home.social

#isomorphism — Public Fediverse posts

Live and recent posts from across the Fediverse tagged #isomorphism, aggregated by home.social.

fetched live
  1. Propositions As Types Analogy • 1
    • inquiryintoinquiry.com/2013/01

    One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy. And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.

    Proof Hint ∶ Proof ∶ Proposition
    ∷
    Untyped Term ∶ Typed Term ∶ Type

    or

    Proof Hint ∶ Untyped Term
    ∷
    Proof ∶ Typed Term
    ∷
    Proposition ∶ Type

    See my working notes on the Propositions As Types Analogy —
    • oeis.org/wiki/Propositions_As_

    #Mathematics #CategoryTheory #ProofTheory #TypeTheory
    #Logic #Analogy #Isomorphism #PropositionalCalculus
    #CombinatorCalculus #CombinatoryLogic #LambdaCalculus
    #Peirce #LogicalGraphs #GraphTheory #RelationTheory

  2. What if universities no longer existed?

    This is a question I’ve been preoccupied by since the last US election. There’s a widespread recognition that ‘alternative media’ displaced ‘mainstream media’, quantitatively and qualitatively, in ways which had a substantial impact on the election. If the first phase of isomorphism through algorithms was homogenisation and hybridisation, the second phase will I suspect be a collapse of the distinction itself such that legacy media only survives to the extent it replicates the style and modus operandi of alt-media.

    What would a parallel process look like in higher education? I thought back to this question when reading Alexandra Mihai’s piece about whether universities are indispensable. She reflects on her reaction to the scenario of no universities, using that as a jumping off point to stimulate analysis of what universities need to do to survive:

    With knowledge becoming more accessible, universities need to come to terms with the fact they so no longer hold the monopoly. Indeed, they should shift the focus towards a more active approach of collaborative knowledge construction. The skills and competences necessary for processing, interpreting and communicating knowledge take centre stage and need to become a more explicit part of the curriculum. To stay relevant, we need to keep asking ourselves “what do our learners need in order to become active and responsible members of society?” and see where in this process we can really bring added value. You may have noticed I did not choose to ask “what do our students need in order to thrive on the labour market?”. This was an intentional choice, as I personally do not believe that universities should profile themselves as a conveyor belt for the labour market (as many of our students see us right now). On the contrary, I feel that job-related knowledge is perhaps one of the first areas that is and will be best tackled outside university. Where we can add value is in developing the right attitudes and competencies that go beyond factual disciplinary knowledge.

    I think we shouldn’t underestimate how robust the intersection between credentialisation and prestige could prove to be. But the unravelling of credentials could come very quickly, if there’s a widespread collapse of trust that outcomes of a degree have anything meaningful correlation to personal attributes. If this left only prestige then it could become a self-reinforcing loop, in which human-centric forms of education are restricted to a small number of elite institutions, retaining the perceived value of a degree from them.

    To go back to the analogy, I don’t think the New York Times or the Guardian are going to vanish from our media landscape. In fact they could become even more influential under these circumstances, even if the scope of that influence takes a different form in a radically changed media ecology. The same would be true of world elite institutions I suspect, even if there is a real and existential threat on the horizon to the majority of universities.

    #AlexandraMihai #DanaBoyd #futures #higherEducation #isomorphism #universities #university

  3. In the Way of Inquiry • Formal Apology 3
    • inquiryintoinquiry.com/2023/01

    Explosional Recombinations —

    Another obstacle to inquiry is posed by the combinatorial explosion of questions arising in complex cases. The embarrassment of riches found here is deceptively deadly to the ends of inquiry in the very measure it appears so productive at first. An eye to form provides a way to manage the wealth of material diversity by identifying formal similarities among materially distinct domains. It allows the same formal answer to unify a host of concrete questions under a single roof, overall reducing the number of distinct topics that need to be covered.

    Overview
    • oeis.org/wiki/Inquiry_Driven_S

    Obstacles
    • oeis.org/wiki/Inquiry_Driven_S

    #Peirce #Inquiry #InquiryIntoInquiry #InquiryDrivenSystems
    #Semiotics #SignRelations #Semiositis #ObstaclesToInquiry
    #Logic #Abduction #Deduction #Induction #ScientificMethod
    #Abstraction #Form #Isomorphism #Combinatorics #Complexity

  4. In the Way of Inquiry • Formal Apology 7
    • inquiryintoinquiry.com/2023/01

    Explosional Recombinations

    An eye to form provides a way to manage the wealth of material diversity by identifying formal similarities among materially distinct domains. It allows the same formal answer to unify a host of concrete questions under a single roof, reducing the variety of topics requiring coverage.

    #Peirce #Inquiry #InquiryIntoInquiry #InquiryDrivenSystems
    #Abstraction #Form #Isomorphism #Combinatorics #Complexity

  5. In the Way of Inquiry • Formal Apology 6
    • inquiryintoinquiry.com/2023/01

    Explosional Recombinations

    Another obstacle to inquiry is posed by the combinatorial explosion of questions arising in complex cases. The embarrassment of riches found here is deceptively deadly to the ends of inquiry in the very measure it appears so productive at first.

    #Peirce #Inquiry #InquiryIntoInquiry #InquiryDrivenSystems
    #Abstraction #Form #Isomorphism #Combinatorics #Complexity

  6. This makes me think a little about #isomorphism in other areas of mathematics. I feel like a lot of folks don't think too much about what the isomorphisms look like.

    "All finite fields of the same size are isomorphic, QED", most people would say, and move on to the next question. "All vector spaces of the same dimension are isomorphic, QED", they continue.

    Yes, but depending on how you represent your field or vector space, it will look very different!

Share on Mastodon

Enter the server where you have an account.