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Metamath! The Quest for Omega
Chaitin's goal is the casual reader's comprehension of an irreducible, uncomputable, and truly random real number. He doesn't actually find one of these numbers, of which there are an indenumerably infinite supply, but he comes as close as a person can to actually referring to it.
Does this sound mysterious (and a little weird)? It is! But this ties in to just the sort of problem mathematicians have been working on for the past hundred or so years. You may be familiar with Goedel's Incompleteness Theorem, in which he proves that no formal axiomatic system (FAS) is powerful enough to prove all of the true statements its notation can express. For a long time, many people were wondering if Fermat's Last Theorem could be one of these statements (although it was finally (and famously) proven by Andrew Wiles about a decade ago). This is the type of "metamathematical" problem Chaitin attacks with his arsenal of complexity and information theory.
Key to understanding the book's premise is understanding the problems involved in defining a truly random number. Chaitin works in binary, so it is easy to find a random number by flipping a coin multiple times, although defining what a random number is supposed to look like (without circularly using the word 'random') is impossible. If you can define exactly what it should look like, then you can use that definition to create (or compress (see below)) a random number. It would not, then, be random.
The next key word is 'reducibility' (or 'compressibility'). If a number is random then it cannot be reduced or compressed into a smaller equation or algorithm. The digits of pi appear to be random, but they are reducible. This entire infinitely long real number can be expressed with just a few symbols- 4*sum_(k=1)^n(((-1)^(k+1))/(2k-1)). The same is true with 'e' or the golden ratio. You might be aware of the distinction between denumerable and nondenumberable infinities-- Chaitin explains this in his book; in short, there are (at least) two kinds of infinite sets, those that map directly to the integers (e.g. the rationals) and those that don't (e.g. the reals). It has been shown that all computer programs may be mapped to integers and hence are denumerable. Any number that can be generated by a computer program (pi, e, etc) therefore is denumerable. For Chaitin's random real number to be truly random, we must look only at real numbers that are indenumerable (cannot be calculated-- otherwise it would be compressible).
Here is where we run into problems-- we can't possibly generate a random real number and we can't even define what it looks like! Chaitin discusses the philosophical arguments for the very existence of such a number, and in the end uses Turing's Halting Program idea to show that a random real number can exist-- and the random real number vaguely referenced in this way, he calls Omega, the halting probability. The probability that an arbitrary program halts is the random real number that Chaitin had been searching for.
But this is not giving away the ending by any means. In fact he tells us this before even embarking upon his journey. What is remarkable about the book is that, in plain English, and using ideas that a non-mathematician like myself can understand, in only 157 pages, Chaitin can explain the grandest ideas on the cutting edge of mathematics. "As you have no doubt noticed," began Chaitin's conclusion, "this is really a book on philosophy, not just a math book. And as Leibniz says... math and philosophy are inseparable."
Although the book can be read quickly and painlessly (there are only a few simple equations in the book), the insights it contains are profound and likely to stick in your brain for some time. Furthermore Chaitin's enthusiastic style is contagious and will leave you on the edge of your seat. He floats through dozens of interesting anecdotes about the great mathematicians-- Leibniz, Newton, Turing, Godel and others--, the process of mathematical discovery from the vantage-point of an actual mathematician, insights into the mind of a working mathematician, and the craft of mathematics, interjecting his own educated thoughts on all of these matters. His style is aimed towards those whose education in mathematics extends only a little past high school and the ideas are simply followed (don't worry if you can't follow my own explanations above; I'm not nearly as skilled an expositer as Chaitin!)
This book is available for free on Chaitin's own website (so why not give it a try?) and also at ArXiv.org. Slashdot welcomes readers' book reviews -- to see your own review here, carefully read the book review guidelines, then visit the submission page.
If this book is reflective of the way I meta-moderate, it should be a breeze!
Hmmm.
Math heart racing? Only on exams
Interesting math books remind me of a book I read a few years ago. Zero: The Biography of a Dangerous Idea by Charles Seife. As fun and interesting as I found math to be, I think that book really did it for me as far as spine tingling mathematics.
All props to the author and review...but this one isn't going to be an up all night page turner for me.
When the people fear their government, there is tyranny; when the government fears the people, there is liberty.
Which actually states that any sufficiently powerful formal system can express true propositions which cannot be proven. Typically, "sufficiently powerful" means self-referential to some degree; the system must be able to refer to a propostion within it, and the truth/falsehood of that proposition.
I am not a mathematician, though, so this may not be completely accurate. However, I am fairly sure that it is not difficult to compose a formal system which is provably complete.
I remember your eyes, on the twelfth of July...
now he needs to release it under the new paradigm :)
of academic textbooks.
like the MIT heat transfer book
i kind of like this idea, that if something was
important enough for you to write down for humanity,
you are just doing it for the sake of society.
that would probably take a huge cut out of the
whole "i wrote a book now buy it for my class"
effect...
Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem is one of those other really really exciting mathematics books.
Comment removed based on user account deletion
"The probability that an arbitrary program halts is the random real number that Chaitin had been searching for."
Perhaps the Slashdotting his ~300KB ebook is about to receive would be a good case study...
[Scene: two children on a playground playing cops+robbers]
Chaitin: I got you!
Milhouse: I got you twice!
Chaitin: I got you [thinks very very fast] Omega!
Milhouse: I got you (Omega + 1)!
Chaitin: AAAARGH!!!
Fin
I want to drag this out as long as possible. Bring me my protractor.
Slashdot, what were you thinking?! I was under the impression that only high ranking Starfleet officials were to be told of Omega, yet you go posting a review of it on the front page!
I was reading a book called "Information and randomness" the other day (highly technical book) and about 50 of the papers in the bibliography are by Chaitin! Seeing how often Chaitin's theorems/discoveries etc. are cited made me realize how vast this guy's contributions are. People so deeply involved in research rarely write popular math books, and so its a pleasant surprise to see that he does, and is quite good at it.
I'm too lazy to look up which mathematician/physicist said this:
"There are only two kinds of math books: those you can't read past the first page, and those you can't read past the first sentence."
Anyway, Chaitin's other books are really interesting too. There is one called "The Limits of Mathematics" which discusses Godel's proof and even "shows" it interactively with some LISP code at the end. The whole book is free online here, which is a great deal for a very interesting Springer text. Some people think Chaitin too arrogant, but there's not denying he's a great mind.
FYI, the continuum hypothesis is neither true nor false (or BOTH true and false, depending on how you think about it :).
It is independent of the rest of set theory... much like Euclid's parallel postulate is to geometry. You can assume it's true, or assume it's false, and you get different versions of set theory in the end. Similar to the existence of both euclidean and non-euclidean geometries.
Many people don't realize that there are multiple versions of something as fundamental to mathematics as set theory! Check out the Axiom of Choice for another example of something that's neither true nor false in set theory.
My favorite proof involving cardinality and set theory is the proof that there are the same number of integers as fractions... so simple that a school kid can understand every step, yet so profound a conclusion!
DiscDividers tabbed plastic CD dividers: divider cards f
We spent about half a semester going from Maxwell's equations to the thin lens approximation.
In a mathematically contiguous manner--no hand waving arguments, all solid derivations and proofs.
With lab.
From electromagnetic theory through to everyday optics. It was fucking beautiful.
Well, I have to go now. I have a date. With my wife.
Nope. Still pegged.
"Reality is that which, when you stop believing in it, it doesn't go away." - Philip K. Dick
I have made a beautiful proof that omega > 0.42
but this comment is to small to hold it.
Knud Sørensen
An interesting take on these incompleteness theories is Jaakko Hintikka's book "The Principles of Mathematics Revisited." He states, among other things, that Gödel only proved the deductive incompleteness of Arithmetics, but his result is really not that important as it says nothing about the descriptive completeness of systems. His (Hintikka's) point is, that deductive completeness (the possibility to deduce all the possible sentences from given axioms), something that mathematicians had always strived for, isn't really that important; more important is a system's descriptive power.
Man is a slave because freedom is difficult, whereas slavery is easy.
Of course I haven't RTFB, so maybe this is answered, but I don't believe that randomness is a property of a number, its a property of the method used to generate the number. The reviewer's example of flipping a coin to generate a random binary number is an example of this. I could flip a coin and generate the number 000000 - the method of generation is random, the number itself is clearly reducible and therefore not "random" in the sense described in the review.
I would reserve the term "random" to talk about the generation method, and use more precise terms like "irreducible" for the numbers themselves.
To go further, it may even be that what we mean by a "random" generation scheme is: "a scheme whose generation method I can't predict". This makes randomness a property of a system's knowledge of the generation system. For example, in many situations a computer's psuedo-random number generator is a sufficiently random generation scheme, in some cases (for example cryptography) it is not. psuedo-RNGs are not random (they are deterministic, thus the use of the term "pseudo") but for some uses they effectively are, because the system using the numbers output from them can't (or doesn't need to) predict the next number generated.
So I would propose that "random" refers to the process of generating a number that is in practice non-deterministic in the specific context in which the number is used.
Sailing over the event horizon
I made a PDF version of the book if anyone's interested.
I realize it takes a while to write a book, but doesn't it usually need to be finished before someone can read and review it?
Well, no, actually. It just needs to be mostly finished. Call it a "release candidate."
Surely it can't take a whole year to setup the press to print the book.
There is considerably more to selling a book than printing up a few copies.
Presumably the publisher has other books it's trying to print and sell as well and this one has to "wait its turn."
We're talking marketing here, not manufacturing. Movies sometimes sit "in the can" for years before being released for various reasons. I believe this is common knowledge. The same is true of books.
Or sound recordings. Or automobiles. Or video games. Or whatever.
KFG
The sorts of people who reject the Axiom of Choice (disclaimer: I'm still undecided on the matter) insist on a "constructive" set theory--meaning you can't pull examples of sets that "ought to exist" out of thin air, you have to build them out of the Zermelo-Fraenkel Axioms (minus the Axiom of Choice, of course).
They have a distinction between truth and provability. A statement is true if no counterexample exists (can be constructed), and a statement is provable if there exists a proof of it using the ZF axioms. Using the words "truth" and "provability" in that way, it's clear that the unprovability of the continuum hypothesis is itself proof of its truth. If a counterexample could be constructed (a set with cardinality greater than that of the integers and less than that of the reals), the hypotheis would be provably false. But since it's known to be unprovable, it must be impossible to construct such a set. And the nonexistence of a counterexample is the definition of truth.
It may not actually be inconsistent to use a version of set theory that includes the negation of the continuum hypothesis as an axiom (I'll call it the NCH axiom for Negation-Continuum-Hypothesis), but very few mathematicians (even those who accept the axiom of choice) would accept such a system. Informally, axioms are supposed to be self-evident truths. Even the Axiom of Choice merely extends a statement that is provably true in the finite case to the infinite case, but the NCH axiom asserts, for no self-evident reason, the existence of an exotic set with properties that aren't even trivial to define. The Continuum Hypothesis is technically unprovable, but unless you're actually doing formal mathematics you can safely think of it as true.
The original Howling Frog is a fictional character and has no UID.
Godel's Incompleteness Theorem does NOT state that no FAS can be complete (any statement that is true under it's notation is provable is true). The first-order propositional calculus & first-order predicate calculus are both complete axiomatic systems (assuming proof of the former, I have done a proof of the latter). It states that any FAS capable of expressing the natural numbers cannot be complete which means no mathematical axiomatic system can yield a complete system. Any system that allow for unrestricted comprehension allows for variants of Russel's paradox - let us have A be the set of all sets that do not contain themselves and only those sets that do not contain themselves. does A contain itself? either answer is contradictory.
I've seen Chaitin present a paper on register allocation. He skilfully held the attention of the entire audience. It wouldn't be surprising if this were a real page turner.
If I ever, ever, get tingly over a math book, someone - I don't care who - shoot me.
The concept that uncountable sets exist is just silly.
The real numbers are an uncountable set. Are you saying that it is just silly to believe that real numbers exist?
How long is the diagonal of a unit square, if sqrt(2) doesn't exist? How long is the circumference of a unit circle, if pi doesn't exist?
The Pythagoreans of ancient Greece believed as you do, and when they found out about the sqrt(2) business, they did their best to hush it up. Unfortunately for them, the truth got out. Ever since, the concept of limiting mathematics to countable sets has been unsuccessful. There are too many inviting pathways into uncountability to put up barriers on all of them.
The reviewer is talking about real numbers. Your intuition about randomness is derived from numbers such as one encounters in a computer or a physical instrument. However, these are not real numbers, they are truncations of real numbers. There are only countably many numbers you can represent on a computer, whereas there are uncountably many real numbers.
There's no such thing as a random number on a computer, because once you single a number out for attention, it isn't random anymore. But, in a technical sense revealed by RTFB, "almost all" real numbers can't be counted. They can't be named exactly, in a way that would allow you to generate them to arbitrary precision. This must be so, because such precise name is a computer program, and there are only countably many computer programs. These numbers are "random" in the sense that it is impossible to single one out for special attention. Although "almost all" real numbers are random, you can't specify a single example!
"The good reader is a rarer swan than the good writer."
It is not the case that the "continuum hypothesis is known to be true". Nor is it the case that it has been proven to be unprovable, though that is closer to being correct.
The continuum hypothesis is a statement about entities which do not exist in the universe. We know what the statement "2+2 = 4" is about; it's about integers, and since we can count, we're pretty sure that integers exist. The statement "the universe is expanding" is a statement about things we can observe. There can be quibbles about how much of the universe we can see, whether our understanding is really great enough to answer such questions, and so on, but in the end, practically everyone would say that the question has meaning and, therefore, has some kind of answer, even if the answer is no better than "the parts we can observe indeed appear to be expanding".
The continuum hypothesis is different. It is a statement about uncountable sets, which are creations of our mind. If we are right about the laws of physics, there are *no* uncountable sets existing as physical entities in our universe. What this means is that the continuum hypothesis is not a statement relevant to physical reality, and therefore is of quite different character than either "2+2 = 4" or "the universe is expanding". It is a completely reasonable belief system to hold that the continuum hypothesis, being entirely about non-existent mentally generated entities, has no meaning, and is therefore neither true nor false.
To believe that the continuum hypothesis has a definite truth value is a strong philosophical statement. The mathematical philosophy called Platonism holds that mathematical objects, such as uncountably infinite sets, actually exist, and therefore that statements about them such as the continuum hypothesis have meaning, and in fact that such statements are either true or false. Another philosophy of mathematics is formalism, which holds that mathematics is a game we play according to rules. If someone proves a complicated mathematical result about uncountable sets, we admire this as brilliant play of the game, but do we "believe" it? We believe it only if we believe those statements from which the reault was proved. To play and appreciate the game, we don't have to believe in the axioms, and in fact may find it entertaining to play the game starting from axioms we believe to be false. A formalist is unlikely to regard the continuum hypothesis as either true or false.
Another poster said that the continuum hypothesis has been proven to be unprovable. This is an oversimplification. What has been proven is that the continuum hypothesis is unprovable from the standard set theoretic axioms, using standard logic. A formalist admires this statement as itself brilliant game play, but understands that it is meaningful only for this game. Add another axiom, and suddenly you can prove CH. Unless you find the axioms compellingly true, you probably regard a claim of the truth (or falsity) of CH as dubious as a claim that one's goal in life should be to own Park Place. Truth is relative to where you started from.
A good Platonist on the other hand, will generally believe that the contiuum hypothesis is meaningful, and either true or false, if only we were clever enough to figure out which. Since we know we can't prove it from the standard axioms using the standard logic, a Platonist must hope for discovery of a new axiom or a new logic which is intuitively compelling, and which will also allow CH to be proved or disproved. So, to ask "Is CH true?" is assuming a Platonic view of the Universe, and can be answered only by mathematical creativity ("I propose Axiom X, which settles it"), not merely by a clever play of the game of mathematical deduction.
It is my understanding that most mathematicians who care about these issues are in fact Platonists.