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Shimura-Taniyama-Weil (STW) Solved
timbo_red writes "The BBC report that an international team of scientists have solved the STW conjecture. I vaguely remember what this is from reading the Fermat book, I'll have to check it again. " This really has me interested in the conjecture. Anyone have any good links for background reading?
I read this a couple of days ago, and understood several of the words involved. Further reading, with decent enough explanations, can be found here.
Comment removed based on user account deletion
Only front-line mathematicians will really understand the STW conjecture. But you could say "there exists a modular form of weight two and level N which is an Eigenform under the Hecke series and has a Fourier series".
Ahh... it's all becoming clear now.
It's not enough to bash in heads, you've got to bash in minds. - Captain Hammer
While I don't pretend to understand the math involved, Simon Singh's book Fermat's Enigma gives a good explanation of why the Shimura-Taniyama-Weil conjecture is interesting and important, even beyond its application in proving Fermat's Last Theorem. It serves to unify two unexpectedly related fields of math. I recommend the book -- although nonmathematical, it gives a feeling and appreciation for the mathematical discovery process, and is a gripping read. It's a midway point between "popular math" and real math.
I remember from documentary on solving Fermat's Last Theorem that Wiles spent ALOT of time grappling with the Shimura-Taniyama conjecture. Although I can't remember if he used a part of it, or managed to circumvent it. Anyone know if Wiles circumvented it? And if so, was Fermat's theorem used in this proof? Seems like a good tact if so.
;)
I find it facinating that math is so deep, that even though I took several math classes after Diff Eq. I still can only barely understand some of the stuff they are talking about.
One professor of mine once remarked that 20th century math concepts aren't really touch upon unless you are pursuing a math degree. The 'newest' math concept for most students being dot/cross-product notation - and I think that was 19th century, if memory serves.
I guess people that think in 9 dimensions scare me.
Tom
Maybe I'm delusional....but..... there are 3 different types of trigonometry, -euclidian (all angles = 180 deg.) --euclidian does not exist in more than 3 dimensions. - and two other kinds...both made with compasses, 3 circles intersecting eachother, one kind has more than 180 deg angles, the other has less - visualize! so heres my point... I was once told by a psycho math/physics professor that if they could actually figure out how to use that info then they could make lots , i mean LOTS of breakthroughs. Hmmn, i am thinking - like 30 years in future or more with the right research - being able to cross dimensions......from 3 to 4. ----- Whats the shortest distance between 2 points? -Straight line? Nope. None at all. Fold the paper from 2 dimensions into 3 and touch the dots. The shortest distance is no distance at all. Just bend 3 dimensions into 4. Sounds easy. Nearly impossible to comrehend with more than 4 dimensions...but...leave that to the people with brains! -What do ya think?
Like the article says, Wiles solved a special case of STW to knock off Fermat's Last Theorem. I guess this is a proof of the general version (but the article is a little vague--any number theorists around who are in the loop?)
As I understand it, Taniyama-Shimura establishes a correspondence between elliptic curves and "modular forms" which are a set of functions that satisfy a certain set of critera, and are based in number theory. Before it was [just] proved, T-S was known to imply FLT, and Andrew Weil's key breakthrough was to prove T-S for the classes of elliptical curves required for FLT. He did this by a novel method of counting both sets (elliptic curves and modular forms), and showing they had the same number of members, hence implying the correspondence. The complete general case of T-S has now been proved. There was a great documentary on FLT a few days ago (PBS I think), which is a must see if it gets reshown.
Disclaimer: IANAL, IANAM.
I thought STW was automatic once we had Fermat.
Yes, STW is intuitively obvious to the casual observer once you have the Wiley proof of Fermat's Last Theorem.
I will leave the details as an exercise for the student.
Leaving aside the question of does the definition of "scientist" include "mathematician"
No. All scientists need to be mathematicians to some extent, but the reverse is not true. Science includes the formal consideration of experimental evidence as part of a model building process, but mathematics can be a purely abstract endeavor without an empirical component.
we McSE's are regularly baffled and defeated,
but we can always resort to lying and obfuscation.
Chuck
try { do() || do_not(); } catch (JediException err) { yoda(err); }
The existence of a proof of the full Taniyama-Shimura conjecture was announced at a conference by Kenneth Ribet on June, 21 1999 (Knapp 1999), and reported on National Public Radio's Weekend Edition on July 31, 1999. The proof was completed by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, building on the earlier work of Wiles and Taylor.
/. posts what we submit. Though, I think monitoring NPR would be a good source for stories -- they reported this one a while back. Perhaps links like this one to the real-audio recordings of their broadcasts might be a nice touch.
Before everybody starts screaming "this is old news" remember,
The Future of Human Evolution: Autonomy
I used to think that math wasn't much of a direct use, but this is incorrect and a lot of it has to do with how mathematics is taught in western culture (something we should be ashamed of; Most people don't do any calculus until senior high school if at all!).
What math does is provide a (perhaps the) universal language with which to describe the universe, science, language, everything. Everything can be represented and manipulated in some form with math - this is what computers do! (discreetly >:).
Discovering relationships between unrelated fields of math allows the scientists and engineers of tomorrow to use these descriptive tools to develop new cool gadgets. ;)
Kudos,
..don't panic
DNA is a Turing machine. You, however, being dynamic and emergent, are not.
It is rather unfortunate that the BBC correspondent has very little idea about the subject he is writing about
It's always the same from the Beeb. "Real" intellectuals have arts or humanities degrees, mathematicians are just geeks and beneath contempt. Did you notice the "related links" they placed on the page ? The top feature was last year's "mathematics of biscuit dunking" story. This just shows what little significance the increasingly dumbed-down BBC now places on science or technology stories.
ObRant: Why is it that at the hypothetical mixed-background middle class dinner party, the scientists are expected to be literate, but the literati still revel in their innumeracy ?
It is true that the subject is too esoteric to be accessible to non-mathemticians,
There's probably at least a dozen people in this room as me who work on elliptic curves on a daily basis. OK, so I work in an unusual environment, but these things do actually have real world applications (crypto, natch) and not just for the NSA.
In an effort to convert the British Broadcasting Company's text to English with Babelfish, I discovered a shortcomming in Babelfish's software. It could not convert this short article into English. DEC (or Compaq or whoever owns Alta Vista this week) really needs to improve their Math-speak to English converter. Maybe, they should OpenSource (tm) it so we could all help?
I'm not trying to be offtopic, and I know I'll probably get moderated down for this, but:
Rob, is there a way to get a math section? I know that crypto is a popular subject on Slashdot, and it's very closely tied with math. I know that a lot of geeks also like to hear about the STW conjecture being solved. It's all very reasonable-- fields as diverse as biology and physics have strong ties with mathematics.
I'm not complaining-- I love Slashdot, and I'm glad that this story was posted. I really think that, while the math is beyond my abilities, it's cool to at least know that the conjecture was proven. It's also pretty neat that I can find out why this is important to the rest of mathematics.
But when I see it posted under the "science" heading, I can't help but cringe a little. It's not likely that this is going to revolutionize science. And there are a lot of geeks who wouldn't care about it because of that fact-- no applications? Why the hell would you bother with it? Giving mathematics articles their own topic heading would most likely be useful to these folks.
I'm also seeing a lot of people joke and complain about the horrible headache that they received just viewing the article. If articles were placed under the mathematics heading, a lot of this can be prevented. This is partially due to the fact that users can filter out the stories, and partially due to the fact that anything under the "Mathematics" section would sort of carry an implied warning-- "Don't read this unless you are *really* in to high-level shit".
So perhaps it's best that a new section, "Mathematics" be created. It would be very much appreciated. I know you're a busy man and all, but it would please a whole lot of us anal retentive blowhards.
This is actually not as obscure as it may sound. Simply put it relates topology to number theory, thus allowing problems in one domain to be translated to the other. That FLT was able to be solved (albeit not in the same way that Fermat did it) using this technique is an indication of the power of being able to do this: suddenly the power of techniques developed in one domain become applicable to problems in another.
As a practical example, I remember once compressing sparse matrices (parser tables) by mapping them to a graph (one line = a node, with node connectivity defined by line "overlap"), then using a minimal graph coloring heuristic.
Alright, I'm no expert on this stuff, but I'm going to take a stab at explaining why anyone would care about the STW conjecture.
First, let's start somewhere seemingly unrelated that may be easier to deal with: Physics; specifically, gravity. I'm working under the assumption that everyone knows what gravity is, so, good. There are other forces that do similar things in Physics, however. The most common are the "Strong" and "Weak" electromagnetic forces. The force that holds electrons close to an atom, and that bonds atoms to each other in molecules are examples of these forces.
Now in Physics, there is a holy grail of theory called the 'Grand Unification Theory'. This is big important stuff. In an amazing oversimplification, it suggests that there is a single formula that relates all of these forces together. We _expect_ this from intuition, we currently just don't have any idea how to prove it, although progress is being made all over the place.
Now, skip back to mathematics. Mathematics is split into tons of different areas. Statistics, Number Theory (the stuff normally used in cryptography), Calculus, and so on. Robert Langlands proposed that there is a Grand Unification Theory (GUT) of sorts for mathematics. This is commonly referred to as the Langlands Proposition (or Program, according to the BBC article).
Some years ago, Yukata Taniyama (The 'T' in STW) asserted a conjecture that did two things. First, if proved, it would bring elliptic curves and modular forms together in the spirit of the GUT, thus giving the Langlands program a big push. Secondly and, while not really more important, at least more interesting to the public, he showed that if his conjecture was proved, the most famous unproved theorem at the time would follow. I speak, of course, of Fermat's last theorem (FLT). This was the holy grail of math at the time.
A few years ago, Andrew Wiles proved enough of Taniyama's conjecture to prove FLT. This was what made STW mainstream; had this not happened, noone would care and the BBC story would probably be overlooked. But it did happen, made lots of papers, was flawed, fixed, flawed again, and currently is believed to be correct.
What recently happened, in the BBC story, is that the _rest_ of the STW conjecture was proved. Not just the part that Wiles used to show FLT, but all of it. In math this elevates STW from a conjecture to a theorem and makes mathematicians everywhere giddy with joy since the Langlands Program is slightly closer to being proved.
And of course, giddy mathematicians are the types who post stuff to Slashdot, which is why this article is here at all.
Was that any better?
Many people believe that Fermat had a flawed proof for his theorem. There are many reasons for this belief, most of which I am entirely unqualified to judge and the rest of which I probably shouldn't judge, but I think it rather likely that his proof was flawed. The sheer number of brilliant minds who attempted to prove it, and the fact that the final proof used such modern techniques, suggests to me that it is unlikely that Fermat had a valid proof.
I know this isn't really related to what you had to say, but I thought it was interesting enough to mention... and maybe someone who knows more about it will have something useful to mention.
---
I hope you're not pretending to be evil while secretly being good. That would be dishonest.
actually, a special case of STW gives you FLT. (if you can show STW for a subclass of elliptic curves, "semistable" ones, then you have proved FLT). this is due to Ribet, Serre, and perhaps others.
Andrew Wiles did this. now, Conrad, Taylor, Diamond, and Bruile (?) have proved STW for all elliptic curves. that's the breakthrough, and the announcement was made earlier this year. it isn't "news" in the popular media sense.
by the way, the BBC has something wrong, the proof is not printed in the Notices Dec issue (ha!), an announcement is. it's been on my desk for about a week now.
- pal
P. Ribenboim, "13 Lectures on Fermat's Last Theorem", Springer-Verlag, 1979, ISBN 3-540-90432-8 (assumes undergraduate maths). You might notice that this book's publication date is way before Wiles, it contains material on which Wiles then expands, e.g. elliptic curves (cf. cryptography too!) and modular forms. A simpler text, still by Ribenboim is "Fermat's Last Theorem for Amateurs", 1999, Springer again, ISBN 0-387-98508-5, which, as the title sort of implies is a tad easier. I wouldn't say it is exactly trivial but it is a very good self-contained book with a number of chapters explaining the number theory you need and a good attempt at explaning Wiles' proof. Borrow this one from your local library if you are really interested and have some mathematical background, the first one if you are into higher mathematics.
I followed a one-semester graduate course (by Laurent Clozel) on the proof of the semistable case of the Shimura-Taniyama conjecture (the case proven originaly by Wiles and which concludes the proof of Fermat's theorem). So I can make a few comments on the subject.
The Shimura-Taniyama conjecture (Weil's name is attached to it for rather dubious reasons: essentially, he mentioned the conjecture — as an exercice for the interested reader! — in a book he published; Serge Lang is always ready to flame anyone calling the conjecture by Weil's name, so let us omit Weil) concerns a correspondance between certain modular forms and certain elliptic curves (actually with Galois representations in between the two). That is, it states that every elliptic curve is associated to a certain modular form (the association can be stated in many different ways: they have the same L function; the eigenvalues of the modular form for the Hecke operators can be deduced from the number of points of the elliptic curve on finite fields, and so on). This conjecture was known (i.e. formulated) long before any relation with Fermat's theorem was observed.
Gerhart Frey had noticed that if a counterexample (A,B,C) (with A+B+C=0, A, B and C being p-th powers) to Fermat's theorem were found it would yield an elliptic curve y=x(x-A)(x+B) having certain miraculous properties, including being ``semistable'' and possibly violating the Shimura-Taniyama conjecture. Using works of Jean-Pierre Serre, Ken Ribet was able to prove this remark of Frey, so that the Shimura-Taniyama conjecture, and in fact even only the Shimura-Taniyama conjecture for semistable elliptic curves, would imply Fermat's theorem.
At that point it became obvious that it would be only a matter of time before Fermat's theorem were proven. Andrew Wiles, was able to complete the task. His first proof contained a flaw (in trying to construct an Euler system), which was noticed by Luc Illusie, but with the help of Richard Taylor, Wiles was able to replace the technique of Euler systems and use Gorenstein rings instead (and some very fine points of commutative algebra) and correct the proof. The full proof (Wiles' ``Modular Elliptic Curves and Fermat's Last Theorem'' and Wiles and Taylor's ``Ring Theoretic Properties of Certain Hecke Algebras'') was published in Inventiones Mathematicæ. Thus, the case of Fermat's theorem was settled.
The general case of the ST conjecture was still unproven. However, soon after Wiles' result, Fred Diamond improvement over it. To understand it, you must know that semistability of an elliptic curve is a ``local'' property, i.e. it can be tested for each prime number. An E.C. is (globally) semistable iff it is semistable at every prime number. (It is always semistable at all but a finite number of primes.) Wiles' result required the E.C. to be semistable at all primes; Diamond refined that and proved the modularity of elliptic curves that are modular at 3 and 5. This was a considerable progress, and it was then pretty obvious that these last conditions would be eliminated. Now they have been (every elliptic curve is known to be modular), but this is more a question of technique than a fundamental improvement.
One might be tempted to think that the proof of the ST conjecture is fascinating. In fact, I found it (or at least the semistable case, which has, it would seem, the gist of the ideas) terribly boring. It is all a matter of controling the behavior of the ramified parts of the cohomology groups of some Galois representations, and it is done in a succession of lemmata, each one seeming exactly the same as the previous one. In fact, the experts' opinion is that the proof of the conjecture is technically difficult but fundamentally trivial in that it does not use any deep results from (algebraic) geometry.
The ST conjecture is part of a more general scheme called the ``Langlands programme''. The Langlands programme is a correspondance (which has not been formulated in a completely satisfactory way, as far as I know, let alone proven) between higher dimensional abelian varieties (elliptic curves are abelian varieties of dimension 1), Galois representations and modular forms (disclaimer: I don't know half of what I'm talking about here). ``Class field theory'', the climax of the number theory of the beginning of the century, is the case ``GL1'' of the Langlands programme (the abelian case). The Shimura-Taniyama conjecture was the case ``GL2'' of the same programme. Some other cases have been proven, such as ``Sp4'' (these funny acronyms refer to certain algebraic groups: GL is the General Linear group, and Sp is the Symplectic group).
The Langlands programme actually splits in two parts: the ``number field'' (or ``global'') Langlands programme, the hard number-theoretic part, of which the ST conjecture is a particular case, and the ``function field'' (or ``local'') Langlands programme, which is an easier analogue of more geometric content.
The major news recently is that the ``function field'' Langlands programme has been proven, by Laurent Lafforgue. This is much more important than the full proof of the ST conjecture. And it also means that Lafforgue will be getting the Fields medal in three years (mark my words).
Simon Singh, Fermat's Enigma : The Epic Quest to Solve the World's Greatest Mathematical Problem. Based on the (excellent) BBC/PBS television show (it was a Nova episode). Highly recommended.
/. get a cut that way?), or here at Amazon.com.
Get it h ere at Fatbrain (does
Stupid job ads, weird spam, occasional insight at
Real-world relevance of higher math:
And you know, I'd hold "furthering the bounds of human knowledge" to be an good thing unto itself, regardless of any real-world applications for, say, generalized statements about Ramsey theory and the Party Problem (to choose something which I've been doing a bit of amateur reading on lately that might well lead to real-world applications).
I didn't know what a meme was, so I asked five friends. They didn't know what a meme was, so they asked five friends.
While we're on the topic of open mathematical conjectures, my favorite still has to be Goldbach's Conjecture. It's tantalizingly simple; it states that any even integer greater than 4 can be expressed as the sum of two prime numbers. It seems intuitive, and it's certainly easy to verify "by hand", at least for relatively small numbers (i.e., 31 = 13 + 17). Indeed, computers have been unable to find a counterexample, regardless of how high they've gone.
Does anybody know the status of this problem? I recall reading something a while back about how somebody determined that this problem is undecidable, though I could be wrong.
When I was in college taking a History of Mathematics class years back, I was fascinated by this one. I even spent a fair amount of time hammering away at it, and while I came up with a few interesting ideas, nothing substantial came out of it. I was working using Euclid's famous proof of the infinitude of the primes as an inspiration. Anybody who's seen that proof knows that in mathematics, sometimes a correct proof can be completely unexpected and yet incredibly elegant and simple at the same time.
We're going down, in a spiral to the ground
But the reason that the keys could be shorter may now be invalidated with the proving of this conjecture, I don't know enough of the math but to quote from the latest Crypto-gram newletter: ... Whether this recommendation makes sense depends on whether the faster algorithms can ever be made to work with elliptic curves. The question to ask is: "Is this lack of smoothness a fundamental property of elliptic curves, or is it a hole in our knowledge about elliptic curves?" Or, more generally: "Are elliptic curves inherently harder to calculate discrete logs in, or will we eventually figure out a way to do it as efficiently as we can in the numbers mod n?" '
'All of the fastest algorithms for calculating discrete logs -- the number field sieve and the quadratic sieve -- make use of something called index calculus and a property of the numbers mod n called smoothness. In the elliptic curve group, there is no definition of smoothness, and hence in order to break elliptic curve algorithms you have to use older methods: Pollard's rho, for example. So we only have to use keys long enough to be secure against these older, slower, methods. Therefor, our keys can be shorter.
Does the proving of this conjecture open the way for a 'smoothness' function to be defined? Crypto-gram can be found at: http://www.counterpane.com/crypto-gram. html
Brian Haskin
Personally I shudder to think what having to take all that math could do to a person if they had to get a job in something like computers!
All that learnin hurts the brain as we all know.
It seems that the higher you go in math the more bland and unapproachable the subject
becoms and the more difficult (difficulty!) it becomes
Well yes it becomes more difficult...just like coding for X is alot more complicated than hello world. However, it actually becomes MUCH more interesting. Think about it...addition and subtraction are pretty fucking boring while higher mathematics gives you stuning results such as the Banach Tarski Paradox (A sphere may be cut up into finitely many pieces and by translating and rotating the pieces reassembled into a two spheres of the orignial size).
Why can't people produce a nice graphical textbook about complex math subjects with plents of examples and
problems to work on?
Math books with examples and problems to work on are fairly common. The reason the textbooks often aren't (and shouldn't be) graphical is because mathematics is not a graphical pursuit. It would be like explaining perl via venn diagrams. Yes, some parts of mathematics MODEL the real world (such as R^3) but all to often people taught via pictures are restricted by them. As soon as they run into a problem without an obvious visual component (say a problem in R^4 (yes it can be useful)) they are stuck.
I have a bias towards things that have a lasting importance versus
things that have a limited appeal
Question who is more famous? Archimdes or the political leaders of athens? It in fact appears mathematics is of much more lasting imprtance than whatever war is occuring at the moment.
How can you tell little Billy about STW?
As we all know little billy is the ultimate judge of these matters. I imagine huffman encodings shouldn't be studied either.
Ohh while not a textbook their is a book On relativity or something either written by einstein or from his notes which is exceptionally good.
Marriage is the "pseudo-ethics" that cloaks the messy truth of sexuality in the raiment of propriety -- it's "Don't Ask,
can get bar none!
:-) I think that this is a very poor take on this situation.
>Personally I shudder to think what having to take all that math could do to a person if they had to get a job in something like computers
Let's see.. impeccable logic... a rock solid understanding of algorithms...Top notch problem *defining* and solving skills.
yeah, not too useful in computers.
>It seems that the higher you go in math the more bland and unapproachable the subject becoms and the more difficult (difficulty!) it become
While there is no arguing that higher mathematics is difficult to wrap your brain around, I would rephrase the first part of this sentence.
I got a BS in mathematics taking several Graduate classes in the process(Real Analysis (The Horror) and Differential Equations/Dynamical Systems) and I would say rather than "bland and unapproachable"
:
Incredibly beautiful, deep, elegant and powerful, but with a much higher price of admission than any other field.
>Why can't people produce a nice graphical textbook about complex math subjects with plents of examples and problems to work on?
Well even in relatively simple math courses i.e. past the basic calc/diffEQ/linear algebra/ 2 year series, you are dealing with n-dimensional spaces.
The fact is there is no way to draw this. That is where the full power of the abstract approach is needed.
For example, take as your space the set of all functions from the real numbers to the real numbers. How the hell do you even draw anything dealing with this? If I remember correctly, this space has a cardinality ("number" of members) greater than that of the real numbers which is strictly greater than the usual "infinity" which is the cardinality of the Natural numbers/integers/rationals
>I think it's only in places in the US where people have the free time and material wealth to do research and be able to have means to feed themselves makes things like this possible
Well given that math and physics were almost completely re written a few hundred years ago by
Newton(England) Leibnitz, 23(?) different Bernoullis(SP),Gauss, Cauchy, Cantor,Riemann (Germany), and a few French people whose names slip my mind
Oops regarding computer theory we can't forget the Russians especially Kolmogorov
Are there any good textbooks (graphical, examples galore, problems) that would make someone an Einstein a little easier? Real world examples?
Einstein was a genius who did very poorly in school. He was not even accepted to any grad schools until he completed his Nobel prize winning work (Not General or Special Relativity either).
There is no easy way to understand the advanced results of mathematics without struggling your way up. Some people will have an easier time than others, but I feel that it is worth it even if I never use the specific facts I learned.
Mathematics has many "real-world" uses that haven't been discovered yet. In general Mathematics is decades and often centuries ahead of the relevant scientific fields. Abstract Algebra (not like in high school) was considered the most esoteric useless field by non-mathematicians until it became indispensible in quantum guage theory.
Superstring theory is built upon Some-old-guy-or-other's Beta function and Symmetry group theory.
General Relativity is written in the language of differential geometry.
To understand some of these theories is a mind blowing experience I would highly recommend.
Seriously though eve if you don't decide to pursue it you will be prepared for anything else you do want to do. You can go to grad school in almost any discipline, and your problem solving skills will exceed those of almost anyone you interview against for a job.
---CONFLICT!!---
from the front page would be a nice addition also. It seems quite often that some good links are provided in the discussion but (even with moderation) that requires some diggin'. Short form: More film and audio links from the front page. Let's *really* smash some servers. :-)
+&x
Not much of anybody in the mathematical community thinks that Fermat had anything resembling a proof to this one. There is a fairly reasonable explanation for where Fermat went wrong.
:)
This is a bit of summarizing and paraphrasing from Joseph A. Gallian's Contemporary Abstract Algebra.
"Most likely, he made the error that his successors made by assuming that the properties of integers, such as unique factorization, carry over to integral domains in general."
In 1839, Gabriel Lame announced a proof to FLT. It involves a fairly simple factorization of x^p+y^p into factors with complex coefficients.
The problem is that in this situation, factorization into irreducibles is not unique. This is a property of the integers (45=3*3*5 and no other primes). This property is only true of certain types of algebras--called unique factorization domains. The algebra (or ring, if you're literate) involved in the factorization used by Lame did not hold the property of unique factorization. The proof is much simpler than Wiles' if you assume the property of unique factorization, which was likely Fermat's mistake.
Anyone who's interested in these terms should pick up a college text on abstract algebra. You'll need to read most of an introductory text....
By the way: MATHEMATICIANS ARE NOT SCIENTISTS. We are theorists. I expected more from the slashdot community.
what is this proven theorem going to allow us to do?
I'd like to hear some examples of how this new technology is going to enable us. Will it allow visualization of data? Will it allow additional methods to be applied to the solution of formerly unsolvable problems?
I'd also like to say that I disagree with a previous poster's assertion that Mathematics and advanced number theory isn't science. A mathemetician see's patterns, theorizes, proves; how is that different from working with physical phenomena? Mathematics MODELS the physical - I believe that there isn't ANYTHING that exists that cannot eventually be modeled using mathematics. There is NO SCIENCE without numbers; ask Lord Kelvin.
The Greeks were right... working with numbers is the closest thing to being a magician; there is magic in it undeniably!
Codifex Maximus ~ In search of... a shorter sig.
That's completely idiotic. Understanding STW on a level deeper than "all chipmunks are really woodchucks in disguise" would require several years of graduate mathematics, and those several years would have to be doing the right type of mathematics. People, even smart ones, need to accept that there are simply some things that they couldn't understand, even if they worked very hard for a very long time.
The most advanced mathematics courses geek types typically take is differential equations, which usually consists of fairly mindless equation manipulation is hence is quite literally nothing like what a typical mathematician does. This is really unfortunate, as much of mathematics is quite beautiful. Great mathematicians are great artists, but appreciating the art has an extremely steep curve.
As for applications, people need to accept that going from understanding something to using an indirect consequence to build a sturdier lunch box could takes hundreds of years. It's a long chain, after all; math to physics to engineering to corporations to consumers.
There are deep and extremely important connections between number theory and physics, e.g. vertex operator algebras, string theories, zeroes of zeta functions, eigenvalues of large random matrices. Understanding these connections, in math as well as physics, is thus key to future progress.
--
Chris Long, Departments of Mathematics & Statistics, Rutgers University
San Diego Padres, 100 Park Blvd, San Diego CA 92101
It is pitch black. You are likely to be eaten by
Although proofs and such can be very comforting to know about, engineers (and some scientists)
routinely used "unproven results" before the mathematical machinery is totally developed...
For example, Heaviside algebraic operator theory was used for solving linear differential equations
before the mathematicians finished proving the domain of applicability (Laplace et al)...
Newton's fluxions were used long before integral calculus formalized the operation of integration.
Not to mention infinite series, asymptotic analysis and the list goes on and on...
The quest for "truth" in mathematics has been a long, unexpected journey... If you haven't studied
up on it, read about Hilbert and his program to formalize math... then read about Godel and how
he showed that sometimes this mathematical foundation is really a mirage.
Sometimes practical use is more satisfying than theoretical comfort... So think about how the
"truth" of the FLT really affects things. I imagine it's a lot less effect than you might think...
It's a bit bold to regard Langland's program (not proposition) as a GUT.
- Shimura-Taniyama originated the idea about a deep connection between modular forms are related to elliptic curves.
- Weil made it plausible and precise but no one likes Weil (PBS) so sometimes his name is not added to the STW conjecture.
- Frey thought that STW-->FLT by using a solution to FLT to create an elliptic curve that probably wasn't modular.
- Serre made the framework of Frey's idea precise in his epsilon conjecture
- Ken Ribet proved Serre's epsilon conjecture establishing that STW-->FLT
- Wiles almost proved STW
- Wiles former student Taylor was brought in to help fill in an essentially small gap (something about deformations of Galois representations, wasn't it?)
- Long refereeing by people like Nick Katz...
Wiles work is a real tour de force. But, I cannot say that all mathematicians care mostly about Langlands program. There are tons and tons of mathematics just as interesting as this topic.We don't call something "a science" because we like it. At least I don't.
A science is a field of study which has a number of characteristics, the main one being that it is based on inductive reasoning from experiment and observation. Mathematics is based on deductive reasoning, not inductive, and therefore is not a science. The entire way we study mathematics is different than how we approach a real science.
Similarly "Computer Science" isn't. A science that is...
Cheers,
Ben
PS Disclaimer: I am not an unbiased observer in this. I am all but dissertation a PhD in mathematics.
My usual seat in the cluetrain is at A HREF="http://pub4.ezboard.com/biwethey.ht
The mathematical association of america has some nice information on this:
http://www.maa.org/mathland/math trek_11_22_99.html
Please moderate this post down for your protection.
--
Industrial space for lease in Flatlandia.
There are many levels of mathematical knowledge. I am sure that this book was great for the complete layman (I did think it did a good job explaining what drives mathematicians to the unenlightened), which is whom it was written for.
However, as a third year mathematics major, I found that it was not suited for me at all (although I am far from good enough to actually pick up the proof and start reading). A lot of Slashdot readers have CS or technical degrees that include quite a lot of mathematics, so I think there are others here who would feel the same.
That was all I was trying to say. Not critisism of Singh, its just the nature of pop-science. Most physisists seem to find _A brief history of time_ appalling...
-
We cannot reason ourselves out of our basic irrationality. All we can do is learn the art of being irrational in a reasonable way.
First, let me say that I heartily agree that math is one of the best degrees you could have to do computer programming or any other kind of work that is heavily oriented toward problem solving.
But I differ with you in your answers about why the explanations/texts/etc can't be easier. There is some truth in what you and (moreso) others in the thread are saying, but there is also a heavy undercurrent of math groupthink. Just because no one takes the time to explain these higher concepts clearly doesn't mean there is not a way to do it. It's a hell of a lot harder to explain this stuff simply, but it could be done.
The fact that it generally isn't done is partly due to how hard it is to explain complex or highly abstract concepts clearly. But it's also partly due to the fraternity/hazing attitude in academia that "they should have to work as hard at it to get it as I did".
I have found that I can, with enough effort, find clear and simple (not necessarily short, though!) ways to explain even highly "esoteric" concepts. This involves the very difficult process of attempting to figure out how a newbie will be thinking about what I am saying, and trying to come up with accurate analogies to things that will already be familiar to them. Inevitably, after a lot of effort in this direction, I end up understanding the subject matter on a much deeper level.
This leads me to think that part of the reason that there are not clearer explanations out there is that you just have to understand it better than most people do before you can explain it that well, and at the same time you have to be thinking about how people outside of your field think.
The union of these two sets (one set being "those with a deep understanding of postgrad mathematics", and the other set being "those who spend a lot of time thinking about how to explain things clearly to newbies) may be vast, but the intersection is damn near the empty set (- that wisecrack is borrowed).
Intersect that with "those who have written math textbooks", and you'll get the picture.
It's not impossible, it's just hard, and, often, our cultural blinders don't let us see the payoff (if you want evidence of that, notice how quickly people reject the notion that more visualization would help--"if you learn with graphics, you'll suddenly quit understanding things when you get to 4-d or infini-d". It's baloney, but it's deeply ingrained baloney.)
Yet another barrier is that mathematicians make excellent use of the economy of notation. You can say a hell of a lot with a few symbols, and the very thought, once you've learned to use these symbols, of actually going back and writing out in english what you just expressed in symbols is anathema.
An analogy, for those who have messed with Perl, is regular expressions. How many people really comment their regular expressions? Once you've said it in such a nice, tight format, it just hurts to think about having to explain it in text.
For example, one of the first RegExp's in the perlre manpage ("man perlre" if you're on unix) looks like this:
s/^([^ ]*) *([^ ]*)/$2 $1/; # swap first two words
okay, that's commented--well, the "effect" is described. But imagine if you were trying to state what that expression does:
"Starting at the beginning of $_ (the default variable for matching), find the longest contiguous block (even if it's a block of length zero) of non-space characters (and store that in a variable called $1), then go past all the contiguous spaces after that, and group together the next contiguous block of non-spaces. Put this block of non-spaces into a variable called $2 [the "store that in $1 and $2" is implied by the presence of the parentheses, by the way]. Replace all of the matched text with a string consisting of the second block, a single space, and the first block."
Now that I've explained it in more excruciating detail, I understand it better. I can see that it won't work as advertised, for example. (try it on
foo bar baz
or even
foo bar
Maybe something like
s/(\S+)\s+(\S+)/$2 $1/;
would be better. Got to be careful with them *'s!
)
But look at the sheer number of characters in the text explanation! To another perlvert, the regular expression says the exact same thing. This is very similar to the situation in math--it's sooooo much easier to get the point across with a few terse symbols and references to theorems that it's really hard to get yourself to go through the effort required to explain it to the uninitiated (oops--pun inintentional).
Again, I'm not meaning to flame you, Darby--you hardly exhibited the problem compared to what other posts did. I'm talking about the general trend.
mike
Liberty uber alles.
check out this link to treasure troves if you want to know more about it... http://www.treasure-troves.com/math/Taniyama-Shimu raConjecture.html
This is kind of a paradoxical position, since the statement itself (the negation of an existential) is universal, and so itself isn't knowledge. Philosophers tend not to like to built upon such self-defeating foundations...
I rather like Popper, and I think his argument (adapted from Hume) about the invalidity of induction is sound. But his alternative I don't think should be adopted uncritically.