The proper precise definition of mathematics can be found at: Section "Formalization of mathematics".

The most beautiful things in mathematics are described at: the beauty of mathematics.

Ciro Santilli intends to move his beauty list here little by little: github.com/cirosantilli/mathematics/blob/master/beauty.md

The most beautiful things in mathematics are results that are:

- simple to state and understand (K-12, lower undergrad), but extremely hard to prove, e.g.
- Fermat's Last Theorem
- transcendental number conjectures, e.g. is $e+π$ transcendental?

- surprising results: we had intuitive reasons to believe something as possible or not, but a theorem shatters that conviction and brings us on our knees, sometimes via pathological counter-examples. General surprise themes include:
- classification of potentially infinite sets like: compact manifolds, etc.
- problems that are more complicated in low dimensions than high like:
- generalized Poincaré conjectures. It is also fun to see how in many cases complexity peaks out at 4 dimensions.
- classification of regular polytopes

- unpredictable magic constants:
- why is the lowest dimension for an exotic sphere 7?
- why is 4 the largest degree of an equation with explicit solution? Abel-Ruffini theorem

- applications: make life easier and/or modeling some phenomena well, e.g. in physics. See also: explain how to make money with the lesson

Good lists of such problems Lists of mathematical problems.

Specific examples:

- from computer science:
- the existence of undecidable problems, especially simple to state ones, e.g. mortal matrix problem

Whenever Ciro Santilli learns a bit of mathematics, he always wonders to himself:

Am I achieving insight, or am I just memorizing definitions?Unfortunately, due to how man books are written, it is not really possible to reach insight without first doing a bit of memorization. The better the book, the more insight is spread out, and less you have to learn before reaching each insight.

Randomly reproduced at: web.archive.org/web/20080105074243/http://personal.stevens.edu/~nkahl/Top100Theorems.html

In mathematics, a "classification" means making a list of all possible objects of a given type.

Classification results are some of Ciro Santilli's favorite: Section "The beauty of mathematics".

Examples:

- classification of finite simple groups
- classification of regular polytopes
- classification of closed surfaces, and more generalized generalized Poincaré conjectures
- classification of associative real division algebras
- classification of finite fields
- classification of simple Lie groups
- classification of the wallpaper groups and the space groups

Oh, and the dude who created the en.wikipedia.org/wiki/Exceptional_object Wikipedia page won an Oscar: www.youtube.com/watch?v=oF_FLN-TmCY, Dan Piponi, aka

`@sigfpe`

. Cool dude.List:

Good place to hunt for the beauty of mathematics.

He's a bit overly obsessed with polynomials for the taste of modern maths, but it's still fun.

Ciro Santilli would like to fully understand the statements and motivations of each the problems!

Easy to understand the motivation:

- Navier-Stokes existence and smoothness is basically the only problem that is really easy to understand the statement and motivation :-)
- p versus NP problem

Hard to understand the motivation!

- Riemann hypothesis: a bunch of results on prime numbers, and therefore possible applications to cryptographyOf course, everything of interest has already been proved conditionally on it, and the likely "true" result will in itself not have any immediate applications.As is often the case, the only usefulness would be possible new ideas from the proof technique, and people being more willing to prove stuff based on it without the risk of the hypothesis being false.
- Yang-Mills existence and mass gap: this one has to do with findind/proving the existence of a more decent formalization of quantum field theory that does not resort to tricks like perturbation theory and effective field theory with a random cutoff valueThis is important because the best theory of light and electrons (and therefore chemistry and material science) that we have today, quantum electrodynamics, is a quantum field theory.

www.youtube.com/watch?v=R9FKN9MIHlE&t=938s Birch and Swinnerton-Dyer Conjecture (Millennium Prize Problem!) by Kinertia (2020)

The summary from www.geeksforgeeks.org/tree-traversals-inorder-preorder-and-postorder/ is a winner:

```
1
/ \
2 3
/ \
4 5
```

- inorder DFS: 4 2 5 1 3
- preorder DFS: 1 2 4 5 3
- postorder DFS: 4 5 2 3 1
- breadth-first search: 1 2 3 4 5

In principle one could talk about tree traversal of unordered trees as a number of possible traversals without a fixed order. But we won't consider that under this section, only deterministic ordered tree traversals.

This is the order in which you would want to transverse

Like breadth-first search, this also has the property of visiting parents before any children.

This is the easiest one to do iteratively:

- pop and visit
- push right to stack
- push left to stack

This is the order in which a binary search tree should be traversed for ordered output, i.e.:

- everything to the left is smaller than parent
- everything to the right is larger than parent

This ordering makes sense for binary trees and not k-ary trees in general because if there are more than two nodes it is not clear what the top node should go in the middle of.

This is unlike pre-order depth-first search and post-order depth-first search which generalize obviously to general trees.

This is a bit harder than iterative pre-order: now we have to check if there is a left or right element or not:

- (START) push current
- if there is left:
- move left

- else:
- (ELSE) pop
- visit
- if there is right
- move right
- GOTO START

- else:
- GOTO ELSE

The control flow can be slightly simplified if we allow NULLs: www.geeksforgeeks.org/inorder-tree-traversal-without-recursion/

Has the property of visiting all descendants before the parent.

This is the hardest one to do iteratively.

Nice result on Lebesgue measurable required for unicity.

As mentioned at Human compatible by Stuart J. Russell (2019), game theory can be seen as the part of artificial intelligence that deas with scenarios where multiple intelligent agents are involved.

The best example to look at first is the penalty kick left right Nash equilibrium.

Then, a much more interesting example is choosing a deck of a TCG competition: Magic: The Gathering meta-based deck choice is a bimatrix game, which is the exact same, but each player has N choices rather than 2.

The next case that should be analyzed is the prisoner's dilemma.

The key idea is that:

- imagine that the game will be played many times between two players
- if one player always chooses one deck, the other player will adapt by choosing the anti-deck
- therefore, the best strategy for both players, is to pick decks randomly, each with a certain probability. This type of probabilistic approach is called a mixed strategy
- if any player deviates from the equilibrium probability, then the other player can add more of the anti-deck to the deck that the other player deviated, and gain an edgeTherefore, using equilibrium probabilities is the optimal way to play

When taking a penalty kick in soccer, the kicker must chose left or right.

And before he kicks, the goalkeeper must also decide left or right, because there is no time to see where the ball is going.

Because the kicker is right footed however, he kicker kicks better to one side than the other. So we have four probabilities:

- goal kick left keeper jumps left
- goal kick right keeper jumps right
- goal kick left keeper jumps right. Note that it is possible that this won't be a goal, even though the keeper is nowhere near the ball, as the ball might just miss the goal by a bit.
- kick right and keeper jumps left. Analogous to above

Related ideas:

Poet, scientists and warriors all in one? Conquerors of the useless.

A wise teacher from University of São Paulo once told the class Ciro Santilli attended an anecdote about his life:

It turned out that, about 10 years later, Ciro ended up following this advice, unwittingly.I used to want to learn Mathematics.But it was very hard.So in the end, I became an engineer, and found an engineering solution to the problem, and married a Mathematician instead.

Ciro once read that there are two types of mathematicians/scientists (he thinks it was comparing Einstein to some Jack of all trades polymath who didn't do any new discoveries):

- high flying birds, who know a bit of everything, feel the beauty of each field, but never dig deep in any of them
- gophers, who dig all the way down, on a single subject, until they either get the Nobel Prize, or work on the wrong problem and waste their lives

TODO long after Ciro forgot where he had read this from originally, someone later pointed him to: www.ams.org/notices/200902/rtx090200212p.pdf Birds and Frogs by Freeman Dyson (2009), which is analogous but about Birds and Frogs. So did Ciro's memory play a trick on him, or is there also a variant; of this metaphor with a gopher?

Ciro is without a doubt the bird type. Perhaps the ultimate scientist is the one who can combine both aspects in the right amount?

Ciro gets bored of things very quickly.

Once he understands the general principles, if the thing is not the next big thing, Ciro considers himself satisfied without all the nitty gritty detail, and moves on to the next attempt.

In the field of mathematics for example, Ciro is generally content with understanding cool theorem statements. More generally, one of Ciro's desires is for example to understand the significance of each physics Nobel Prize.

This is also very clear for example after Ciro achieved Linux Kernel Module Cheat: he now had the perfect setup to learn all the Linux kernel shady details but at the same time after all those years he finally felt that "he could do it, so that was enough", and soon moved to other projects.

If Ciro had become a scientist, he would write the best review papers ever, just like in the current reality he writes amazing programming tutorials on Stack Overflow.

Ciro has in his mind an overly large list of subjects that "he feels he should know the basics of", and whenever he finds something in one of those topics that he does not know enough about, he uncontrollably learns it, even if it is not the most urgent thing to be done. Maybe everyone is like that. But Ciro feels that he feels this urge particularly strongly. Correspondingly, if a subject is not in that list, Ciro ignores it without thinking twice.

Ciro believes that high flying birds are the type of people better suited for venture capital investment management: you know a bit of what is hot on several fields to enough depth to decide where to place your bets and how to guide them. But you don't have the patience to actually go deeply into any one of them and deal with each individual shit that comes up.

Cosmos: A Personal Voyage (1980) episode 1 mentions as quoted by the Wikipedia page for Eratosthenes:

According to an entry in the Suda (a 10th-century encyclopedia), his critics scorned him, calling him beta (the second letter of the Greek alphabet) because he always came in second in all his endeavours.That's Ciro.

This dude looks like a God. Ciro Santilli does not understand his stuff, but just based on the names of his theories, e.g. "Yoga of anabelian algebraic geometry", and on his eccentric lifestyle, it is obvious that he was in fact a God.

Good film about him: Blaise Pascal (1972).

Good quote from his Les Provinciales (1656-57) Letter XII, p. 227:

French version reproduced at: www.dicocitations.com/citation/auteurajout35106.php.The war in which violence endeavours to crush truth is a strange and a long one.All the efforts of violence cannot weaken truth, but only serve to exalt it the more.The light of truth can do nothing to arrest violence; nay, it serves to provoke it still more.When force opposes force, the more powerful destroys the less; when words are opposed to words, those which are true and convincing destroy and scatter those which are vain and false; but violence and truth can do nothing against each other.Yet, let no one imagine that things are equal between them; for there is this final difference, that the course of violence is limited by the ordinance of God, who directs its workings to the glory of the truth, which it attacks; whereas truth subsists eternally, and triumphs finally over its enemies, because it is eternal, and powerful, like God Himself.

A way to defined geometry without talking about coordinates, i.e. like Euclid's Elements, notably Euclid's postulates, as opposed to Descartes's Real coordinate space.

Postulates are what we now call axioms.

There are 5: en.wikipedia.org/w/index.php?title=Euclidean_geometry&oldid=1036511366#Axioms, the parallel postulate being the most controversial/interesting.

With major mathematicians holding ideas such as:

Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done.it is not surprise that the state of STEM education is so shit as of 2020, especially at the the missing link between basic and advanced! This also implies that the number of people that can appreciate any advanced mathematics research is tiny, and consequently so is the funding.

Ciro Santilli's wife, who was frustrated with academia at some point, admires the fact that Simons managed to make infinite money, and then invested back in actual science, e.g. through the Simons Foundation.

It is hard to decide what makes Ciro Santilli more sad: the usage of Greek letters, the invention of new symbols, or the fifty million alternative font styles used.

Only Chinese characters could be worse than that!

These are not in the Greek alphabet:

Definition: "easy" number theory learnt in primary school, notably the operations of addition, subtraction, multiplication and division.

Can be calculated efficiently with the Extended Euclidean algorithm.

They come up a lot in many contexts, e.g.:

Two numbers such that the greatest common divisor is 1.

Techniques to get numerical approximations to numeric mathematical problems.

The entire field comes down to estimating the true values with a known error bound, and creating algorithms that make those error bounds asymptotically smaller.

Not the most beautiful field of pure mathematics, but fundamentally useful since we can't solve almost any useful equation without computers!

The solution visualizations can also provide valuable intuition however.

Important numerical analysis problems include solving:

Selected answers by Ciro Santilli on the subject:

All those dedicated applied mathematicians languages are a waste of society's time, Ciro Santilli sure applied mathematicians are capable of writing a few extra braces in exchange for a sane general purpose language, we should instead just invest in good libraries with fast C bindings for those languages like NumPy where needed, and powerful mainlined integrated development environments.

And when Ciro Santilli see the closed source ones like MATLAB being used, it makes him lose all hope on humanity. Why. As of 2020. Why? In the 1980s, maybe. But in the 2020s?

Used a lot in quantum mechanics, where the equations are really hard to solve. There's even a dedicated wiki page for it: en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics). Notably, Feynman diagrams are a way to represent perturbation calculations in quantum field theory.

Let's gather some of the best results we come across here:

Sometimes mathematicians go a little overboard with their naming.

Open as of 2020:

- $e+π$

Generalization of systems of polynomial equations.

mathoverflow.net/questions/11540/what-are-the-most-attractive-turing-undecidable-problems-in-mathematics/11557#11557 contains a good overview of the decidability status of variants.

mathoverflow.net/questions/11540/what-are-the-most-attractive-turing-undecidable-problems-in-mathematics/103415#103415 provides a specific undecidable equation over the integers with only one missing constant

A polynomial of degree 1, i.e. of form $ax+b$.

A polynomial with multiple input arguments, e.g. with two inputs $x$ and $y$:
as opposed to a polynomial with a single argument e.g. one with just $x$:

$f(x,y)=x_{2}+2x+y_{3}+1$

$f(x)=x_{2}+2x+1$

By default, we think of polynomials over the real numbers or complex numbers.

However, a polynomial can be defined over any other field just as well, the most notable example being that of a polynomial over a finite field.

Given this, we could evaluate the polynomial for any element of the field, e.g.:
and so on.

$P(0)=2(0×0×0×0)+(0×0)+2=2P(1)=2(1×1×1×1)+(1×1)+2=2(1)+1+2=2P(2)=2(2×2×2×2)+(2×2)+2=2(16$

We can also add polynomials as usual over the field:

$P(x)+Q(x)=2x_{4}+x_{3}+(1+2)x$