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: https://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
- 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:
- 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.
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: http://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 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
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!
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 value
https://www.youtube.com/watch?v=R9FKN9MIHlE&t=938s Birch and Swinnerton-Dyer Conjecture (Millennium Prize Problem!) by Kinertia (2020)
Nice result on Lebesgue measurable required for unicity.
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
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: https://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.
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.
Good film about him: Blaise Pascal (1972).
Good quote from his Les Provinciales (1656-57) Letter XII, p. 227:
French version reproduced at: https://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.
Postulates are what we now call axioms.
There are 5: https://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.
Only Chinese characters could be worse than that!
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.
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.
Used a lot in quantum mechanics, where the equations are really hard to solve. There's even a dedicated wiki page for it: https://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.
Generalization of systems of polynomial equations.
https://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.
https://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 .
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.:
We can also add polynomials as usual over the field:
The only thing is that then we would need to differentiate between different orderings of the terms of multivariate polynomial, e.g. the following would all be potentially different terms: ring we end up with a lot more more possible terms.
Amazing graphs and formulas.
Python graphics engine open sourced at: https://github.com/3b1b/manim "Animation engine for explanatory math videos". But for some reason there is a community fork: https://github.com/ManimCommunity/manim/ "This repository is maintained by the Manim Community, and is not associated with Grant Sanderson or 3Blue1Brown in any way (though we are definitely indebted to him for providing his work to the world). If you want to study how Grant makes his videos, head over to his repository (3b1b/manim). This is a more frequently updated repository than that one, and is recommended if you want to use Manim for your own projects." what a mess.
He got so old from 2012 to 2021 :-)
This dude did well. If only he had written a hyperlinked wiki rather than making videos! It would allow people to jump in at any point and just click back. It would be Godlike.
Robert Donley received his doctorate in Mathematics from Stony Brook University and has over two decades of teaching experience at the high school, undergraduate, and graduate levels.
- The best articles by Ciro Santilli
- Ciro Santilli's bad old event memory
- Computer science
- Eli Benderski
- Existence and uniqueness
- Fields Medal
- High budget movies are shit
- High flying bird vs gophers
- High-frequency trading as a form of Nirvana
- Magic: The Gathering
- Mathematics illustration software
- Minimal working example
- Molecular Sciences Course of the University of São Paulo
- Numerical analysis
- How to convince teachers to use CC BY-SA
- Quantum electrodynamics
- Lecture 2
- Why you should give money to Ciro Santilli
- The beauty of mathematics
- There is value in tutorials written by early pioneers of the field
- Video game
- Website front-end for a mathematical formal proof system