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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.
Have you ever read a math book that was able to carry you through its proofs, heart racing, and make your skin tingle upon reaching its philosophically astounding conclusion?
I can honestly answer this question: no
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...
That's a relief, I was starting to think poorly of you.
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...
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? If it is already finished, why is it taking so long to publish it? Surely it can't take a whole year to setup the press to print the book.
Edward Burr
Having a smoking section in a restaurant is like having a peeing section in a swimming pool.
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!
*Publisher not responsible for any mental or physical anguish caused by this book.
I Am My Own Worst Enemy
Comment removed based on user account deletion
He doesn't actually find one of these numbers, of which there are an indenumerably infinite supply...
What the heck does that mean?
--
As a matter of fact, I am a lawyer. But I play an actor on TV.
I've been outnerded.
Man, I thought I was a dork, but I stand humbled in the presence of a math book reviewer. I thought my star trek fanzine and degree in robotics had me prepared, but in one swift book review, all my nerdly accomplishments are like ashes in the wind.
Best Windows Freeware
...Chaitin tell you his own work is the most important work in mathematics? If he does it less than 100 times in the book I may consider reading it.
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
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.
It is rarely so technical as to terrify and his sense of humor and careful exposition makes reading his stuff enlightening and fun all at once.
Highly recommended.
is somebody going to need to understand the book? Unfortunately for myself I was not born with the genius gene, and being in highschool (just finished sophmore year, taking alg3 next year), I don't understand a lot of the more advanced pyshics and math discussed on /.
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.
Even if the writer's conclusion is true, it is not so obvious as to justify stating it without argument.
Those who would give up essential liberty to purchase a little temporary safety, deserve neither liberty nor safety.
He mentions that the book is about defining true random numbers. Now then, this suggests that randomness is a matter of degree, that is, some numbers are more random than others. I suppose the best way to determine the most random number is to poll people to pick a random number and see what the most common choice is.
Unknown host pong.
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
Man, I thought I was a dork
:)
Well, not to pussyfoot around the issue, you are by virtue of having a Trek fanzine. Any dork can have a Trek fanzine, even one without a degree in robotics, or a brain.
Math is about geekyness, not dorkyness.
But hey, if you weren't such a dork you'd already know that.
KFG
Hmmmm ! I don't know about maths or geometry but some of the best kicks I got was from a book called "The Cosmological Argument from Plato to Liebniz" by William Craig Lane. In particular the chapter on Don Scotus' form of the argument was a logical tour de force. Really blew my mind. Of course these arguments are proofs not demonstrations so if you like logical forms with a theological bent check this out its last imprint was 2001.
Actually, if this book is compelling, I hope that some of the academic book authors take an example and figure out a way to make math interesting and compelling for children to learn in schools. It is a real shame that most of the public school system in the U.S. makes math seem so boring (the memorization of formulas and crap, rather than learning something that is truly useful and learning how to apply concepts to solve real life problems) that most kids do poorly in math. This, in my opinion, is part of the reason that a lot of the programmers being turned out by schools suck, but think they're hot stuff because they can turn out word processors with VB#.NET or whatever. They really don't have a good solid foundation in math, logic, and science to make really good software. The same problem applies to other areas as well, which is why a lot of U.S. jobs are being outsourced to other countries. I strongly believe that if the public education system here in the U.S. were improved drastically, a lot of employers would see a compelling reason to pay the higher price for domestic workers, because they would get increased value out of their investment.
Anyway, that was a rant, but I think a lot of technical subjects, like math, tie into the greater overall problem of teaching children how to think, how to apply concepts, how to learn something when they don't know the answer, rather than how to memorize the steps to accomplish a particular task, and fail when the task doesn't exactly match, they fail...
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.
he's described this number in a book with a finite number of numbered pages ..... methinks something's fishy here ....
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.
> pick up a non-nerdy book like some good manga or something
/. phrase of the week.
This has to be the
I'll try to explain it in laymans terms for you...
He means that he did not find a number which is part of an infinite quantity of infinite supply and for which there is also an infinite number of. Get it? Good.
Now, for my part I do not give much credibility to a guy who can't even find a number for which there is an infinite quantity. F*ck, just pick one and there you are! But again, I must concede that to find a number (for which similar number exists in an infinite supply) must be harder to do if you look for ONE specific number and you need to look for it thru an indenumerably infinite supply of those. I imagine the complexity of it must be an indenumerably infinite order of magnitude harder to do then to find the bug I am actually tracking which also exist in an indenumerably infinite supply of in the application I am currently working on.
Now I think I've done my fair share of productivity in this world today and I'll just go back to sleep, thank you.
I'd rather be sailing...
This book is... interesting. Really. Stuffing Franz Kafka, Leibniz and Mahabharata to a math book and ending it with poems, that's a piece of artistic achievement.
Perhaps, should we start some C++ coding in verses?
There you are, staring at me again.
I thought "omega" was already taken in number theoretical circles--the surreal number consisting of up-up-up-up-... ("up-hat")? Hell, Cantor broke out the Hebrew numbers to express his weird idea. That link uses omega in its own way. This guy really should have tried a little harder.
blarg.
Random technically means non-deterministic. If you want to nitpick, yes numbers generated by an algorithm aren't random, but some algorithms are "cryptographically secure" which means given the previous numbers, one cannot predict the next number w/ certainty or even high probability. There's also the concept of probability distributions which are random in the sense that one cannot for sure know the value but one can know the expected value and the probability of anything being the value.
In sum, random has very different connotations depending on it's use. I have been under the impression that omega in math refers to the first uncountable ordinal number (cite: wikipedia.org) which by definition can have no numeric value associated to it.
Have you ever read a math book that was able to carry you through its proofs, heart racing, and make your skin tingle upon reaching its philosophically astounding conclusion?
Slashdot: News for Nerds. Stuff that matters.
The concept that uncountable sets exist is just silly. The sets are simply not well defined. If you can't define something well enough for it to be calculated, then it is not mathematics. Just as I can describe "love" or "happiness", but I cannot give a formal definition of them... they are not math.
These supposed mathematical objects are claimed to exist because someone came up with a formal axiomatic system which assumes they exist. It is a self-fulfilling prophecy.
The problem is that such assumptions result in foundational or metamathematical problems. Formally you can prove the existence of uncountable sets, but semantically all sets are countable. So within the formal system you have one thing, while outside of the formal system you have another... its a sort of semantic inconsistency.
For example, in ZFC set theory you can easily prove that the set of all functions on natural numbers is uncountably infinite. However, the fact that ZFC is a formal system tells us that we can count every function on natural numbers that can be proven to exist in ZFC. This second part cannot be proven within the system, but it is immediate from the fact that finite strings have a one-to-one correspondence with the naturals. So if we assume that ZFC set theory is a formal language for describing the mathematical concept of sets, then we see that an inconsistency exists between the formalism and the mathematical concepts.
Many people, including mathematicians, only think it is necessary to avoid simple inconsistencies... while allowing semantic inconsistencies.
Others, including some of the pioneers of axiomatic set theory, realized that a more constructive foundation was required for mathematics. There are many variations of constructive mathematics. One such branch roughly states that something is mathematical if and only if it can be computed. So mathematical objects are algorithms. This is an interesting formulation of mathematics because all of math is complete, computable, and consistent.
Formal axiomatic mathematics is flawed. In it only guarantees that you have a system for deriving strings in a formal language. It cannot guarantee that these strings have any mathematical meaning. Hence you can derive meaningless things such as a number that cannot be written down or computed to a sufficient decimal expansion.
Omega is not math, its just words. Math invovles precise, absolute concepts. Omega is nothing ore than a formal gesticulation.
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.
Chaitin's ideas are quite profound both in expanding upon theories of computation:
Goedel -> Turing -> Chaitin
and also in opening up new areas of mathematics and physics, much as chaos theory did.
Understand - the randomness Chaitin is dealing with is NOT the pseudo-random output of a transcendental equation, or of finite state automata (ala Wolfram) - these are truly random numbers that are not being computed, but rather revealed.
Chaitin is also a lisp hacker who (at least in previous books) includes lots and lots of code so you can play with the numbers yourself.
His writing style is a little bit too casual for me (lots of exclamation points), but if you want to learn more about TRUE randomness then go to the source.
Also let me add that Chaitin is a really nice guy - sent him some questions after reading one of his essays several years ago and he answered them straight away.
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.
It is also noteworthy that his contributions aren't solely in the field of mathematics - he has contributed some groundbreaking work in the area of compiler research, such as this paper.
"I love my job, but I hate talking to people like you" (Freddie Mercury)
A word of warning: Chaitin is an entertaining writer, but he is not a careful writer. His purely mathematical theorems and proofs are perfectly fine, of course, but when his thoughts turn philosophical, he is prone to fairly idiosyncratic and dubious thinking.
For example, in one article he inexplicably quotes Einstein to make a point about philosophy of math. In the quote, Einstein alleges that mathematical axioms are invented by humans. Chaitin proudly proclaims that this shows Einstein is an "empiricist". This is a very unusual use of the term "empiricist", not at all consistent with what philosophers of mathematics would mean if they used the term.
Chaitin also defines technical terms (like random) and then pretends he uses them in their usual, non-technical sense. But his definition of random is not the same as its usual sense. For Chaitin, there is a non-zero probability that a random source of 0's and 1's produce a "random" string. This probability goes to 0 as the length of the string goes to infinity, but even then the random source may produce a non-random string (it is a possible event with probability 0).
Finally, Chaitin produces his "random" number Omega, and proudly proclaims that he has proven some mathematical claims are "true for no reason". I don't really know what this would even mean, but unless it means "some equations involve random numbers" then it's not clear how he's proved it.
Anyway, my comments are not referring to this new book, which I have not read, but only to a few articles of Chaitin's that I've read in preparation for a course. For a coherent and clear criticism of Chaitin's work, see Panu Raatikainen's articles.
Phiwum's law: anyone that names an obvious law after himself and then puts it in his own sig is just pathetic.