#analogy — 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

Mechanical Logic Gates With Amplification https://hackaday.com/2024/09/20/mechanical-logic-gates-with-amplification/ #digitallogic #electronics #leverlogic #mechanical #amplifier #logicgate #teaching #Science #analogy #capstan #analog #pulley

Mechanical Logic Gates With Amplification - One of the hardest things about studying electricity, and by extension electronics... - https://hackaday.com/2024/09/20/mechanical-logic-gates-with-amplification/ #digitallogic #electronics #leverlogic #mechanical #amplifier #logicgate #teaching #science #analogy #capstan #analog #pulley

In the Way of Inquiry • Reconciling Accounts
• https://inquiryintoinquiry.com/2023/01/24/in-the-way-of-inquiry-reconciling-accounts-a/

The Reader may share with the Author a feeling of discontent at this point, attempting to reconcile the formal intentions of this inquiry with the cardinal contentions of experience. Let me try to express the difficulty in the form of a question:

What is the bond between form and content in experience, between the abstract formal categories and the concrete material contents residing in experience?

Once toward the end of my undergrad years a professor asked me how I'd personally define mathematics and I told him I saw it as “the form of experience and the experience of form”. This is not the place to argue for the virtues of that formulation but it does afford me one of the handles I have on the bond between form and content in experience.

I have no more than a tentative way of approaching the question. I take there to be a primitive category of “form‑in‑experience” — I don’t have a handy name for it yet but it looks to have a flexible nature which from the standpoint of a given agent easily passes from the “structure of experience” to the “experience of structure”.

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
#Experience #Expectation #EffectiveDescription #FiniteMeans
#Abstraction #Analogy #Form #Matter #Empiricism #Rationalism
#Concretion #Information #Comprehension #Extension #Intension

In the Way of Inquiry • Material Exigency 2
• https://inquiryintoinquiry.com/2023/01/20/in-the-way-of-inquiry-material-exigency-a/

A turn of events so persistent must have a cause, a force of reason to explain the dynamics of its recurring moment in the history of ideas. The nub of it's not born on the sleeve of its first and last stages, where the initial explosion and the final collapse march along their stubborn course in lockstep fashion, but is embodied more naturally in the middle of the above narrative.

Experience exposes and explodes expectations. How can experiences impact expectations unless the two types of entities are both reflected in one medium, for instance and perhaps without loss of generality, in the form of representation constituting the domain of signs?

However complex its world may be, internal or external to itself or on the boundaries of its being, a finite creature's description of it rests in a finite number of finite terms or a finite sketch of finite lines. Finite terms and lines are signs. What they indicate need not be finite but what they are, must be.

Fragments —

The common sensorium.

The common sense and the senses of “common”.

This is the point where the empirical and the rational meet.

I describe as “empirical” any method which exposes theoretical descriptions of an object to further experience with that object.

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
#Experience #Expectation #EffectiveDescription #FiniteMeans
#Abstraction #Analogy #Form #Matter #Empiricism #Rationalism

In the Way of Inquiry • Material Exigency 1
• https://inquiryintoinquiry.com/2023/01/20/in-the-way-of-inquiry-material-exigency-a/

Our survey of obstacles to inquiry has dealt at length with blocks arising from its formal aspects. On the other hand, I have cast this project as an empirical inquiry, proposing to represent experimental hypotheses in the form of computer programs. At the heart of that empirical attitude is a feeling all formal theories should arise from and bear on experience.

Every season of growth in empirical knowledge begins with a rush to the sources of experience. Every fresh‑thinking reed of intellect is raised to pipe up and chime in with the still‑viable canons of inquiry in one glorious paean to the personal encounter with natural experience.

But real progress in the community of inquiry depends on observers being able to orient themselves to objects of common experience — the uncontrolled exaltation of individual phenomenologies leads as a rule to the disappointment and disillusionment which befalls the lot of unshared enthusiasms and fragmented impressions.

Look again at the end of the season and see it faltering to a close, with every novice scribe rapped on the knuckles for departing from that uninspired identification with impersonal authority which expresses itself in third‑person passive accounts of one's own experience.

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 #Analogy #Form #Matter #Empiricism #Rationalism

In the Way of Inquiry • Formal Apology 4
• https://inquiryintoinquiry.com/2023/01/12/in-the-way-of-inquiry-formal-apology-a/

Interpretive Frameworks —

Iterations of the recombinatorial process generate alternative hierarchies of categories for controlling the explosion of parts in the domain under inquiry. If by some piece of luck an alternative framework is uniquely suited to the natural ontology of the domain in question, it becomes advisable to reorganize the inquiry along the lines of the new topic headings.

But a complex domain seldom falls out that neatly. The new interpretive framework will not preserve all the information in the object domain but typically capture only another aspect of it. To take the maximal advantage of all the different frameworks that might be devised it is best to quit depending on any one of them exclusively. Thus, a rigid reliance on a single hierarchy to define the ontology of a given domain passes over into a flexible application of interpretive frameworks to make contact with particular aspects of one's object domain.

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 #Analogy #Form #Matter #Paradigms #Pragmatics
#Aristotle #Categories #Complexity #InterpretiveFrameworks

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

Conceptual Extensions —

The second use of the formal apology is to permit the tentative extension of concepts to novel areas, giving them experimental trial beyond the cases and domains where their use is already established in the precedents of accustomed habit and successful application.

This works to dispel the “in principle” objection that any category distinction puts a prior constraint on the recognition of similar structure between materially dissimilar domains. It leaves the issue a matter to be settled by after the fact judgment, a matter of what fits best “in practice”.

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 #Analogy #Form #Matter #Paradigms #Pragmatics

In the Way of Inquiry • Formal Apology 1
• https://inquiryintoinquiry.com/2023/01/12/in-the-way-of-inquiry-formal-apology-a/

Using “form” in the sense of abstract structure, the focus of my interest in this investigation is limited to the formal properties of the inquiry process. Among its chief constituents are numbered all the thinking and unthinking processes supporting the ability to learn and to reason. This “formal apology”, the apologetics of declaring a decidedly formal intent, will be used on numerous occasions to beg off a host of material difficulties and thus avoid the perceived necessity of meeting a multitude of conventional controversies.

Category Double‑Takes —

The first use of the formal apology is to rehabilitate certain classes of associations between concepts otherwise marked as category mistakes. The conversion is achieved by flipping from one side of the concept’s dual aspect to the other as the context demands. Thus it is possible in selected cases to reform the characters of category mistakes in the manner of categorical “retakes” or “double‑takes”.

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 #Analogy #Form #Matter #Aristotle #Categories

#Semiositis • 1
• https://inquiryintoinquiry.com/2015/07/10/semiositis-1/

There is a deep and pervasive #Analogy between systems of #Commerce and systems of #Communication, turning on their near-universal use of #Symbola (images, media, proxies, signs, symbols, tokens, etc.) to stand for #Pragmata (objects, objective values, the things we really care about, or would really care about if we examined our values in practice thoroughly enough).

#Peirce #Pragmatics #Semiotics #Semeiotics #SignRelations #Triadicity