If there is one thing that makes Ciro Santilli learn German, this is it (the Romance language are all the same, so reading them is basically covered for Ciro already).

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:

Good lists of such problems Lists of mathematical problems.

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

Mathematics is a beautiful game played on strings, which mathematicians call "theorems".

Here is a more understandable description of the semi-satire that follows: math.stackexchange.com/questions/53969/what-does-formal-mean/3297537#3297537

You start with a very small list of:

Using those rules, you choose a target string that you want to reach, and then try to reach it. Before the target string is reached, mathematicians call it a "conjecture".

Mathematicians call the list of transformation rules used to reach a string a "proof".

Since every step of the proof is very simple and can be verified by a computer automatically, the entire proof can also be automatically verified by a computer very easily.

Finding proofs however is undoubtedly an uncomputable problem.

Most mathematicians can't code or deal with the real world in general however, so they haven't created the obviously necessary: website front-end for a mathematical formal proof system.

The fact that Mathematics happens to be the best way to describe physics and that humans can use physical intuition heuristics to reach the NP-hard proofs of mathematics is one of the great miracles of the universe.

Once we have mathematics formally modelled, one of the coolest results is Gödel's incompleteness theorems, which states that for any reasonable proof system, there are necessarily theorems that cannot be proven neither true nor false starting from any given set of axioms: those theorems are independent from those axioms. Therefore, there are three possible outcomes for any hypothesis: true, false or independent!

Some famous theorems have even been proven to be independent of some famous axioms. One of the most notable is that the Continuum Hypothesis is independent from Zermelo-Fraenkel set theory! Such independence proofs rely on modelling the proof system inside another proof system, and forcing is one of the main techniques used for this.

Not to be confused with algebra over a field, which is a particular algebraic structure studied within algebra.

Well summarized as "the branch of mathematics that deals with limits".

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.

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.

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!

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:

Ciro Santilli is actively looking for donations and contracts so he can continue to work full time on OurBigBook.com sustainably, and develop free hardcore university-level STEM education for all ages!

For 1M USD I will quit my job to work on OurBigBook for three more years to try and kickstart The Higher Education Revolution. Status: ~44k / 1M USD. MacArthur fellowship wink. At 4M USD I retire and work on open STEM forever.

I first quit my job 1st June 2024 to work on the project for 1 year after I reached my initial 100k goal mostly via a 1000 Monero donation. In this first year I improved project tech, but didn't go and solve university courses to create super targeted content, and didn't obtain a single contributing user except myself, see a summary at Section "OurBigBook Project Update March 2025". Day job total compensation as of August 2026 for reference: 205k USD/year. This may be inflated by the AI bubble. I try to keep the value up to date on LinkedIn: www.linkedin.com/in/cirosantilli/

For a followup, it would be an interesting experiment to spend on year full time solving as many courses as I can from a world class university in the city where I live in the UK, and adding that as content to the platform to see if that would attract interest, and reaching out directly to course takers at their university environment to try and help them. I intended to do that in year one but my got distracted by tech. It is quite possible that no one has ever done that before in history: a highly motivated technical person with the time and opportunity to do one single thing: help top university students learn their STEM courses better and have more fun doing so. I have in particular identified one course where this would be particularly feasible: the mathematics course, given that much of their course materials, and also their building are quite open. I do sometimes wonder if doing this would be just a waste of my life. But part of me tells me it could generate big interest and is worth a try. It would also be fun for you to watch me continue to commit career seppuku.

At 4M USD I retire and work on open STEM projects forever. At these timelines, I can't guarantee it will be specifically on education technology specifically all the way, but I guarantee that whatever it is it will be open and extremely well explained as usual.

Total donations to date ~144k USD. Donation breakdown:More details: Section "Accounting method"

Note to potential anonymous crypto donors: I live in the UK, and after some fun back and forth that included Barclays freezing my account for one week at one point, Barclays finally decided that they do not allow me to receive further anonymous donations. And I'm pretty sure that the same would happen on any other British bank sooner or later. Therefore, while I continue to accept anonymous donations, they will not count towards my "if I get X I do Y"-type donation goals, since I can't reliably spend that money. If you want to donate in crypto, and clearly give me your real identity and an explanation of how you got the money, then that is fine, it can count. Just be warned that I will need to give that information to my bank and clear it with them beforehand, and if they are still not happy with it, I'll just give it all back. Your identity does not have to be publicly disclosed, only the dates and amounts. But my bank has to know. Now, if you still want to donate a "retirement level amount" and treat me as a guinea pig in "find the best country for money laundering", well, go for it. But I make no promises! I don't want my children to grow up in shitcountries.

How to give:And if you have a different preferred payment mechanism not listed above, please contact Ciro, and he will set it up.

Ciro's current ambitions require him to remain in developed countries, because Ciro wants to document advanced science and technology by liaising with top universities, and there is not nearly as much high technology in poor countries. Remaining in developed countries is also a required due to family reasons.

If you would like public acknowledgement for your support, Ciro will very gladly give it, just let Ciro know how you'd prefer it. Due to Ciro Santilli's campaign for freedom of speech in China, many supporters have chosen to be anonymous, and that is totally fine, not everyone is interested in politics, or has a situation where going public is acceptable, so we don't have a standard setup yet, let's build it together. A acknowledgement section at the bottom of this page would be a minimum, but I for larger donations we could add a your advertisement in a locations such as:

Ciro Santilli has sometimes wasted time with low impact projects such as those listed at Ciro Santilli's minor projects instead of doing higher impact projects such as those mentioned at: Section "The most important projects Ciro Santilli wants to do".

But maybe "Everything you did brought you where you are now." applies, maybe it is during the "low impact activities" that one gets the inspiration and experience required for the "high impact ones".