(High School student here) In physics there is the wave-particle duality among others, but in mathematics what are some examples and concepts of duality?

For example in Terence Tao's Analysis 1 he talked briefly about the duality in De Morgan Laws.

I will appreciate any advanced explanation even if i don't fully understand it. Thanks 😊

I am a Bachelor's student studying Physics and Math ,My knowledge of Mathematics is upto the level of Multivariable Calculus(Yet to complete) ,I know beginner level linear algebra (From matrices till Eigen vectors and values ,no abstract linear spaces) ,I know the basics of solving Differential Equations ,and a bit of stuff in various topics ,I have tried to learn all these with either some kind of Physical (or) Geometric Intuition ,I really love searching the why's and how's of a concept ,the satisfaction that I get from understanding these concepts that feeling of finally clicking as to why we do the math ,is ... I don't have words to describe it.

Now as I go on to Higher level math ,is it possible to get a physical and Geometric meaning or some kind of intuition for advanced topics ?

Will I be able to continue to get that level of satisfaction ,if I proceed with math?

What field of math ,is kind of rich with physical intuition or geometric interpretations ?

I guess Complex Analysis is great(I have no knowledge of Real Analysis ),I came to this conclusion by skimming through Tristan Needham's Visual Complex Analysis .Pls forgive me if I am wrong .

Any other fields ?

Alight, I usually do not bother with starting serious discussion topics myself as I usually find it pretty srressful to do so and would much rather stick to posting memes, but... I feel like I had to get this out. So here it is.

This was originally going to be about an analysis on how Shinji ended up being hated on so much compared to all the other antagonists and what could have been done to improve his reputation, but... That would be way too long of a post, and most of my thoughts that I've gathered up ended up becoming comparisons between these two characters instead.

DISCLAIMER: This has nothing to do with trying to influence the current holy elimination war game that's been going on. I've been working on this for a long time way before it ever began. I personally have not even touched the polls and still do not intend anytime soon. In fact, I was thinking of waiting until the game was over before releasing this in order to see how it pans out first-but not only am I not a patient guy, but have to go for an eye surgery soon plus dealing with severe keratocronus so I wanna just finish this and get this out of the way as soon as possible.

This is mostly going to explore a lot of their antagonistic side as the duality theme between them as the white and black grails mostly comes from that, unfortunately. Also, this takes a lot from the VN, so if you're going into this from anime-only, a lot of this probably won't make sense to you.

Sakura and Illya are both two of the most polarizing characters within Fate/stay night-a characteristic that likely stems from them being absolutely terrifying as antagonists yet very wholesome when they're not. For now, let's just jot down the basic characteristics that makes them foils to each other on a surface level.

Both Illya and Sakura has been messed with and forced against their wills for something neither of them wished for. Neither of them had a choice to make and had their fates imposed upon them by someone they refer to as their "grandfather" and forced to become lesser grails. Sakura on one hand was messed with for the purpose of longevity (of Zouken that is), while Illya was for intents and purposes messed with to die. One ends up being a foster elder sister (Illya) while the other ends being an adopted younger sister (Sakura). Illya is the white grail while Sakura is the black grail. Illya makes her debut as a major antagonist at ethe beginning of F/SN as a whole while Sakura takes the stage as the major a

I'm either paralyzed by indecision (overthinking) or I'm being too impulsive (underthinking).

I'm either talking too much ("just get to the point" "let someone else talk" "you're being annoying") or not enough ("why are you so quiet?" "is something wrong?" "you're being weird").

I'm distractable and obsessive, scatter-brained and a one-track mind, understimulated and overwhelmed.

I can't maintain a routine and I'm terrified of change.

I'm a perfectionist who can't do anything right.

I'm not enough and I'm too much to handle.

0. Preface — Consciousness is the One and only Reality.

>"As above, so below; as below, so above; as within, so without; as without, so within", "Correspondence", the second of The Seven Principles of Hermes Trismegistus

The system of philosophy based on the teachings of Hermes Trismegistus became known as Hermeticism; these ideas became prominent during the Renaissance & Reformation periods of history. Once it was realized that the Egyptian god Thoth & the Greek god Hermes were the same being, he then began being referred to as Hermes Trismegistus. The written works of Hermeticism were created over many centuries, from 300 B.C to 1200 A.D.

In Hermetic philosophy there is the microcosm, the word of the individual, and the macrocosm, which is the universe, or the sum total of all that is. The word macrocosm comes from Greek makros kosmos, “the great world”; the universe as a whole. Microcosm comes from Greek mikros kosmos, “the small world”; the human being (understood as a miniature universe). In other words, the phrase can be interpreted to say that “Above” is referring to nature and the cosmos, and the “Below” part refers to what happens inside of us as humans, or our personal subjective experience of being alive. The microcosm – macrocosm relationship is wonderfully described in the following quote:

>“To truly know the world, look deeply within your own being, To truly know yourself, take real interest in the world.” ~ Rudolf Steiner

This relational dynamic can be seen anywhere there is duality, where one half is above, and the other below, and there is an established relationship between the two. This is the same duality that the very physicists I have discussed previously in this post and in my previous discussions describe (Discussion 1, Discussion 2, & Discussion 3), which explains how our classical world (in physics) can stem from a subatomic quantum world.

See—we mustn’t scrutinize everything as Einstein did with this partic