Mathematics is Representation

This page is a sub-page of our page on What is Mathematics?

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Related pages:

Numbers and their Digits in different Bases
• Vectors and their Coordinates in different Bases
Linear Transformations and their Matrices in different Bases
Musical Tunes and their Score in different Arrangements
Shift of Basis for a Representation

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Representation and Reconstruction of a Presentant with respect to a Background

Representation: \, [ \, p_{resentant} \, ]_{\, B_{ackground}} \, \equiv \, \left< \, r_{epresentant} \, \right>_{\, B_{ackground}}

Reconstruction: \, \left( \, \left< \, r_{epresentant} \, \right>_{B_{ackground}} \, \right)_{B_{ackground}} \mapsto \,\, p_{resentant}

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\, O_{bjects} \, \xrightarrow [ \; on \; the \; background \; B \; ] { \; have \; the \; representations \; } \, [ \, O_{bjects} \, ]_B \,

\, o_{bject} \, \shortmid \xrightarrow [ \; on \; the \; background \; B \; ] { \; has \; the \; representation \; } \, [ \, o_{bject} \, ]_B \,

\, R_{epresentations} \, \xrightarrow [ \; on \; the \; background \; B \; ] { \; are \; the \; representations \; } \, \left< \, R_{epresentations} \, \right>_B \,

\, r_{epresentation} \, \shortmid \xrightarrow [ \; on \; the \; background \; B \; ] { \; is \; the \; representation \; } \, \left< \, r_{epresentation} \, \right>_B \,

/////// Abbreviated notation:

\, O_{bjs} \, \xrightarrow [ \; on \; the \; background \; B \; ] { \; have \; the \; representations \; } \, [ \, O_{bjs} \, ]_B \,

\, o_{bj} \, \shortmid \xrightarrow [ \; on \; the \; background \; B \; ] { \; has \; the \; representation \; } \, [ \, o_{bj} \, ]_B \,

\, R_{eps} \, \xrightarrow [ \; on \; the \; background \; B \; ] { \; are \; the \; representations \; } \, \left< \, R_{eps} \, \right>_B \,

\, r_{ep} \, \shortmid \xrightarrow [ \; on \; the \; background \; B \; ] { \; is \; the \; representation \; } \, \left< \, r_{ep} \, \right>_B \,

/////// In Swedish:

Representation och Rekonstruktion av en Presentant med avseende på en Bakgrund

Representation: \, [ \, p_{resentant} \, ]_{B_{akgrund}} \, \mapsto \, \left< \, r_{epresentant} \, \right>_{B_{akgrund}}

Rekonstruktion: \, \left( \, \left< \, r_{epresentant} \, \right>_{B_{akgrund}} \, \right)_{B_{akgrund}} \mapsto \,\, p_{resentant}

/////// Back to English:

So, what is mathematics? Here is a representation of mathematics that takes the form of a category which in turn is represented as a (binary) hypergraph:

\, [ \, [ \, m_{athematics} \, ]_{C_{ategory}} \, ]_{H_{yperGraph}} \, \equiv \,

\, \equiv \, \left< \, {\mathbb 2}^{P_{ropositions}} \ni A \, \stackrel {e_{ntails}} {\longrightarrow} \, B \in {\mathbb 2}^{P_{ropositions}} \, \right>_{H_{yperGraph}}

With natural abbreviations of names, the same configuration can be expressed as:

\, [ \, [ \, m_{ath} \, ]_{C_{at}} \, ]_{H_{yper}} \, \equiv\, \left< \, {\mathbb 2}^{P_{rops}} \ni A \, \stackrel {e_{ntails}} {\longrightarrow} \, B \in {\mathbb 2}^{P_{rops}} \, \right>_{H_{yper}}

A category consists of objects and arrows, with two conditions on the arrows:
1) associativity of compositions, and
2) existence of an identity arrow for each object.

Let us specify \, P_{ropositions} to denote the set of expressions that humans have ever made in writing. Although this set is rather large, it is important to recognize that it is (and always will remain) finite.

In this representation, the category of mathematics, \, [ \, m_{ath} \, ]_{C_{at}} , consists of:

Objects: Subsets of \, P_{ropositions} , i.e., elements of the power set \, {\mathbb 2}^{P_{rops}} .

Arrows: Logical implications between some pairs of objects, i.e., between some pairs of subsets of propositions. If there is an arrow from an object \, A \, to an object \, B \, it means that the set of propositions \, B \, are a logical consequence of the set of propositions \, A .

This situation is normally denoted by \, A \implies B
and it is verbalized as “ A \, implies \, B ” or “ A \, entails \, B “.

A proof that A \, implies \, B \, is a sequence of propositions
that logically justifies this implication.

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\, [ \, C_{ommunism} \, L_{ost} \, ]_{A_{ristotelian} \, L_{ogic}} \, \equiv \, \left< \, C_{apitalism} \, W_{on} \, \right>_{A_{ristotelian} \, L_{ogic}}

\, [ \, P_{eircean} \, L_{ogic} \, ]_{T_{ernary} \, H_{yperGraph}} \, \equiv \, \left< \, P_{ragmatism} \, \right>_{T_{ernary} \, H_{yperGraph}}

Ternary operation
Trichotomy (philosophy)
• Charles Sanders Peirce
Categories (Peirce)
Existential Graphs, comments by John F. Sowa on a paper by C.S. Peirce
Conceptual Graphs
• An introduction to Conceptual Graphs, by John F. Sowa, 2008
Conceptual Graph Examples
• Pragmatism
Model Theory
• Type Theory

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