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The "Omega Number" & Foundations of Math

Hemos posted more than 13 years ago | from the my-brain-hurts dept.

Science 247

speck writes "Here's a link to an article in New Scientist about mathemetician Gregory Chaitin, who seems to have thrown some of the basic foundations of math into question with his work on the 'omega number.' Among the more provocative statements in the article: '[Chaitin] has found that the core of mathematics is riddled with holes. [He] has shown that there are an infinite number of mathematical facts but, for the most part, they are unrelated to each other and impossible to tie together with unifying theorems. If mathematicians find any connections between these facts, they do so by luck.' Also of interest is the transcript of a lecture Chaitin gave at CMU, which explains some of the theory in quite accessible language."

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Re:This is not surprising (2)

Anonymous Coward | more than 13 years ago | (#355160)

The only thing which could really throw the foundations of mathematics into an uproar would be to show that some hypothesis can be proven to be both true and false

Not at all! If you found a system of mathematics in which some theorem could be proven both true and false, mathematicians would just say that your system was inconsistent (italicised 'cos I'm using that word in it's maths jargon sense, not it's everyday sense) and be uninterested in it. They'd be uninterested 'cos in a formal system, if any false theorem can be shown to be true, then any theorem can be shown to be true (if 2=1, it logically follows that I'm the Pope).

The basic foundation of maths that Chaitin's work questions is the belief that mathematics is a solid body of knowledge waiting to be discovered, and that we can "get to" proofs in any part of it from where we are now. His work shows that maths is more like the distribution of matter in space: lots of lumps all over the place, with huge gaps between them and no way to travel from one to the other.

Re:This IS surprising! (1)

gavinhall (33) | more than 13 years ago | (#355161)

Posted by ahbbuddha:

Are you sure they're not compressible? I'm just thinking of a trivial example like gzip. Every algorythm can be expressed as an implementation of gzip and along with a string of 0's and 1's fed into it that unpacks into the original algorthm. If the two together are smaller than the original algorythm, then it is compressible. This obviously isn't always the case, but is certainly true some of the time.

Re:2 + 2 = ? (1)

gavinhall (33) | more than 13 years ago | (#355162)

Posted by _LFTL_:

You could also talk about rings where such as the integers mod 3 where 2+2=1. (the integers mod x can be thought of as the set of all possible remainders when you divide a positive integer by x. For example 7 mod 3 = 1.) So maybe freedom should be the freedom to say 2+2=1 :).

This IS surprising! (5)

Paul Crowley (837) | more than 13 years ago | (#355164)

If you don't think this is surprising, maybe you haven't fully understood it. He defines a number, W_UTM, which must have some perfectly ordinary value - it's not one of these weird, undetermined things, like the continuum hypothesis or something. It's just the probability that a random Turing machine will halt. Every Turing machine either halts or doesn't halt, so if only you could solve the halting problem you could get a good approximation to W_UTM in a moment. Since you can't, W_UTM is unknowable. But it's *much more unknowable than you might expect*. To quote the lecture:
So this becomes a specific real number, and let's say I write it out in binary, so I get a sequence of 0's and 1's, it's a very simple-minded definition. Well, it turns out these 0's and 1's have no mathematical structure. They cannot be compressed. To calculate the first N bits of this number in binary requires an N-bit program. To be able to prove what the first N bits of this number are requires N bits of axioms. This is irreducible mathematical information, that's the key idea.
Dwell on that a little. That is serious weirdness.
--

Re:Two to One odds (1)

singularity (2031) | more than 13 years ago | (#355166)

Showing that 1.999... == 2 is not a "parlour trick." Indeed, the foundation on which such proofs lie is one of the basis for the set of real numbers. The real numbers allows for irrational numbers such as the square root of 2.

If you want to study this idea, I would suggest a good undergraduate-level Analysis class. The ideas you would want to pay attention to are least upper bounds, greatest lower bounds and the least upper bound theorem.

As far as the original article - there are always going to basic, unprovable ideas behind mathematics.

Actually, if you want to bring up a point with one assumption that math relies on, ask why math must assume that 1 does not equal 0.

-Hank, mathematics major

Re:broken link? No, it's more proof... (2)

freeBill (3843) | more than 13 years ago | (#355167)

...that New Scientist uses coin flips to generate the programs which run their site.

Would you define "random Turing machine"? (3)

roystgnr (4015) | more than 13 years ago | (#355168)

A Turing machine can require an arbitrary amount of data to encode. If you encode them as integers, then the numbers W_1000 (probability that a random Turing machine from 1 to 1000 will halt), W_10000, W_100000, etc. are well-defined for that encoding. But how do you know that these probabilities converge?

Like Any Another (1)

jjr (6873) | more than 13 years ago | (#355170)

science Mathematics has things that can not be explained and may never be explained. Even some of the "fundamental truths" in math have no proofs. We are just scratching the surface of what we know in math and maybe someday we will be able to prove it all but that is a long way off.

Re:Like Any Another (2)

jjr (6873) | more than 13 years ago | (#355171)

Science

Any branch or department of systematized knowledge considered as a distinct field of investigation or object of study; as, the science of astronomy, of chemistry, or of mind.

From That I believe you can mathematics a science

This stuff is constructive and fun (1)

Nelson Minar (7732) | more than 13 years ago | (#355175)

I was fortunate to learn bits of this theory from Chaitin about five years ago, when he was visiting Santa Fe. As folks here have noted, the work has its roots in Godel's Incompleteness Theorem, but there's a huge amount more detail in it. In particular, his work is highly specific and constructive, boiling down some very abstract concepts into specific machinery that is graspable. Definitely worth time if you are interested. I hope that logic courses will use this material as a basis for instruction.

Re:too is wun (2)

PD (9577) | more than 13 years ago | (#355177)

1) a = b
2) a^2 = ab
3) a^2 - b^2 = ab - b^2
4) (a - b)(a + b) = (a - b)b
5) a + b = b
6) 2b = b
7) 2 = 1

The problem with this is that to go from step 4 to step 5 you need to divide by zero. If you had a step 4.5 in there, it would look like this:
(a-b)(a+b) = (a-b)b
---------- ------
(a-b) (a-b)

a-b = 0

What would Penrose say about this? (3)

PD (9577) | more than 13 years ago | (#355178)

Penrose wrote a series of books (3 last I counted) which basically made the same claim: because humans have a special insight into Mathematics which computers provably do not, computers cannot be intelligent and computation is not an appropriate model for a theory of intelligence.

Well, if human mathematicians are basically wandering around the landscape digging up theorems at random, that sort of blows Penrose out of the water, doesn't it? It would mean that the special "human insight" into Mathematics was essentially a large sequence of random discoveries.

Misconceptions (5)

PureFiction (10256) | more than 13 years ago | (#355181)

There are a lot of posts here about how this is simply a rehash of Godel's theorem.

This is partly true, but not point. Godel showed that incompleteness exists in any type of formal system capable of self reference. Ala the infamous "This sentence is false" translated into an equivalent in a formal system. The original is rather obscure and reads:

On Formally Undeciable Propositions in Principa Mathematica and Related Systems

  • "To every w-consistent recursive class 'k' of formulae there correspond recursive class-signs 'r', such that neither v Gen r nor Neg (v Gen r) belongs to Flg(k) (where 'v' is the free variable of 'r')


This is all well understood and old news at this point. What the author has done is taken Godel's theorem, and the halting problem, and turned them around a different way.

The essence of what he is trying to say is summarized nicely in this paragraph of the conference log:

  • "So I had a crazy idea. I thought that maybe the problem is larger and Gödel and Turing were just the tip of the iceberg. Maybe things are much worse and what we really have here in pure mathematics is randomness. In other words, maybe sometimes the reason you can't prove something is not because you're stupid or you haven't worked on it long enough, the reason you can't prove something is because there's nothing there! Sometimes the reason you can't solve a mathematical problem isn't because you're not smart enough, or you're not determined enough, it's because there is no solution because maybe the mathematical question has no structure, maybe the answer has no pattern, maybe there is no order or structure that you can try to understand in the world of pure mathematics. Maybe sometimes the reason that you don't see a pattern or structure is because there is no pattern or structure!


He then describes how randomness would indicate an irreducible statement of truth that could not be compressed by finding a 'proof' that proves this truth. The idea being that a 'proof' is a program or function that generates truths, or verifies the truth of a statement.

Again, this is not groundbreaking, Godel proved essentialy the same thing with his proof. The turing halting problem is another variation, but this is where it gets interesting.

The author takes the halting problem and instead of determining whether the program halts or not, determines the probability of the program halting given a random program produced by flipping coins.

The equation to solve this is straightforward, the the 'proof' which is used to determine whether the program halts or not is the computer itself, and the statement is a program produced by random bits from a coin toss. Each bit determined by an individual coin toss.

What you then end up with is a statement that is well defined in number theory terms, but maximally unknowable. Every sample program produced from the random coin toss is a straight forward sequence of 1s or 0s, but the statement as a whole is irreducible.

Again, this seems rather unrelated, until you consider proofs as the computers which calculate the truth or non truth of a given statement.

It then becomes obvious that the truth or non truth of the statement requires a proof that can reduce the statements into a true or non true result. And there are a huge number of situations where such a proof can not exist.

So, godel's theorem deals with incompleteness in a formal system where a single proof cannot encompass the entire set of true and and un-true statements.

The authors work deals with the vast number of statements that are true or un-true, but for which no 'proof' can ever be discovered. They are simply true by random chance.

Which holds a lot of interest for physicists because they have been dealing with truths that are random and true for no provable reason for decades...

Re:Would you define "random Turing machine"? (3)

Lamesword (14857) | more than 13 years ago | (#355189)

The definition is oversimplified in the lecture transcript. (I couldn't read the article.)

What you do is you fix a self-delimiting universal Turing machine M. This is a machine that takes its input, interprets it as another Turing machine, and simulates that other machine. Self-delimiting here essentially means that if it interprets "100011101" (or some other string) as a program, then it won't interpret any extension of that string as a program. In particular, if M halts on input "10001101", it won't halt on any extension of that string.

Define Omega_M (the halting probability of M) to be the sum of (1/2)^(length(x)) over all inputs x on which M converges. Because M was self-delimiting, this series will converge to some number between 0 and 1. (You can prove by induction on n that the sum restricted to x of length <=n is bounded by 1.)

This number depends on your choice of M, but that's no big deal.

So, to address your question a little more directly, we're calculating this probability by averaging over infinitely many Turing machines (as inputs to our universal Turing machine), and we're doing this by weighting the Turing machines with short codes more heavily--Turing machines of length n get weight (1/2)^n, and the self-delimiting nature of our universal TM makes the sum of these weights converge.

Re:Old news... (1)

Kimble (17437) | more than 13 years ago | (#355195)

That list left out some of my favorites! What about:
The Montessori Method
The Fibonacci Sequence
The Zimmerman Telegram
The Ostend Manifesto

(with an assist from an old LA Times Sunday crossword)
--

Re:Old news... (2)

Mike Schiraldi (18296) | more than 13 years ago | (#355197)

But it was oddly prophetic

Of what?

--

It's not always so hard (1)

GlobalEcho (26240) | more than 13 years ago | (#355201)

Just make sure your Turing Machine is running Windows 95.

Prob(halting) = 1.0

It's very easy to encode!

Re:This is not surprising (1)

alkali (28338) | more than 13 years ago | (#355202)

I understand that Chaitin's work builds on Gödel's work in interesting ways -- so it's not just a "restatement" of Gödel's work -- but you're right that it hasn't "thrown some of the basic foundations of math into question."

(For those who don't know, Gödel proved that there are some mathematical hypotheses that can't be proven true or false. That was very surprising, but it didn't call into question any theorem which has been proven true. The only thing which could really throw the foundations of mathematics into an uproar would be to show that some hypothesis can be proven to be both true and false.)

Re:This is not surprising (1)

alkali (28338) | more than 13 years ago | (#355203)

Obviously anyone could invent a formal system which generates inconsistencies, and if you did, no one would care. But if one of the formal systems in common use -- say, algebra or analysis -- were found to generate inconsistencies, I think "uproar" would not be an inappropriate term.

Re:Aleph-Zero is the Omega number?! (1)

Dougan (30082) | more than 13 years ago | (#355206)

If I recall correctly, Aleph-Zero is the same as the omega number...

Different omega. And in Cantor's transfinite cardinality theory aleph-null is not the same as omega; aleph-null is a cardinal, omega is an ordinal.

Cheers,
Greg

Re:Isn't this already known? (2)

spectecjr (31235) | more than 13 years ago | (#355207)

Goedel proved back in the 30's that there were many things (an infinite number?) which were true but for which proofs cannot be provided. OTOH Chaitin is a well known mathemetician (in some circles, anyway). Presumably he has something interesting to say, but I doubt it's as revolutionary as the post makes it sound.

Oh, I dunno. I'd say that it's the equivalent of Quantum Mechanics, but for math.

Think about it... the Omega number puts a limit on the accuracy to which we can know mathematical theorems... Maybe it's the equivalent of the Heisenberg Uncertainty Principal?

... well, maybe. :)

That might actually be a good way to use the Omega number... build it in, turn everything to probabilities. *sigh*

Simon

Re:Misconceptions (2)

HackLore (31416) | more than 13 years ago | (#355209)

Actually, there are more irrational numbers than rational ones. And, moreover, between any two rational numbers there exists at least one irrational number.

The irrational numbers that you mention are drops in the bucket of irrational numbers.

Re:Okay, so we didn't invent/create math. (2)

SpinyNorman (33776) | more than 13 years ago | (#355212)

Exactly so!

Math is a bunch of islands (number theory, topology, ...) of theory, that arn't necessarily related.

In fact, usually when someone such as Andrew Weil DOES build a bridge between a couple of these islands then we herald that as a rare triumph!

Also, as others have stated, it's a bit late to be worried about Godel's incompleteness theorem... kind of old news!

Popstar? (Was: Re:Look ma, karma whoring!) (1)

wabewalker (42099) | more than 13 years ago | (#355226)

I haven't read the article yet (can't get through), but I usually take Chaitin's claims cum grano salis (with a grain of salt, for the rest of you), for, among other things, the following reason:
IANACS, but I get the impression for example that he has sometimes "forgotten" other people's work: if you look at prefix Kolmogorov (algorithmic) complexity, in Chaitin's books he usually describes it as his "brilliant insight" (paraphrased), whereas a quick look in Li and Vitanyi's "Introduction to Kolmogorov Complexity and it's applications" attributes it to Levin, Gacs, and Chaitin, based on work by Solomonoff.
Oh, and he may be a very smart bloke, but he ain't no popstar in maths -- in CS maybe?

Re:This IS surprising? (2)

kaphka (50736) | more than 13 years ago | (#355227)

Every Turing machine either halts or doesn't halt, so if only you could solve the halting problem you could get a good approximation to W_UTM in a moment. Since you can't, W_UTM is unknowable.
In other words, W_UTM is non-computable? So what? There are lots of non-computable numbers. For example, take the irrational number in which each binary digit n is equal to 1 if TM(n) halts, 0 if it doesn't. An algorithm that "compressed" that number would also solve the halting problem; therefore, it does not exist.

So, explain to me again why this particular non-computable number is special?

Re:We create math everyday. (2)

1010011010 (53039) | more than 13 years ago | (#355230)

> THERE ARE ALTERNATIVE MATHEMATICAL PHILOSOPHIES!

Conveniently priced at $146.50 [barnesandnoble.com] ... for suckers. "Logic not necessary! Give me $150.00 and I'll show you why!"

- - - - -

Re:This IS surprising! (2)

vectro (54263) | more than 13 years ago | (#355231)

You could always use HTML character entities: For example, omega would be entered as &omega;, which appears as .

This works in any browser compliant with (I think) HTML since version 3, and XHTML 1.0. It at least works in Mozilla 0.8; I can't speak as to other browsers.

Re:Like Any Another (1)

ErikZ (55491) | more than 13 years ago | (#355232)

Maybe because instead of actually SAYING what your point was, you ended up being a sarcastic smart-ass in every sentance your wrote.

I wouldn't even know where to begin writing a counter point to your post.

So, today's lesson is that talk down to the moderators, they're gonna drag you down.

Later,
ErikZ

Re:Misconceptions (1)

Voltage_Gate (69001) | more than 13 years ago | (#355235)

Is that an explanation of irrational numbers, like e, Pi, and phi? If so, why are most of them so close to zero (2.7xxx, 3.14xxx, 1.6xxx), given that they could be between + and - infinity? Makes me think that there are really no bigger numbers than maybe 4 or 5 or 6, around there... hmm.

The main point (1)

Smthng (71777) | more than 13 years ago | (#355236)

His main and coolest point comes close to the end of the paper. He shows that there is a well-defined mathematical number which is "maximally knowable", completely random and uncompressible.

I'm not sure how powerful or fundamental this result is, but it is definitely a new result and provides a new and neat way to think about things from a computer science point of view !

Disprove this math: (1)

PhatKat (78180) | more than 13 years ago | (#355240)

I don't need to see his blackboard. 200,000 clicker happy slashdotters is still 200,000 clicker happy slashdotters, and there's one more broken link to prove it.

You know I have always been suspicious of zero (2)

Camel Pilot (78781) | more than 13 years ago | (#355241)

Other than the fact that the intelligent content around here approaches zero at times.

Isn't this already known? (1)

randombit (87792) | more than 13 years ago | (#355243)

(I can't read the article right now, it's /.ed, so this may be totally wrong).

Hasn't this sort of thing already been known? Chaitan's Omega Number stuff has been known for some time (he's written a book about it, which I think is this one [fatbrain.com] , but Fatbrain doesn't have descriptions so I could be wrong).

Goedel proved back in the 30's that there were many things (an infinite number?) which were true but for which proofs cannot be provided. OTOH Chaitin is a well known mathemetician (in some circles, anyway). Presumably he has something interesting to say, but I doubt it's as revolutionary as the post makes it sound.

Re:Misconceptions (2)

Fnkmaster (89084) | more than 13 years ago | (#355244)

I believe the posters point was the the members of the set of _significant_ irrational numbers (i.e. those that occur in "fundamental" mathematical proofs) are mostly on the order of magnitude of 1. But this itself might just be one of those random, proof-averse facts that this theorem theorizes about itself. Enough to give me a headache in any case.

Re:Would you define "random Turing machine"? (3)

Fnkmaster (89084) | more than 13 years ago | (#355245)

I believe that's a big part of the point of the theorem:

You can get it in the limit from below, but it converges very, very slowly --- you can never know how close you are --- there is no computable regulator of convergence, there is no way to decide how far out to go to get the first N bits of W right.

So it looks like it appears to converge, but you can't really know whether it's converging or not. :) Or something along those lines.

Chaitin's homepage (2)

magi (91730) | more than 13 years ago | (#355247)

Chaitin has quite a lot of stuff in his homepage: http://www.cs.umaine.edu/~chaitin/ [umaine.edu]

Some entire book texts there, etc.

Quite difficult stuff, even for a CS major. Having at least familiarity with automatas and formal languages is recommended, although still far too little.

There's some quite weird stuff in some of his books. I can't say I would recommend reading his stuff without healthy sceptic attitude...

interesting stuff (4)

molo (94384) | more than 13 years ago | (#355253)

I don't have a big formal math background, but I think i was able to pick up what he says in the lecture transcript.

The interesting point of the matter deals with Turing machines and the halting problem. If you have a well defined turing machine, it will either halt or not depending upon its input (the program). Turing's idea was that you can't determine beforehand whether a given program will halt (for all possible programs). That is, the only guranteed way to see if a program halts or not is to run it. If it halts in the time you observe it, good. If not, then will it halt in n+1 time? Unknown.

Chaitin defines W as "the probability that a program generated by tossing a coin halts." And he says that this W will be a real number between 0 and 1 that is well-defined. He says once you define the language of the turing machine, W becomes well defined. He then claims that W is 'maximally unknowable' - that is, it is irrational like PI and e, having no mathematical structure. But it is not just irrational, he says that you can't generate W like e or PI from a formula.

You can get it in the limit from below, but it converges very, very slowly --- you can never know how close you are --- there is no computable regulator of convergence, there is no way to decide how far out to go to get the first N bits of W right.

He also claims that W is 'irreducible information' - it cannot be compressed because it is truly random.

From here it gets pretty complex, but my understanding of it is that this introduces true randomness into pure mathematics, which people think shakes things up quite a bit. He compares it to the introduction of quamtum mechanics into Physics.

Greg Egan's "Luminous" (1)

Nova Express (100383) | more than 13 years ago | (#355276)

This reminds me of Greg Egan's short story "Luminous." In it two mathamaticians deduce that there may be "bubbles" of alternate mathmatics left over from the initial chaotic state of the Big Bang, bits of an alternate, alien system of math incongruent with our own. The protagonists find such a flaw and figure out that it can be used to wrest short-term gains from economic markets. Unfortunately, the downside seems to be the possible destruction of the universe. And then it gets weird. Well worth checking out.

Philosophical interest (the future of science) (2)

higgins (100638) | more than 13 years ago | (#355279)

I happened to sit in on the lecture at CMU. Certainly Chaitin's results do owe a lot to Godel and Turing, but it's not just a rehash.

Here's what's interesting to me:

First, it's mysterious that mathematical truths are applicable to the "real world". This is a philosophical question that people have struggled over for a long time. Why is it that abstract mathematical structures discovered without any reference to physics often later turn out to be useful in physical theories?

Now consider what Chaitin is saying. Very few mathematical truths have any structure at all. That means that we can't prove the vast majority of true theorems, and if you were to pick a mathematical truth out of the air it is unlikely in the extreme that you could find a proof for it.

Put these two facts together. Isn't it awfully surprising that mathematics is so successful at describing the real world? Math is full of unproveable truths, and yet, we seem to be able to prove a bunch of really useful things.

Now why should that be? I don't know.

If you're an optismist, you might say, how lucky! It's a good thing that the universe is structured in a way that's mostly congruent with the proveable sections of mathematics.

If you're a pessimist, you might wonder how long our luck is going to last.

There's a solution, sort of... (2)

Tom7 (102298) | more than 13 years ago | (#355280)

Sounds like another good reason to be a constructivisit. In Constructive mathematics, numbers are defined in terms of a total function which computes them (or, for a "real" number, a function which can get you arbitrarily close to them). None of this "let n = 1 if the continuum hypothesis is true, 0 otherwise" stuff! Constructive mathematics is pretty nice, though some "obvious" stuff is not provable.

Here's some links:

http://plato.stanford.edu/entries/mathematics-cons tructive/ [stanford.edu]

http://www.cs.cmu.edu/~fp/courses/logic/ [cmu.edu]

Of course, some classicists find delight in how insanely undecidable their mathematics is, and that's fine, too. =)

\Omega_{UTM} is a pretty cool idea, though, much worse than the standard trick of defining which has decimal digit n = 1 if turing machine n halts, 0 otherwise (also undecidable, but not as hopeless as his!). I wish I hadn't missed the lecture.

Re:Like Any Another (1)

snarkh (118018) | more than 13 years ago | (#355289)

The academic definition of science is: that which follows the philosophy of positivism and uses the scientific method. Mathematics does not fit this definition.

Science is that which follows the scientific method. Hm...

Positivism had not existed before early 19th century. Are you saying there had been no science prior to that?

Oxford English Dictionary:

6.b.In modern use, often treated as synonymous with 'Natural and Physical Science', and thus restricted to those branches of study that relate to the phenomena of the material universe and their laws, sometimes with implied exclusion of pure mathematics. This is now the dominant sense in ordinary use.

Whether you consider math to be a science or not is really a question of whether mathematical knowledge has a separate reality.

Ah, mathematicians... (2)

smallstepforman (121366) | more than 13 years ago | (#355291)

Reading about eternal mathematical problems reminds me of a joke I once heard.

Two baloonists (the baloon with a basket) are lost and decide to land and ask for directions. They find an elderly gentleman strolling through the countryside and land the baloon next to him.
"Excuse me, sir, can you tell us where we are?"
The gentleman crossed his arms, scrathed his head, rested his chin in his palm and then gently said - "Why, you're in a baloon." and walked off.
The two baloonists just shrugged their shoulders and took off again. Once high up in the air, one of the baloonists turned to his friend and said:
"You know, I think that the elderly gentleman was a Mathematician."
"Really, how can you tell?"
"First, he was intelligent because he thought long before answering. Second, his answer was correct. Lastly, his answer hasn't helped us at all."

i like this guy (1)

Lord Omlette (124579) | more than 13 years ago | (#355293)

"So my talk is very impractical. We all know that you can have a start-up and in one year make a million dollars if you're lucky with the web. So this is about how not to make any money with the web. This is about how to ruin your career by thinking about philosophy instead."

because he's worried more about the pursuit of knowledge than the pursuit of cash. big ups for that...
--
Peace,
Lord Omlette
ICQ# 77863057

Re:This IS surprising! (1)

Lord Omlette (124579) | more than 13 years ago | (#355294)

you know, one time a teacher failed me saying "W is NOT omega!!" Wish there was an easy way to put Greek letters in these boxes.
--
Peace,
Lord Omlette
ICQ# 77863057

Re:broken link? (1)

dcshoes (127523) | more than 13 years ago | (#355295)

The first link in the story also does not work for me.

Corollary (3)

ozbird (127571) | more than 13 years ago | (#355296)

'If mathematicians find any connections between these facts, they do so by luck.'

And if Slashdot posts a connection to these facts, the mathematicians website is out of luck.

Okay, so we didn't invent/create math. (1)

crashnbur (127738) | more than 13 years ago | (#355297)

Thanks, mr. genius, but we already knew that we didn't create math. We are not in control of how we discover our mathematical theorems, and we are not in control of how they relate. Whether lucky or not, the laws of mathematics stand, and this is one case that I'll rely on hundreds of years of mathematicians and philosophers instead of one brainy information age scientist/mathematician.

There ARE unifying mathematical axioms! (1)

T. (128661) | more than 13 years ago | (#355298)

In fact, the following two should seem both fundamental and self-evident: 1. unity; 2. zero. The natural consequence of these are, in a rigorous sense, the notions of order and truth. From these we may then construct sets and operations. Chief among operations seems to be union. From this we arrive at the additive inverse. No mathematics exists beyond the space defined in the commonplace transform (1) --> (-1).

Re:This is not surprising (1)

Jagasian (129329) | more than 13 years ago | (#355300)

Of course it has helped throw more doubt on the current popular foundations of mathematics, just as Godel, Church, and Turing threw doubt on the quality of our popular mathematical foundations.THERE ARE ALTERNATIVE MATHEMATICAL PHILOSOPHIES! [barnesandnoble.com]

Re:Like Any Another (1)

Jagasian (129329) | more than 13 years ago | (#355301)

Please tell me I didn't have a point! I can't believe that I got modded down for making a good point that there are many things that are not science.

The moderation is really bad on for this article. Earlier I saw something get a +3, which was complete and total garbage about some prophecy of this by a writer.

Don't mod posts unless you understand the topics!

Re:i like this guy (1)

Jagasian (129329) | more than 13 years ago | (#355302)

Yeah, but try convincing the majority of people in the USA that greed is a bad thing, and that understanding is a good thing.

Look at Universities for instance. Most people are in Univeristy so that they can make more money in the industry. Most of these people can't even see another good reason for being in a University, besides: "It will get me more pay."

Why oh why did you have to get me started on this topic. I better stop typing before I get modded down even more. (If you go against the norm, you risk getting modded down.)

Re:Like Any Another (1)

Jagasian (129329) | more than 13 years ago | (#355303)

I have been studying the foundations of mathematics as a special person interest of mine. I am a CS major, so we aren't forced to learn such interesting things, which is sad.

Well, if you really think that I am right, can you spare some mod points ;-) These guys are putting me into a hole. This always happens when mathematical topics come up.

Re:We create math everyday. (1)

Jagasian (129329) | more than 13 years ago | (#355304)

Nice troll. While you may believe these things, and I am not saying they are not true (your beliefs)... I am saying that the Platonic Idealism is a belief system... a religion. Just as the existance of God cannot be proved, neither can an extrenal objective absolute reality of ideal forms.

Re:We create math everyday. (1)

Jagasian (129329) | more than 13 years ago | (#355305)

*cough* *cough* *choke*

Ok, while readers of the book may be crappy programmers :-) it doesn't mean that they don't know their math.

Re:We create math everyday. (1)

Jagasian (129329) | more than 13 years ago | (#355306)

Who ever said that the good books were cheap. One of the reasons that the book is so expensive is because it uses a sew-threw-the-fold binding, which is the most high quality book binding and the most desired too because the book lays flat when opened to any page. It is also the strongest binding.

In addition, considering that there are only 1 or 2 different books available on Brouwer's Intuitionism, other economic factors come into play.

If you disagree with the price, then ask your library for a copy. If they don't have it, they can interlibrary loan it for free. Remember, knowledge doesn't necessarily come easily.

Finally, Intuitionism is not a hoax. Search the net for info on "Brouwer Intuitionism". It is a very well established mathematical philosophy/system, which has been around for almost 100 years.

Re:Like Any Another (1)

Jagasian (129329) | more than 13 years ago | (#355307)

The academic definition of science is: that which follows the philosophy of positivism and uses the scientific method. Mathematics does not fit this definition.

Now, if you use the "joe blow" defintion for many different words, you will end up with totally different things. Take the joe blow definition of "hacker" for exmaple. How about the joe blow definition of "operating system", or "computer", or the "internet"? Do you get my point? Pulling out the dictionary only verifies that joe blow uses the particular word in that way.

Furthermore, I figured that we are discussing an academic topic, and therefore, I assumed the academic definition of "science".

Re:The main point (1)

Jagasian (129329) | more than 13 years ago | (#355308)

... and what did he say the number was exactly or did he never give an exact number because he made reference to non-constructive methods which have been questioned by mathematicians for 100s of years (Kronecker and Brouwer for example).

Re:There's a solution, sort of... (1)

Jagasian (129329) | more than 13 years ago | (#355309)

Actually, not all of constructive mathematics is as you say. You have described a small subset of constructive mathematics called "constructive recursive mathematics and Markov's School" (page 25 in Constructivism in Mathematics, An Introduction by Troelstra and van Dalen). Constructive recursive mathematics formulates everything as an algorithm. However, there are other constructive mathematical systems such as the most famous Brouwer's Intuitionism and lesser systems such as Finitism, Bishop's Constructive Mathematics, etc...

You do bring up a good point though, that there are alternative mathematical foundations which start from a different philosophy and therefore aren't plagued with such problems pointed out by this topic. The problem is that your average joe blow never questions what he is taught, and therefore, we end up with masses of sheep who whole-heartedly believe that Platonic mathematics is absolutely and undeniably correct. What is even better is when you try to point out people's dogmatic mathematical beliefs, they call you a troll and mod you down into nothingness. It is too overwhelming for some people to realize that the mathematics that they have been taught is horribly flawed, and that better foundations/philosophies have been around for quite some time.

Re:i like this guy (1)

Jagasian (129329) | more than 13 years ago | (#355310)

*cough* case in point *cough*
Greed implies taking more than your share (see Marxism). Knowledge is created within a person's mind, and therefore labelling such activity as greed is an abuse of the word. If I create it within myself, then how could it be taking away from somebody else's share?

I also never implied that I wanted to stop people from living their lives, as long as them doing so does not harm others. However, if gaining monopolistic power over an industry makes a person happy, yet harms thousands of people... I say that is a "wrong" way to live. If running around killing people makes someone happy, well I am sorry, but that is a "wrong" way to live.
Most people call my philosophy "libertarianism". I claim that the greedy violate the libertarian law by taking away from others.

We create math everyday. (2)

Jagasian (129329) | more than 13 years ago | (#355314)

What you have just touched on is a philosophical issue. You for some reason believe in the Platonic idealism, that mathematical concepts exist independently of the mind. Without the existance of humans, you believe that mathematics still exists.

However, this belief has never been proven. It is nothing more than a belief, just like many people agree that their exists a God. Therefore, just as the belief in the existance of a God turns something into a religion; the belief in the Platonic idealism turns mathematics into a religion - rife with all of the problems associated with religions!

Great mathematicians such as L.E.J. Brouwer argued that such dogmatic beliefs should not be used within mathematics, because it causes horrible foundational problems of paradox, undecidability, and incompleteness. Brouwer went on to establish the mathematical philosophy of intuitionism, and then built an entire mathematical system ontop of that. In effect, he created mathematical intuitionism, just as each mathematician creates (or recreates depending on how you look at it) mathematical concepts in their mind.

The Platonic idealism has been a cancer on the foundation of mathematics for thousands of years. Please, stop and realize that the Platonic idealism is nothing more than a belief system, and witness how it has partially destroyed mathematics.
THERE ARE ALTERNATIVE MATHEMATICAL PHILOSOPHIES! [barnesandnoble.com]

Occam's Razor (2)

Jagasian (129329) | more than 13 years ago | (#355315)

This could be used to argue against the principle of Occam's Razor (which says that the simplest theory that fits the facts of a problem is the one that should be selected), because science is based on the belief that mathematical concepts can be usefully projected onto the perception of nature. If it turns out that the mathematics used is horribly complex and disconnected, then Occam's Razor could cause a scientist to turn from the truth more often than he/she is turned towards the truth by the principle.

Note that I use the term "truth" with regards to scientific "truth", realizing that science can never in fact portray any absolute truth, as is the normal definition of truth (i.e. undeniable truth). This is why science has evolutionary mechanisms built in like peer review and disproving old theories.

Re:Situation of modern mathematics ;-) (2)

Jagasian (129329) | more than 13 years ago | (#355316)

Actually, you couldn't be more wrong. If integer arithematic cannot be proven to be consistant (free from contradiction), then there is the possibility that you could wake up one day and have 2+2=5. I am not claiming that any of this will happen, because I have no mathematical proof, but the whole problem with a lack of consistancy proof is the problem you have mentioned... waking up only to realize that your math was nothing more than a "Matrix" (in the movie sense) so to speak.

Time and time again, the chosen philosophy of mathematics used for a foundation of mathematics as shown to cause huge differences in the actual mathematical system. Bertrand Russell's paradoxes (A set which contains all sets that do not contain themselves. This set both contains itself and doesn't contain itself.), Brouwer's Intuitionism (mathematics is complete, and undeniably consistant in with this philosophy), the Platonic Idealism (you have foundations that are of the quality of the foundations of Christianity), Formalism (Godel's Incompleteness Theorem says that we can never know if this system is flawed or not), etc...

Saying that philosophy does influence mathematics down to a consistancy level ignores hundreds of years of mathematical history! Philosophical foundations can lead to actual contradictions. This is why philosophy has an extremely important role in mathematics.

Re:This IS surprising! (2)

istartedi (132515) | more than 13 years ago | (#355323)

They cannot be compressed

Well, we could agree to call it W_UTM. Then all we would have to do to compress it is send W_UTM and people would know what it was.

Of course the codec would eventually become infinitely large, but as long as our pace of discovering stuff like this doesn't outpace Moore's law, we are fine.

I'm only semi-serious.

Re:i like this guy (1)

Zebbers (134389) | more than 13 years ago | (#355324)

ahh yes, because greed for knowledge is so much better than greed for money...sorry bub. Let people live their lives. Whatever their main focus will be, it should make them happy. If that means studying and becoming more knowledgeable so be it. If that means earning enough cash to have 15 cars, so be it. there is no "right" way to live. If you think there is, please tell me. I'm sure it will have an extreme impact on the whole universe and not just itself ;)

Re:Wrong (1)

kfg (145172) | more than 13 years ago | (#355329)

Of course it's a parlour trick. I've performed it in a parlour many, many times, always with amusing and " parlour trickish" results.

A mathmatical proof may be quite legitimate and based on very sophisticated mathmatical principles and still be a parlour trick.

What MAKES it a parlour trick is the aforementioned ability to perform it at a party for non mathmaticians to achieve a certain entertaining result.

Try THAT with the binomial theorem.

Perhaps what you call a parlour trick would come in the rounding down of 1.9999. . . to 1.

YOU think of the numeral as equal to 2. People in a parlour would have no problem with the idea of rounding down 1.9999. . . to 1.

Please also bear in mind that my orginal post to which you responded was not intended seriously, but was a JOKE.

KFG

Re:Two to One odds (1)

kfg (145172) | more than 13 years ago | (#355330)

For my take on the "parlour trick" aspect of this proof see my reply above.

As for the rest, I took those undergraduate courses 25 years ago.

KFG, Physics and math major in decades past

Two to One odds (2)

kfg (145172) | more than 13 years ago | (#355331)

To turn the old saw on its head:

Two does equal one for sufficiently small values of 2.

Actually, using the old mathmatical parlour trick of showing that 1.99999. . . = 2 one could at least show that 2=1 when rounded to the lowest integer.

KFG

Re:2 + 2 = ? (1)

Glowing Fish (155236) | more than 13 years ago | (#355334)

I am so glad that I live in a world where a persons intelligence can so adequatly be judged by an AC reading one comment that I wrote in an off hand moment of boredom.

Damn, Mr. Coward (are you any relation to Noel?), maybe you should get a job at a psychiatric facility diagnosing peoples sanity by their ability to play ping-pong. A person with your talents has many routes open to them in life.

2 + 2 = ? (2)

Glowing Fish (155236) | more than 13 years ago | (#355335)

Freedom is the freedom to say 2 and 2 is 4, everything else follows from there.

Old news... (2)

Saint Aardvark (159009) | more than 13 years ago | (#355337)

From "The Omega Number", by Robert Ludlum:

"Do you realize what this means?" Johnson looked at the mathematician worriedly. "I have to report this to the CIA. I'm sorry."
"But why? What does this have to do with national security?" asked Thomas.
"I can't tell you. In fact, it's--" Suddenly a shot rang out, and Thomas watched in horror as the Dean of Mathematics slumped forward, a surprised look on his face. He caught Johnson in his arms as half a dozen more shots were fired into the office, and dragged him frantically behind a desk.
He looked down and saw that the shirt was red. That was bad. Then he saw that the redness was spreading. That was very bad. The shots stopped, but Thomas' ears kept ringing.
"The...Omega number..." gasped Johnson.
"Don't talk! Save your strength!"
"I'm dead...anyway...you have to...tell the CIA...can't let...the Soviets...know...about the hole...the weaknesses...in our mathematical...model..."
The dean stopped, gave a pitiful little gasp, and went limp in Thomas' arms.

It's not his best work by any means. But it was oddly prophetic.

The ultimate program of life, the universe and 42? (3)

TeknoHog (164938) | more than 13 years ago | (#355340)

Just gunzip the hex representation of the Omega number.

--

Re:2 + 2 = ? (1)

ebyrob (165903) | more than 13 years ago | (#355341)

Freedom is the freedom to say, "I don't know, and here's why I don't think you do either."

I think intellectual honesty is very important. When we teach 2+2=4, we should also teach the underlying assumptions.

Disagreeing with the assumptions, does not excuse one from responsibility for the subject matter. But, it should not be a reason to flunk, browbeat or otherwise attack the party in disagreement either.

Perhaps another point.... (1)

ebyrob (165903) | more than 13 years ago | (#355342)

Granted, this stuff has been around a while, (since 1933) but I think the point perhaps is that one fundumental tennant of math/science is on it's heals.

The idea of Scientific Naturalism. The idea that math/science/human brains will eventually be able to "understand" everything "meaningful" in the universe.

The fact of the matter is there are things that can be demonstrabably "proved" that we don't know, and we will NEVER know(because they can't be known!!)! So, faith in human understanding of EVERYTHING is at best misplaced.

The first step to solving a problem is admitting you have it. If science/math/colleges everywhere can admit they have a problem, that of bad philosophy. Perhaps they can begin to reverse the damage to creative minds, deflate egos, and begin to get some useful work done.

Does this mean 2+2=4 is not a useful concept? No. Does this mean 2+2=4 is TRUTH? I'm only human, what's TRUTH?

Re:We create math everyday. (1)

ebyrob (165903) | more than 13 years ago | (#355343)

I'd email you but you don't seem to have an email address posted.

You seem to have rendered my comment following redundant. Guess I should read more thoroughly before posting.

I'm curious if there are places to get more information on Brouwer without buying a $150 book.

I'm very interested in good philosophy, and the lack thereof in our educational system as well as my country (USA) as a whole.

My email is "beby@leveltwo.com" for response (if any).

Re:These Statements need proof to back them. (1)

xigxag (167441) | more than 13 years ago | (#355345)

First of all, the statement that mathematical theory is riddled with holes is questionable. Finding an unprovable statement is a rarity that happens once every so often.

That sounds more like an indication of our own limited way of experiencing the universe. For example, if you ask a child to name a number at random, they will come up with a natural number very near zero. But that hardly proves that other numbers are rarities. It merely means that children, by and large, don't have the vocabulary to express "complicated" numbers. Even if you get that smartass junior genius child who says "a googolplex," he will still be lower than virtually all natural numbers, which themselves are an insignificant subset of R (of which the majority are nonconstructible transcendental numbers which could not be named even in principle.)

Mathematical statements are the same. Due to our own concrete finite minds, we could not possibly state, experience or comprehend the vast majority of them. And it is in that vast solution space that lie the unprovable statements Chaitin alludes to.

Re:2 + 2 = ? (2)

logicnazi (169418) | more than 13 years ago | (#355346)

"Disagreeing with the assumptions...."

Mathematics doesn't really have the kind of assumptions you can disagree with. Two, Four and plus are not the same things in mathematics that we may call two and four in the outside world. Two is by definition the successor of the succesor of 0 and four has a similar definition. Therefore the conclusion that two and two is four follows inevitably from the definitions of two, four and plus because by definition these are the things obeying the appropriate axioms (this may be confusing because we actually use the same words to refer to real world concepts, and concepts in various axiomatic systems which arent always the same thing).

Re:Isn't this already known? (3)

logicnazi (169418) | more than 13 years ago | (#355348)

Yes its been known for some time (studied it in class two years ago so it must have been around for a good deal of time before then).

His stuff is certainly interesting, and his results about the omega number are bizarre but you are right it isn't THAT revolutionary. Once you accept the results of Godel's theorem the fact that you can somehow concentrate all that unprovability in one place drives the strangeness home but isn't fundamentally upsetting.

Re:provability (1)

fatphil (181876) | more than 13 years ago | (#355352)

As commonly used by mathematician /theorem/ must be not just provable, but proved.
Conjectures and postulates can get by without proof though (for different reasons though).

FatPhil
--

These Statements need proof to back them. (4)

Syllepsis (196919) | more than 13 years ago | (#355355)

I think that based on the lecture notes, the New Scientist is just trying to make a sensational article out of a nice lecture on a few of the more surprising highlights of 20th century math.

First of all, the statement that mathematical theory is riddled with holes is questionable. Finding an unprovable statement is a rarity that happens once every so often. Most things can either be proven or separated out as an axiom (such as the Continuum Hypothesis). Granted, every formal algebraic system is going to have at least one unprovable theorem, but no one has attempted to count out the percentage of theorems that are provable.

The New Scientist is claiming that the percentage of provable theorems is small compared to the number of theorems in any given system. This is akin to the idea that the rational numbers are "sparse" on the real number line. Such a statement about a formal system, such as the ZFC axioms of set theory, needs to be proven.

It would be an interesting study to Godelize ZFC in an intuitive way and use automated theorem proving to see what percentage of statements of a given length are theorems, and what percentage of those could be proven with proofs of less than a certain length. Then by asymptotic analysis one might be able to make a statement to see if "most" theorems could be proven.

Such a study would be similar in method to Graphical Evolution, but would require quite a bit of supercomputer time. Even then, some really difficult proofs would have to be made. However, one does not know if the statement is provable :)

Re:Like Any Another (1)

arnald (201434) | more than 13 years ago | (#355356)

Never mind - those of us with half a clue about mathematics (including those of us who have spent the past week revising their Foundations of Mathematics work :-) ) know that you are right.

Re:Like Any Another (1)

arnald (201434) | more than 13 years ago | (#355357)

Nothing doing, I'm afraid - no mod points today. You have my sympathies, though. :-)

Here at Oxford the Maths faculty give an excellent course on the Foundations of Mathematics, available to Computation students as well as Maths and Maths + Comp students. Covers axiomatic set theory, mathematical logic, and basic computability (although the latter is barely more than an introduction).

I personally do joint schools Maths + Comp, but I may have taken the course if I just did straight computing, as it's very interesting. Shame your place doesn't let you do it...

Re:2 + 2 = ? (1)

Vuarnet (207505) | more than 13 years ago | (#355358)

Freedom is the freedom to say 2 and 2 is 4, everything else follows from there.
Depends on your definition of "2" and "4". For instance, 2 cups of sugar added to 2 cups of water don't produce 4 cups of sugared water.

Besides, as George Orwell said, "there are 5 fingers". Mostly offtopic, but then I don't have anything to do on a Sunday afternoon.

Re:This IS surprising! (1)

rabidcow (209019) | more than 13 years ago | (#355360)

the probability that a random Turing machine will halt

The problem I have with this is that you're starting with randomness. I fail to see why it's shoking that the result is random.

It seems to me that the probability of any truely random event (whether or not a truely random turing machine will halt) will have the same properties. This is getting chaos out of chaos, not chaos out of order.

Re:provability (1)

Beatlebum (213957) | more than 13 years ago | (#355362)

duh, this is exactly what Godel proved- there exist theorems for which it can be proved that a solution can never be found. An example is the problem of tiling the plane.

Re:What would Penrose say about this? (1)

Beatlebum (213957) | more than 13 years ago | (#355363)

If anyone is interested in further reading, I would highly recommend The Emperor's New Mind : Concerning Computers, Minds, and the Laws of Physics by Roger Penrose [amazon.com] . However, be prepared to devote some serious time to this book, I have spent days pondering single paragraphs.

Re:What would Penrose say about this? (1)

NonSequor (230139) | more than 13 years ago | (#355375)

Isn't Penrose the one who believes that consciousness is the result of quantum effects in electrons moving through microtubes in the neurons? As far as I know there is no current evidence of these quantum effects and until there is this is just speculation. The work is good, just not my pet theory. I kind of like the idea that the hippocampus (or was it hypothalamus, whichever one controls concentration) is the center of consciousness. This matches Sartre's idea that the mind is made up of not one but many consciousness (he may have gotten that from Husserl and some others though). Sartre rejects Descarte's "I think therefore I am" (although he also says that the existence of oneself and others is a "factual necessity") because the consciousness that does the thinking isn't the one that says "I am" (I believe that the consciousness that says "I am" corresponds to the hippocampus (or hypothalamus or whatever).


"Homo sum: humani nil a me alienum puto"
(I am a man: nothing human is alien to me)

Situation of modern mathematics ;-) (4)

m51 (255152) | more than 13 years ago | (#355380)

As things get circulated like this in spells every now and then, it becomes time to recirculate an important theme: philosophical problems do not equate to mathematical inconsistencies. By standards of purely mathematical order, there aren't holes such that you might wake up tommorow to discover that 2+2 suddenly equals 5. ^_^ A parallel to this type of discussion can be given from quantum mechanics; the Schrodinger's Cat paradox. While it does present serious philosophical and logical problems, what it does not do is poke any actual holes into quantum mechanics. Anyone particularly interested in this topic should check out the work of Godel, who did some very intriguing work earlier in the 20th century.

Aleph-Zero is the Omega number?! (1)

doctored (257519) | more than 13 years ago | (#355381)

If I recall correctly, Aleph-Zero is the same as the omega number...

Re:We create math everyday. (2)

Anoriymous Coward (257749) | more than 13 years ago | (#355382)

I followed your link. Here's what I found:

bn.com customers who bought this book also bought:
Programming in Visual Basic: Version 6.0

I rest my case!
--
#include "stdio.h"

Re:We create math everyday. (2)

Anoriymous Coward (257749) | more than 13 years ago | (#355383)

Right. And we can compute W_UTM with a SAFEARRAY of VARIANTs.
--
#include "stdio.h"

Maxwell's Demon? (1)

servasius_jr (258414) | more than 13 years ago | (#355384)

In the lecture transcript some mention is made of Maxwell's Demon; I've heard about this before (through Thomas Pynchon) but I'm still pretty vague on the details. Is there anybody out there with enough background in hard science and free time to explain this or point me in the direction of good references? Thanks . . . .

Does 'lucky' mean NP-hard? (1)

paranormalized (278300) | more than 13 years ago | (#355386)

Now, it's already been proven that a single 'unified theory of mathematics' is a boondoggle, (see previous posts referencing Godel's work) so the first part of this idea (i.e., 'holes' in mathematics) is nothing new... However, the site has been /.-ed and I can't find out how much he describes the connections between facts that mathematicians make...

The difficulty of the connections is what interests me... can we set up computers and programs to search for such connections, or is the problem too big or difficult (like NP-complete, or something) to be automated? And if it is, what philosophical implications does that hold? For instance, if finding Godel's Theorem was NP-complete or hard, what does that imply about human intelligence? I doubt that is the case, but if it is, well, the implications are pretty darned impressive... Some would seize upon it as proof of a human soul, others would start looking for a quantum computer implementation in the human brain, etc...

On a related note, can quantum computers solve NP complete problems in P time? I haven't kept up w/ quantum computers, so I don't really know their full possibilities or shortcomings.... If I were still considering a mathematics major I might want to delve into these issues personally, but I'm not, so I'll just ask the rest of the /. crowd and see what happens...

-----
IANASRP- I am not a self-referential phrase
-----

Look ma, karma whoring! (1)

Heidi Wall (317302) | more than 13 years ago | (#355394)

A quick search [google.com] on google [google.com] turns up the masters homepage [auckland.ac.nz] . There.

The guys seems to be something of a pop star among mathematicians.

And I'm now looking forward for the obligatory halfdozen proofs that 2=1 in the next fifty comments. Yay for Slashdot...


/* And you'll never guess what the dog had */
/* in its mouth... */

"Then some magic happens" (2)

BillyGoatThree (324006) | more than 13 years ago | (#355397)

This reminds me of that cartoon where all these equations are on the left and on the right, but in the middle is a box with "then some magic happens" written in it.

I understand your argument that W_UTM is "unknowable" (non-computable, anyway--surely GOD knows it). And I understand how to write a number in binary. Then we hit this: "Well, it turns out these 0's and 1's have no mathematical structure. They cannot be compressed."

Yeah the old "it turns out" argument. I guess he's leaving that part up to the students? Or maybe the proof is "trivial"? I can understand leaving out details, but for crying out loud this is the whole POINT of the research. WHY don't they have structure?

In any case, I still say it's unsurprising. I would have been skeptical of a claim (Godel's) that there are an infinite number of theorems without there ALSO being an infinite source for those theorems to theorize about.
--

This is not surprising (4)

BillyGoatThree (324006) | more than 13 years ago | (#355398)

As someone else mentioned, this sounds like a pretty simple application of an (admittedly difficult) earlier result by Godel: Given a formal system of "sufficient power" there will always be theorems expressible but unprovable in that system. (And if you add the "missing theorem" to the system, the resulting system has the same "problem", ad infinitum)

Considering that Godel's stuff came out in 1933(?) and if this work really IS just a restatement of that fact then I doubt the "foundations of mathematics are in an uproar".
--

too is wun (1)

scorcherer (325559) | more than 13 years ago | (#355400)

Here goes:

a = b
a^2 = ab
a^2 - b^2 = ab - b^2
(a - b)(a + b) = (a - b)b
a + b = b
2b = b
2 = 1

QED/QCD.

--

provability (1)

TrollFeeder (396384) | more than 13 years ago | (#355401)

Sure, we've known for a while that at any point in mathematics there will be theorems which are expressable but not provable.

However, as mathematics advances, do we know if there are theorems which can never be proven?

--

broken link? (1)

jdschmid (398680) | more than 13 years ago | (#355402)

...hmm...well...atleast for me it's broken
:(

cya
jds

Re:Situation of modern mathematics ;-) (1)

Vornzog (409419) | more than 13 years ago | (#355404)

If integer arithematic cannot be proven to be consistant (free from contradiction), then there is the possibility that you could wake up one day and have 2+2=5.

Not quite. Learn a bit of math, check out Godel's Incompleteness Theorem. Then learn a bit more math, and check out Godel's Completeness Theorem.

The first deals with the general case axiomized mathmatical system. These have problems with - suprise - incompleteness!

The second allows you to get around the first, at least enough to do calculus. Nice how that works out.

And so, in spite of Godel, or maybe because of him, freedom is still the freedom to say 2 + 2 = 4. Now if only Free Beer followed from that...

Vornzog

Who can decide a priori? Nobody.

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