Metamath! The Quest for Omega 338
| MetaMath! The Quest for Omega | |
| author | Gregory Chaitin |
| pages | 157 |
| publisher | Self-published e-book |
| rating | Excellent! |
| reviewer | Joe Kauzlarich |
| ISBN | n/a |
| summary | Limits of computability |
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.
Mods (Score:5, Funny)
heart racing (Score:4, Funny)
Zero (Score:5, Interesting)
Re:Zero (Score:5, Interesting)
I agree. That's a great book.
Even though I know calculus pretty darn well, after reading Seife's discussion of the development of 'limits', I realised that I hadn't truly 'grokked' it as well as I'd thought.
The book includes a fascinating account of just how tantalisingly close the Greeks came to inventing calculus. One can only wonder what would've happened if they'd done it.
Re:Zero (Score:5, Funny)
The Greeks would have sat around having endless philosophical discussions about the ethical significance of the relationship between differentiation and integration. The Romans would have taken it from the Greeks and used it to build things. By now, we'd be speaking Latin in orbit around Alpha Centauri.
Re:Zero (Score:3, Interesting)
Brad...are you out there? (Score:5, Funny)
All props to the author and review...but this one isn't going to be an up all night page turner for me.
Re:Brad...are you out there? (Score:2, Funny)
Are our RR packets
the class was laughing by that part in the sentence.
ever had a lab TA named "bin-bin" and asked to dispose of something: May I throw this in the trash bin, bin-bin?
what was the sentence that was lke "fish" 7 times?
Re:Brad...are you out there? (Score:2, Funny)
what was the sentence that was lke "fish" 7 times?
Can't help you there, but I do remember a contest on NPR a few years back. The winning entry was the following:
My apologies to both the noted cellist [yo-yoma.com] and the Duncan corporation [yo-yo.com] for bringing this up.
Re:Brad...are you out there? (Score:3)
Misstatement of the Incompleteness Theorem (Score:5, Interesting)
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.
[OT] Incompleteness (Score:5, Interesting)
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.
Re:[OT] Incompleteness (Score:3, Informative)
No - it is more general than that. To quote Nagel and Newman:
"He proved it impossible to establish the internal logical consistency of a very large class of deductive systems.."
Arithmetic was only one example of this.
Re:Misstatement of the Incompleteness Theorem (Score:2)
Propositional logic (Score:2)
Re:Misstatement of the Incompleteness Theorem (Score:2)
Re:Misstatement of the Incompleteness Theorem (Score:4, Funny)
Dang. No good in math OR spelling?
[OT] umlaut (Score:2)
If you post HTML Formatted, the entitywill generate a "ö" for you.
You may notice that it is an "o" and the characters "uml" inbetween an ampersand and a semicolon.
I'm sure you can figure out how to umlaut the character "u" with that information.
pdf availability (Score:4, Interesting)
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...
Re:pdf availability (Score:2)
Here are some more links [google.com].
coming in September 2005?? (Score:4, Informative)
Re:coming in September 2005?? (Score:5, Interesting)
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
Taking about exciting math books (Score:4, Interesting)
Comment removed (Score:5, Informative)
Re:btw, on Infinite sets the reviewer talks about. (Score:2)
Re: (Score:2)
Re: (Score:2)
Re:btw, on Infinite sets the reviewer talks about. (Score:5, Informative)
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 [wolfram.com] 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 [mathforum.org] 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!
Re:btw, on Infinite sets the reviewer talks about. (Score:5, Informative)
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 [wolfram.com] (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.
Re:btw, on Infinite sets the reviewer talks about. (Score:2)
Re:btw, on Infinite sets the reviewer talks about. (Score:5, Interesting)
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.
Slashdot Effect (Score:4, Funny)
Perhaps the Slashdotting his ~300KB ebook is about to receive would be a good case study...
Chaitin as a child... (Score:5, Funny)
[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
Re:Chaitin as a child... (Score:5, Funny)
Slashdot: Where even geeky jokes are incomparably geeky.
Re:Chaitin as a child... (Score:2)
Thanks!
The sad thing is it didn't occur to me that there might be another way to write it. I mean, uhhh, I figured Milhouse would be at least as math-oriented as Chaitin for them to hang out together, and he would definitely use the parens. Or something. D'oh.
PS I just hope he wouldn't say "I got you open paren Omega plus one close paren!" It is quintessential geek but detracts from the rhythm a bit.
Re:Chaitin as a child... (Score:3, Insightful)
Oh no (Score:5, Funny)
Seen on the back cover (Score:4, Funny)
*Publisher not responsible for any mental or physical anguish caused by this book.
Comment removed (Score:3, Interesting)
Re:is Halting number really random? (Score:2, Interesting)
Re: (Score:2)
Re: (Score:2)
Re:is Halting number really random? (Score:2)
It remains _impossible_ to compute it, since the question of whether or not an arbitrary program will halt is uncomputable. The shortest program that can compute
Read the book -- it's free, and proves all of this VERY beautifully (much easier than Turing's original proof).
And it also proves
indenumerably infinite supply? (Score:2, Insightful)
What the heck does that mean?
A quick trip to Dictionary.com... (Score:2)
reveals the following:
denumerable
adj.
Capable of being put into one-to-one correspondence with the positive integers; countable.
[From denumerate, to count, from Late Latin dnumerre, dnumert-, alteration of Latin dnumerre : d-, dis-, dis- + numerre, to number; see numerate.]
In keeping with the higher math theme, the definition of 'indenumerable' is left as an exercise for the reader.
Explanation (Score:4, Funny)
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.
Re:indenumerably infinite supply? (Score:2)
The book explains this brilliantly and very simply, and it's free. Read it!
-Billy
Re:indenumerably infinite supply? (Score:2, Informative)
How many times does... (Score:4, Funny)
Re:How many times does... (Score:4, Informative)
Chaitin (Score:5, Interesting)
Re:Chaitin (Score:2)
It's not surprising that Chaitin was cited so often in the book--he's one of the three co-authors. [amazon.com]
Which is not to suggest that I don't think his work is brilliant; it's just that the sampling represented in the references is not entirely unbiased.
Chaitin (Score:3, Informative)
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.
What level math... (Score:3, Insightful)
Re:What level math... (Score:3, Funny)
Re:What level math... (Score:2)
This country's education system is in the crapper. Algebra 3! WTF! As a Junior!? No offense to the parent poster, but there is no reason why algebra can't be mastered in the 7th and 8th grades.
Re:What level math... (Score:3, Insightful)
There is accordingly absolutely no justification for your assumption that an Algebra III class covers the material that should have been mastered
Re:What level math... (Score:2, Interesting)
Jumping to a Conclusion (Score:3, Informative)
Even if the writer's conclusion is true, it is not so obvious as to justify stating it without argument.
Re:Jumping to a Conclusion (Score:2)
Re:Jumping to a Conclusion (Score:4, Informative)
Consider all of the numbers that can be generate by a computer program. (Let's talk for the moment just about terminating computer programs.) You can denumerate them by listing the computer programs that generate them, in order. You can order the computer programs by treating their sequence of bytes in machine code (for any machine you choose) as an N-byte number.
If you want to extend this to nonterminating programs with multiple outputs, we can work in pairs: program N running 1 step, program N running 2 steps, etc. You can then order those pairs.
Note that we're not actually running the programs. The program that generates pi runs infinitely, but it's expressed by a fairly short program.
Or look at it from the other direction: consider the set of all computer programs, in order. Aggregate the numerical output from each program (terminating or not), and you have a denumerably infinite set of numbers (one for each program. Let's assume that each program has only one output; you can always transform a program with multiple outputs to several programs with a single output.)
So there you go, the author's conclusion: any number that can be generated by a computer program is denumerable (that is, it's denumerated by the machine code for the program itself).
Which leaves you, bizarrely, with an uncountably infinite set of numbers, otherwise indistinguishable from the infinitely smaller first set, which do not fit into this denumeration, none of which you can name.
Re:Jumping to a Conclusion (Score:2)
Here's the proof.
Since we accept that all computer programs can be mapped to integers, it follows that every program that computes a number (a strict subset of all pro
Re:Jumping to a Conclusion (Score:3, Informative)
Even if the writer's conclusion is true, it is not so obvious as to justify stating it without argument.
All physical computers are a finite implementation of a register machine. All programs for a register machine are equivalent to programs for all other known kinds of computers (Turing Machines, single stack machines, functional systems, even Quantum Mechanical devices), so changing the architecture is not going to change the argument. All programs for register machines can ultimately be described as
Re:Jumping to a Conclusion (Score:2)
"It can clearly be shown by diagonalization that this is true." is clearly in order.
true random numbers (Score:4, Funny)
quote on math books (Score:5, Informative)
"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 [auckland.ac.nz], 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.
Yes, I have... (Score:5, Informative)
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.
omega 0.42 (Score:5, Funny)
but this comment is to small to hold it.
Knud Sørensen
Philosophical satisfaction (Score:2, Informative)
We need more math!!! (Score:3, Interesting)
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...
Just a dog garned moment .....l (Score:3, Insightful)
What do we mean by "random"? (Score:5, Insightful)
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.
Not What You Mean (Score:5, Informative)
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!
PDF (Score:5, Informative)
This book is art moderne extravaganza itself (Score:2, Interesting)
Perhaps, should we start some C++ coding in verses?
Slashdot moderation scores (Score:2)
isn't "omega" spoken for? (Score:3, Informative)
Never has it been more true ... (Score:4, Funny)
Slashdot: News for Nerds. Stuff that matters.
Not Math, Just Words (Score:3, Interesting)
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.
Re:Not Math, Just Words (Score:5, Interesting)
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.
Re:Not Math, Just Words (Score:3, Insightful)
Re:Not Math, Just Words (Score:3, Interesting)
You are wrong that such formulations of mathematics have not been successful. Do a little research into Constructive Recursive Mathematics, for example. You probably haven't heard of it because it was
Re:Not Math, Just Words (Score:3, Interesting)
Not to put words in the gradparent poster's mouth, but I'd be willing to answer yes to that. I don't think real numbers really exist. There's a saying that "God created the integers, all else is the work of man."
How long is the circumference of a unit circle, if pi doesn't exist?
Well, since spacetime is not flat, it's some number slightly different from pi (or way different from pi, if you happen to be reading slashdot from just ou
Re:Not Math, Just Words (Score:3, Interesting)
The concept of a square is itself an idealised one, i.e. that all four sides are *exactly* equal. The concept that you can measure something *exactly* is idealised.
In the 'real' world you're never going to have such a thing as a 'square', so you're never going to be physically confronted with square root of 2.
Even the concept that you can count past 1 is an idealised concept as it assumes that you idealise certain objects as essentially
Clarification (Score:5, Informative)
Please, shoot me... (Score:5, Funny)
Recommended (Score:2, Informative)
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) inc
Chaitin = not just a weird mathematician ... (Score:4, Informative)
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 [acm.org].
The answer (Score:3, Funny)
I can honestly answer this question: no
Re:Groovy (Score:3, Funny)
Re:Only on slashdot (Score:2, Funny)
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.
Re:Only on slashdot (Score:2, Insightful)
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
not surprised if Chaitin did it (Score:4, Interesting)
Re: (Score:2)
Re:Huh-huh! (Score:3, Funny)
This has to be the
Re: (Score:2)
Re:Defining a random number (Score:3, Informative)
In sum, random h
Re:Defining a random number (Score:2)
No.
"Omega" represents a number which cannot be generated by a computer algorithm. All a computer is capable of doing is computations. Therefore, any result a computer can generate is denumerable. This includes what you're thinking of as random numbers. The random numbers a computer generates are sufficiently random for most of our computational uses, however, they are not truly random in the mathmat
Re:pi as denumerable? (Score:2)
Denumerable and rational are not synonymous. Denumerability is a property of sets, rationality is a property of numbers. To make denumerability applicable to number then, one considers whether a program exists that can generate an arbitrary digit of that number, which is true for pi [uwaterloo.ca]. If such a program exists, then because the set of all programs is denumerable we can say that the number is denumerable. There is no program of finite size which can compute an arbitrary digit of Chaitin's constant, therefo
Re:pi as denumerable? (Score:2)