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More on Riemann Hypothesis
Anonymous Coward writes "The NYTimes has a little story on a recent conference at New York University's Courant Institute where mathematicians gathered to discuss potential attacks on the Riemann hypothesis. The Clay Mathematics Institute had announced an award of a million dollars for a proof (or refutation) of the Riemann hypothesis during the millenial celebrations. That million dollars won't be worth much if it takes as long as that Last Theorem by Fermat to solve. There were some interesting observations such as the statistical distribution of the zeros looked just like calculations on the energy levels of large atoms." We did a related story on hard math problems two years ago.
We're already being searched at airports, now mathematicians can't carry a protractor or a compass without being looked as being suspicious. When will terrorists learn that attacking math problems never solves anything. Wait, maybe it does...
Who stands a better chance at proving/refuting the hypothesis: mathematicians pushing chalk or an NSA supercomputer using brute force? After all, this has an application in encryption...
I do hope that a mathematician wins the prize money though.
"What is the sound of one belly slapping?"
I have discovered a truly remarkable proof which this post is too small to contain.
Can someone explain exactly what this is and what it means in very small words?
My understanding of the article is that:
A) You can't predict prime numbers.
B) That guy predicted prime numbers.
C) Alot of money goes to whoever proves how the hell he predicted prime numbers.
Ca)If we know how he predicted them we can crack old codes and make new ones?
Never confuse volume with power.
Who is responsible for this? I ask you, slashdot just never seems to have any decent stories anymore, just keeps recycling stuff over and over... ;-)
n cm
And what is n cm?
the length of the piece of string
He wrote a function called the "zeta function." Any number, when fed into this equation, will yield a result somewhere on a plane. For some reason, primes always plot along one of the axes. No one can figure out why.
Nope, no sig
I mean, they're just cheap noodles. No reason to attack them. They are very popular with poor college students and geeks, and .... oh, wait...
For a second there I thought that said 'Rainman Hypothesis.' Somethine to do with counting cards maybe?
I will believe it as soon as I see some sort of news link. You hear all kinds of crap on the radio, fm/am, anymore (mmmm XM radio yummy) that until I see a news link from a reputable source I won't believe it.
:)
Besides he needs to finish the Dark Tower series before he can die.
J
Can you say, AC Troll?
Fuck Off!!
Why did I even bother?? (my sig sucks, but it's better than yours!!)
As ever with the NYT, log in with:
: nospamnospam : nospamnospam
User
Password
I am a Karma Library.
Wha-wha? I was under the impression that proofs are rock-solid demonstrations of a particular fact given a set of well-defined mathematical laws . . .
With highly visible scientists like our own PhysicsGenius, SIGFPE, and others here on Slashdot, along with this site's tendency towards science and mathematics, I doubt that anyone will be losing faith in science any time soon.
That said, biologists have a long way to go in getting themselves to the stature of mathematicians and physicists. Those groups deal with laws and pure theories. Biologists can't even prove evolution to a good enough degree to refute intelligent design theory. If you follow the physics, I don't see how you could believe anything else, there's simply too much beauty and "intelligence" for this all to have happened by chance.
"We did a related story on hard math problems two years ago."
I know various people complain when Slashdot re-posts stories, and it's good to see you're taking note and warning us, but that's going a little too far back.
Maran
Things have changed since my day. It used to be that anyone who was capable of a serious attempt on the Riemann hypothesis would get to work because the problem needed solving and the kudos would be immense. Any monetary reward would be incidental. I suppose a million dollars will attract mathematicians who are today working on other topics. Presumably the Clay Institute feels that Riemann has to compete hard against other avenues of research. Or maybe they are targetting all the rich quants who were just laid off On Wall Street.
http://www.bearnol.pwp.blueyonder.co.uk/Math/riema nn.htm
I'll be in New York City this weekend. Meet me in front of the Museum of Modern Art at 3:00 PM this Saturday. I'll be wearing olive nylon hiking shorts and a tan t-shirt if it's warm, if it's raining I'll have on jeans and a red pullover. I'll be holding a pack of Lucky Strikes in my hand. Just walk up to me and say "I'm from slashdot." I'll fucking smash your mouth in right then and there. Then I'll take you up to the burrito place on the corner, buy you a burrito, and then punch you in the stomach and watch it fly all over the street. I'll beat you so bad your innards will turn to liquid, and drain out of your bunghole. I'll make you cry right in the center of the biggest city in the world, you little half-man.
Time to put up or shut up, punk. MoMA, 3:00, Saturday. Be there.
What an attitude! You are so full of yourself that all you have really done is prove that you don't know jack.
If you can't explain something in ordinary words to a layman, then you really don't understand it. It isn't until you teach something that you really begin to understand it.
Infuriate left and right
The mathematician stands a better chance of proving the hypothesis, but the NSA supercomputer stands a better chance of refuting the hyposthesis.
If a computer disproves it by finding a prime that happens to map wrong on the zeta theorem, mathemeticians will still want to know why this one didn't work, when all the others have.
BTW You have also determined a relative probability -- "better chance" -- of something that may be undefined. If the theorem is in fact true, then a computer's chance of disproving it is exactly equal to a mathemetician's chances: zero.
Nope, no sig
I'm still waiting for someone to prove this: Step One: Steal Underpants Step Two: ...
Step Three: Profit.
When they can prove that, THEN I'll be impressed. :)
This is my sig. Its pathetic.
I'm still waiting for someone to prove this: ... :)
Step One: Steal Underpants
Step Two:
Step Three: Profit.
When they can prove that, THEN I'll be impressed.
This is my sig. Its pathetic.
Explain sight to the blind.
Explain sound to the deaf.
Explain intuitive leaps of any kind.
Not every concept maps to a clean explanation in a few simple words. That's why we have the different words. True, most concepts can be mapped somewhat to common language, but come on...give the guy a fucking break. We're talking about advanced mathematics.
Get off YOUR high horse, bubby.
Keep in mind this proof looks much better if you can actually use the square root symbol
The problem:
Prove that women are all evil.
(With written proof, men don't have to worry about women arguing this fact anymore
The proof:
Given that:
Proceede with the proof:
See what an undergrad in Mathematics, an undergrad in C.S., and a Master's in C.S. gets you
Seriously, I wish someone could prove that P=NP. I hated graduate Algorithms! This would have eliminated a portion of my least favorite topic in that course (NP and NP-completeness). If this world is not truely hell, someone will prove that and share it to help prevent the suffering of innocent C.S. graduate students.
HallmarkOrnaments.Com
While I admit that it's certainly possible that Riemann's Hypothesis may, God willing, be proven or disproven, isn't it also possible that it cannot be either proven or disproven under the applicable mathematical system? Gödel's Theorem means that that's a possibility, doesn't it? Not everything has to necessarily be true or untrue...
Oh cruel fate, to be thusly boned! Ask not for whom the bone bones; it bones for thee. -Bender
Apparently there's a distributed computing project called ZetaGrid which has calculated the first 50 billion zeros out ... if you're bored of SETI@Home, this might be a nice change of pace.
Riemann Hypothesis
Riemann Zeta Function
Also, there's some rather technical details on the subject, from Stephen Wolfram's (A New Kind of Science) pet site.
We can neither love nor pity nor forgive. If you make a slip in handling us you die!
Comment removed based on user account deletion
...that these proofs will not be solved using conventional methods, but they will eventually be solved using SMALL PROGRAMS with SIMPLE RULES. These rules can be run on a simple computer using my program, Mathematica. Easy!
Either that, or you can solve them by buying REAL ESTATE with NO MONEY DOWN! or by placing SMALL ADS in NEWSPAPERS with your own 900 NUMBER!!!!!
"I'm an old-fashioned type of guy. I worship the Sun and Moon as gods. And fear them."
Please ignore me. I haven't had my coffee yet. I don't know what I'm saying.
I am totally wrong.
Hardly a blip on the radar screen... now, if it were in the billions we could finally have a mathematician in the Forbes 400 ... that would be signficiant. A million dollars is puny; hardly worth the time. Hell, even Lotto winners get more money. Picking random numbers in a lotter must be more important.
Love of money is the root of evil. Introducing evil breaks the proof.
"that God -- with whom he waged a very personal war -- would not let Hardy die with such glory."
That has to be the funniest things I've read, today.
Is it me or does it seem that all "hard" mathematicians are either at war with God or trying to "refute"/"prove"/divide/discover/humiliate him/her/it/Taco?
Get your Unix fortune now!
Even if you are able to get into a cell it can be extremely difficult to stay in and keep your sanity. Many people who do get in just sort of drift off from society and are all but lost. Those few that make it often end up working alone, late at night in the back of dimly lit coffee houses.
There is simply no way to stop someone who is willing to make such sacrifices.
Whether it was strictly-speaking accurate, it was simpler and more understandable, and basicaly got across the idea of the theorem.
... on the Riemann Hypothesis:
Riemann Hypothesis
"I have never let my schooling interfere with my education." - Mark Twain
Here's a brief explaination of the Zeta function given by mathworld...
a super computer to play with for awhile...
get ascii white in on it and even the earth simulator, if anything can prove/refute this it could be that massive thing (it can simulate the earth right? haha) it could be interesting tho.
The greatest right given is the right to be wrong...
Actually, I was just about to post with the same idea (consequence of Godel's incompleteness theorum.)
When people were trying to solve Fermat's Last Theorem (essentially proving the grander Taniyama-Shimura Conjecture), I thought that people worried that Godel's incompleteness theorem could have applied and thus no solution was possible. (Simon Singh's book "Fermat's Engima" gave this impression.)
Likewise, my understanding is that Godel applies to any axiomatic system. Since our number theory is built with a finite number of axioms, it should apply.
indeed..
does Dave Sim post to /.?
Researchers at a leading US university have made an astounding discovery. They have found that the square length of the hypotenuse of a right angled triangle triangle can be found by adding the squares of the hypotenuses of the other two sides.
Dr. P Thagoras explains: "we've experimented with many kinds of right angled triangle it it seems to hold in all situations." Prof. E Clid is enthusiastic about the applications "for example a builder can predict the length of the diagonal of a plot of land withput actually measuring it. We can run the software to compute it from the sides on something as small as a laptop. A builder could easily have one of these on the actual building site."
Of course the discovery is not without skeptics. "They haven't tested every triangle", says Dr. P Appus, professor of post-modern sociology, a researcher who studies scientists themselves. "These researchers have only picked those triangles that fir the pattern. It's a kind of unconscious Freudian repression where triangles that don't fit are collaboratively eliminated from the field of view in a reactionary social construct".
But Thagoras isn't disheartened. He believes gis result might hold even for really big triangles. "I think you could use this when urban planning. I bet it'd hold for triangles miles across".
A bold claim, and only time will tell whether these claims will hold. But don't expect to see builder wielding those laptops any time soon!
-- SIGFPE
Curb CO2 emissions: Kill yourself today!
Now we have to worry about "potential attacks on the Riemann hypothesis" during the holidays...
I have discovered a truly remarkable proof of the Riemann Hypothesis...
But this margin is too small to contain it.
Remember "Bring 'em on"? *sigh
Once upon a time (1/t), pretty little Polly Nomial was strolling across a field of vectors when she came to the edge of a singularly large matrix.
Now Polly was convergent and her mother had made it an absolute condition that she must never enter such an array without her brackets on. Polly, however, who had changed her variables that morning and was feeling particularly badly behaved, ignored this condition on the grounds that it was insufficient, and made her way in amongst the complex elements.
Rows and columns enveloped her on all sides. Tangents approached her surface. She became tensor and tensor. Suddenly two branches of a hyperbola touched her at a single point. She oscillated violently, lost all sense of direction, and went completely divergent. As she reached a turning point she tripped over a square root that was protruding from the erf, and she plunged headlong down a steep gradient. When she was differentiated once more, she found herself, apparently alone, in a non-Euclidean space.
She was being watched, however. That smooth operator, Curly Pi, was lurking inner product. As he numerically analyzed her, his eyes devoured her curvilinear coordinates, and a singular expression crossed his face. Was she still convergent, he wondered. He decided to integrate improperly at once.
Hearing a common fraction behind her, Polly rotated and saw Curly approaching her with his power series expanding. She could see by his degenerate conic that he was up to no good.
"What a symmetric little polynomial you are," he said. "I can see that your angles have lots of secs."
"Oh sir," she protested, "keep away from me. I haven't got my brackets on."
"Calm yourself, my dear", said our suave operator. "Your fears are purely imaginary."
"I, i," she thought. "Perhaps he's homogeneous."
"What order are you?" the brute demanded.
"Seventeen," replied Polly.
"I suppose you've never been operated on?"
"Of course not," Polly cried indignantly. "I'm absolutely convergent."
"Come, come," said Curly. "Let's go off to a decimal place, and I'll take you to the limit!"
"Never!" gasped Polly.
"Abscissa!" he swore, using the vilest oath he knew. His patience was gone. Coshing her over the head with a log until she was powerless, Curly removed her discontinuities. He stared at her significant places and began smoothing her points of inflection. Poor Polly. She felt his hand tending to her asymptotic limit. Her convergence would soon be gone forever.
There was no mercy, for Curly was a heavyside operator. Curly's radius squared itself. Polly's loci quivered. He integrated by parts. He integrated by partial fractions. After he cofactored, he performed Runge-Kutta on her. The complex beast even went all the way around and did a contour integration. Curly went on operating until he satisfied her hypothesis, then he exponentiated and became completely orthogonal.
When Polly got home that night her mother noticed that she was no longer piecewise continuous, but had been truncated in several places. As the months went by, Polly's denominator increased monotonically. Finally she went to l'Hospital and generated a small but pathological function which left little surds all over the place and drove Polly to deviation.
The moral of the story is, "If you want to keep your expressions convergent, never allow them a single degree of freedom."
One can clearly see that the answer is 42.
Power Corrupts, But Absolute Power is Kinda Neat!
roads!, where we're going we don't need roads...
How is it you buffoons can remember what stories
.. i dunno .. use your .. just stop plaguing the world.
you ran two years ago, but don't seem to recall
what stories you ran last week?
Do us all a favor and
imagination
Number theory isn't axiomatic. I've never heard of any axioms for number theory except for Peano-style axioms for simple arithmetic. However, they aren't really used for number theory.
Gödel's theorem applies to axiomatic systems of a certain* complexity. For simple axiomatic systems Gödel's theorem doesn't apply.
* I don't think anyone has proven how complex things have to be for Gödel's theorem to apply.
OK. I need to correct myself. There are axioms of number theory, but they really aren't used by number theorists. They are mostly of interest to set theoreticians and logicians (like Gödel). I have two number theory books on my shelf and neither of them have a list of axioms.
Why is this a troll? It was a poor attempt at humor but it was no worse than the original Eureka post. The Eureka poster said that he/she had a proof but there wasn't enough space to write. The above AC said that Slashdot actually gives you more than enough space, so there are no restrictions. Of course, he doesn't have a proof, and *that's* the joke.
I'd love to see the look on Fermat's face if were challenged to show us his proof. Did Fermat actually have a proof or was he just playing with us because he could never solve the problem and thought no-one else did?
Anyway, I doubt anyone would find it funny now. Trying to analyze humor is like trying to disect a frog. The stuff you're disecting dies in the process.
Well reading thought the article, they seem to miss? a few things.
Of course primes have a generally log distribution, because every prime you find provides a factor later on down the line so the primes become more sparse.
Then there's the atoms thing, sfaik shells/energy levels are basically harmonic and a harmonic is more-or-less the opposite of a prime.
since harmonics and the increasing sparseness of primes could be taken as identical you're going to get the same distribution patterns out.
here goes
primes v harmonics
2 is prime and a harmonic root
3 is prime and a discord (root)
4 is non prime, and the second octave of the first root
5 is prime and a discord (root)
6 is non prime, and cord of the first and second roots
7 is prime and a discord (root)
8 is non prime, and third octive of the first root
9 is non prime, and first octave of the second root
etc....
thank God the internet isn't a human right.
I'm only on chapter 4 of Wolfram's opus 'A New Kind of Science' but reading about the Riemann Hypothesis just screams out connections with Wolfram's work. ANKOS is littered with these odd little diagrams of cellular automata, many of which exhibit prime number relationships.
That was an absolutely brilliant explanation. I have degrees in Economics, so I have background in mathematics, but I know basically nothing about number theory. Nonetheless, I understood your explanation of Reimann. You are going to be a good prof (of course, you'd probably prefer if I told you that you're going to be published in .
A
If you look at the page you specified: "Number theory is a theory about the integers, a set which we call Z where..." Obviosly to represent integers you first need the Peano axioms to get the natural numbers. Therefore Gödel's Incompleteness Theorem applies (you only need the Peano axioms).
Best post on the thread so far...
Now that's the basic idea, but it doesn't explain *why* it's so difficult that some of the greatest minds of the past 150 years have failed to prove it, and why the Clay institute are willing to pay a million dollars to someone who can.
Is it considered likely that our (our, as in people smarter than me) current understanding of mathematics is inadequate to explain this? Would the proof necessarily change our models? Or is it more likely that everyone is missing something obvious?
My God, it's Full of Source!
OUTSIDE_IP=$(dig +short my.ip @outsideip.net)
In my original post I was making two comments. My second comment was a reply to Dr. Molf's comment that "Godel applies to any axiomatic system". My comment was that Gödel's theorem only applies to axiom systems of a certain level of complexity.
My original point is that number theoreticians don't use axioms and don't use formal systems to get their results. Unless you can prove mathematicians are Turing machines then Gödel's result doesn't apply.
I am sorry to downplay you but apparently you do not have a strong background in math and maybe some knowledge in algorithm theory? In short: Gödel's theorem applies to all axiomatic systems containing Peano axioms. The system that number theoreticians study contains the Peano axioms (obviously as it is a theory of numbers). Therefore the Gödel theorem applies as considering the Riemann hypothesis.
You'll find the Peano axioms here. The most important thing about the Peano axioms is that they state the existence of natural numbers {1,2,3,...} (note the axioms give out an infinite number of such objects). So no Peano axioms => no theory of numbers...
See also Some Theorems Derivable from Peano's Axioms. It should help to understand what signigicance these axioms have. Also, all mathematics is axiomatic. For mathematicians if a claim is not based a axiomatic system then it is just speculation...Better place to look for Peano axioms is here. It should give the Axiom of Induction in a more sensible manner.
To the best of my knowledge, there still isn't a primality test which is provably polynomial without any additional hypotheses; all of them require the Extended Riemann Hypothesis, which is a statement about versions of the zeta-function, known as L-functions, over other number fields and in particular elliptic L-functions. There are actually some ties to Wiles' proof of Fermat's Last Theorem here too, but I won't claim to understand them, much less be able to explain them. It seems unlikely to me that any proof of the (base) RH could be expanded to a proof of the ERH, but they're often lumped together.
Even if the ERH were proved it would be more of a theoretical breakthrough than a practical one; it wouldn't make primality testing any faster, only allow us to prove that (versions of) the current primality tests are 'fast' in some suitable sense. Still, it's a convenient way of linking this admittedly fairly obscure mathematics to something people are likely to have a little more practical knowledge of.
I recall a handfull of economics students in calc 185 or whatever at U-Mich, but that was solely so that, theoretically, for one brief moment in their life they would understand the derivation of the continuously compounded interest formula. That's not really a background in mathematics as far as this is concerned.
Ohhh...if only I wasn't so lazy to fake up a quick login ID for NYT.
I'd like to believe I have a good background in mathematics. I'm not a practising mathematician, but I have a degree in mathematics and I specialized in set theory, but I'll let you be the judge.
Gödel's theorem applies to formal axiomatic systems which contain the Peano axioms, but mathematicians in practice don't limit themselves to these axioms or the rules of derivation.
Note that while Peano's axioms might be used as the basis of some presentations of number theory, number theory existed long before Peano's axioms which are only about 100 years old.
While you can of course get valuable results from Peano's axioms those results were also available before Peano. All mathematics is not axiomatic. Geometry is the only branch of mathematics with a long history of axiomatization. The rest of mathematics has existed without axiomatization until the 19th century or so. It is only because contradictions were found in mathematics that mathematicians started developing axioms and working out more explicit rules of derivation.
My view is that although we have axioms, they are only mathematical tools, not the foundations of mathematics. Mathematicians developed the axioms in order to provide a foundation, but with Gödel's results it appears that axiom systems can not provide a complete foundation.
I do actually practice mathematics (research and also application to practice) so maybe I'm a step ahead here. I would say that your view of mathematics (as a partially non-axiomatic science) is flawed in a quite a common way (for example for people in engineering, physics and computer science). I hope you will read my lengthy comments below.
It is true that a lot of early mathematics was not first developed based on a set of axioms. You refer to axiomatization that was started by Hilbert; what you in some way fail to see is reason for the the need of axiomatization (alltough you point out contradictions etc.). Mathematics not based on axioms (or this actually means: based on wrong axioms; axiom = some basic assumption) is just speculation and can lend to wrong results (some examples below). Number theory however can be and fortunately is based on axioms; they are not the tools but instead the basis on which number theory is built (logically, not in terms of development time-line). Alltough a lot of number theory was developed before complete axiomatization (actually these basic axioms are really intuitive so it's not easy to go wrong), number theory is still an axiomatic science and the Gödel result applies. If it were not so (that number theory can be based on axioms), number theory would not be the 'queen of mathematics' but just a lot of worthless speculation. E.g. (as you might know) propability theory was considered just as play before it was axiomatized (your typical text book in propability theory usually forgets a rigid axiomatization and just goes on to display things in an intuitive manner; ok for text book but not for research work).
Also, obviously you can use basic axioms such as Peano's without your knowledge of doing so. But if you were to choose a wrong set of axioms you might end up with wrong results. Let us propose for example an axiom: there exists a largest natural number. We can then easily go on to prove that '1' is this number (work it out, if you will).
There's actually a quite famous case in the 19th century where a famed mathematician called Frege started his work on set theory. He chose a set of axioms which is now known as 'naive set theory' as it is flawed. Frege however did not understand this and proceeded to work on the naive set theory for some dozen years. When his work was finally published an other famous mathematician (philosopher, logician) Bernard Russell wrote to Frege and pointed out that his set of axioms is flawed. Frege was despaired as now the major body of his life's work might be worhtless. (It turned out that many parts of his work were usable but some major conclusions were wrong; As I remember Frege stopped publishing for several years because of this).
So the point that I'm trying to carry out here is that if a theory is based on wrong assumptions it might be worthless and therefore axiomatization is integral for mathematics. The problem is to choose the right axioms, however even now we sometimes do not know if we have chosen correctly (e.g. Axiom of Choice). Somebody pointed out that mathematics could be seen as an empirical science; it is a science where one empirically tries out different thoughts and chooses the ones that are correct.
One said to the other, "Hey, Bill, I haven't seen our friend Joe for a while. I wonder what he is up to."
Just then, Joe ran up to his two atom friends, and declared, "Oh no! Hey - you have to help me!! I've lost an electron!!"
"Are you sure?" Bill asked.
"Yes," Joe answered, "I'm positive!"
Okie doke, forgive me if I'm missing something here, but is Fermat's Last Theorem the same as the conjecture mentioned in the article? The one that took Dr. Andrew Wiles seven years to solve? If so, why would $1 million not be worth much in 7 years?
There's two ways to look at this. The first is, how much money do I expect to make in the next seven years? I calculated mine, assuming I continue to get the same percent pay raise for each of the next seven years, and let's just say, I won't have made my first million for a few years after that unless we get another dot-com boom or some other such aberation.
The other way is, how much will a million dollars in today's money be worth seven years from now? Assuming the inflation rate for the next seven years matches that of the previous seven years, it'll be worth approximately $850,000 (see this inflation calculator).
So, why will $1 million dollars be a paltry sum in seven years?
There ain't no rules here; we're trying to accomplish something.
I think your conclusions are problematic. If you believe everything that is done without axioms is "worthless speculation" and yet that we still don't know if we've chosen all the correct axioms, then all mathematics based on these axioms is potentially worthless (in your words).
However, I don't believe that. I believe you can have valuable mathematics without axioms. Think about what would happen if a contraction was found in these axioms, would mathematicians throw out all their work? No, they would develop a new set of axioms. The rest of mathematics stays much the same.
I'm not saying that axioms should be ignored. Again I'm just saying that they are tools, not the foundations of mathematics.
I'm very familiar with the work of Frege and Russell. It is only through studying them that I've come to my conclusions. I originally agreed with them that mathematics requires axioms, but the process of developing the axioms makes me conclude that axioms are not foundations. Instead I think axioms (+ definitions, etc.) serve as tools for clarification, clearing up the kind of confusion that lead to contradictions in 19th century calculus.
Back to my original point, the reason I don't think Gödel's result applies is that I don't think mathematicians are bound by finite and formal rules of deduction (that is, like a Turing machine). I don't think mathematicians just go around making stuff up either, just that I believe that mathematics is more than just a set of axioms and rules of deduction.
I think your mentioning of mathematics as an empirical science shows that we might not be disagreeing that much. When you do an experiment in physics we are using observation of the world as a guide to show we're right or wrong. What do we use as a guide for mathematics? I'm not suggesting here that there is some sort of mathematical reality, I'm just suggesting our guide is the existing mathematics and that the goal of axioms is to provide a clarification for what we already have, but might not be clear about.
As for number theory being the 'queen of mathematics' I think it would still be so with or without axioms - it would remain a field that is central and beautiful to the rest of mathematics.
If some university did do this, then all attempts to resale old mathbooks would fail as everyone switched the Pythagorean Theorem to the Pythagorean Proof, thus doing inestimable harm to the pocketbooks of mathmatics students.
Is anyone sufficiently familiar with tax law to make a comment about the tax benefits that could accrue to the founder of this non-profit organization by structuring a "gift" to the mathematical community in the form of several $1M prizes for solving some of the hardest problems in all of mathematics, most of which are very unlikely to be solved in any of our lifetimes?
... a.k.a the Clay Math Institute.
...
In particular, is Landon Clay free to spend some of the interest on the millions he has supposedly "donated" to math through these prizes in any way that he pleases, so long as a fraction of the interest is spent on some tenuous connection to "promoting mathematics". (Check out the link on the CMI webpage to the Clay-sponsored yacht cruise in the Boston Harbor.)
Rumor has it that the president of the Clay Math Institute was fired by the Harvard Math Department for spending too much time shaking Clay down for umpteen millions, and not enough time doing research. Can anyone provide a confirmation of this rumor? Furthermore, after being "dismissed" from the Harvard Mathematics Department, the president of CMI mysteriously popped up across the river at Boston University. Does anyone know how much the Clay Math Institute has donated to Boston University in the process?
Finally, more to the point of the Riemann Hypothesis, which we all want to see solved, what are folks' opinions about whether a $1M prize on the problem is likely do more to decrease the likelihood that a solution is found sooner than later, given that the money will create less incentives for researchers to share their insights and conferences or publish partial results in journals?
Personally I think the prizes smell too much of Clay's past career in the actively-managed mutual fund business, where it's all about out-performing the index for that bonus at the end of the year. Perhaps the first bit of math that Clay should learn (he supposedly dropped out of Harvard himself and never learned anything beyond high-school algebra) is a little statistics, which would show how an active manager's "ability" to beat the index has more to do with luck than business acumen. (Read the famous book A Random Walk Down Wall Street, or check out the site www.indexfunds.com). Then maybe he might realize the right place to "donate" his money is in the form of a refund to investors who got jipped by the front-end 5% loads they paid supposedly for Clay's investment genius. Clay's fund specializes in tax-managed investments, so I guess we can be sure that those skills for dodging the watchful eye of the IRS sure came in handy when setting up his retirement tax shelter
Anyway, I'm starting to ramble now
Yes! The first part in your response about the axioms is what I meant. Choosing different axioms yeilds different theory (and possibly rubbish); for example the Axiom of Choice is necessary for basically all modern analysis, but you can have a lot of classical analysis without it. It is a requirement for measure theory (a measure cannot be constructed without it). Still the Axiom of Choice allows for some very non-intuitive results: for example you can break the unit sphere (3d) into finitely many (however immeasuralbe) pieces and then proceed to construct two unit spheres out of those (Banach-Tarski Decomposition). The axiom itself is however very intuitive and is part of established mathematics (from 1920s on I think). One can only wonder... Anyway excellent page here. Includes comment by Jerry Bona: The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma? (the three are equivalent). Luckily the mathematics we now have seems to portray nature rather well, so I think we can rest assured.
You still fail to understand the meaning of axioms. Let's forget the name 'axiom' and talk about just assumptions. For example you might implicitly use some basic assumption when calculating 2+2=4 (at least, you apparently are calculating in Z, not in Z_2 for example). You see, every time you try to set up some proposal or theorem you need to assume something. Without assuming some underlaying construct what is there to deduce (based on nothing)? The reason we talk about 'axioms' is because we wish to emphasize the importance of these basic assumptions. You should go to some mathematician you respect and discuss the matter with him, if cannot convey it over here.
The point however is indisputable: all mathematics is logically based on some set of axioms (or assumptions, if you will). These assumptions need to be correct for the mathematics to be correct. The actual process of axiomatization has got nothing to do with this; here you are mixing history with mathematical constructs to prove something. In mathematics, the most important thing is to completely understand what you are doing. It may be your intuition that is guiding you: intuition is necessary but can just as easily lead you to wrong theorems. Only by complete understanding and carefull verifiying should you be confident on your results. This is however very difficult; recently a friend of mine had to 'cancel' several of his published articles, because he was using an established ten year old result that was proven to be wrong. So mathematics (remember the 'empirical' point) is not unerring.
With your deduction about Gödel Incompleteness theorem you are also mixing things; namely mathematical constructs and mathematicians themselves. As with mathematics (and with all kinds of logic), if you choose a set of assumptions which is allready conflicting within itself you can prove anything. This will have nothing to do with nature however. So, the Gödel Incompleteness (GI) result applies because of the following first-order logic: GI applies to all mathematical constructs which include at least the Peano axioms AND Number Theory as a mathematical construct includes Peano axioms => GI applies to number theory. It couldn't get more simple! (hope I got my assumptions right...)
As to Number Theory being the 'Queen of Mathematics'; this is the general opinion. I myself do think that Complex Analysis is the most beautiful part of mathematics (eloquent proofs, non-intuitive results (at first), all accessible to a first or second year student). Anyway I've allways disliked purely discrete things (such as integers). I don't study complex analysis by the way; I've done research on Markov operators (stochastics stuff) and now I'm back to basic applied stuff (cutting and packing; you even get to see actual results!).
The Riemann Hypothesis (RH) is either true or false. If it is false if and only if there is a counterexample. It is the mathematician's job to show if it true or false.
RH is not on the same plane as the Axiom of Choice, which is independent of the standard (ZF) axioms of set theory. This means that there are set theories (mathematical objects) with the Axiom of Choice true and others with it false. However, there is only one set of complex numbers (or the natural numbers).
The complex numbers or the natural numbers are mathematical objects. Godel's (second?) incompleteness theorem says only that we will never be able to come up with finitely many (actually, recursively enumerable) axioms that will have the natural numbers or the complex numbers as their only model. This is a failure of first-order logic, not of complex number theory or number theory.
Actually, he is quite correct that it continues to be difficult to prove. Even if the proof can be contained in a statement that's Einteinianly simple (E=m_0c^2), the road to reach that proof has still proven to be phenomenally difficult.
I've been trying to publish my proof of Goldbach's conjecture, and it's just 12 pages long. (I'm serious.) I'm discovering a lot of barriers in Academia to getting heard. But as simple as the result sounds, the road to get there took weeks for direction and years for refinement... And I doubt it could be attained by someone who weren't a multidisciplinary scientist-and-artist, becuase the problem-solving required both the logic and the thinking-outside-the-box to deconstruct known methods into untried ones.
Yes, I totally agree. Should you read my comment and the thread carefully (yes, a lot of text...) you'll see that the AoC was presented just as an example for an axiom and how the axioms affect mathematics. The discussion was about whether math can be done without basing it on some formalized assumptions or axioms. Obviosly you can do something just based on your gut feeling, but you will not know whether your work is worth anything, agree?
Also you were pointing out that if ZF is ok, then it is so still if AoC is included. Still the beginning of the previous centure saw a lot of discussion whether AoC should be included as one get's some apparently unreasonable results. The point being two-fold: 1) do we get contradictions because of AoC 2) do we get just an abstract mathematical construct or then again something that really can be used (analysis being closer to applications at that time; now of course almost all math is applied). To my mind mathematics as a whole should always bear in mind that it's primarily purpose is to aid other sciences, not to do art for art's sake.Although I know that Platonism (art for art's sake) is regarded as naive, I suspect that a lot of pure mathematicians are Platonists at heart. I don't know if it is such a bad thing.
Most mathematical advances at the theoretical level (starting at the concept of zero and then the negative integers) have been considered "way too abstract" at some time or another. Time and time again, the rest of science catches up and finds use for these things.
Just to continue the timeline, consider complex numbers (electrical engineering) and group theory (chemistry). Even things that were considered abstract nonsense even a few years ago are now finding application (for example 2-category theory in physics). So I'm more of the opinion that we should let mathematicians do whatever they find interesting, it's worked up to now and I think that it will continue to do so.
As for the AoC, I think that most mathematicians don't consider it right or wrong, natural or unnatural. As Hilbert famously once said, "It's not mathematics, it's theology". But they are happy to use it if it will help them prove theorems in their mathematical world which consists of real things like the natural or the complex numbers. To most mathematicians, questions like the validitity of the continuum hypothesis are simply not interesting. I'm a computer scientist, but I do have some knowledge of maths departments. How many set theorists do you know of? I have never met one.
Ok, I'll try to give out a dummy proof for Gödel's Incompleteness theorem (the whole thing is apparently about 30 pages, I'll admit I haven't read the whole thing; I've read a partial proof in Russel's and Norwig's 'Artificial Intelligence'). This should clear things out a little bit and give insight to the discussion.
We'll start with the observation that in number theory we have names for all the natural numbers. This is seen as follows: let's say we have the successor function S and a single constant 0; then let S(0) denote 1, S(S(0)) denote 2 etc. By induction we have names for all the natural numbers.
Gödel also included the following function symbols: +, * and Exp and also the usual set of logical connectives and qualifiers in first-order logic. It is now obvious that that the set of sentences we can write in this language can be enumerated (order the symbols in alphabetical order, then do the same with sentences of lenght 1, then with 2 and so on). We can therefore number any sentence a with a unique natural number #a (the Gödel number). Therefore: Number theory contains a name for each of it's own sentences!!! In the same way we can number each possible proof P with a Gödel number G(P) because a proof is a finite sequence of sentences.
Then let us assume that we have an arbitrary set A of true statements about natural numbers. Because A can be named by a given set of integers we propose that it is possible to write the following sentence in our language: a(j,A) =
All i for which i is not the Gödel number of a proof of the sentence whose Gödel number is j, where the proof uses only premises in A.
Furthermore, let r be the sentence r(#r,A) i.e. a sentence that states its own unprovability from A. Can such a sentence exist for all A? Don't ask me, but apparently Gödel would have said that the answer is yes.
The rest is rather simple alltough rather ingenious. We need to prove that r is true. We'll go with reductio ad absurdum: Let's first suppose that r is provable from A (that r actually is false statement! remember that r was stating it's unprovability from A). But this would mean that we have a false statement provable from A. Therefore A cannot consist of only true sentences. This is a contradiction since according to our premises A consists of only true sentences! Therefore r must not be provable from A which is exactly what r claims.
So from the above (assuming that we believe the sentence r can be constructed) we have seen that for any set A of true sentences in number theory we have statements that cannot be proven from A. As a special case we can choose A = axioms of the number theory. Hence number theory containts statements that cannot be proven!
Feel free to complain about the inaccuracies in the above; all I can do is to suggest you get Gödel's proof into your hands. Anyway to my mind (if I do not miss any subtleties) the above goes on to establish that we can never prove all the theorems of mathematics within any given system of axioms (as the above problem appears allready with the natural numbers). This is apparently why Hilbert was pissed about Gödel's proof.
What you say is true. However it refers to proofs given a particular set of axioms. Godels sentence :)
is not provable in the system but it is true. He proved it. Outside the system. In the unlikely case that ZF+AoC is not enough to prove Riemanns Hypothesis, mathematicians will step outside that formal system. In fact, if this is the case, we should get some more set theorists real soon.
I personally like the Turing proof of Godels theorem. The set of all theorems is recursively enumerable, but the set of all true statements about the natural numbers is not. The latter is a reduction from the complement of the halting problem, using computation histories of turing machines. It's really cool, the details can be found in Kozen's "Automata and Computability" which by the way is an excellent textbook on introductury computability theory.
Your points are absolutely true and I agree. However if all the mathematicians would go and claim that: "We don't care about your stupid applications; we're doing this for our own fun! Go stuff you apps!" the rather quick implication would be that the guys with the applications would take their money and and put it somewhere else (in computer science?).
The point is that let's say Wiles has just proven Fermat's last theorem and a guy comes up and says: 'that's neat, I think I can use this in my device'. Then Wiles should go: 'Very interesting. Can I help you with that?'. Actually, however unpropable the above would be I hope/think Wiles would respond just so. So I think we should continue developing math as we do but never forget that the final reason is not to do it just for the heck of it. Maybe we should even venture so far as to look for applications after creating something totally new? Anyway, I think you would be in your right to say that this it not a problem today.
As considering AoC I'm a believer. That's propably because I did some research on stochastics at the university (markov operators; the asymptotic properties thereof) and you will not get anywhere without Measure and Integration theory which is actually the basis for the whole thing. In my mind most of the stuff that follows from AoC seems to be natural (barring the unit ball problem but that can be explained too...).
I don't think I've ever met anybody claiming to study set theory per se (those guys are all dead by now?). Some guys however studied areas that to my understanding were closely related to set theory. Nothing I would understand however...
Ok maybe you will clarify this for me (remember stochastics and analysis not in-depth computational/CA/related theory ok)? If we add axioms to Peano axioms and use maybe a higher order logic the Gödel theorem will still hold, true? So if we are working just based on a set of axioms needed for Riemann Hypothesis we do not yet know that a proof exists, do we? I would say that it is likely that we will someday find a proof, but to my understanding as of yet there is not any proof that such a proof actually exists (we'll actually if we had that we would be done :> ). What would it mean to study Riemann hypothesis outside the system? To start from scratch? And could we still claim to be studying the Riemann hypothesis?
uh, yeah, this would seem reasonable, until it's logical consequence emerges:
if you can't prove it's true, then it's true
Incompleteness doesn't require unprovable statements to be "true", only unprovable.
"I assumed blithely that there were no elves out there in the darkness"
See http://zetafunctions.coolissues.com/zeta.htm and http://fermat.coolissues.com/fermat.htm
The Peano axioms admit models other than the standard model (interpretation) of number theory. When Gödel's theorem states that there are "true but unprovable" propositions what that means is that there are propositions that are true in the standard model, but unprovable from the axioms. So one interpretation of Gödel's incompleteness theorem is that any set of axioms will fail to include number theory while excluding other models. There will always be alternative models.
One conclusion that could be drawn here is to say that the structure of the integers is not uniquely captured by a finite set of axioms.
I don't think it is indisputable that all mathematics is based on axioms, because outside of geometry the use of axioms is a late 19th century development of mathematics. Are you saying that before that people weren't doing mathematics but something else? Euler and Riemann did their number theory without axioms.
[By axiom I'm talking about a proposition statable using first-order logic (+ symbols etc). The term "assumption" is to me a bit more vague.]
History is important. The way things are now are due to certain historical developments. In the future mathematics might be different.
Li(x) is defined as the integral from 0 to x of (1/t) dt. And apparently the number of primes below x (usually denoted pi(x)) is pretty well approximated by Li(x).
Not quite: note the obvious initial logarithmic divergence. Informally, you can just change the integrand from what you had to dt/(log t), but you really ought to work around the singularity at 1. Some people change the bounds of integration to start at 2 to avoid this. It simply shifts the function by a small constant (about 1.05)
I'm a bit surprised no one here has mentioned Pierre Deligne's 1974 proof of the Weil conjectures, in particular the analogue of RH for smooth projective varieties over finite fields (for which he was awarded a Fields medal in 1978). This is perhaps the strongest "evidence" for the original hypothesis (unless you find the brute force calculation convincing), and it has other interesting consequences, for example the resolution of Ramanujan's tau conjecture (ref: Hartshorne's Algebraic Geometry).
There is a nice discussion of potential avenues of attack on the Riemann Hypothesis at the end of chapter 5 in Patterson's text on the Zeta Function (Cambridge Studies 14), including some vague ideas on why a purely analytic strategy is not likely to be successful.
"Your notation sucks!" -- Serge Lang (1927-2005)
Theoretically in the sense that it's useful for many other theorems. Quite a few things are based on numbers having a unique (down to the order) factorization as a product of primes. That proof is reasonably simple, and we can easily find pseudoprimes (which for most intents and purposes are fine) and that covers what we mostly need. The Riemann hypothesis is more of purely theoretical and cryptological interest though.
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