#universeofdiscourse — Public Fediverse posts

New blog post: “There are two kinds of theorems”.

Mathematics alternates between model-building and model-using. After the model building comes a series of theorems showing that the model works and makes sense. And then come the real theorems, the new ones about the thing that was being modeled.

This is perhaps the most important mathematical methodology, and it is never explained to the students, who are left wondering why Euclid proves a lot of theorems about things that are obvious (“vertical angles are equal”) or why we show that the Peano axioms can prove the commutativity of addition.

https://blog.plover.com/math/two-kinds-of-theorems.html

#math #pedagogy #mathEducation #geometry #universeOfDiscourse

From the archives: “Decomposing a function into its even and odd parts” (July 2016).

Every continuous function f can be decomposed as \[ f(x) = f_e(x) + f_o(x) \] where \( f_e \) and \( f_o \) are even and odd functions, respectively. But what do those even and odd component functions actually look like? Often, it's interesting and sometimes rather surprising.

https://blog.plover.com/math/even-odd.html

#math #universeOfDiscourse

New blog post “‘The road to ε₀: Coin-moving games with no coins”

I continue to approach ε₀ in small stages. We take the previous model of \( ω^ω \), an infinite-dimensional array of cells, and reinterpret it as a much simpler model about finite sequences of numbers.

I also get to write the expression \[ ω^{ω^{ω^{ω^⋰}}}. \]

https://blog.plover.com/math/ordinals/04-coordinates.html

#math #blog #setTheory #countableOrdinals #universeOfDiscourse

New blog post “Starting to understand epsilon-zero”.

A long time ago I said that \( ω^ω \) was the place where the countable ordinals start to get scary, and then about 18 months ago I realized I could think about it in a different way that made it not scary at all.

Then I realized that by taking that insight a little farther I could really get my head around \( \epsilon_0 \) for the first time.

This isn't either of those articles, it's an introduction that explains infinite ordinals and introduces what \( ω^ω \) and \( \epsilon_0 \) are, so that I can move on to those other two articles without explaining everything from first principles.

https://blog.plover.com/math/epsilon-zero.html

#math #setTheory #ordinals #infinity #universeOfDiscourse

Logic Syllabus • 3
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Concepts
• https://oeis.org/wiki/Logic_Syllabus#Logical_concepts

Ampheck • https://oeis.org/wiki/Ampheck
Boolean Domain • https://oeis.org/wiki/Boolean_domain
Boolean Function • https://oeis.org/wiki/Boolean_function
Boolean-Valued Function • https://oeis.org/wiki/Boolean-valued_function
Differential Logic • https://oeis.org/wiki/Differential_logic
Logical Graph • https://oeis.org/wiki/Logical_Graphs
Minimal Negation Operator • https://oeis.org/wiki/Minimal_negation_operator
Multigrade Operator • https://oeis.org/wiki/Multigrade_operator
Parametric Operator • https://oeis.org/wiki/Parametric_operator
Peirce's Law • https://oeis.org/wiki/Peirce%27s_law
Propositional Calculus • https://oeis.org/wiki/Propositional_calculus
Sole Sufficient Operator • https://oeis.org/wiki/Sole_sufficient_operator
Truth Table • https://oeis.org/wiki/Truth_table
Universe of Discourse • https://oeis.org/wiki/Universe_of_discourse
Zeroth Order Logic • https://oeis.org/wiki/Zeroth_order_logic

#Logic #LogicSyllabus #Ampheck #BooleanDomain #BooleanFunction #BooleanValuedFunction
#DifferentialLogic #LogicalGraph #MinimalNegationOperator #MultigradeOperator
#ParametricOperator #PeircesLaw #PropositionalCalculus #SoleSufficientOperator
#TruthTable #UniverseOfDiscourse #ZerothOrderLogic

#LogicSyllabus
• https://inquiryintoinquiry.com/logic-syllabus/

This page serves as a focal node for a collection of related resources.

#Logic #LogicalOperators

#LogicalConjunction #LogicalDisjunction
#ExclusiveDisjunction #LogicalEquality
#LogicalImplication #LogicalNegation
#LogicalNAND #LogicalNNOR

#LogicalConcepts

#Ampheck #SoleSufficientOperator
#BooleanDomain #UniverseOfDiscourse
#LogicalGraph #PropositionalCalculus
#MinimalNegationOperator #TruthTable
#BooleanFunction #BooleanValuedFunction

#DifferentialPropositionalCalculus • 7.2
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

In a #UniverseOfDiscourse based on 3 #BooleanVariables \(p,q,r\) there are \(2^3=8\) #SingularPropositions. Their #VennDiagrams are shown in Figure 10.

\(\text{Figure 10. Singular Propositions} : \mathbb{B}^3 \to\mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-singular-propositions.jpg

Related Subjects —
#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#CactusSyntax #MinimalNegationOperators
#DiscreteDynamics #QualitativeDynamics

#DifferentialPropositionalCalculus • 7
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

In our #Model of #Propositions as #Mappings of a #UniverseOfDiscourse to a set of 2 values, in other words, #IndicatorFunctions of the form \(f:X\to\mathbb{B},\) #SingularPropositions are those singling out the #MinimalDistinctRegions of the universe, represented by single cells of the corresponding #VennDiagram.

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#ModelTheory #ProofTheory #Semiotics

#DifferentialPropositionalCalculus • 6.1
• https://inquiryintoinquiry.com/2020/03/02/differential-propositional-calculus-6/

In a #UniverseOfDiscourse based on 3 #BooleanVariables \(p, q, r\) there are \(2^3 = 8\) #PositivePropositions. Their #VennDiagrams are shown in Figure 9.

\(\text{Figure 9. Positive Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-positive-propositions.jpg

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#CactusSyntax #MinimalNegationOperators
#DiscreteDynamics #QualitativeDynamics

#DifferentialPropositionalCalculus • 5.2
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/

In a #UniverseOfDiscourse based on three #BooleanVariables \(p, q, r\) the #LinearPropositions take the shapes of the #VennDiagrams shown in Figure 8. Equivalent verbal & variant logical expressions are given in the next few posts.

\(\text{Figure 8. Linear Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/02/venn-diagrams-e280a2-p-q-r-e280a2-linear-propositions.jpg

Related Subjects —
#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions

#DifferentialPropositionalCalculus • 1.5
https://inquiryintoinquiry.com/2020/02/18/differential-propositional-calculus-1/

Figure 3. Back, To The Future
https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-figure-3.jpg

Figure 3 preserves the initial #UniverseOfDiscourse \(X\) and extends the basis of discussion to a set of two qualities \(\{q, \mathrm{d}q\}.\) In corresponding fashion, the initial #PropositionalCalculus is extended by means of the #EnlargedAlphabet \(\{``𝑞", ``d𝑞"\}.\)

#DifferentialPropositionalCalculus • 1.3
https://inquiryintoinquiry.com/2020/02/18/differential-propositional-calculus-1/

Figure 1 represents a #UniverseOfDiscourse \(X\) together with a basis of discussion \(\{q\}\) for expressing propositions about the contents of that universe. Once the quality \(q\) is given a name, say, the symbol \(``q",\) we have the basis for a #FormalLanguage specifically cut out for discussing \(X\) in terms of \(q.\) This language is more formally known as the #PropositionalCalculus with #Alphabet \(\{``q"\}.\)

#DifferentialPropositionalCalculus • Overview 🧵
• https://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-overview/

By way of introduction to #DifferentialLogic, here begins a chapter on #DifferentialPropositionalCalculi, which augment #PropositionalCalculi with terms for describing aspects of change and difference, for example, processes taking place in a #UniverseOfDiscourse or #LogicalTransformations mapping a #SourceUniverse to a #TargetUniverse.

#Logic #LogicalGraphs #DifferentialLogic
#DiscreteDynamics #QualitativeDynamics

#DifferentialPropositionalCalculus • #Overview

• https://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-overview/

By way of introduction to #DifferentialLogic, here begins a chapter on #DifferentialPropositionalCalculi, which augment #PropositionalCalculi with terms for describing aspects of change and difference, for example, processes taking place in a #UniverseOfDiscourse or #LogicalTransformations mapping a #SourceUniverse to a #TargetUniverse.

Applications —
#Cybernetics #SystemsTheory
#QualitativeDynamics
#DiscreteDynamics