but there's only so much space to study the core Philosophy ideas in the undergrad time allotted, and what you're suggesting sounds like they'd be getting a math degree rather than a phil degree. It would be more appropriate for graduate phil courses to consider the ideas you highlighted.
- nbulka 5y ago
  In my undergrad experience, there was way more choice in a PHIL degree than Math/Engineering or even English. My suggestion is to jump on the CT train early because it is new, but truly enlightening. It's the best framework we have for ontological assessment, and that is one of the three major areas of undergrad philosophy.
- myWindoonn 5y ago
  Yes; why should we give degrees in understanding the parts of philosophy which have been disproven or refuted? The word "mathematics" literally means things which humans must teach each other. Mathematics is philosophy, but with proofs. Look at ethics for an example. Mathematics is the venue for arguing against utilitarianism. Putting those together with category theory, we get a mathematical explanation for why pragmatic ethics is preferable to utilitarian ethics.

nbulka 5y ago

Thanks I should clarify. I don't mean understanding all of the details. I mean from a high level. To be clear, I mean the incredibly useful notion of realizing that cross-domain knowledge can share structure. Logic and geometry are in some sense related in a very formal way. I was never introduced to that notion in the required logic class.

nomoreplease 5y ago
> To be clear, I mean the incredibly useful notion of realizing that cross-domain knowledge can share structure. Is there somewhere you can suggest that I can read more about this notion?
- Koshkin 5y ago
  By "share structure" they mean (the various degrees of) analogy - a widely-known concept that has been made rigorous within the framework of category theory (via the notion of a functor).
- randcraw 5y ago
  Check out this section of the SEP entry on Category Theory: 3. Philosophical Significance https://plato.stanford.edu/entries/category-theory/ https://plato.stanford.edu/entries/category-theory/ I suspect a lot of the formal work on ontology involves CT.
platz 5y ago
Do you think there could be a risk telling the eager-eyed philosophy students that there are truths that they should grant on principle but that they don't need to understand the details? What do they think when they take the epistemology course next?
- nbulka 5y ago
  Yes the details are important; if you are so inclined to trace the reasoning all the way back to the axioms, thankfully they are very easy to understand! And doing so would definitely prepare them for epistemology. There's a reason that physics leans heavily on category theory.