Mapping the dualities in math

[Andrius Kulikauskas]

Hi everybody,

Every so often I've been studying math, listening to videos, trying to
understand the big picture.

Today I made a list of dualities in math:
http://www.ms.lt/sodas/Book/Duality

I sorted through the dualities listed in Wikipedia and nLab:
https://en.wikipedia.org/wiki/List_of_dualities
https://ncatlab.org/nlab/show/duality

Next, I will make a diagram to see what kinds of dualities are truly basic.

I suspect that this diagram will actually help to organize the various
branches of Math.  It also seems to be bringing up the crucial theorems
in Math.  So that's helping me realize what I should learn.

A recent idea that inspired me is that the foundation of logic is its
pure duality.  Namely, all of the true statements in logic
(equivalently, in set theory and Boolean algebras, thanks to Stone's
duality) are the same if you swap intersections and unions and also swap
negatives and positives.  What this means is that logic supremely
balances what is (say, things) and what is not (circumstances).  Or
perfectly balances the rational and irrational points of view, the
conscious and the unconscious, the unknown and the known, the left and
right hemispheres, "male" and "female" roles etc.  The point being that
such pairs of voices need to be balanced in order to have a free
choice.  And also that logic has no bias as regards the "content"
(positive or negative) but is focused on formal issues.

Areas of math seem to be slight deviations from such duality. For
example, in topology you define open sets which can be generated by
arbitrary unions but only finite intersections.

This summer I wrote a research proposal to look for the big picture in math:
http://www.ms.lt/derlius/AResearchProgramForABigPictureOfMathematics.pdf
I didn't win, though.

I've started to give talks in Europe on my philosophy.
http://www.ms.lt/sodas/Book/Book

I am grateful for any thoughts you may have.

Andrius

Andrius Kulikauskas
VGTU Lecturer