1/ Chaitin's Meta Math is an important book that distinguishes itself not only in the erudition with which it treats the most fundamental questions of our time, but also in the clarity of its prose. The book is about the limits of reason, the nature of mathematics,...

2/ ...the importance of information and complexity theory, discrete and continuous mathematics, and most importantly randomness. These topics are at the basis of our most important disciplines: math, physics, computers science, epistemology, morality and politics.

3/ The books starts with a dedication to beauty. An affirmation of the fact that from Pythagoras to Euler and Ramanujan, the driving force behind the discovery of new math has been an aesthetic desire. https://twitter.com/rdntola/status/1041878788722704385

4/ Mathematicians “like many creative artist, are passionate and emotional people who deeply care about their art, they are unconventional eccentrics motivated by mysterious forces, not by money nor by a concern for the “practical applications” of their work”.

5/ The essence of math resides in its freedom to create. The value of mathematical theories is judged by their fertility “the extent to which they illuminate other mathematical ideas or the physical world.” Only the beautiful concepts survive.

6/ This is why Chaitin thinks the best way to study math is through the developmental history of these ideas: https://twitter.com/rdntola/status/1031029628205326336

7/ So, let’s see what we learn when we look at the history of our conception of prime numbers. Primes are irreducible numbers that seem to exhibit some order in the way they are distributed.

8/ ( My cousin @p_tereziu tells me that primes do exhibit some predictability, because there exist prominent diagonal (dense) lines on the Ulam cloth that correspond to, for instance, quadratic polynomials. Among other things, this makes primes very interesting.)

10/ Although we don’t know their precise distribution we do have a couple of proofs that there exist infinitely many primes. These proofs vary based on our historical knowledge.This leads to the question: “How much of our current mathematics is habit, and how much is essential?”

11/ For example, instead of primes perhaps we should be concerned with the opposite, with “maximally divisible numbers”!” “If the history of math would rerun, would primes reappear?” Probably primes would, but what about other more esoteric mathematical objects?

12/ To Chaitin and @stephen_wolfram the answer seems to be “that current mathematics is much more arbitrary than most people think.” “Mathematical style and fashion varies substantially as a function of time…”

13/ Let’s look at the proofs for the existence of prime numbers and how they change in time.

14/ Euclid’s Reductio ad absurdum Proof:, he proved there are infinitely many primes by assuming there are finitely many and deriving a contradiction.

15/ Euler’s Analytical Proof: found a connection between the harmonic series and prime numbers.

16/ Chaitin Complexity-based Proof: because the idea is that if there were finitely many primes, then the prime factorization of a positive integer would provide a lossless compression algorithm so strong that it can't possibly exist.

17/ (If you are curious you can find other proofs here: https://www.maths.tcd.ie/~vdots/teaching/files/MA2316-1314/Primes1.pdf )

18/ But why is the variety of proofs important?“If a mathematical statement is false,there will be no proof but if it’s true,there will be an endless variety of proofs. Proofs are not impersonal, they express the personality of the creator/discoverer just like literary works do."

19/ “Mathematical facts are not isolated, they are woven into a vast spider web of interconnections.”

20/ This made me think of Wittgenstein "...we cannot think of any object apart from the possibility of its connection with other things." And also this article from Wolfram: “ “ http://blog.stephenwolfram.com/2018/01/showing-off-to-the-universe-beacons-for-the-afterlife-of-our-civilization/”

21/ “Whether a piece of math is correct isn't enough, the real question is whether it’s interesting and that is a absolutely and totally a matter of opinion, and one that depends on current mathematical fashions.”

22/ If this line of thinking gives you vertigo. You are in good company. Bertrand Russell wanted to discover the foundations of Mathematics precisely to avert future Chaitin from making this comments.

23/ Agrippa's Trilemma would show its ugly face wherever he searched for the foundations. It think it made him feel as if all the edifice of mathematics really rested on the reptilian brain of the humans who created it.

24/ This was troubling for Russell because Physics was doing quite well. Thomson and Rutherford were bringing the world closer to the Democritean vision of atoms. Alas, Russell would also bring the modern mathematical world closer to the understanding of some dead Greek thinkers.

25/ In his quest he discovered a set theoretical paradox that bears his name, which was an eco of a previous paradox known to the greeks as the paradox of the liar or Epimenides paradox. https://www.wikiwand.com/en/Russell%27s_paradox

26/ This brought a crisis in the world of logic, leading the logician Gottlob Frege in an act of supreme intellectual honesty to print in his magnum opus “Foundations of Arithmetic” that;

27/ “Hardly anything more unfortunate can befall a scientific writer, than to have one of the foundations of this edifice shaken after the work is finished.

28/ The collapse of one of my laws, to which Mr. Russell’s paradox leads, seems to undermine not only the foundations of my Arithmetic but the only possible foundations of Arithmetic as such.”

29/ David Hilbert tried to escape this crisis of logic in formalism. As I understand it, the feeling was that this paradox might emerge from the fact that language is imprecise and thus might lead to paradoxes.

30/ Hilbert wanted to build a “perfect artificial language for reasoning, doing mathematics and deduction.” He wanted to be precise about the definitions, elementary concepts and grammar. And to find a mechanical procedure to decide if a proof obeys the rules or not.

31/ “Hilbert envisioned creating rules so precise that any proof could always be submitted to an unbiased referee” “He did not think mathematics should be done this way but rather that if you could take mathematics and do it this way, you could then use mathematics to study math.

32/ His ethos was immortalized in the reply he gave to "Ignoramus et ignorabimus":

“Wir müssen wissen.

Wir werden wissen.”

“Wir müssen wissen.

Wir werden wissen.”

33/ Russell inspired by the same animus of Hilbert thought he would be able to go around his own paradox by creating Set Types. He and his collaborator went to work on the famous Principia Mathematica which again tried to create a foundation for arithmetic, trying to prove 1+1=2

34/ Alas, the day before Hilbert pronounced the ethos above, a young Austrian named Godel announced his first expression of the Incompleteness Theorem.

35/ Godel ended up proving that there is no way to have a formal axiomatic system for all of mathematics in which it is crystal clear whether something is correct or not. “Actually, it will either be inconsistent or incomplete.”

36/

Inconsistent( a = b),

Incomplete (can’t prove the truth of its own theorems )

Inconsistent( a = b),

Incomplete (can’t prove the truth of its own theorems )

37/ Yet, another english chap called Turing started working on Hilbert’s machine and build the first abstract universal computer. “In his 1936 paper, Turing claims that such a machine should be able to perform any computation that human beings can carry out.

38/ And then he proceeds to ask what can’t this machine do? Enter the Halting problem: “This is the problem of deciding in advance whether a computer program will eventually find its desire solution and halt.”

39/ He showed that if you don’t impose a time limit on the program there is not way to know if it will halt or not.

40/ This meant that there’s no way you can use a program to decide if another program will halt, which in turn means that there’s no way to decide this issue using reasoning. No axiomatic system can help you deduce this. The paradox is similar to Godel’s.

41/ But Godel’s proof applied primarily to Number Theory. Turing showed that no formal axiomatic system can be complete. This made Chaitin understand that:

42/ “No mechanical process (rules of the game) can be really creative because in a sense anything that comes out was already contained in your starting point. Does this mean that physical randomness, something non mechanical, is the only possible source of creativity?!

43/ " At least from this ( enormously over-simplified!) point of view, it is. "

44/ "So the world of mathematical truth has infinite complexity, even though any given Formal Axiomatic System only has finite complexity. I therefore believe that we cannot stick with a single FAS, as Hilbert wanted, we've got to keep adding new axioms, new rules of inference.."

45/And where can we get new stuff that cannot be deduced from what we already know? Well, I'm not sure, but think that it may come form the same place that physicist get their new equations: based on inspiration, imagination and on the case of math, computer experiment."

46/ This leads Chaitin to exclaim that "in the conflict between Hilbert and Poincare over formalism versus intuition, I am now definitely on the side of intuition." The same thing @nntaleb said in the Black Swan.

47/ From this conflict between Hilbert’s and Poincare’s ethos something beautiful emerged, the computer. And this is what the book is about. Chaitin, using the practical and the philosophical results of this conflict creates something new, and shows us what creation feels like.