I've always been a mathematician, ever since I was a child I've been doing research. And I'm self-taught: my first published paper was written when I was in high-school, and my first major paper, on my definition of randomness, was written when I was still a teenager, and an undergraduate at the City University of New York.
When I was a child Shannon's theory of information was part of the zeitgeist. It had just been invented, and people still thought that it might apply universally. But in fact Shannon's theory is just a narrow technical subject dealing with communications engineering. On the other hand, my algorithmic theory of information does have broad philosophical consequences.
Most computer programming languages are messy and complicated, reflecting human society, which is messy and complicated. But in LISP I could perceive a core of beautiful, elegant, powerful mathematical ideas. I feel in love with LISP for the same reasons that I fell in love with the beautiful world of mathematical ideas, that is so different from the ugly messy real world!
Since I didn't think that I could earn a living with my philosophical mathematical ideas, and I liked playing with computers, I got a job with IBM in Buenos Aires, and developed my theory of randomness as a hobby. Later I transferred from IBM Buenos Aires to the IBM Watson Research Center in New York, where I also did some practical work, but where for almost twenty years they have let me pursue my theoretical ideas. I am greatly indebted to IBM for giving me carte blanche to investigate whatever I want.
The computer captures the mechanical part of mathematics, it corresponds to systematically deducing the consequences of known principles. The computer, however, does not capture the creative part of mathematics, in which we discover new principles, new axioms.
In molecular biology, information is contained in DNA, and is measured in kilo, mega and giga-bases of information. (Three bases determine one amino acid.) In my theory of algorithmic information, information is contained in computer programs, and the quantity of information is measured in 0/1 bits, bits of software.
Actually my theory is a fresh, new viewpoint, completely different from that of Gödel and Turing. My goal was to understand what it means for something to be random, to have no structure, to obey no law, to be incomprehensible, to be logically, algorithmically and conceptually irreducible. It was only after developing my theory that I slowly realized how it can provide a dramatic new view of the work of Gödel and Turing on the limits of mathematical reasoning. And it was only recently that I discovered that Leibniz in 1686 had similar ideas, but never developed them.
Ω is my most dramatic discovery, it is a place where mathematical truth has absolutely no structure or pattern. It's a place where God appears to play dice, a place where mathematical truth is completely incomprehensible, because the bits of Ω are mathematical truths that cannot be deduced from any principles simpler than they themselves are. In other words, this is an area in which mathematical truth is irreducible, in contrast with the normal view of math that everything can be reduced to simple commonly agreed-upon axioms. Ω shows that what Gödel discovered in 1931 was only the tip of the iceberg, it shows that there are real limits to what reason can accomplish. Ω shows how complicated math can really be, infinitely complicated.
I define something to be random if the smallest computer program that calculates it has the same size that it has. In other words, something is random if it is incompressible, if the simplest theory that explains it has the same size in bits that it does. Such information has irreducible complexity, that's what randomness is.
Well, we don't think that the universe is random, on the contrary, we think of the universe as a giant computer that continually calculates how things behave. We think of the laws of the universe as algorithms or computer programs, procedures for calculating what is going to happen in the future if we know how things are right now. Because the universe obeys such laws, it is highly compressible, very understandable, not at all random.
Well, physicists think so, they believe in a "theory of everything", a set of equations that would fit on a T-shirt and explain everything. That theory has not been discovered yet, but fundamental physics does obey simple elegant laws. On the other hand, in the world of biology things are much more complicated. DNA is very complicated, there is no simple equation that explains a human being or a human society. And what I've discovered, surprisingly enough, is that the world of pure math behaves, in this regard, more like biology than like physics, because pure math is actually infinitely complex, it's not simple like we all thought that it was.