I'm exploring a possibility to define an #isomorphism between #FunctionalProgramming and #ObjectOrientedProgramming #OOP based on prev. defined dimensions of simplicity. We've possibly "established" and isolated an inverse relation between deviating memory behavior in FP and OOP based on calling in-place updates an optimization that can be undone for base-case.

Grok is now calling FP "historical" programming and OOP "present-oriented".

I'm liking this AI. 😋

Propositions As Types Analogy • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-analogy-1/

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 —
• https://oeis.org/wiki/Propositions_As_Types_Analogy

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

#GeneralSystemsScience #Isomorphism: The same #Structure in different systems, eg #NegativeFeedback loops in fetal organ development, policy development, environment cycles #EnergyNetworkScience #SharedConcepts

SOFT AXIOM a surreal prose-poem from ATHENALOGIA by C. A. Corbell

#poetry #poetrycommunity #prosepoem #fossils #sundial #isomorphism #starward #pnwpoet #poetryvideo #video #poetryreading #newpoem #poetrybook #surrealism #surrealpoetry

https://www.youtube.com/shorts/Ho70brmsuLM?si=HJjPTKJ-R8nTZTA5

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

In the Way of Inquiry • Formal Apology 3
• https://inquiryintoinquiry.com/2023/01/12/in-the-way-of-inquiry-formal-apology-a/

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
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Obstacles
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_5#Obstacles

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

Isomorphism and Legal Knowledge Based Systems
(1992) : Bench-Capon, T J M Coenen, F P
DOI: https://doi.org/10.1007/BF00118479
#rule #isomorphism #maintenance #logic #KBS #legal_knowledge_based_systems #my_bibtex

In the Way of Inquiry • Formal Apology 7
• https://inquiryintoinquiry.com/2023/01/12/in-the-way-of-inquiry-formal-apology-2/

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

In the Way of Inquiry • Formal Apology 6
• https://inquiryintoinquiry.com/2023/01/12/in-the-way-of-inquiry-formal-apology-2/

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

#LogicalGraphs • 5
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Abstract_POV

#AbstractPointOfView —

We may note in passing historical details like the fact Charles Sanders #Peirce used a #StreamerCross symbol where George #SpencerBrown used a #CarpentersSquare marker but the themes of primary interest at the abstract level of form are indifferent to variations of that order.

#Logic #PropositionalCalculus #BooleanFunctions
#Form #Idea #Isomorphism #MathematicalPerspective
#LawsOfForm #GraphTheory #ModelTheory #ProofTheory

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!

- the functor T is represented by (S, {1}) where S is the #Sierpiński space. That is, T is naturally isomorphic to the Hom #functor Hom(–, S) with the natural #isomorphism determined by the universal element {1} ∈ T(S). This is generalized by the notion of a #presheaf

infinite structures can never be discriminated in FO: Löwenheim– #Skolem theorem, => no fo theory w infinite model can have a unique model up to #isomorphism.

The most famous example is probably Skolem's theorem, that there is a countable non-standard model of arithmetic.

#Fourier transform is a *-#isomorphism from C^{*}{{R} (1) to C_{0}({R}(2) C^{*}-algebra of continuous C Z -> 0 on R -> \inf. So 1:1-> bw strongly C 1-param unitary groups and *-repr R of (1) As every *-R of (2)
-> uniquely to a self-adjoint operator, Stone's Theorem holds.

#Hodge considered new proof much superior, a serious flaw was discovered by Bohnenblust. Independently, #Weyl and Kunihiko Kodaira modified Hodge's proof to repair error. This established Hodge's sought-for #isomorphism between harmonic forms and #cohomology classes.

#Hodge considered new proof much superior, a serious flaw was discovered by Bohnenblust. Independently, #Weyl and Kunihiko Kodaira modified Hodge's proof to repair error. This established Hodge's sought-for #isomorphism between harmonic forms and #cohomology classes.

a #morphism between smooth varieties is #étale at a point iff the differential between the corresponding tangent spaces is an #isomorphism. This is in turn precisely the condition needed to ensure that a map between manifolds is a local #diffeomorphism

a #morphism between smooth varieties is #étale at a point iff the differential between the corresponding tangent spaces is an #isomorphism. This is in turn precisely the condition needed to ensure that a map between manifolds is a local #diffeomorphism

Up to #isomorphism, the octonions and the split-octonions are the only two 8-dimensional composition algebras over the real numbers