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Thread: fermat's last theorem

  1. #1 fermat's last theorem 
    Forum Freshman BitterSweet's Avatar
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    So I was just at the General Discussion's "you know you're a geek when..." and i was wondering... has anyone here ever tried to prove fermat's last theorem? Because I've tried it before, and I'm only a high school student......


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  3. #2  
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    It's been proven, but IIRC the proof is very complicated and took some extremely high level maths.


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  4. #3  
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    it was proved true in 1993, and the proof is EXTREMELY complicated, I understand only bits and pieces, but I've read the whole thing, quite long too. The proof is, itself, a book.
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    Not hard to grasp??? You have got to be kidding me.
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    there's a lot that went into it too, not just the indivual proofs f the individual theoroms that he used. The ties are rather complex too, but understanding a few of the PRE-cepts isn't too hard. the whole thing is nice and difficult though
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  7. #6  
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    Quote Originally Posted by frank
    The proof was solved by the reknown mathematician Andrew Wiles. It's really not too difficult to grasp; just continue to read bits and and pieces of the theorem until you get a portion of the jist, than read it from top to bottom.
    salsaonline is correct. Utterly correct.

    Here is link to Wiles paper.

    If this is not a difficult paper, I don't know what one would look like. They don't give out Fields Medals for the obvious.

    http://math.stanford.edu/~lekheng/flt/wiles.pdf
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  8. #7  
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    Quote Originally Posted by frank
    Quote Originally Posted by DrRocket
    salsaonline is correct. Utterly correct.

    Here is link to Wiles paper.
    Salsaonline hardly made a statement; to suggest he is "correct, 'utterly' correct'" hardly makes sense and in general is characterized by flawed logic. For one, the difficulty level of something is relative, and what might be difficult for you or I to grasp probably won't be so for the likes of von Leibniz, Newton, or da Vinci. Secondly, Fields Medals aren't given out for their degree of "obviousness," nor are they dependant on the length of a paper for that matter. I could similarily post a paper on this forum for my general formula for calculating the diagnols of any polygon or one on supergravity and make the concepts more difficult than they really are intentionally, according to you that would, at least, make me a nominee for a Fields Medal. Than again, I haven't read Wiles full paper (again, I read that of a different theorem Wiles proved); I'll read the paper when I find the time and comment on its difficulty, but simply because you or Salsaonline suggest it's difficult doesn't make it so.
    No logical flaw. Perhaps a bit of license with the wording. But, nevertheless an opinion based on considerable background and experience, as is the case with salsaonline.

    I am quite certain that you will find that paper tough sledding. Unless, perhaps, your name is Deligne. I suspect even he would admit that Wiles's proof is a tour de force, and not at all easy to follow.

    Fields Medals are awarded for work that is groundbreaking in mathematics, and that develops radically new and deep concepts or methods. That in and of itself obviates the obvious.

    There is quite a difference between lack of obviousness and intentional obfuscation. It is not sufficient that work not be obvious in order to mertit a Fields Medal, but it is most certainly necessary. Since you seem to fail to grasp the nature of this implication, the logical fallalcy is yours.

    The remark by salsaonline expressisng incredulity at your comment, was quite appropriate. I'll add my own harumph!!
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    To add to my comment:

    A few weeks ago I was hanging out at the math dept with a number theory professor and an undergraduate. From what I gather, the undergraduate student is rather strong--at the very least he seems to be taking grad level classes and attends number theory seminars hosted by this professor. Somehow, the topic of the Taniyama-Shimura conjecture came up, and the student suggested that the professor put together a seminar next semester to go over Wiles' proof. The professor politely rejected this suggestion, saying something along the lines of, "I think you may not realize how technical Wiles' proof actually is".

    As for "what's hard for some might be easy for others," I sincerely doubt that anyone, especially Wiles, would characterize his proof as "easy". In fact, I suspect he'd find a statement like that really insulting. Brilliant as some mathematicians are, they work their asses off to get their results.

    By the way, Wiles never received the Fields Medal, because of the age restriction. Although a special award was created in Wiles' honor to compensate for this unfortunate technicality.
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  10. #9  
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    Quote Originally Posted by salsaonline
    By the way, Wiles never received the Fields Medal, because of the age restriction. Although a special award was created in Wiles' honor to compensate for this unfortunate technicality.
    That is right. For some reason I had thought that he had received the traditional Fields Medal, but that is not correct.

    He has, not surprisingly, received a number of other awards.
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  11. #10  
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    Quote Originally Posted by salsaonline
    To add to my comment:

    A few weeks ago I was hanging out at the math dept with a number theory professor and an undergraduate. From what I gather, the undergraduate student is rather strong--at the very least he seems to be taking grad level classes and attends number theory seminars hosted by this professor. Somehow, the topic of the Taniyama-Shimura conjecture came up, and the student suggested that the professor put together a seminar next semester to go over Wiles' proof. The professor politely rejected this suggestion, saying something along the lines of, "I think you may not realize how technical Wiles' proof actually is".
    That sounds about right.

    With regard to people who think they understand complicated mathematics, I recall an incident that occurred years ago.

    We were having one of the usual colloquim lectures, with a very distinguished speaker. Rene Thom was talking about his work in the classification of singularities of manifolds. His talk was quite difficult for most of us to follow. At one point he showed a slide with a surface containing many cusps. Another fellow and I were staring intently at this surface when Thom said something like, " And here we have 'le chicone" (spelled more or less phonetically to try to impart the French accent). And sure enough, the slide was a close up of the skin of a plucked chicken. At that pont the guy next to me nearly fell out of his chair.

    In any case the talk was quite technical for the most part (plucked chickens aside) and very hard to follow. There was only one person who claimed to have completely understood the talk -- the nineteen year-old departmental secretary.
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    Wow, that's so cool that you met Rene Thom.

    Btw, what area of research are you in?
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  13. #12  
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    Quote Originally Posted by salsaonline
    Wow, that's so cool that you met Rene Thom.

    Btw, what area of research are you in?
    I'll send you a PM. No need to bore the world.
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    At this point I just have to shrug my shoulders and say, hey, do whatever you want. It's your time.
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  15. #14  
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    Quote Originally Posted by frank
    Quote Originally Posted by DrRocket
    I am quite certain that you will find that paper tough sledding. Unless, perhaps, your name is Deligne. I suspect even he would admit that Wiles's proof is a tour de force, and not at all easy to follow.
    I disagree. I can think of a lot of highly intelligent, eminent people in history that I'm quite sure wouldn't have too much difficulty comprehending Wiles' proof.
    OK let's make it simple. Put your money where your mouth is. Go read the paper and then prepare and publish a brief synopsis explaining the proof in simple terms, say something easily understood by a third year graduate studient in algebraic geometry.

    I would anticipate little trouble getting it published if you have the understanding that you seem to think you can attain. In fact, if you can put together a really good explanation and have trouble getting it reviewed, post it here. I can forward it to a friend who sits on a board with Andrew Wiles himself. I am sure that Wiles would be interested.

    What is a reasonable time for you ? Say one month from today ?
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    Except that no one here is a number theorist, so who's going to be able to read it? The true conditions should be that it should be understandable to a 3rd year number theory grad student, not algebraic geometry.

    Anyway, I have to interject that this is probably an impossible task even for an expert. If the subject were explainable at that level, then almost by definition, someone would have written that book already. Moreover, what you're asking is for Frank to not only understand the proof, but to understand it so well that he can explain it better than Wiles. Not exactly fair terms.

    But if someone really wants to take up that challenge, far be it from me to interfere.
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  17. #16  
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    Quote Originally Posted by frank
    Quote Originally Posted by DrRocket
    OK let's make it simple. Put your money where your mouth is. Go read the paper and then prepare and publish a brief synopsis explaining the proof in simple terms, say something easily understood by a third year graduate studient in algebraic geometry.
    I like the way you think . But, I haven't suggested that I could easily understand the paper, nor have I looked at it for that matter. Still, I wouldn't want to give the impression that I'm not willing to take the risk.

    Quote Originally Posted by DrRocket
    I would anticipate little trouble getting it published if you have the understanding that you seem to think you can attain. In fact, if you can put together a really good explanation and have trouble getting it reviewed, post it here. I can forward it to a friend who sits on a board with Andrew Wiles himself. I am sure that Wiles would be interested.

    What is a reasonable time for you ? Say one month from today ?
    I'm currently busy preparing with finals and must read the books Ulysses and Under the Volcano and write essays on both. Do you think I could start this "project" after school ends (a month from now-lol)? Of course, I'll understand if this "violates the rules" in some way.
    OK. Done. I'll expect you to have a paper, as specified above, suitable for publication in a professional mathematics journal (Notices of the American Mathematical Society would be a good candidate, but another if you prefer) in two months. That would be July 11, 2009.
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  18. #17  
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by frank
    Quote Originally Posted by DrRocket
    OK let's make it simple. Put your money where your mouth is. Go read the paper and then prepare and publish a brief synopsis explaining the proof in simple terms, say something easily understood by a third year graduate studient in algebraic geometry.
    I like the way you think . But, I haven't suggested that I could easily understand the paper, nor have I looked at it for that matter. Still, I wouldn't want to give the impression that I'm not willing to take the risk.

    Quote Originally Posted by DrRocket
    I would anticipate little trouble getting it published if you have the understanding that you seem to think you can attain. In fact, if you can put together a really good explanation and have trouble getting it reviewed, post it here. I can forward it to a friend who sits on a board with Andrew Wiles himself. I am sure that Wiles would be interested.

    What is a reasonable time for you ? Say one month from today ?
    I'm currently busy preparing with finals and must read the books Ulysses and Under the Volcano and write essays on both. Do you think I could start this "project" after school ends (a month from now-lol)? Of course, I'll understand if this "violates the rules" in some way.
    OK. Done. I'll expect you to have a paper, as specified above, suitable for publication in a professional mathematics journal (Notices of the American Mathematical Society would be a good candidate, but another if you prefer) in two months. That would be July 11, 2009.
    Ok. It is now July 12.

    Where is the paper ?
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  19. #18  
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    LOL!!! That's awesome.
    Wise men speak because they have something to say; Fools, because they have to say something.
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  20. #19 Fermat - did he ask the right question? 
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    Hi,
    I have been considering Fermat's last theorem and find it fascinating that anyone should try to answer it.

    To me, I think Fermat was joking when he asked the question and never expected anyone to try and solve it.

    As everyone knows, Fermat suggests that there is no solution such that:
    X^n + Y^n = Z^n Where n is greater than 2

    When I look at this formula it immediately strikes me as true without any high genius Maths proof!

    What is X^2 + Y^2 = Z^2 but Pythagoras' theorem...
    This deals with the proof that the sum of the square of the two right-angled sides of a right-angled triangle is equal to square of the hypotenuse.
    This has been solidly proved and is an extremely useful formula.

    So what if we try the formula with a power of 3?

    Try as you might you will find no solution.
    And 4th power - nope
    Fifth power - no no ...nothing - not even using a Super-Computer (Big Blue even!)

    If it is so obvious, what was Fermat asking? Was he Really asking?

    Something is wrong... Can anyone see it?

    Here is a clue (To my mind!)
    X^2 + Y^2 = Z^2 is a 2-Dimensional formula.
    X^3 + Y^3 = Z^3 is an incomplete and mismatched 3-Dimensional formula.

    There are no two cubed values that will produce a cuberooted integer value...
    The real question is WHY... Now that is an answer that can be shown...!

    It is no wonder that Fermat did not show a proof - and that none of credible and fully qualified value... I mean, has anyone ACTUALLY PROVED that Wiles' proof IS valid... Or just best Unproved so far!!

    Readjust Fermat/Pythagoras:
    (X^n + Y^n)^(1/n) = (IntValue) ' ( ^(1/n) = the nth Root )
    Write a computer program with n=2... great, easy... Many values
    Now plug in n=3 .... close but nothing....
    So, how long did you run the program and what values did you reach for X and Y before giving up...
    Ok, what about n=4 .... n=5? 6? 7?...
    Why? Can anyone answer?
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  21. #20 Re: Fermat - did he ask the right question? 
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    Quote Originally Posted by Hermes11
    !!

    Readjust Fermat/Pythagoras:
    (X^n + Y^n)^(1/n) = (IntValue) ' ( ^(1/n) = the nth Root )
    Write a computer program with n=2... great, easy... Many values
    Now plug in n=3 .... close but nothing....
    So, how long did you run the program and what values did you reach for X and Y before giving up...
    Ok, what about n=4 .... n=5? 6? 7?...
    Why? Can anyone answer?
    Your algorithm will search forever.

    Why ? Because there is no solutiom

    Can anyone answer ? Wiles answered.
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  22. #21 Fermat 
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    Hi,

    What was Wiles' answer.

    My point was : does ANYONE understand it or can PROVE it?

    It seems everyone in the Mathematics world is doing an 'Emperor's Clothes' thing and make-believing they understand - or just feel sorry for Woies for spending so much time trying to prove what was obvious...

    They all say he proved it and gave him a prize but not one single person can show HOW!!

    Everyone is just PRETENDING that they understand his proof so as to not be the one one who 'looks ignorant' just in case he IS right!!

    And, yes, the program will run forever - did I not say, 'Where did you stop it before giving up?'

    Moreover, it would run forever because there potential could have been an infinite number of integer solutions. The program, in full, would be designed to stop at the first solution because that first solution would be all that was required to disprove Fermat... i.e. that there WAS an integer that satisfied the equation.
    Re: Fermat was out to prove there WAS NOT!
    And he IS right but he didn't show HOW!

    Run the program for each value of n from 2 to say 7.
    All values of X and Y from 3 upwards (Use shortcuts: discard obvious repeated values and so on) - the program is simple and neat - I used Excel vba. But any language or system with good long precision decimal calculator would do - my iPhone wouldn't - it gave the cuberoot of 125 as 5.0000000001!!
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  23. #22 Re: Fermat 
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    Quote Originally Posted by Hermes11
    Hi,

    What was Wiles' answer.

    My point was : does ANYONE understand it or can PROVE it?

    It seems everyone in the Mathematics world is doing an 'Emperor's Clothes' thing and make-believing they understand - or just feel sorry for Woies for spending so much time trying to prove what was obvious...

    They all say he proved it and gave him a prize but not one single person can show HOW!!

    Everyone is just PRETENDING that they understand his proof so as to not be the one one who 'looks ignorant' just in case he IS right!!
    Wrong.

    His proof of the Taniyama-Shimura conjecture has been reviewed thoroughly by experts. In fact his first attempt contained a subtle error and only in a later attempt, using different methods did he succeed. It is still a very difficult proof.
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  24. #23 Re: Fermat 
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    Quote Originally Posted by Hermes11
    Hi,

    What was Wiles' answer.

    My point was : does ANYONE understand it or can PROVE it?

    It seems everyone in the Mathematics world is doing an 'Emperor's Clothes' thing and make-believing they understand - or just feel sorry for Woies for spending so much time trying to prove what was obvious...

    They all say he proved it and gave him a prize but not one single person can show HOW!!

    Everyone is just PRETENDING that they understand his proof so as to not be the one one who 'looks ignorant' just in case he IS right!!
    This proof is very well covered by the media, and well explained by experts in the field. I've seen at least 4 different programs on it, and read 2 books. I may not understand the fine points, but I do understand the general idea and the popularizations. It's not so far out there that no one can understand it, I'm sure anyone well versed in the topics necessary for the proof can at least understand the bulk of proof, if not the entirety of it as Wiles does.

    HE showed how he did it, in a very long and massive proof, which has been checked by the experts and validated as true.
    Wise men speak because they have something to say; Fools, because they have to say something.
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  25. #24  
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    If you want a good understanding of Fermat's last theorem, I suggest that you read the book by that title authored by Simon Singh. It does not explain the proof as much as the people and the contributions that they made to solving the last theorem. It is made clear how Andrew Wiles work stands on the shoulders of the work done by brilliant mathematicians before him. It gives an excellent overview of just how difficult the problem of proving the last theorem was and how we know it is correct. Although I can not understand it, there are people that can and they spent the time to ensure it is correct through very precise methods. They did this not because it proves Fermat's Last Theorem but because it proves the Taniyama-Shimura conjecture which is much more important to mathematics.

    It also explains in very clear terms what it means to prove something mathematically as opposed to "knowing it". What infinity means, and why computer proofs are not trusted or used by mathematicians.
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  26. #25 Fermat proof 
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    Ok,

    So Wiles' solution has been checked by experts and verified as 'Proved' - is that what you are saying?

    I accept what I believe you are saying, then.

    But that I believe you believe what you are saying is not 'Proof' the fact of what you believe.

    As such, please can you explain, by your own understanding, what/how Wiles' solution works. Any amount will do! Thanks.

    What I am saying is that at present all over the Internet there are claims that there is a solution but NO ONE has shown it in any way that is understandable (and yes, I've seen the programmes, too)

    Sorry if I seem to be giving you all a hard time over this (I'm sure I'm not the only one) or seem naive but I really think this IS an Emporer's Clothes scenario.

    For even here, in this thread, the responses seem lacklustre on committed proof![/quote]
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  27. #26 Re: Fermat proof 
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    Quote Originally Posted by Hermes11
    For even here, in this thread, the responses seem lacklustre on committed proof!
    Given that Wiles worked on the problem for such an extensive period then produced a lengthy, intricate, complex proof, just how do you expect anyone (including Wiles, should he have this misfortune to stumble on this thread) to provide a brilliant and committed proof in a post or two? Your entire argument appears to be "I can't understand it, so all these other people must be fooling themselves." This is the logical fallacy of argument from ignorance.
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  28. #27 Re: Fermat proof 
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    Quote Originally Posted by Hermes11
    Ok,

    So Wiles' solution has been checked by experts and verified as 'Proved' - is that what you are saying?

    I accept what I believe you are saying, then.

    But that I believe you believe what you are saying is not 'Proof' the fact of what you believe.

    As such, please can you explain, by your own understanding, what/how Wiles' solution works. Any amount will do! Thanks.

    What I am saying is that at present all over the Internet there are claims that there is a solution but NO ONE has shown it in any way that is understandable (and yes, I've seen the programmes, too)

    Sorry if I seem to be giving you all a hard time over this (I'm sure I'm not the only one) or seem naive but I really think this IS an Emporer's Clothes scenario.

    For even here, in this thread, the responses seem lacklustre on committed proof!
    All you show in this post is your own ignorance and stupidity. As ophiolite stated, just because you don't understand it doesn't mean others don't understand it, and just because you believe everyone is pretending and its not a 'proof' doesn't mean you're right. You expect entirely too much. Learn the relevant mathematics and then ask the question again of a professor with knowledge of the proof.
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  29. #28 Fermat proved! 
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    Editing error - please see next post...
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  30. #29 Fermat proved? 
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    Sorry for double post - pressed send by accident!

    Thank you both for your responses - it was as I suspected...

    Please forgive my 'Ignorance' - it is exactly that which I seek to overcome!

    Again, forgive my arrogance if that is what it is even, but aren't your responses EXACTLY what the Courtiers told the 'ignorant little boy' who couldn't see the 'fool' proof idea that the Emporer's clothes (Wiles' Solution) were so transparent as to be completely see-through - but couldn't explain HOW?

    I asked an open, diminished, question seeking an open, diminished (Simplified) answer and neither of you could provide one - you both only proffer abuse!

    I am not one for retaliation here and will not but only restate my question hoping to gain an understanding from such wise ones as yourselves:

    Please can you show 'Any kind' of indication of Wiles' proof from your own understanding - this means it need not be protracted formulae nor in high minded professor level language.

    Fermat is seeking to show that the formulae has no solutions for powers greater than 2... I see this as 'Obvious' because the formula is dealing with a 2-dimensional right-angled triangle when the power is 2.

    When the power is 3, this requires a Cubic formulae similar to:
    W^3 + X^3 + Y^3 = Z^3
    which will then yield solutions...

    X^3 + Y^3 = Z^3 cannot yield a solution... Nor any other higher power!

    P.s. 'Solution' means 'a positive Integer' value for Z.

    My 'Assumption' of 'Cannot' is borne out by the fact that no one until Wiles provided any kind of proof and even Wiles' 'Proof' is so complexed as to be incomprehensible to all but those who are 'Gods of Mathematics'.

    I put it to you that if Eistein can put the whole of the form and nature of the universe into one simple easily explainable formula (E=MC^2) then proving that you can't put a Cube into a Square should be child's play!

    Ok, to give you something to kick me for (Ouch!) please show me a use for the formula: X^3 + Y^3 = Z^3

    Ok (again) Here is a simpler one: how about : X^2 + Y^2 = Z^2

    No, no... The latter is just a teaser - I use that formula every day in my job (We all do, actually, without knowing it - hands up anyone who says they are good at Maths!!)

    Oh, and one other thing: length of time working on something does not mean that the outcome should be made 'proof'!
    'Credit' of course, should be given for effort and tenacity but if the outcome was always going to be 'Unknown' or a waste of time: prove that water is wet, for instance, what value should be put on the judgement of the person who pursued the solution (No pun intended) such that his health and sanity were almost at stake?

    Please read that last part as 'concern' for anyone who should put so much effort into anything outside of Godly matters such that they question their own sanity...
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  31. #30  
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    Please explain what level of schooling you have achieved.

    You obviously do not have a degree in mathematics since you are missing the meaning of proof completely. Mathematical proof is the most rigorous proof possible and is built upon logic. If you doubt this explain why 1 + 1 does not equal 2 by using logical argument. "Obvious" is not a logical argument. Your "proof" for the n=3 variation is not a proof nor does it prove anything beyond that.

    You stating that Andrew Wiles proof is not valid because you can not understand it is not a valid argument. You stating that "Eistein can put the whole of the form and nature of the universe into one simple easily explainable formula (E=MC^2)" just goes to show you do not actually understand Einstein's theory. Your "My 'Assumption' of 'Cannot' is borne out by the fact that no one until Wiles provided any kind of proof" only shows that you are not thinking logically. Just because no one provided a proof did not mean that one did not exist. You have clearly not paid attention to the "programmes" on the problem. Andrew Wiles proof is built on math that was not discovered until after Fermat's death, the critical section (the Taniyama-Shimura conjecture) was only proposed in the 50's.
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  32. #31 Re: Fermat proved? 
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    Quote Originally Posted by Hermes11
    Sorry for double post - pressed send by accident!

    Thank you both for your responses - it was as I suspected...

    Please forgive my 'Ignorance' - it is exactly that which I seek to overcome!

    Again, forgive my arrogance if that is what it is even, but aren't your responses EXACTLY what the Courtiers told the 'ignorant little boy' who couldn't see the 'fool' proof idea that the Emporer's clothes (Wiles' Solution) were so transparent as to be completely see-through - but couldn't explain HOW?

    I asked an open, diminished, question seeking an open, diminished (Simplified) answer and neither of you could provide one - you both only proffer abuse!

    I am not one for retaliation here and will not but only restate my question hoping to gain an understanding from such wise ones as yourselves:

    Please can you show 'Any kind' of indication of Wiles' proof from your own understanding - this means it need not be protracted formulae nor in high minded professor level language.

    Fermat is seeking to show that the formulae has no solutions for powers greater than 2... I see this as 'Obvious' because the formula is dealing with a 2-dimensional right-angled triangle when the power is 2.

    When the power is 3, this requires a Cubic formulae similar to:
    W^3 + X^3 + Y^3 = Z^3
    which will then yield solutions...

    X^3 + Y^3 = Z^3 cannot yield a solution... Nor any other higher power!

    P.s. 'Solution' means 'a positive Integer' value for Z.

    My 'Assumption' of 'Cannot' is borne out by the fact that no one until Wiles provided any kind of proof and even Wiles' 'Proof' is so complexed as to be incomprehensible to all but those who are 'Gods of Mathematics'.

    I put it to you that if Eistein can put the whole of the form and nature of the universe into one simple easily explainable formula (E=MC^2) then proving that you can't put a Cube into a Square should be child's play!

    Ok, to give you something to kick me for (Ouch!) please show me a use for the formula: X^3 + Y^3 = Z^3

    Ok (again) Here is a simpler one: how about : X^2 + Y^2 = Z^2

    No, no... The latter is just a teaser - I use that formula every day in my job (We all do, actually, without knowing it - hands up anyone who says they are good at Maths!!)

    Oh, and one other thing: length of time working on something does not mean that the outcome should be made 'proof'!
    'Credit' of course, should be given for effort and tenacity but if the outcome was always going to be 'Unknown' or a waste of time: prove that water is wet, for instance, what value should be put on the judgement of the person who pursued the solution (No pun intended) such that his health and sanity were almost at stake?

    Please read that last part as 'concern' for anyone who should put so much effort into anything outside of Godly matters such that they question their own sanity...
    You have no idea how complex relativity is... E=mc^2 is simply the equivalence term, and is only a small part of the whole that makes up his theory. He couldn't explain relativity in a way that the layperson could easily understand, it just so happens that that one bit, much like specific bits of Wiles' proof, was an easy rendition to explain to the masses. Time dilation and length contraction? Not so much. General relativity? I highly doubt any layperson has any rudimentary understanding of GR, really. Popularizations of relativity aren't the best at explaining it and even they aren't too easy for a layperson to understand. QED, as well, is something that goes far beyond the heads of the layperson and even the experts who aren't trained specifically in QED. Thats the way it works in science and mathematics. Not everyone can know everything we know. We simply know too much at this point.

    If you want to learn, you need to drop the pomp and arrogance.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  33. #32 Fermat proved? 
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    Guys,

    I see from your none response on what Wiles' solution actually says, not even in part, let alone whole, that you, too, do not know!

    Can you guys really not just give me some idea of his proof from your own understanding?

    Only because you asked - I hold a degree in Computer Science.

    I use the formula in 2-D space mapping to find objects that are within a given radius from another given object by virtue of their grid coordinates.

    By the way, neither of you mentioned a use yourselves.

    It seems almost like you two just defend a solution without giving any indication or attempt to show anything yourselves.

    And you are selective in which parts of my discussion you reply to - do I take it then that you agree with the rest, then?

    All you do is proffer abuse. Are you two really mathematicians?
    what are your qualifications, Please...?

    Oh, and Computers HAVE been used in an attempt to find the solution.

    Nearly everything I've read so far credits my thoughts on the solution down to the fact that only Prime numbers need be used for X and Y.

    In fact, Z is not actually required in the formula seeing that the only outcome is to prove that Z is NOT an Integer, therefore the formula could be rewritten as:
    (X^n + Y^n)^(1/n) <> INT

    By the way, you do understand that I AGREE that there is no solution, don't you?

    What this problem seems to be is similar to if DesCarte said, 'I think therefore I am - prove it!'

    I look forward to your next abusive reply...
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  34. #33 Fermat proved by Computer and other other issues... 
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    Guys,

    I found this website and found it useful...

    http://www.cs.rug.nl/~wim/fermat/wilesEnglish.html#a3

    I won't say, 'Enjoy'!
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  35. #34 Re: Fermat proved? 
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    Quote Originally Posted by Hermes11
    Guys,

    I see from your none response on what Wiles' solution actually says, not even in part, let alone whole, that you, too, do not know!

    Can you guys really not just give me some idea of his proof from your own understanding?

    Only because you asked - I hold a degree in Computer Science.

    I use the formula in 2-D space mapping to find objects that are within a given radius from another given object by virtue of their grid coordinates.

    By the way, neither of you mentioned a use yourselves.

    It seems almost like you two just defend a solution without giving any indication or attempt to show anything yourselves.

    And you are selective in which parts of my discussion you reply to - do I take it then that you agree with the rest, then?

    All you do is proffer abuse. Are you two really mathematicians?
    what are your qualifications, Please...?

    Oh, and Computers HAVE been used in an attempt to find the solution.

    Nearly everything I've read so far credits my thoughts on the solution down to the fact that only Prime numbers need be used for X and Y.

    In fact, Z is not actually required in the formula seeing that the only outcome is to prove that Z is NOT an Integer, therefore the formula could be rewritten as:
    (X^n + Y^n)^(1/n) <> INT

    By the way, you do understand that I AGREE that there is no solution, don't you?

    What this problem seems to be is similar to if DesCarte said, 'I think therefore I am - prove it!'

    I look forward to your next abusive reply...
    You don't know what in the hell you are talking about.


    Your "proof" in the case n=3 is just plain ludicrous.


    http://mathworld.wolfram.com/Taniyam...onjecture.html
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  36. #35 Re: Fermat proved by Computer and other other issues... 
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    Quote Originally Posted by Hermes11
    Guys,

    I found this website and found it useful...

    http://www.cs.rug.nl/~wim/fermat/wilesEnglish.html#a3

    I won't say, 'Enjoy'!
    Anyone that believes that a computer program is less fool proof than a team of the best mathematicians in the world is an idiot. Computer programs have bugs.

    As a person that has a computer science degree you should be very aware that it is impossible to create a computer program of any complexity without it also having errors.

    Please prove to me that these validation programs do not have bugs.
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  37. #36  
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    As a computer scientist, I agree. I think computerized or computer-assisted proofs are a valid technique, but that the resulting programs need just as much validation as a classic proof in the end. (As someone that just spent most of a week grading computer science homework, I can tell you that that's a lot of work.)
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  38. #37 Re: Fermat proved by Computer and other other issues... 
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    Quote Originally Posted by Darkhorse
    Quote Originally Posted by Hermes11
    Guys,

    I found this website and found it useful...

    http://www.cs.rug.nl/~wim/fermat/wilesEnglish.html#a3

    I won't say, 'Enjoy'!
    Anyone that believes that a computer program is less fool proof than a team of the best mathematicians in the world is an idiot. Computer programs have bugs.

    As a person that has a computer science degree you should be very aware that it is impossible to create a computer program of any complexity without it also having errors.

    Please prove to me that these validation programs do not have bugs.
    There is a ongoing project to reduce mathematical proofs to formal symbolic logic. This involves computers.

    http://www.ams.org/notices/200811/tx081101395p.pdf

    Here is the complete volume of the Dec 2008 Notices dedicated to formal proof http://www.ams.org/notices/200811/

    It is in principle possible to do this, and it has been done for quite a few theorems. But it is hardly central to the progress of mathematics, and does nothing for either simplicity or clarity -- quite the contrary.

    Very few theorems in mathematics are proved by computer. One notable exception is the Appel and Haken proof of the 4-color theorem, which was rather a disappointment since it offered little new insight beyond reduction of the problem to a (large) finite number of cases. More recently a large project resulted in the computation of the character table for E8.

    Wiles proof has been gone over carefully by the experts in algebraic geometry. A "computer verification' will be greeted with a big yawn, unless the technique can yield new insights.
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  39. #38  
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    I don't disagree that certain problems lend themselves to having computers do the heavy lifting, however I agree with MagiMaster that they require as much validation as the proof. Proving a conjecture through peer review in my opinion is a more fool proof solution than a computer validation. With peer review you have multiple people validating it, with computer validation you have one program that may have one bug in one critical area. Furthermore the act of transcribing the proof to be tested into the computer could also be a source of errors.

    Both the examples you provided deal with finite sets of solutions. I take it that the next big leap taken by computer validation will be to deal with conjectures that involve infinite possible solutions? It will be interesting to watch this develop but as a computer "scientist" (though personally I that that term should be struck from our vocabulary) and software quality engineer I would back the mathematicians if I was going to make a bet in the near term.
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  40. #39 Re: Fermat proved? 
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    Quote Originally Posted by Hermes11
    I asked an open, diminished, question seeking an open, diminished (Simplified) answer and neither of you could provide one - you both only proffer abuse! ...
    I should just like you to confirm that I was not one of the posters (neither of you) who you feel proffered abuse. Stating that someone has practiced the logical fallacy of Argument from Ingorance is not abuse. It is not a personal attack. Therefore I feel it likely you were addressing Arcane and Dr. Rocket. I should just like that confirmed.
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  41. #40  
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    Quote Originally Posted by Darkhorse
    I don't disagree that certain problems lend themselves to having computers do the heavy lifting, however I agree with MagiMaster that they require as much validation as the proof. Proving a conjecture through peer review in my opinion is a more fool proof solution than a computer validation. With peer review you have multiple people validating it, with computer validation you have one program that may have one bug in one critical area. Furthermore the act of transcribing the proof to be tested into the computer could also be a source of errors.

    Both the examples you provided deal with finite sets of solutions. I take it that the next big leap taken by computer validation will be to deal with conjectures that involve infinite possible solutions? It will be interesting to watch this develop but as a computer "scientist" (though personally I that that term should be struck from our vocabulary) and software quality engineer I would back the mathematicians if I was going to make a bet in the near term.
    You misunderstand me. I am not advocating the use of computer programs for either theorem proving or theorem validation. I merely pointed out two isolated examples where computers did in fact turn out to be useful and an ongoing project that uses computers to translate proofs into symbolic logic. When one is forced to use such tactics much of the understands that comes with elegant arguments is lost.

    Like most other professional mathematicians, I think that computers are useful in their place. But theorem proving is not very often that place.

    The validity of proofs, even computer-based proofs, is accomplished by expert review, and for important theorems that review is extensive (not just a referee report for a journal publication, but usually several long seminars involving real experts).

    The usual methods of validating software are not adequate for verification of the logic of mathematical proofs. The standard of mathematical rigor is much higher than that.
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  42. #41 Re: Fermat proved? 
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    Quote Originally Posted by Ophiolite
    Quote Originally Posted by Hermes11
    I asked an open, diminished, question seeking an open, diminished (Simplified) answer and neither of you could provide one - you both only proffer abuse! ...
    I should just like you to confirm that I was not one of the posters (neither of you) who you feel proffered abuse. Stating that someone has practiced the logical fallacy of Argument from Ingorance is not abuse. It is not a personal attack. Therefore I feel it likely you were addressing Arcane and Dr. Rocket. I should just like that confirmed.
    Confirmed... And thank you for pointing it out...
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  43. #42 Fermat proved? 
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    I don't think I was saying that Computers would/Should/will or can show all proofs because I am well aware that transcribing pure mathematics to a computer program is problematic (I showed early on that my own iPhone that I write these posts on even gives a computational error for something as simple as the cuberoot of 125!)

    What seems to have occurred here is that certain posters have taken exception to my request for information from themselves concerning my dismay at the actual lack of actual proof from Wiles that is in any way understandable by anyone other than Math Gods... My detractors have sieved my posts and only responded to the parts they can dispute over... Which then becomes the focus of further detractions...

    Each reply I posted I reemphasised my request but nothing formal was forthcoming...

    One thing I notice though is that many forget that the formula is only asking to prove that NO INTEGER VALUE can result from any power greater than 2.

    Power 3 is easy to prove, 4 is harder but then, after experimentation, it turns out that ONLY PRIME POWERS need be checked... How did I arrive at conclusion and I'm only a Computer Scientist??

    When I woke this morning HITCH HIKERS GUIDE TO THE GALAXY came into my head where the lowly lab cleaner stumbled on the solution to the long sort after Infinite Improbability Drive... Only to later find himself being lynched by rampaging mob of [Mathematicians...] who had come to the conclusion that the one thing they couldn't stand was a smartypants...! (How dare you discover that which WE, who have searched for 385 years, are the discovery custodians of!!)

    Oh, I don't say that I have discovered anything except that perhaps the solution was not as cumbersome as it was made out to be!

    Remember Rubics cube... Ha ha... A genius mathematician friend at my university was in our group of Cubies and discovered a pattern on the cube... True story... He twisted away on a 'clean cube' for nearly 100 move and said 'See this - great eh.'
    I laughed and asked for the cube and untwisted in about 20 moves...
    Yes, me.. The Computer Scientist beat the Genius mathematician...

    But hey, give him credit... Yes, his method long term was better and many things were discovered in his method where as mine was pure limited block pattern based.

    Same as Wiles, who progressed Math models on his quest in a way that a simple solution would not have touched... There's always an easier way!!

    So, exactly what was Wiles' solution... I have yet to read Anyones report here!

    I mean, after his travails around the universe, did he sign of with:
    '... and therefore this proves that for any integer power (n) greater than 2 there is no solution that satisfies the equation: X^n + Y^n = Z^n'
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  44. #43 Re: Fermat proved? 
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    Quote Originally Posted by Hermes11
    So, exactly what was Wiles' solution... I have yet to read Anyones report here!
    http://math.stanford.edu/~lekheng/flt/wiles.pdf

    Primary sources are always preferable.
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  45. #44 Re: Fermat proved? 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by Hermes11
    So, exactly what was Wiles' solution... I have yet to read Anyones report here!
    http://math.stanford.edu/~lekheng/flt/wiles.pdf

    Primary sources are always preferable.
    DrRocket,

    what you really saying then is 'I don't know either but I know a man who says he does!'
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  46. #45 Re: Fermat proved? 
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    Quote Originally Posted by Hermes11
    Quote Originally Posted by DrRocket
    Quote Originally Posted by Hermes11
    So, exactly what was Wiles' solution... I have yet to read Anyones report here!
    http://math.stanford.edu/~lekheng/flt/wiles.pdf

    Primary sources are always preferable.
    DrRocket,

    what you really saying then is 'I don't know either but I know a man who says he does!'
    You asked for "exactly what was Wiles'ssolution". That is precisely what was provided. The fact that it takes specific expertise to understand is not relevant to your request.

    I am a PhD mathematician, whose area of expertise is analysis. I am neither an algebraic geometer nor a number theorist. I do not have the expertise to understand Wiles's paper. Of course I know someone who does. I know, professionally and personally, several people who do.

    Apparently you are not interested in the mathematics, but rather are simply a troll, and a fool to boot, seeking controversy where none exists.

    Please go straight to hell.
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  47. #46  
    Forum Radioactive Isotope MagiMaster's Avatar
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    Quote Originally Posted by Darkhorse
    computer "scientist" (though personally I that that term should be struck from our vocabulary)
    I'm just curious what else you'd call the field?

    Anyway.

    @Hermes, the point is that the proof cannot be boiled down to a point that would be both correct and still understandable to a layperson like you or me. Having said that, there's no reason we have to take it on faith that this proof is real. The scientific process is well proven and works in math too (if it doesn't in fact work better, with the rigor of mathematical proofs), and other experts in the field have given it the thumbs up when they could easily tear it apart if it were wrong.

    Seriously, if you (or anyone else) found something wrong with the proof, you could publish that and be famous. Your name would definitely become attached to the story in a fairly permanent way. There are a lot of people who are experts in the field that would do just that given the chance.
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  48. #47  
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    Based on your background I figured you might question that statement, please don't take it as an insult. :-) Personally I have a very picky definition of scientist and in commercial software and in parts of this thread I have seen very little scientific process or thought. Yes, in some area of research and leading edge development there is science but most computer science graduates become programmers. I would prefer to reserve the term scientist for individuals that have at least a masters degree and are actually moving the field forward. If your think this is too harsh, ask me about the social sciences (I dare you :-D).

    Personally I would call it a degree in computer programming, but like I said I have a pretty narrow view of what a scientist is. I just feel we are diluting the meaning of the word and therefore diluting the understanding of the scientific process. JMHO.

    @Hermes
    As MagiMaster indicated proving a theorem false is just as well respected as proving a theorem true. Mathematicians and scientists are like everyone else and there is a long history of personal feuds between individuals. Nothing pleases people more than to prove a rival wrong.

    Additionally proving a theorem is an adversarial based process . You have to PROVE you are right by defending your theory, no one will blindly take your word for it. In this way the process is the exact opposite of the judicial system where someone has to prove you are wrong and the courts assume you are right.
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  49. #48  
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    Quote Originally Posted by Darkhorse
    ..., ask me about the social sciences (I dare you :-D).
    Ask you about an oxymoron ?

    Go right ahead and state your opinion anyway. I suspect you will not get much of an argument.
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  50. #49  
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    Dark Horse and MagiMaster,
    Thank you for your honest responses. I agree with what you say.

    Happier I would have been had your response come earlier but it has served to expose unwarranted animosity in others here.

    I asked for proof in posters own understanding but got back abuse frommsome who cherrypicked my post. Then, when I post without the 'your understanding' I am given a link that I could well have found myself and called names or such... So, that poster could not give an account of their belief so refused to answer but managed to respond when they were not called to account... Does that indicate something about that poster? My summary concerning their response (or lack, thereof) was then correct!

    I see FLT as an overblown puzzle that was a joke to Fermat...
    This is purely my thought - not a judgement...

    Here is another: prove that the squareroot of 1 is 1... Silly, eh?

    But none the less, what I am trying to say is that perhaps there is a good reason why it took 385 years from it's inception. And it is still not completely proved even now - it has only been ACCEPTED AS PROVED by eminent peers given the complexity of the 'proof'.
    Sentiment as to the length of time Wiles spent on the problem nor that a limit was put on the prize should in any way influence an acceptance or ratification of a proof!

    My university friend must have spent many hours over many days working out the symmetery of the Rubics Cube before producing the pattern and must have been incredibly farsighted (he was a Maths genius) yet, my solution was incredibly simple and achieved exactly the same pattern in one fifth the number of moves (did I mention he actually hated me for a short time after - if I mentioned others in the small group you may get a better sense of the level of persons engaged (there were no written guides in those early days - we worked it out by ourselves from our own discovered principles)

    However, I do give high credit to my esteemed colleague, as I do to Wiles because their methods provoked research into areas that would not have been touched by a simpler method...
    Are not both method then valid even though lesser mortal are the solution finders.

    Can I also mention that a place I worked, my line manager had a Topology puzzle. He couldn't do it having spent hours in secret trying. He eventually gave it out to the staff (It was a quite friendly office really) his favourite ones first hoping to glorify them... None could do it. Eventually one of them brought it to me and I solved in less than five minutes... What was their reaction... Yes you guessed it!! Why? I was given the puzzle last exactly because he thought I MIGHT do it - but MIGHT NOT... thereby gaining a laugh to him (I was known for providing solutions where even the line manager had failed ... Our job was resolving customer problems in and on their computer systems, hardware and software so 'Line manager' does not imply he knew everything but only that he supervised a team whose role it was to find solutions - and indeed I do pride myself as a 'Solution Provider' in Computer System)

    Computer Science... Yes. It does seem a little dated. The idea at the time was that all aspects of the computer system was up for 'Sciencing' it being in flux of development.
    Well, that was the title that was used at the time - today, hardware and software are virtually split and other areas have evolved into their own 'Science': Databases, Internet and Web, etc., a wide variety which cannot be forced under one title.

    Ok, o agree that computers should only be used for testing a theory with reservations and that the final proof should be academical based for good reasons already stated by others.

    My motives are only to suggest that there may be a solution in a far more simplified form.

    My feeling is that X^n+Y^n=Z^n was never meant to be provocative such that anyone should spend virtually their whole life trying to resolve it - and it alone unless it was originally meant to be part of a wider, and ongoing, research project.
    Fermat did not publicise it as such and only the idea of the prize money (and fame... Always a dangerous couple) could have driven anyone to such lengths... Everything will be discovered in it's own time and researchers can sleep soundly in their beds at night. 'I've found it - Eureka moments' are by sparse revelations.

    Can i ask a mathematician here:
    Why is the formula never written as: (X^n + Y^n)^(1/n) = INT

    I was drawn to FLT due to using the formula in my everyday work on maps where n=2 for 2-Dimensional mapping (essentially, Pythagoras' theorem)

    I had toyed with 3-D mappings by extending the formula and started writing a program to check the values. Then I saw a program on FLT and thought, hey...looks like what I'm doing - what's the problem! (Ok, I hear what you saying - just be happy with me in any possible dilussion and naivety - humour me!)
    My reaction was that X^3 + Y^3 = Z^3 couldn't possibly compute an Integer solution - and so it has (virtually) proved to be by pure mathematical analysis.
    My analysis is that the formula is wrong... That the formula is 2-D but the values are n-D (2-D formula with say, Cubic values).
    Ok, the resulting values may be interesting (As ANY and EVERY formula and results thereof, are to a mathematician) but I believe that a normal and simple explanation could have resolved the problem long ago (Being too clever is not a failing reserved only for fools!)

    And one last thing... What is FLT used for? Or is it just a math conjecture?
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  51. #50  
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    I have taken the time to read your post. Please return the same courtesy to my reply since I am going to give you some advice.

    No one is going to take you seriously if you do not provide explanations for your theories. Saying n=3 is obvious is not proof, you need to explain it in formal logic. See http://en.wikipedia.org/wiki/Formal_...ogical_systems for details.

    You appear to want people here to say you are smart or be impressed with your intelligence. No one cares about the puzzles you have solved. The most you can hope for is to have someone on this forum say "you appear to know your stuff", and that will only come after you have posted scientific reproducible solutions to questions other individuals ask.

    You really need to read some books on mathematics or the scientific method and what is meant by proof. You appear to not understand what the word means and how rigorous the process is.

    You really need to read about Fermat and his last theorem since your statements about him and his work are incorrect. Your question about what "FLT" is used for is a clear example of this. Fermat's last theorem is not used for anything, however the pursuit of it has created some ground breaking mathematics. Again I suggest again that you read "Fermat's Last Theorem" by Simon Singh.
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    Hi Darkhorse,
    I don't think you realise that you cannot rub me up because I have no desire for a fight or to return negativity to you.

    You ask me to show proof of n=3 under your terms.. Terms of which you yourself do not even seem qualified to put in front of me.

    In any case I thank you for answering the most vital question: FLT is not used for anything... Exactly what I imagined and as I alluded to in my last post that the output would only be interesting to pure mathematicians as they don't seem to represent anything worthwhile ( Yes, I said that before you did).

    I also alluded to the fact that Wiles and the maths world benefited from the pursuit of FLT, so once again I was there before you.

    Do you see that every time you think you are pushing me down you are actually squeezing more truth?

    Back to n=3. Are you saying that you do not know how to prove that FLT for n=3 is true (That there are no integer solutions)?

    Are you really a mathematician or someone who just like being vociferous against others?

    Have you even tried a basic systems of practical trial as a start of:
    X=2 Y=2 .: Z=16^(1/3)
    X=2 Y=3 .: Z=35^(1/3)
    X=2 Y=4 .: Z=72^(1/3)
    X=3 Y=3 .: Z=18^(1/3)
    X=3 Y=4 .: Z=91^(1/3)
    Repeat...ad nauseum
    Or, casual analysis shows that certain values need not be considered - in fact only Prime numbers need be considered for X and Y as all values in between are simply repeats of lesser values (Basic numbers theory)

    I find it strange that you should ask me to justify an elementary value (n=3) for proof - are you just testing me - or did you not know how to validate it yourself. From what I have read from you so far, you just seem to say, 'You don't know what you talking about...there are cleverer people than you so get off your podium'
    You don't show any interest personally except to point me to what other people say.. I'm interested now even more in what YOU say concerning FLT... What is your choice view on it and opinion or plain interest - Why are you responding concerning this matter but have no actual contribution other than to ost hyperlinks to someone roses opinion?
    Thank you for any honest, forthright and enlighting response.
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    Ah yes the classic ad nauseum method of solving mathematical and scientific theories.

    Tell you what, you prove by ad nauseum that Fermat's Last Theorem is true for all prime numbers where the power is > 2. Report back to this thread when you are done.

    I am done feeding this troll.
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  54. #53  
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    Quote Originally Posted by Darkhorse
    Based on your background I figured you might question that statement, please don't take it as an insult. :-) Personally I have a very picky definition of scientist and in commercial software and in parts of this thread I have seen very little scientific process or thought. Yes, in some area of research and leading edge development there is science but most computer science graduates become programmers. I would prefer to reserve the term scientist for individuals that have at least a masters degree and are actually moving the field forward. If your think this is too harsh, ask me about the social sciences (I dare you :-D).

    Personally I would call it a degree in computer programming, but like I said I have a pretty narrow view of what a scientist is. I just feel we are diluting the meaning of the word and therefore diluting the understanding of the scientific process. JMHO.
    Well, I'm in the CS graduate program here, and at that point it's somewhere between math and science. There's a lot that can be done with rigorous mathematical proofs (P=NP, for example) and there's a lot of experimentation with new algorithms.

    The undergraduate CS degree felt very similar to the undergraduate math degree. There was a lot of learning the tools of the trade. I guess the CS tools are just more directly applicable to normal jobs, so it ends up feeling a bit more like the programming is the important bit.
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    GH Hardy said of his work:
    "I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."

    This is kind of how I approach pure mathematics, however this doesn't mean there aren't applications. You can never look to closely at (virtually) anything before you see mathematics. So yes some obscure number theoretic problem could have fundamental applications to computer science or whatever, but thats not why I think many of us do mathematics.
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    Magi and Boson,

    Thank you for your beneficial contributions. I don't think either of those other two (and latterly, one) would ever give me anything of a credible response or sympathy for lateral thinking.

    I tried and succeeded in not returning like for like to show my desire was quest for alternate argument on FLT and not an attempt to refute it.
    The 'refute' established in my post was that it could be shown categorically that Wiles had found THE SOLUTION but just that it was INFERRED because of it being ACCEPTED as THE SOLUTUON (for the present time)
    I asked for their understanding of what that proof was but they could not answer except to post links to other peoples writings on the subject. Didnt exactly inspire me with confidence in the validy of their outpouring of venom towards me.

    They asked me to show HOW I proved n=3 and when I do this is still not acceptable - why? Because they thought I could not?
    Whatever my practical model - spin it into a formula - once it is seen that all non Primes can be ignored the rest is almost childsplay...
    And what of n=4, 5,6,7.... These have all been proved by 'non-Wiles' methods and again it has been shown that all non-prime 'n's can be ignored.

    So all that is left is to prove prime x, prime y and prime n. Z is not actually required as the formula can be rewritten to avoid z as I have shown remembering that it need only be shown that nth-root(x^n + y^n) /= Integer.
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    Has anyone else worked out that:
    a^3 + b^3 + c^3 = z^3
    has integer solutions?

    For instance:
    6^3 + 8^3 + 1^3 = 216 + 512 + 1 = 729 (Which is 9^3)

    And there are many more such integer solution for cubes.

    I put it to the forum that Fermat's Last Theorem was never meant to cause such dilemmas. It is virtually obvious that a Square Formula (x^n + y^n = z^n), constrained by the request for an INTEGER solution, would result in a conclusive 'No'.

    My belief is that Fermat was just toying with the formula.

    Now, I don't take anything away from anyone who wants to toy with irrational formulae as irrational solutions can result which push back the bounds of possible reality.

    Andrew Wiles' solution(?) certainly broke new grounds in the areas that he researched in his pursuit of his FLT.

    Question: Is it possible for an integer solution to result from the formula for powers greater than 2?

    Answer: No.

    Ok, prove it...

    Ok, there are no two Cube values x and y such x+y cube-rooted results in an Integer value - simple... And this is also true of ALL POWERS greater than 2.

    The Sum of TWO Cubes is always another unique Cube (A Cube Prime) therefore that unique Cube cannot be cube-rooted to an base Integer value.
    And this is true of all higher powers given the formula.

    Why is the answer so simple and takes only moments (Well, ok, few weeks, I'm not a mathematician!) to work out?
    Because the formula is SQUARE and the Powers are CUBED TIME CUBES...

    The formula works for SQUARE VALUES because the formula is SQUARE.

    Use a CUBE FORMULA with CUBE VALUES and everything works.

    Please bear in mind that it is the constraint of seeking an INTEGER SOLUTION that is the cause of the problem that most people keep forgetting.

    The formula using powers greater than two obviously gives solutions - and extremely interesting they are too. Incredible irrational objects can result from structures designed from the results - but, that was not the question...it was simply asking if an INTEGER solution could be forthcoming - simple: No!

    Obviously, someone is going to say that I only showed my thinking on Cubed values - then what about 4th powers - show us how that should work...?

    Ok, watch this space!
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  58. #57  
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    This has solutions.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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    Hi Arcane,
    Thank you for your input.

    Your formula is, of course, not the same as x^4 + y^4 = z^4 and goes a long way to prove the point I was making.

    Reading back, someone said asked me to show x^3+y^3 = z^3 Has no integer solution and when I did all I got was abuse.
    I also showed that the reason was that the FORMULA was wrong and should be: w^3+x^3+y^3 = z^3 in similar manner to your formula for 4th power.

    I do understand (See very near the beginning of this thread) that the question was as stated by Fermat - and answered that it was TRUE but it was TRUE (I.e. That there were no integer solutions for powers greater than 2) because the formula is a square formula and therefore could not possibly yield an integer solution.

    My Cubed formula was dismissed as 'Does not make sense'... Yet, check it yourself, three Cubed values CAN result in another Cube value and therefore an integer singleton (Is that what it's called?)
    Again:
    1, 6, 8 = 9
    1, 9, 10 = 12
    2, 12, 16 = 18
    2, 18, 20 = 24
    3, 18, 24 = 27

    I'm sure you can see the pattern and can write a formula to show all values ...
    I will expand on all this later... Suffice it to say, larger powers are harder to prove with examples but, given an emerging pattern, can a formula be deduced to works for all powers... A resounding 'Yes'.

    And if 'Yes' then this is sufficient to show that there are NO 'Integer' SOLUTIONS for powers greater than 2.

    !!Boom!!! FLT solved.
    (Ps. I leave the accolades open to those that create the formula as suggested above because ... it is too hard to type in on my iPhone that I am using to write this!)
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    FLT is specifically has no integer solutions for not as you now put forward, which may or may not work, personally I'm not aware of any postulate that there is always a rational answer solution for x,y in
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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    Thank you all for you contributions to my question concerning Fermat's Last Theorem.

    In all the time I have been looking at this issue I have failed to come across a single person who actually understands what it is that Andrew Wiles has given as a proof of what to me looks like a simple problem that was never meant to be an issue.

    Ok, I can't produce a hundred page 'Mathematician's proof' document that only a God can understand - but I do know that it doesnt take a huge amount of brain work to 'show' that there is no INTEGER solution to powers greater than two up to say, six...
    List the multiples of the powers and you will see repeated patterns of parts of the numbers which even casual Maths will show cannot possibly produce an Integer solution...
    Innother words, any multiple two powers must add up to another power of a similar multiple - it's hardly rocket science (ooops... Maths!) to show this.

    And with the reduction of possible numbers to check by virtue of only needing to check PRIME numbers of powers it is just as easy to show from a limited set of numbers that the pattern leads to IMPOSSIBILITY of an Integer solution.

    Please keep in mind always... Integer solution... Not simply any solution for there always IS a solution - just not integer.

    The benefit of the research is enormous - but the reality of the proof that there is no solution - was always certain!
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  62. #61  
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    Quote Originally Posted by Hermes11
    Thank you all for you contributions to my question concerning Fermat's Last Theorem.

    In all the time I have been looking at this issue I have failed to come across a single person who actually understands what it is that Andrew Wiles has given as a proof of what to me looks like a simple problem that was never meant to be an issue.

    Ok, I can't produce a hundred page 'Mathematician's proof' document that only a God can understand - but I do know that it doesnt take a huge amount of brain work to 'show' that there is no INTEGER solution to powers greater than two up to say, six...
    List the multiples of the powers and you will see repeated patterns of parts of the numbers which even casual Maths will show cannot possibly produce an Integer solution...
    Innother words, any multiple two powers must add up to another power of a similar multiple - it's hardly rocket science (ooops... Maths!) to show this.

    And with the reduction of possible numbers to check by virtue of only needing to check PRIME numbers of powers it is just as easy to show from a limited set of numbers that the pattern leads to IMPOSSIBILITY of an Integer solution.

    Please keep in mind always... Integer solution... Not simply any solution for there always IS a solution - just not integer.

    The benefit of the research is enormous - but the reality of the proof that there is no solution - was always certain!
    If the problem is so simple, then by all means write up a simple proof and send it to a journal. I'm sure a simple proof of FLT would easily make it into The Annals of Mathematics.

    Seriously, I can't think of a single mathematician who wouldn't leap at the opportunity to publish something like that.
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    Yes, only the fake mathematicians post it on a forum where they can glibly ignore everyone saying they are wrong. Peer reviewed journals are much less forgiving
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    Quote Originally Posted by salsaonline
    If the problem is so simple, then by all means write up a simple proof and send it to a journal. I'm sure a simple proof of FLT would easily make it into The Annals of Mathematics.

    Seriously, I can't think of a single mathematician who wouldn't leap at the opportunity to publish something like that.
    Yes, a proof published in the margin of a page of The Annals would be just the ticket.
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    I nearly died looking at that paper proof by Wiles. :-D

    But that's understandable seeing as I'm only doing AS Level Mathematics at the moment- which is just things like- trigonometric graphs, geometric series, logarithms, binomial expansion, integration etc.

    Anyway, I'm sure one would have a very difficult time writing a synopsis of that paper in a much simpler way- but I'm not going to stop someone who tries...!
    "Nature doesn't care what we call it, she just does it anyway" - R. Feynman
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    If the proof is so long and complicated and hardly anyone can understand it, how do we know it is really a proof?

    I mean there must be a lot of pressure on people to say 'oh yea I understand it'.
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    Quote Originally Posted by HotJoe
    I mean there must be a lot of pressure on people to say 'oh yea I understand it'.
    Nope.

    The paper has been reviewed by specialists, experts whose reputation is so solid that they cannot be pressured like that.
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  68. #67  
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    More cynically/pragmatically, experts who's reputations would be bolstered more by pointing out flaws than by accepting the paper.
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    Yes but what if they could not understand it?
    Would they admit that and risk losing stupid?
    Could be a bit of a case of the emperor's new clothes, could it not?

    Anyway they will all look rather stupid if some computer simulation proves it wrong.
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    Hotjoe i'm just gonna throw this out there and you can take it or leave it. Go somewhere and study mathematics for 7years, specialize in modular forms and elliptic functions, study the proof (by Wiles) and then you decide. Your politicizing mathematics that should be a sin!
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    If you decide to reply to this I would greatly appreciate if you replied to all the points with which you do not agree, otherwise I will assume that you do agree. I also took the liberty to number my points and questions so accidental skipping on your part won't lead to assumptions on mine.

    (1)A child comes up to a computer scientist.
    The child says: "teach me how to write programs"
    To which the computer scientist replies "first you will need to learn the fundamentals; commands, functions..
    "No, I don't want to learn any of that, just tell me how to write a program" the boy Interrupts.
    "It's not that simple, you need to learn the fundamentals first so you can piece it all together into the bigger picture" replies the computer scientist.
    "What a hoax! you obviously don't even know how to write a program since you can't just explain it to me!" the child exclaims.

    Guess who is who.
    You may react to this analogy as you have to most things said in this forum not favorable by you; by victimizing yourself, but hopefully you will realize that I'm simply painting a picture of the situation in which you can see things from our perspective. Afterall, winning without understand your opponents position is just overpowering with brute force.

    (2)You claim everything can be put into simple terms, but given that simple is relative your statement could only mean one of two things

    1. Everything can be simplified to what you consider simple
    2. Everything can be simplified to anyone's standard of simplicity.

    (2 cont'd)Think of some of the things that you consider simple as a computer scientist that below average laymen couldn't understand even if you explained it to them. Just as your intelligence towers over some, so does the intelligence of some tower over yours and so we have the same case with you and the below average laymen except you are in the shoes of the laymen and certain mathematicians are in your shoes.

    (3) Although Wiles is exceptionally intelligent, there are people who match and even exceed him in intelligence and capability in the field and they would "bust" Wiles if his proof didn't make any sense. Wiles isn't a lone god above everyone else, he doesn't follow the principles of many monotheistic gods that work in ways incomprehensible to anyone. You just need to study the field to understand it (and of coarse be fairly intelligent). (4) Do you believe everything which cannot be explained to you without some or lots of background is a hoax? Do you believe most things in theoretical physics and high level mathematics are a hoax as well?

    You seem to have a problem with the fact that the truth to FLT is seemingly obvious yet mathematicians "supposedly" use such complex and pointless mathematics, but although it is seemingly obvious, you cannot know unless you account for all possibilities (which are in this case infinite) or use some form of logic to prove that the theorem is without question (as without question as things can get in mathematics) true. It is true that is extremely unlikely that some set of integers actually prove the theorem wrong (forgetting about Wiles' proof for a minute) and if the theorem was needed for practical application then no one would think twice about that possibility, but as you stated yourself; theoretical mathematics is mostly useless in the practical world, it is more of an art form than anything, you could even go as far as to call it a hobby and where's the beauty in just testing some cases and then slapping a label on the theorem? this isn't Mythbusters.(5) What exactly is your problem with mathematicians using mathematics that proves something to the greatest extent that it can be proved, whether that mathematics is simple or complex. There is no for them to hurry.
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  72. #71  
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    I think there is some confusion here regarding understanding and checking a proof. Now these are my opinions so caveat emptor.

    You do not need to understand a proof to check the validity of a proof. The understanding comes in the grand ideas, how the individual lemmas slowly fit together to get you to the conclusion. I think you can say with confidence that you understand a proof if given enough time and paper you can hash it out in your own words and style, and at each point knowing why you need to prove that step and where you are heading (ie. its not mere mechanical regurgitation of unconnected lines)

    To check a proof though is something else. In effect this is entirely mechanical and very local. Does step B follow logically from step A? rinse and repeat. Given a list of definitions and sufficient mathematical maturity and time, anyone can check a proof. Its something computers can do.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  73. #72  
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    Quote Originally Posted by river_rat
    I think there is some confusion here regarding understanding and checking a proof. Now these are my opinions so caveat emptor.

    You do not need to understand a proof to check the validity of a proof. The understanding comes in the grand ideas, how the individual lemmas slowly fit together to get you to the conclusion. I think you can say with confidence that you understand a proof if given enough time and paper you can hash it out in your own words and style, and at each point knowing why you need to prove that step and where you are heading (ie. its not mere mechanical regurgitation of unconnected lines)

    To check a proof though is something else. In effect this is entirely mechanical and very local. Does step B follow logically from step A? rinse and repeat. Given a list of definitions and sufficient mathematical maturity and time, anyone can check a proof. Its something computers can do.
    While I certainly that checking a proof is much easier than creating one, it is not as easy or mechanical as all that.

    If it were there would be no mistakes in the literature.

    The Tomita decomposition theory was wrong, and a chapter disappeared from Naimark's book. Smale's proof of the Poincare conjecture in dimension five and above had errors, but the ideas were worth a Field's Medal. Romanov has retracted the claim to a proof that P=NP. Kirillov's book on representation theory has so many mistakes that it is pretty much useless.

    There are methods for translating some proofs to symbolic logic, but not all major theorems, by a long shot, have been so reduced.

    Very difficult proofs still require experts. But experts are certainly up to the task.
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  74. #73  
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    It is not as easy as I claim more due to our human nature and less to the nature of the mathematics involved. We are imprecise in our language and very few "higher level" proofs are really written in full mathematical rigor (for obvious reasons, see Principia Mathematica). We leave things to the reader or make claims like : It is easy to see that OR This follows trivially from without actually checking that it really is easy to see or follows trivially. Every step left out makes the mathematicians job easier but the proof checkers job more difficult. If proofs were written in a certifiable way (i.e. every result listing the previous results required to prove it) then proof checking would not be limited to experts in the field in question. But those experts would probably only publish one paper a decade

    As an interesting side note, work is underway to machine check fermat's last theorem (see http://www.cs.rug.nl/~wim/fermat/wilesEnglish.html)
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Quote Originally Posted by river_rat
    It is not as easy as I claim more due to our human nature and less to the nature of the mathematics involved. We are imprecise in our language and very few "higher level" proofs are really written in full mathematical rigor (for obvious reasons, see Principia Mathematica). We leave things to the reader or make claims like : It is easy to see that OR This follows trivially from without actually checking that it really is easy to see or follows trivially. Every step left out makes the mathematicians job easier but the proof checkers job more difficult. If proofs were written in a certifiable way (i.e. every result listing the previous results required to prove it) then proof checking would not be limited to experts in the field in question. But those experts would probably only publish one paper a decade

    As an interesting side note, work is underway to machine check fermat's last theorem (see http://www.cs.rug.nl/~wim/fermat/wilesEnglish.html)
    There is a project to convert a host of major theorems to pure symbolic logic. I don't know the details but there was a piece on it in The Notices a couple of years ago.

    You left out something: If proofs were written out in gory detail (certifiable), no human would be inclined to read them , but journals would weigh a couple of hundred pounds per volume. Instead of being enlightening, proofs would be only burdensome and boring. There would be no insight provided and research would likely grind to a halt. People just don't think like that.


    Speaking of "it follows trivially" there is an old math joke (but read on for the new punch line).

    A professor is giving a mathematics lecture when a student asks how one line on the board follows from the previous assertion. The professor says "It is obvious". Then he steps back and stares at the blackboard.

    After staring at the board for a few minutes he turns and leaves the lecture hall. Some time passes and one student leaves to find the professor. He finds him in his office working on a blackboard which he is filling with symbols, but does not interrupt and returns to report to the class.

    Near the end of the assigned class period the professor returns and announces to the class, "Yes, it is obvious."

    I told that joke to a now emeritus and very well known engineering professor. He replied, " That is not a joke. The professor was Norbert Weiner. I was in the class."
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    When I was elbow deep in set theory and topology before my career change I used to dread the words this follows trivially

    I know of the Mizar project, they have got quite far with their proof checking but they are using a stronger set theory then ZFC as their base (which may or may not horrify you depending on your persuasion).

    They actually publish a journal of human and machine readable proofs and have got quite far in covering most of undergrad level mathematics as well as a couple of other gems (Hahn-Banach and Godel's theorems I believe).
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  77. #76  
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    Quote Originally Posted by river_rat
    When I was elbow deep in set theory and topology before my career change I used to dread the words this follows trivially
    Career change ?

    When you last were active I was under the impression that you were an advanced grad student specializing in topology. Did you graduate or change fields ?
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  78. #77  
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    hi Doc Rocket

    I got my MSc and then went off into the high flying world of quantitative finance. But my belated Phd studies start in January.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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