rating: "mathematically mature"
I'm trying something different: squeezing a post, or part of one, in my morning routine, before I see Kimmie off to work and get on with my breakfast and writing. Blog-posts have gotten lost in the shuffle as I have adjusted my lifestyle around power-walking in the afternoons, causing untold suffering and anxiety to my loyal readers, who find themselves suddenly in cold-turkey withdrawal. Take heart! I do want to write my posts like a good chap.
Last night I finished reading Meta Math! by Gregory Chaitin. In some places this small book was beyond me. Or at the very least, I did not have the passion or perseverance to attempt some of the exercises he eagerly suggests, such as coming up with my own proofs for certain things to do with real numbers or Diophantine equations (really, Gregory!). My sense with the book is that Chaitin doesn't realize how hard some of the ideas in it are, and he spends more time explaining relatively simple things (like how DNA is based on four bases) than he does on certain very (to me) difficult things. The image that came to my mind was of wading through a lake where the bottom is very uneven: shallow in places, and in others dotted with deep holes down which one might disappear at any moment.
The book is aimed at the "mathematically mature". (This expression stuck in my mind from my days as a janitor at Vancouver General Hospital. One summer, one of the vacation-relief workers was a math student, and had brought a textbook to read on his breaks. One of the other guys, an artist named Doug, was flipping through it as we sat at a table at coffee-break. Looking on, I could see that it was an advanced textbook on topology, with many arcane equations and references to things called "sheafs" and "manifolds". Doug asked the student, "Do you think I could read this book?" The student, choosing his words, said thoughtfully, "You would need to be mathematically mature to read this book." His choice of words stuck with me. I was, I'm sure, one of the few janitors at VGH who had a topology text of his own at home: a book called Structural Stability and Morphogenesis, the seminal work of a mathematical field called catastrophe theory by the French mathematician Rene Thom. I'd bought it as a research book for a would-be novel, but could barely make head or tail of any of it. If you'd like to read a truly hard book, I suggest giving that one a try.) But its main argument can still be grasped clearly by the mathematically less educated, as long as you have the interest.
And I do have the interest. Chaitin's book is essentially a description of the discovery and significance of a certain real number that Chaitin found, which he dubbed Omega. (The symbol, uppercase omega, I noted, is the same as that used for Absolute Infinity, which I read about in Rudy Rucker's Infinity and the Mind.) So-called real numbers, by definition, have infinite decimal expansions. An example is pi, the ratio of a circle's circumference to its diameter. The digits of its decimal expansion go on and on, never repeating, forever, the digits tumbling on in no predictable order. No one can tell you what the trillionth decimal of pi is, or the quadrillionth, or the quintillionth. They could be 4, 9, and 2 respectively--or any other digits.
But they could be worked out from a formula to calculate pi. That formula is relatively simple. Finding the trillionth digit simply means going though a lot of repetitive but well-defined steps. In Chaitin's terms, that means that pi is very compressible: the infinite series of digits can be worked out from a relatively simple formula. The formula functions as a simple program to generate the digits. Even though we can't predict pi's digits at a glance, pi is in no sense a random number, because we can so precisely calculate it from a small program.
Chaitin's number Omega is the antithesis of pi: it is a truly random real number. It is devised so that its sequence of digits is finitely inexpressible; it cannot be compressed; no program can be created to generate its digits that is any smaller than Omega itself. He has found something that is truly and fundamentally random.
What's the significance of it? It actually says something about how knowable the universe is. All scientific inquiry is a search for laws, which are simple ways of accounting for complex phenomena. The existence of Omega proves that there are specific objects, at least in mathematics, that cannot be compressed--cannot be derived from a simpler formula or law. As I understand it, he's found a specific thing that is fundamentally indescribable and unknowable. That means it's at least possible that the universe shares those properties, and can never be completely understood in a scientific sense, even in principle.
Needless to say, there's a great deal more to this topic. I've been fascinated by it ever since reading Godel, Escher, Bach by Douglas Hofstadter back in 1981. This nexus of ideas, in my opinion, is the most important of the 20th century, and I have enormous admiration for the few brilliant minds who have been able to peer into this mysterious void and descry the outlines of order.
But I'm on the outside looking in on the mathematically mature.
Last night I finished reading Meta Math! by Gregory Chaitin. In some places this small book was beyond me. Or at the very least, I did not have the passion or perseverance to attempt some of the exercises he eagerly suggests, such as coming up with my own proofs for certain things to do with real numbers or Diophantine equations (really, Gregory!). My sense with the book is that Chaitin doesn't realize how hard some of the ideas in it are, and he spends more time explaining relatively simple things (like how DNA is based on four bases) than he does on certain very (to me) difficult things. The image that came to my mind was of wading through a lake where the bottom is very uneven: shallow in places, and in others dotted with deep holes down which one might disappear at any moment.
The book is aimed at the "mathematically mature". (This expression stuck in my mind from my days as a janitor at Vancouver General Hospital. One summer, one of the vacation-relief workers was a math student, and had brought a textbook to read on his breaks. One of the other guys, an artist named Doug, was flipping through it as we sat at a table at coffee-break. Looking on, I could see that it was an advanced textbook on topology, with many arcane equations and references to things called "sheafs" and "manifolds". Doug asked the student, "Do you think I could read this book?" The student, choosing his words, said thoughtfully, "You would need to be mathematically mature to read this book." His choice of words stuck with me. I was, I'm sure, one of the few janitors at VGH who had a topology text of his own at home: a book called Structural Stability and Morphogenesis, the seminal work of a mathematical field called catastrophe theory by the French mathematician Rene Thom. I'd bought it as a research book for a would-be novel, but could barely make head or tail of any of it. If you'd like to read a truly hard book, I suggest giving that one a try.) But its main argument can still be grasped clearly by the mathematically less educated, as long as you have the interest.
And I do have the interest. Chaitin's book is essentially a description of the discovery and significance of a certain real number that Chaitin found, which he dubbed Omega. (The symbol, uppercase omega, I noted, is the same as that used for Absolute Infinity, which I read about in Rudy Rucker's Infinity and the Mind.) So-called real numbers, by definition, have infinite decimal expansions. An example is pi, the ratio of a circle's circumference to its diameter. The digits of its decimal expansion go on and on, never repeating, forever, the digits tumbling on in no predictable order. No one can tell you what the trillionth decimal of pi is, or the quadrillionth, or the quintillionth. They could be 4, 9, and 2 respectively--or any other digits.
But they could be worked out from a formula to calculate pi. That formula is relatively simple. Finding the trillionth digit simply means going though a lot of repetitive but well-defined steps. In Chaitin's terms, that means that pi is very compressible: the infinite series of digits can be worked out from a relatively simple formula. The formula functions as a simple program to generate the digits. Even though we can't predict pi's digits at a glance, pi is in no sense a random number, because we can so precisely calculate it from a small program.
Chaitin's number Omega is the antithesis of pi: it is a truly random real number. It is devised so that its sequence of digits is finitely inexpressible; it cannot be compressed; no program can be created to generate its digits that is any smaller than Omega itself. He has found something that is truly and fundamentally random.
What's the significance of it? It actually says something about how knowable the universe is. All scientific inquiry is a search for laws, which are simple ways of accounting for complex phenomena. The existence of Omega proves that there are specific objects, at least in mathematics, that cannot be compressed--cannot be derived from a simpler formula or law. As I understand it, he's found a specific thing that is fundamentally indescribable and unknowable. That means it's at least possible that the universe shares those properties, and can never be completely understood in a scientific sense, even in principle.
Needless to say, there's a great deal more to this topic. I've been fascinated by it ever since reading Godel, Escher, Bach by Douglas Hofstadter back in 1981. This nexus of ideas, in my opinion, is the most important of the 20th century, and I have enormous admiration for the few brilliant minds who have been able to peer into this mysterious void and descry the outlines of order.
But I'm on the outside looking in on the mathematically mature.
Labels: books by others, my life history, philosophy, science
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