Postscript---Letter to a daring young reader

Now that you, dear reader!, have worked your way through this difficult book, and some of you have probably worked your way through all three of my Springer volumes, I would like to state my conclusions, my views, much more emphatically. I would like to summarize as forcefully as possible my new viewpoint. These three books are my justification for these strong claims! I'll make my polemical points in haphazard order:

1. AIT is tremendously revolutionary, it is a major paradigm shift, which is why so many people find the philosophical conclusions that I draw from my theory to be either incomprehensible or unpalatable.

2. Time complexity, a highly fashionable field of theoretical computer science, is of great technological interest, but not of deep conceptual interest. Information complexity has no practical applications, but it connects with epistemology in a way that time complexity does not. Yes, fast algorithms are useful, very useful! But it is asking about the size of programs that leaves you staring into the darkness at the limits of mathematics!

3. The purpose of mathematics and of empirical science is to detect and describe pattern in the Platonic world of pure mathematics and in the physical world that we normally inhabit, when we're not doing calculations. [The patterns that we believe in are our theories---that's how we cope.] For the first time, AIT says what pattern or structure is, and what lack of pattern or structure is. This is tremendously deep, and was never achieved nor attempted by any previous theory.

4. Some physicists I know, e.g., Karl Svozil, [Author of Randomness and Undecidability in Physics, World Scientific, 1993.] follow Einstein and believe that the physical universe is actually deterministic, with the apparent randomness of QM [Quantum mechanics, quantum physics.] as a temporary aberration, one that will be removed by future, deeper theories. Karl, however, acknowledges that I have demonstrated the existence of total randomness in the mental mindscape (in the imaginary world) of pure mathematics.

5. Kolmogorov envisioned AIT as an alternative formulation for probability theory. This is not particularly interesting: all you get is a set of random real numbers that must have any properties that you can prove a real number has with probability one. So ordinary probability theory and this new probability theory are not that different.

   It is in the field of epistemology and in metamathematics, that AIT makes a fundamentally new contribution. K. did not realize how revolutionary AIT is, that AIT states that some mathematical truths are fundamentally probabilistic, that there is randomness in the foundations of mathematics, that the truth content of some statements in pure mathematics is grey, not black or white, that sometimes truth is probabilistic, not sharp and clear. And that this occurs even in the world of pure mathematics, not just in empirical sciences like physics.

6. The leading new ideas of the 20th century are information and randomness, just as set theory was the leading new idea of late 19th century mathematics: Randomness coming from statistical mechanics, QM and chaos theory in physics, and information coming from computer technology and DNA technology (molecular biology). AIT brings these foreign new ideas into the foundations of pure mathematics. It is a miraculous interdisciplinary unification.

   AIT has shown that randomness arises in the very heart of pure mathematics, and is the real reason behind Gödel incompleteness and Turing uncomputability. According to AIT, probability theory is no longer a branch of applied math, in which probabilities come only from physics. AIT shows that probabilities and randomness arise naturally in pure mathematics. Ω demonstrates this, and the diophantine equation for Ω demonstrates a much more devastating kind of randomness in number theory than was envisioned in any of the work on probabilistic number theory.

7. It's totally miraculous that Ω, merely defined as the halting probability, turns out to have all possible constructive statistical properties of randomness. It's totally amazing that algorithmic incompressibility yields statistical randomness as a corollary. This provides a deep explanation for randomness, a reason that randomness appears in the foundations of pure mathematics. The classical logical effort to concentrate axioms into a minimal irredundant set was an attempt to achieve incompressibility and therefore randomness! Occam's razor applied in physics is the same story! The most concise theory that accounts for a given set of data must necessarily be highly incompressible.

8. AIT unifies mathematics, theoretical computer science, information theory, and probability theory, logic, recursive function theory, metamathematics! It's tremendously interdisciplinary! It's miraculous that such a unification was possible! There are even connections, at more than the level of poetic metaphor, with the notion of entropy or disorder in statistical mechanics and thermodynamics.

9. AIT is the true foundations of mathematics: Mathematics deals with mathematical information, which is what axioms contain, just as biological information is contained in DNA, which is the foundation of biology.

10. Hilbert's attempt to entomb mathematical truth in a fixed, formal system was completely misguided. Mathematics, like biology, is dynamic, not static. At any moment in time our mathematical knowledge consists of only a finite amount of information, but the Platonic world of pure mathematics contains an infinite amount of information.

11. Gödel and Turing were only the tip of the iceberg. AIT provides a much deeper analysis of the limits of the formal axiomatic method. It provides a deeper source of incompleteness, a more natural explanation for the reason that no finite set of axioms is complete.

12. Martin-Löf's and Solovay's statistical definitions of a random real are beside the point. They are superficial, external characterizations of randomness, but they do not indicate where randomness comes from, how it arises, what is its source. For that, you need program-size complexity. And it is a miracle that my incompressibility definition yields statistical randomness as a corollary in the infinite case. My theory provides the only definition in the finite case, which was my main concern until I discovered Ω and became interested in its asymptotic properties.

13. AIT will lead to the major breakthrough of 21st century mathematics, which will be information-theoretic and complexity based characterizations and analyses of what is life, what is mind, what is intelligence, what is consciousness, of why life has to appear spontaneously and then to evolve.

14. AIT is a drastic paradigm shift, and as such, obeys Max Planck's dictum that major new scientific ideas never convince their opponents, but instead are adopted naturally by a new generation that grows up with them and takes them for granted and that have no personal stake nor have built careers on older, obsolete viewpoints.

15. AIT combines practical computer programming with deep theory; it's simultaneously deep philosophy and computer technology, clever hacking! It is a totally new kind of mathematics! And it was a lot of fun for me, over the soon to be four decades that I've had this adventure, and the more than three decades that IBM has funded it!

16. Science is not static, old theories are of little interest, the fun is making new ones. I've done it, and you can do it! Get to work! Tell me how the mind works, what intelligence is, why life must evolve, what consciousness is, why information and complexity are the fundamental notions of biology. I want to know. It's time for you to do your share! Guts and imagination!

    You have to imagine the possibility of success before you can achieve it. You have to dream of a more beautiful new viewpoint before you can confirm it! You have to want to understand with all your soul, you have to be obsessed with that! A lifetime is barely enough time to begin to make a dent in any significant problem. You have to want to more than anything!

    Without a passion for understanding, without impatience with received majority views, it's completely hopeless. Chance favors a prepared mind! Major new ideas do not simply drop onto the lap of scientists, it takes years of work and dreams for that to happen. It just looks like chance when it finally happens! And you forget the years of study and toil that prepared for that magical moment of enlightenment.

    Be prepared to have many false breakthroughs, which don't survive the glaring light of rational scrutiny the next morning. You have to dare to imagine many false beautiful theories before you can hit on one that works; be daring, dare to dream, have faith in the power of new ideas and hard work.

(Signed)
Gregory Chaitin

September 2000