Greater achievement  Perelman's or Wile's proof?
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Greater achievement  Perelman's or Wile's proof?
Both of these guys produced stunning results, rife with original thinking, that solved incredibly difficult problems. Both worked on the cutting edge of modern mathematics, innovating not merely in the fact that they created these proofs, but in that the techniques they used and subproblems they solved have reshaped their fields.
However, I ask you, XKCD, which do you think was the greater achievement, the one which represents a more herculanean intellectual feat?
Also, say if you feel there are any results of comparable magnitude that have occured in recent times  I don't wish to exclude.
However, I ask you, XKCD, which do you think was the greater achievement, the one which represents a more herculanean intellectual feat?
Also, say if you feel there are any results of comparable magnitude that have occured in recent times  I don't wish to exclude.
 skeptical scientist
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Re: Greater achievement  Perelman's or Wile's proof?
I don't think it's really the kind of thing you can evaluate without being an expert in both areas, and understanding both proofs. That definitely doesn't include me, and I would be very surprised if anyone on these fora meets the criteria.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Greater achievement  Perelman's or Wile's proof?
I would guess that a grad in math, maybe a PHD, would probably be able to do it, but maybe you're right.
Where would I be able to find a place with people who have such mastery of modern mathematics as you think are needed to weigh this question?
Where would I be able to find a place with people who have such mastery of modern mathematics as you think are needed to weigh this question?
Re: Greater achievement  Perelman's or Wile's proof?
I've never seen the point of asking questions like this; "greater" is a matter of opinion. It's more interesting to talk about meaningful qualitative differences: which proof is more abstract? Which result is farther away from having a practical application?
As for your other question, Terence Tao taught a course on Perelman's proof of the Poincare conjecture, so presumably he thought that his students could handle it. On the other hand, Wiles' field is notorious for requiring intense study as it requires an enormous amount of background, so I think you're less likely to find grad students who could thoroughly understand Wiles' proof than Poincare's. Of course, I'm just guessing.
As for your other question, Terence Tao taught a course on Perelman's proof of the Poincare conjecture, so presumably he thought that his students could handle it. On the other hand, Wiles' field is notorious for requiring intense study as it requires an enormous amount of background, so I think you're less likely to find grad students who could thoroughly understand Wiles' proof than Poincare's. Of course, I'm just guessing.
Re: Greater achievement  Perelman's or Wile's proof?
HalfThere wrote:I would guess that a grad in math, maybe a PHD, would probably be able to do it, but maybe you're right.
Where would I be able to find a place with people who have such mastery of modern mathematics as you think are needed to weigh this question?
There might be some professors out there, but I think it would be unlikely that someone would be very familiar with both. Their fields are not very closely related, but I'm sure there are some people out there. It would be extremely uncommon for a math grad student to be familiar with both, or even close to being able to approach them.
 Gaydar2000SE
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Re: Greater achievement  Perelman's or Wile's proof?
No idea, I don't think I understand either's.
I just hate Wiles because he did it for the glory with his working in secrecy business, serves him right that he erred on the first go.
Also, I share Perelman's idea that it should be about the results and not about the people. Also, every mathematical theorem proven is essentially built upon other theorems, lemmata and otherwise, surely then the contribution is vast, the one who lays the last block just gets the prize? I mean, what if Wiles first proved that elliptic curve thing, forgot how it was called. And then some other person came with 'Yeah, but if fermat's is true, then some elliptic curves aren't modular, didn't some guy 50 years back prove that ALL elliptic curves are?', would they then posthumously award Wiles the Fields and the Millennium and whatnot? It's just a matter of subjection who has done the most important part. I mean, obviously the most important thing was the guy who invented the concept of axiomatic proving, shouldn't he get ALL the awards?
And also, the point of mathematics that a statement does not become a theorem when it's proven. It's a theorem if it is provable. To say 'Now a proof exists because some one has found it' isn't a thing that can just formalized. Saying 'a proof exists because it's provable' is quite easy to formalize in proof theory. Thus, it is the investigation if it is a theorem or not, it was 'always' a theorem or never will be. The most common way to investigate if a proof exists is just to find it of course. Generally humans do this because humans do not know other things yet that can, but really, couple of lose from the human ego. Ego's are only good for Freudian bad humour about explaining all things by saying it's a subconscious desire to have sex with your mother. And then take that ego away by saying 'Hey, but if people believed that crap yesterday, surely most things we believe today that aren't maths are probably crap too, yeah, let's just close psychiatry as we know it caused nothing but harm yesterday, so probably today and tomorrow too.'
I just hate Wiles because he did it for the glory with his working in secrecy business, serves him right that he erred on the first go.
Also, I share Perelman's idea that it should be about the results and not about the people. Also, every mathematical theorem proven is essentially built upon other theorems, lemmata and otherwise, surely then the contribution is vast, the one who lays the last block just gets the prize? I mean, what if Wiles first proved that elliptic curve thing, forgot how it was called. And then some other person came with 'Yeah, but if fermat's is true, then some elliptic curves aren't modular, didn't some guy 50 years back prove that ALL elliptic curves are?', would they then posthumously award Wiles the Fields and the Millennium and whatnot? It's just a matter of subjection who has done the most important part. I mean, obviously the most important thing was the guy who invented the concept of axiomatic proving, shouldn't he get ALL the awards?
And also, the point of mathematics that a statement does not become a theorem when it's proven. It's a theorem if it is provable. To say 'Now a proof exists because some one has found it' isn't a thing that can just formalized. Saying 'a proof exists because it's provable' is quite easy to formalize in proof theory. Thus, it is the investigation if it is a theorem or not, it was 'always' a theorem or never will be. The most common way to investigate if a proof exists is just to find it of course. Generally humans do this because humans do not know other things yet that can, but really, couple of lose from the human ego. Ego's are only good for Freudian bad humour about explaining all things by saying it's a subconscious desire to have sex with your mother. And then take that ego away by saying 'Hey, but if people believed that crap yesterday, surely most things we believe today that aren't maths are probably crap too, yeah, let's just close psychiatry as we know it caused nothing but harm yesterday, so probably today and tomorrow too.'
^ :/
Re: Greater achievement  Perelman's or Wile's proof?
Gaydar2000SE wrote:I just hate Wiles because he did it for the glory with his working in secrecy business, serves him right that he erred on the first go.
Also, I share Perelman's idea that it should be about the results and not about the people. Also, every mathematical theorem proven is essentially built upon other theorems, lemmata and otherwise, surely then the contribution is vast, the one who lays the last block just gets the prize? I mean, what if Wiles first proved that elliptic curve thing, forgot how it was called. And then some other person came with 'Yeah, but if fermat's is true, then some elliptic curves aren't modular, didn't some guy 50 years back prove that ALL elliptic curves are?', would they then posthumously award Wiles the Fields and the Millennium and whatnot? It's just a matter of subjection who has done the most important part. I mean, obviously the most important thing was the guy who invented the concept of axiomatic proving, shouldn't he get ALL the awards?
Let's get a few things straight. Wiles did not win the Millennium Prize, it was not offered for Fermat. Wiles did not win the Fields Medal, as he was too old. Also, the other people who made substantial contributions (Ribet, Frey, Taylor, Serre, to name just a few) have had their own share of glory and are mostly household names for mathematicians. Sure, there may not have been an article about them in the New York Times, but I highly doubt that was Wiles's motivation.
 Gaydar2000SE
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Re: Greater achievement  Perelman's or Wile's proof?
Well, I know a mathematician would have to be under 40 for the Fields, also I think it to be conceivable that if it wasn't proven by then Fermat would be in the Millennium Prizes no? I was referring to that, Wiles actually formulated a specification for the Prizes, they were made after he got to the front for Fermat, I know this.auteur52 wrote:Gaydar2000SE wrote:I just hate Wiles because he did it for the glory with his working in secrecy business, serves him right that he erred on the first go.
Also, I share Perelman's idea that it should be about the results and not about the people. Also, every mathematical theorem proven is essentially built upon other theorems, lemmata and otherwise, surely then the contribution is vast, the one who lays the last block just gets the prize? I mean, what if Wiles first proved that elliptic curve thing, forgot how it was called. And then some other person came with 'Yeah, but if fermat's is true, then some elliptic curves aren't modular, didn't some guy 50 years back prove that ALL elliptic curves are?', would they then posthumously award Wiles the Fields and the Millennium and whatnot? It's just a matter of subjection who has done the most important part. I mean, obviously the most important thing was the guy who invented the concept of axiomatic proving, shouldn't he get ALL the awards?
Let's get a few things straight. Wiles did not win the Millennium Prize, it was not offered for Fermat. Wiles did not win the Fields Medal, as he was too old. Also, the other people who made substantial contributions (Ribet, Frey, Taylor, Serre, to name just a few) have had their own share of glory and are mostly household names for mathematicians. Sure, there may not have been an article about them in the New York Times, but I highly doubt that was Wiles's motivation.
And errr, Wiles essentially worked on it in total secrecy together with Katz, and when the error was found, even then he refused to make it public to any one else and wanted to repair it himself. He wanted to be the one that solved the problem al right.
Also, glory sucks, I'm with Perelman that all these 'awards' really need to GTFO out my science. It's getting more of a popularity contest really. Like a commission awarding Perelman the Fields if none of them had even read the proof themselves or were specialized in his field and really had no grasp of Poincaré but just believe those that had when they said it was a daring mountain to cross with no way to even start. That Leibniz and Newton battle about who invented calculus is also sickening, come on, all that matters is that it's there, especially because both of them divided by zero. They just called it 'the infinitesimal', that doesn't change a darn thing, they had no concept of hyperreals and in the framework they placed it in it's damned trivial to prove that the infinitisimal is the same object as 0. The unique additive identity of the reals.
That theorems and laws are named after people is also really just a sign of human lust for glory and popularity. I would just go there being Perelman, not accept it but throw a bunch of rotten tomatoes on the rest, that would be so damned awesome, ahahah, and they can't really get you then either because you just proved Poincaré and refused the Fields.
^ :/
Re: Greater achievement  Perelman's or Wile's proof?
Gaydar2000SE wrote:That theorems and laws are named after people is also really just a sign of human lust for glory and popularity.
How do you figure? Wouldn't it make more sense that we are just trying to honor people that we respect? People do not (usually) name theorems and definitions after themselves.
Edit: Plus the fact that there are only so many words to describe things. If we didn't name things after people, we would have a hard time coming up with so many snappy names.
 Gaydar2000SE
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Re: Greater achievement  Perelman's or Wile's proof?
Newton's first law <=> The Law of equality of force to the derivative of momentum to time., or shorter: The Law momentum and forceauteur52 wrote:Gaydar2000SE wrote:That theorems and laws are named after people is also really just a sign of human lust for glory and popularity.
How do you figure? Wouldn't it make more sense that we are just trying to honor people that we respect? People do not (usually) name theorems and definitions after themselves.
Edit: Plus the fact that there are only so many words to describe things. If we didn't name things after people, we would have a hard time coming up with so many snappy names.
I find the latter case much more descriptive, after all, it's called 'The prime number theorem' and 'The main theorem of arithmetic', neither are named after Euclid which proved them. It is called 'The Pythagorean Theorem' and 'Fermat's last theorem' though, and the latter didn't even prove it?
And no, it doesn't make sense, in fact, it makes no sense to respect some one you haven't even spoken to or at least seen videos of and read into the ideas of said. Discovering that force is equal to the derivative of momentum with respect to time does not mean I respect some one. After all, Wiles did daunting mathematical work, can't say I respect him for putting his lust for fame and spotlight above mathematical advancements.
I don't have any ambition to 'honour' people, especially if they are dead, then you just do it for yourself and not them. And to that last point, AngloSaxon and western naming structure in general is inconvenient because in formal situations one is referred to by the name of one's family, and not the name of one's own. There are a lot of Robinsons, Dawkinses, Johnsons, and Smiths, thus.
Actually, most theorems and/or laws are so extremely simple to state they don't even need a name. F(t) = dp/dt is quite easy to write.
It's hero worship, it's nothing different to teenage children honouring people like Alexi Laiho for one part of him, because they feel he's a good guitarist, they've never spoken to him, they don't know him. But that one thing is enough because they're really obsessed with guitars in that phase of their life, mathematicians can be no different it seems if they are obsessed enough with mathematics to honour a person based on that one facet of them. They even have the urge to meet that person because of that, with no guarantee Wiles'll make any good conversationalist. Just because he proved Fermat, what's next? Signing your breasts? Human need to have idols and to look up to things, dangerous urges like this created strange myths and Stalinic personality cults. Also, the fact that mathematics is a popularity contest trying to get the most acclaim (like the citations system, Christ, that just means that the easiest to comprehend your work is the better it is?) results into things like this, that the proving of Fermat was delayed for a significant time because Wiles wanted all the honour.
^ :/
Re: Greater achievement  Perelman's or Wile's proof?
I don't think Euclid first proved either of those, the Pythagoreans did prove the pyhtagorean theorem and the FermatWiles Theorem is frequently named after the man who devoted his whole life to proving it and eventually did.
The citation system is not just for honoring the people on whose work yours is built. It's there so that everyone can make references to that supporting work so that they don't have to include the details of it.
The citation system is not just for honoring the people on whose work yours is built. It's there so that everyone can make references to that supporting work so that they don't have to include the details of it.
 Gaydar2000SE
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Re: Greater achievement  Perelman's or Wile's proof?
Euclid essentially gave a proof that was not completely rigorous to them by today's standards. Though note that today's rigour has it's roots in only a 100 years back, essentially nothing Euler or Newton did was the mathematical rigour they ask of undergraduate students today. Newton divided by zero quite a lot.
And with the citation system, I meant that often people give 'scores' (no idea why that needs to be there any way), based on the number of citations.
It's a typical human thing to bend it to justify his needs. Man wants to be able to put a 'rating' on things like music and the 'quality' of a research and 'intelligence' while clearly such things cannot just have a total order put unto them, so they come with shaky things to realize it like 'expert opinions' (argument to authority) or citations and topforty (argumentum ad populum), all chiefly logical fallacies of course.
Humans are the masters of deceiving themselves into believing that things are there which are not there, but they want to be there. Dawkins was quite correct in his delusiontypology, he just forgot to class the entire human species in it.
And with the citation system, I meant that often people give 'scores' (no idea why that needs to be there any way), based on the number of citations.
It's a typical human thing to bend it to justify his needs. Man wants to be able to put a 'rating' on things like music and the 'quality' of a research and 'intelligence' while clearly such things cannot just have a total order put unto them, so they come with shaky things to realize it like 'expert opinions' (argument to authority) or citations and topforty (argumentum ad populum), all chiefly logical fallacies of course.
Humans are the masters of deceiving themselves into believing that things are there which are not there, but they want to be there. Dawkins was quite correct in his delusiontypology, he just forgot to class the entire human species in it.
^ :/
Re: Greater achievement  Perelman's or Wile's proof?
Man, you need to relax. We name theorems after people for a lot of reasons. Respect is a big part of it. But a lot of the time it is just for ease of communication. I really can't think of a snappy name for Green's Theorem for instance. But I would immediately understand what you meant if you said it.
To say Wiles proved Fermat's Last Theorem only for the glory seems completely unfounded. I have read that he came across the theorem in his youth, it was obviously a personal obsession of his. It seems naive to say that he was only in it for the fame, although I am sure that was a part of it.
Also, Perelman is surely much more famous for rejecting his Field's medal rather than accepting it.
To say Wiles proved Fermat's Last Theorem only for the glory seems completely unfounded. I have read that he came across the theorem in his youth, it was obviously a personal obsession of his. It seems naive to say that he was only in it for the fame, although I am sure that was a part of it.
Also, Perelman is surely much more famous for rejecting his Field's medal rather than accepting it.
Re: Greater achievement  Perelman's or Wile's proof?
Gaydar2000SE wrote:I mean, what if Wiles first proved that elliptic curve thing, forgot how it was called. And then some other person came with 'Yeah, but if fermat's is true, then some elliptic curves aren't modular, didn't some guy 50 years back prove that ALL elliptic curves are?', would they then posthumously award Wiles the Fields and the Millennium and whatnot? It's just a matter of subjection who has done the most important part.
Fermat's Last Theorem was a very hard proposition to prove, and nobody succeeded for 300 years, but that's about the extent of its relevance. There were no developing fields full of theorems of the type "if FLT is true, then...". FLT seems to be no more than an arithmetical statement.
However, the TaniyamaShimura conjecture ("all elliptic curves are modular") was a natural development of that branch of number theory, and ripe with consequences. It was a conjecture that connected several things. Arguably, TS was harder to prove than FLT, since FLT fell out of TS + FreyRibet. TS is clearly much more relevant to mathematics than FLT, and moreover FreyRibet was not felt to be a major piece of work (even by Ribet himself). However, TS required the introduction of several new ideas, and would be a landmark in number theory even if FLT had never been formulated.
Gaydar2000SE wrote:That Leibniz and Newton battle about who invented calculus is also sickening, come on, all that matters is that it's there, especially because both of them divided by zero. They just called it 'the infinitesimal'
Newton never talked about infinitesimals; he focused on the concept of derivative, which he called "fluxions". In the Principia, he doesn't even use these; it's all Euclidean geometry and limiting arguments. There is not a single mathematical gap in the first book of the Principia (which is the more mathematical part). Even in his other, more liberal work on infinite series, saying he "divided by zero" is a gross misrepresentation of the semiformal arguments he presented, and anachronistic to the point of irrelevance.
Gaydar2000SE wrote:The prime number theorem' and 'The main theorem of arithmetic', neither are named after Euclid which proved them.
The Prime Number Theorem was proved in the 19th century by a large collaboration, the final touches being given by Hadamard and de la Vallée Poussin. The Fundamental Theorem of Arithmetic was only recognized as something to be proved by Gauss, which he did in the 1801 Disquisitiones Arithmeticae (in the first couple of chapters).
Gaydar2000SE wrote:Euclid essentially gave a proof that was not completely rigorous to them by today's standards. Though note that today's rigour has it's roots in only a 100 years back
Euclid proved that there is not a finite number of primes, and his proof stands perfectly today. This is not what is called the "Prime Number Theorem".
Euclid's "lack of rigor" stems only from some unstated continuity assumptions on lines and circles, which leads back to analysis. If you grant him this, all the rest (of which there is a lot) is flawless. In particular, the discrete aspects, like the number theory, has not required any modification. It is mathematically and historically misinformed to state that "today's rigor has its roots only a 100 years back". It smells of shallow popularization articles rather than scholarly study.
Gaydar2000SE wrote:essentially nothing Euler or Newton did was the mathematical rigour they ask of undergraduate students today. Newton divided by zero quite a lot.
Euler was a bit hasty in analysis, but his other work, of which there is quite a lot (he is "everywhere dense" in mathematics) is not lacking in rigor. I defy you to point a single flaw in Euler's work in graph theory, elementary number theory, combinatorics or algebra. There should be a good 2000+ pages for you to try your luck.
I also defy you to produce a single instance of Newton "dividing by zero" in his published work.
You seem quite eager to talk about math and its history, but you don't seem to have anything but a very tenuous grasp of either.
I always wondered about the meaning of life. So I looked it up in the dictionary under "L" and there it was  the meaning of life. It was not what I expected. (Dogbert)

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Re: Greater achievement  Perelman's or Wile's proof?
Pietro wrote:(he is "everywhere dense" in mathematics)
Thanks, you just made my day. Unfortunately I don't really have anything else to add to the discussion.
 Gaydar2000SE
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Re: Greater achievement  Perelman's or Wile's proof?
Ease of communication? How's saying 'Newton's fist law of motion' any easier than 'The Law of momentumandforce'? The latter is surely quite a lot more descriptive, strangely. 'Euclid's theorem' is a lot shorter than 'The fundamental theorem of arithmetic' and both are equally vague in name. Naming theorems after people essentially causes confusion if people had more than one interesting result. The 3 and the 1 are quite arbitrary of course in Newton's case.dissonant wrote:Man, you need to relax. We name theorems after people for a lot of reasons. Respect is a big part of it. But a lot of the time it is just for ease of communication. I really can't think of a snappy name for Green's Theorem for instance. But I would immediately understand what you meant if you said it.
Just saying: The law of momentumandforce, the law of gravity, and the law of conservative force is quite a lot more descriptive and stops confusion. But then you aren't idolizing your mathematical idols like a teenage girl idolizing Laiho do you? Be it God, Raymond van Barneveld, Newton, Knuth, or Fabio Lanzoni, people always need some thing to idolize. You could also ask yourself why people tend to idolize people they don't really know, could it be that.. ghasp.. if you do know them from a close distance they're not gods any more but just random people?
I never said it was the only part, I said it mattered to him that he was the one that proved it. Not that it was proven in the end by some one. Your sentence seems to imply we agree here. Typical really, one sees it every where, vegetarians do not care for that meat isn't eaten, they care for it that they aren't the ones that eat it. I have it too, I find it a lot worse if for instance a friend of mine cries and I was the one that caused it, it's disgusting, that said cries is all that matters, not who caused it, the damage is done and that I was the one that caused it isn't relevant. Truly disgusting aspect of people here. And in the end we all try to delude ourselves that we're caring to heal it.To say Wiles proved Fermat's Last Theorem only for the glory seems completely unfounded. I have read that he came across the theorem in his youth, it was obviously a personal obsession of his. It seems naive to say that he was only in it for the fame, although I am sure that was a part of it.
True, ahaha. Poor guy though, he retired from maths after that, the pictures taken of him in the trains and all, ahah, he's apparently scared to go out of (his mother's) house.Also, Perelman is surely much more famous for rejecting his Field's medal rather than accepting it.
Newton quite did divide by zero to produce his derivatives and his methodology often returned simply absurd errors and the approach was then to simply go another path until it's no longer absurd. Derivative functions in the days of Leibniz and Newton were never rigorous mathematics, in fact, the concept didn't even exist really then as it was never quite clearly stated what axioms are used. They used naïve imported explicitly and implicitly axioms they saw around them and deemed 'common sense'. There was no definition of what 'real numbers' really were back then but Newton used the standard addition and added a certain constant p a lot to a given x to leave x. That's an additive identity per definition, and where he operated the additive identity is the only element that does not have a multiplicative inverse. Yet he still divided by that p a lot of times to make his derivatives. That p is often just called 0, in Newton's day people had the idea that it was a different object, often called such things as 'the smallest possible thing next to zero' or 'infinitely small', real numbers are dense, there is no 'smallest possible next to zero', if there is no other element between p and 0 then p = 0.Newton never talked about infinitesimals; he focused on the concept of derivative, which he called "fluxions". In the Principia, he doesn't even use these; it's all Euclidean geometry and limiting arguments. There is not a single mathematical gap in the first book of the Principia (which is the more mathematical part). Even in his other, more liberal work on infinite series, saying he "divided by zero" is a gross misrepresentation of the semiformal arguments he presented, and anachronistic to the point of irrelevance.
The free encyclopaedia will all love and hate says:
'The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics. In his publications, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica. In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.'
The 'naïve' concept of the infinitesimal in newton's time equals zero. The infinitesimal cannot exist as a real number, this is quite trivial. Essentially Newton's derivatives usually worked but not always and to that it's not mathematical, but rather engineering.
Though it is true that that Newton divided by zero is as much naïve as his work, as he never explicitly said that dx was a real number, or that it was a number or what it really was. He never really said what he was doing, as was the norm in those days. Russell tore it down at the fundamental when he showed that things can go wrong if one is not extremely precise about what one is actually doing. It's just to assume that dx was to be a real number, because it really couldn't be any thing else. But there's no way to proof that it wasn't or was a real number, as he never really defined dx.
Euclid proved that there is not a finite number of primes, and his proof stands perfectly today. This is not what is called the "Prime Number Theorem".
I know, I meant that Euclid gave a 'rough' proof of the fundamental theorem of arithmetic, the prime number theorem came quite some time later. Also, his prove doesn't stand, his conclusion stands. It was dumb luck actually. I could point out one flaw in his proof. To begin with his proof depends on the existence of a total order on the naturals and he didn't prove it was there. He just naïve assumed that if one has two natural numbers n,m, unequal, that one is larger than the other, he never proved it. It makes sense to human intuition, naïve mathematics opposed to axiomatic.
Euler's lack of rigour is far deeper, in fact, it's essentially what filled the entirety of mathematics until Russell tore it all down, the point is that Euler never was explicit about what axioms precisely he proved from. And a lot of axioms in those days were implicitly assumed because they just 'made sense' to the common naïve realist perception. There was really no hard way in that time to why one couldn't implicitly assume the theorem to prove to be true and then beg the question because from what axioms it was being proven was not set in stone. The only thing that stopped people was the subjective line of 'that it just didn't feel good', hardly mathematics of course.
^ :/
Re: Greater achievement  Perelman's or Wile's proof?
I thought that the reason why Wiles hide from everyone was because that if he says it out, people would have think he's nuts. Another thing, just because you do not respect the work that came before you, it does not mean that the rest of us do not. It's true that some of the things are considered "obvious" now, but to come up of it in the first place requires pure ingenuity.
The only thing I hate about the citation system is if the person that is being cited is either Erdos, Euler, Gauss, or Cauchy. They have too many theorems named after them that it is impossible to tell which one is which.
The only thing I hate about the citation system is if the person that is being cited is either Erdos, Euler, Gauss, or Cauchy. They have too many theorems named after them that it is impossible to tell which one is which.
Find me a few example of his graph theory that has problems, then. I want specific examples, not hand waving.Gaydar2000SE wrote:Euler's lack of rigour is far deeper, in fact, it's essentially what filled the entirety of mathematics until Russell tore it all down, the point is that Euler never was explicit about what axioms precisely he proved from. And a lot of axioms in those days were implicitly assumed because they just 'made sense' to the common naïve realist perception. There was really no hard way in that time to why one couldn't implicitly assume the theorem to prove to be true and then beg the question because from what axioms it was being proven was not set in stone. The only thing that stopped people was the subjective line of 'that it just didn't feel good', hardly mathematics of course.
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Re: Greater achievement  Perelman's or Wile's proof?
Gaydar2000SE wrote:Ease of communication? How's saying 'Newton's fist law of motion' any easier than 'The Law of momentumandforce'? The latter is surely quite a lot more descriptive, strangely.
Newton's First Law of Motion is often called the Law of Inertia. Just FYI. The "Law of momentumandforce" sounds like a better name for Newton's Second. But actually, it would be confusing to refer to any of them as being laws of force (or motion), because all three are fundamental laws of force and motion in Newtonian mechanics.
I guess you could go the thermodynamics route, and just call them The First Law of Motion, the Second Law of Motion, and the Third Law of Motion. I like that better than the Law of MomentumandForce, certainly.
Re: Greater achievement  Perelman's or Wile's proof?
Gaydar2000SE wrote:I know, I meant that Euclid gave a 'rough' proof of the fundamental theorem of arithmetic, the prime number theorem came quite some time later. Also, his prove doesn't stand, his conclusion stands. It was dumb luck actually. I could point out one flaw in his proof. To begin with his proof depends on the existence of a total order on the naturals and he didn't prove it was there. He just naïve assumed that if one has two natural numbers n,m, unequal, that one is larger than the other, he never proved it. It makes sense to human intuition, naïve mathematics opposed to axiomatic.
You have a strange definition of "flaw".
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Re: Greater achievement  Perelman's or Wile's proof?
Token wrote:Gaydar2000SE wrote:I know, I meant that Euclid gave a 'rough' proof of the fundamental theorem of arithmetic, the prime number theorem came quite some time later. Also, his prove doesn't stand, his conclusion stands. It was dumb luck actually. I could point out one flaw in his proof. To begin with his proof depends on the existence of a total order on the naturals and he didn't prove it was there. He just naïve assumed that if one has two natural numbers n,m, unequal, that one is larger than the other, he never proved it. It makes sense to human intuition, naïve mathematics opposed to axiomatic.
You have a strange definition of "flaw".
Yes, that is a ridiculous idea of a flaw. Of course the natural numbers are ordered. The concept of an ordering was only ever posited because folks wanted to isolate and generalize a particular property of the naturals to other contexts.
Also, gold star for Pietro's post.
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Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Greater achievement  Perelman's or Wile's proof?
Token wrote:Gaydar2000SE wrote:I know, I meant that Euclid gave a 'rough' proof of the fundamental theorem of arithmetic, the prime number theorem came quite some time later. Also, his prove doesn't stand, his conclusion stands. It was dumb luck actually. I could point out one flaw in his proof. To begin with his proof depends on the existence of a total order on the naturals and he didn't prove it was there. He just naïve assumed that if one has two natural numbers n,m, unequal, that one is larger than the other, he never proved it. It makes sense to human intuition, naïve mathematics opposed to axiomatic.
You have a strange definition of "flaw".
That, and I'm quite certain that if you pressed him, he could give you a proof that the naturals are ordered.
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Re: Greater achievement  Perelman's or Wile's proof?
Pietro: You forgot the "Summer Glau"
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Re: Greater achievement  Perelman's or Wile's proof?
Gaydar2000SE wrote:I find the latter case much more descriptive, after all, it's called 'The prime number theorem' and 'The main theorem of arithmetic', neither are named after Euclid which proved them.
The infinitude of primes is actually called Euclid's Theorem (check Wikipedia if you don't believe me). As has been pointed out, the Prime Number Theorem is something very different and much deeper, go look it up.
You keep bringing up the example of Newton's laws, but seriously I do not think it is reasonable to always think up a concise name for every theorem. What would you call the Sylow Theorems? Galois groups? ArzelaAscoli Theorem? Poincaré Duality? I can't think of any snappy names for those, naming them after the people responsible is convenient, respectful and aesthetically pleasing.
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Re: Greater achievement  Perelman's or Wile's proof?
dissonant wrote:Man, you need to relax.
Yes, seriously. This is a discussion of which proof is the greater achievement, anyway. Not about whether either or both men deserve the recognition they've gotten as a result of said proof. Start another thread for that if you want, because further discussion of it in this thread will be deleted and perhaps accompanied by a warning to whoever engages in it.
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