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Book: Duality

See: Math Notebook, Category duality


Investigation: Understand mathematics as the discrimination of a variety of dualities.

See: John Baez: Duality in Logic and Physics

I am studying the various cases of duality in math. I imagine that at the heart is the duality between zero and infinity by way of one as in God's Dance. Duality is the basis for logic, and mathematics gives the ways of deviating from duality. Duality is also the structural mirror established within the foursome, fivesome, sixsome and sevensome.

Logic: Duality

Duality arises from a symmetry between two ways of looking at something where there is no reason to choose one over the other. This is driven by the sevensome in defining logic as the balancing of the conscious (not known) and the unconscious (known), what is and what is not (but complements it).

Math is subtle deviations from pure duality

These subtle deviations seem to leverage infinity.

Logic in Mathematics: The Kinds of Duality

Sources of examples of duality


The fundamental anti-duality (Schur-Weyl duality) between external representations and internal structure

Anti-duality: Internal structure and external relationships

Dualities in the symmetric functions: Elementary and homogeneous; Schur and power; monomial and forgotten?

Anti-Duality: Symmetry and Structure

Duality: Translating structures

Internal, Implicit Dualities

Duality: Conjugation

Duality: Halving space: Rotation: Reversing orientation

Duality: Reflection

Duality: Reversing order of operations

Duality: Bottom-up and Top-down

Duality: Complements

Duality: Complements: Plane duality

Duality: Existing and nonexisting

Intersections and Unions

Generated by complements

External, Explicit Dualities

Duality: Functionals

Duality: Actions and Objects

Duality: Adjunction

Duality: Reversing the maps

Duality: Reversing the ordering


Mathematics: Almost Duality - Duality Breaking

The duality between zero and infinity, between nothing and everything, is broken in many subtle ways. Here are some examples:

Dual values

Mathematical Tension: Equivalence and Uniqueness

In Math, there is an everpresent tension between the notions of equivalence class and uniqueness. If something is mathematically significant, it should in some sense be unique. But math is a model and so, as such, can never be entirely unique but represents a variety of cases. Thus it is ever natural to define equivalence classes, especially in math itself. For example, a rational number is an equivalence class that establishes a proportion.

Ways of figuring things out

Duality in Physics

Duality in Mathematics and Physics by Sir Michael Atiyah





The transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces

Duality breaking

Note that turns left and right are conjugates but not a division of everything because there can be no turn. Instead, the twosome is "turn" (left/right) and "no-turn". "Turn" does not need to be marked, but "no-turn" needs to be marked ("no"). Although, content-wise, not-turning is unmarked and turning is marked.

This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry. Wikipedia: List of simple Lie groups: Symmetric spaces

Tannaka duality

Cohomology at nLab: Cohomology is dual to homotopy (as an operation): the cohomology of X with coefficients in A is the homotopy of A with co-coefficients in X. Cohomology should be thought of as "cohomotopy". For any {$A\in H$} the set {$H(S^n,A)$} is equivalently:

Examples in Math Companion:

Another key concept for me is the idea of an "unmarked opposite" vs. a "marked opposite".

So I'm very interested where such dualities and opposites come up in math.

Unmarked opposite




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