#isomorphism — Public Fediverse posts

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

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

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

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

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

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

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

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

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

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

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

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

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 (Math)

https://en.wikipedia.org/wiki/Isomorphism

#Math #Isomorphism #Articles

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

#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

#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

#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!

a #book for the week?

https://en.wikibooks.org/wiki/Haskell/The_Curry–Howard_isomorphism Haskell/

The Curry–Howard #isomorphism