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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.
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.
The mathematician stands a better chance of proving the hypothesis, but the NSA supercomputer stands a better chance of refuting the hyposthesis.
With current technology, it's extremely unlikely that the mathematician would refute the hypothesis or the computer might prove it (although it is possible).
Finally, props goes out to the Courant Institute of Mathematical Sciences. The best, my favorite, and my current graduate school (@ nyu).
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 . . .
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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.
Thanks. Great explanation.
Could you elaborate and tie this in with the number of primes between m and n?
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!).
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.