I never watched any documentaries before going to college and this was about a century and a half ago I am getting old But yeah, 2009 to be precise I was always interested in NatGeo and History Channel but they never showed the real deal on television The documentaries would be mostly half assed, and at worst, total crap That s also how Indian television landscape can be broadly categorized too, give or take a few exceptions ofcourse And so I grew up loving the sciences based on w I never watched any documentaries before going to college and this was about a century and a half ago I am getting old But yeah, 2009 to be precise I was always interested in NatGeo and History Channel but they never showed the real deal on television The documentaries would be mostly half assed, and at worst, total crap That s also how Indian television landscape can be broadly categorized too, give or take a few exceptions ofcourse And so I grew up loving the sciences based on what was taught in school curriculum, and elsewhere what I read on the slow 64 bit internet connection.And then college happened Parents got me my own laptop and the college intranet had a ton of stuff that other students shared That place and that time was where my love for documentaries was born I had never been so fascinated with anything before And the first two that I watched in a long line of them were Einstein s Biggest Blunder and BBC s Fermat s Last Theorem The memory of that sunday afternoon is still pretty fresh Back then, I only had a casual interest in astronomy and cosmology, and Einstein s theories were still something exotic And so I basically understood jackshit from the first documentary Evenintrigued than before, I started the second one Fermat s Last Theorem was muchrelatable I had known the theorem, and understood the concept too Years later when I joined goodreads, I found out that there was a book too Keeping in with the tradition of firsts, it became the first book on my TBR pile too Where it stayed until a few days ago and I finally marked it as read last night To be honest, this isn t the greatest book ever It isnt even Simon Singh s best, who delivered the goods the much better in Big Bang But it surely captures the essence of all mathematical and scientific endeavor very well That every once in a while, in the middle of an ordinary life, science gives us a fairytale Simon converts what could have been a dry chronicle of proofs into an ode full of excitement, inspiration and intrigue worthy of a gothic love affair Full review to follow. #FREE BOOK Õ Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem ê xn yn zn, where n represents , no solution I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain With these words, the seventeenth century French mathematician Pierre de Fermat threw down the gauntlet to future generations What came to be known as Fermat s Last Theorem looked simple proving it, however, became the Holy Grail of mathematics, baffling its finest minds for thanyears In Fermat s Enigma based on the author s award winning documentary film, which aired on PBS s Nova Simon Singh tells the astonishingly entertaining story of the pursuit of that grail, and the lives that were devoted to, sacrificed for, and saved by it Here is a mesmerizing tale of heartbreak and mastery that will forever change your feelings about mathematics I guess the author does a reasonable job But when I reached the end, I still didn t feel I understood at all how the proof worked Probably that s just because it s so bloody hard I got a lotthough out of Prime Obsession, Derbyshire s book on the Riemann Hypothesis, where the author opens up the box and shows you some of the actual math What a fun book this was thanks, Trevor, for the recommendation There are many reasons I think I like good nonfiction a sense of direct relevance, gravitas, frequent insights into the workings of the universe and people , but mostly for knowledge narcs high levels of information density served up into an intriguing package by someone else who has undertaken the heavy lifting research, organization, thinking So, here in Singh s work I get a solid lay understanding not only of the p What a fun book this was thanks, Trevor, for the recommendation There are many reasons I think I like good nonfiction a sense of direct relevance, gravitas, frequent insights into the workings of the universe and people , but mostly for knowledge narcs high levels of information density served up into an intriguing package by someone else who has undertaken the heavy lifting research, organization, thinking So, here in Singh s work I get a solid lay understanding not only of the proof to Fermat s Last Theorem, but of much of mathematics and the lives of mathematicians since the seventeenth century.I ve been thinking also about what attracts me to books on mathematical topics the works by Martin Gardner, William Poundstone, and the various other authors in the company of whose thoughts I ve had pleasure to spend a week orWhat I ve come away with, is that the best of them feed off surprises, those bits of counterintuitive anecdotes that leave you blurting out, Huh How about that, and then looking madly around for someone to tell Like a book of jokes, riddles, or puzzles that provides immediate gratification in the back of the book, Fermat s Enigma plugs at least ten conundrums and their easy to understand, logical solutions into its appendices For example say you re unlucky enough to be forced into a three way duel If everyone gets to take turns in order of their skill such that worst shoots first, what should the worst do Aim at the best in the hopes of getting lucky and eliminating the most dangerous gunsel Nope, the correct answer is to pass up the turn in the hopes that your first shot will get to be expended against only one remaining combatant That way, even if you miss, you at least had a chance to take aim at the only person able to shoot back.Pierre Fermat turns out to have been quite the prankster, often tweaking professional mathematicians and academics by mailing them problems they knew full well he had already solved For those who don t keep this type of trivia at the forefront of their brains, Fermat s the French recluse and hanging circuit jurist who once famously scribbled in a copy of Diophantus Arithmetica that x n y n z n for any number n greater than 2, a propostion for which he had a truly marvelous demonstration which this margin is too narrow to contain This gets to be Fermat s Last Theorem, simply because it ends up being the last of his conundrums to be proven not necessarily the last one he wrote Just think, were it not for the scrupulous care taken by Fermat s son to go through and publish all of Fermat, Sr s writings, the world would not have been tantalized by this particular gem for over 350 YEARS.Andrew Wiles published the first and only proof in 1994, and Enigma does a tremendous job of walking the reader through not only the stunning depth of his intellectual achievement, but its significance as well Suffice it to say that I was happy here to read that Taniyama Shimura get their well earned due and that modular and elliptical equations can finally be understood to be mathematically analogous whether or not I have any idea what modular equations actually are Still, all of this leads to what I think is an eventantalizing problem We now know that all of Fermat s conjectures ultimately proved to be solvable and that Fermat s own notes would seem to indicate that he had indeed apparently found ways to solve each of them But there is no doubt that Fermat s solution could not have relied on the up to the minute maths Wiles employs over 200 pages So if it was really the limitations of the margin and not of Fermat that inhibited publication what was Fermat s proof This is the kind of book that we non mathematical minds can easily digest and love It gives you an epic scope of the number of minds that it takes to build new ideas I doubt if Fermat had actually solved this theorem correctly, but this is impossible to prove Fermat s theorem however was not impossible to prove It was solved Thanks to the efforts of many men and women over many lifetimes and one final man who had the determination and persistence to finish the unthinkable This book has This is the kind of book that we non mathematical minds can easily digest and love It gives you an epic scope of the number of minds that it takes to build new ideas I doubt if Fermat had actually solved this theorem correctly, but this is impossible to prove Fermat s theorem however was not impossible to prove It was solved Thanks to the efforts of many men and women over many lifetimes and one final man who had the determination and persistence to finish the unthinkable This book has a lot of wonderful elements, and really exemplifies a love of mathematics Although if you want to actually understand the theorem this book may not be for you I can honestly say reading it did not put the theorem in anydigestible light than it started out with Perhaps it was to the authors advantage to skip the technicalities and focus on the enjoyment of the journey I personally loved this approach, but it may not be for everyone, especially if you are actually looking to understand the theorem a massive undertaking that is really not in my repertoire to comment on Strap in, guys I m going to walk you through the history of how Fermat s Last Theorum was proved, all in one little okay, big review And I can do this because of this awesome, semi accessible, frequently tangent taking, but mostly, this deeply fascinating book STEP ONE THE THEORUM For the unenlightened, Fermat s Last Theorum is this you probably know the Pythagorean theorum, a b c , which explains that if you square the shorter two sides of a right angled triangle Strap in, guys I m going to walk you through the history of how Fermat s Last Theorum was proved, all in one little okay, big review And I can do this because of this awesome, semi accessible, frequently tangent taking, but mostly, this deeply fascinating book STEP ONE THE THEORUM For the unenlightened, Fermat s Last Theorum is this you probably know the Pythagorean theorum, a b c , which explains that if you square the shorter two sides of a right angled triangle and add them together, you get the value of third side squared This is easily proved that is, demonstrated to be completely, utterly, logically true via a mathematical proof, using axioms known to be true, which is just like a logic proof if you ve done philosophy Or a geometry proof, if you went to 9th grade Take any triangle, and this will be true There are infinite solutions to this equation literally infinite values of A, B, and C which will render this solution true However, Fermat discovered that the formula an bn cn where n 2 has NO whole number solutions Go ahead, give it a whirl I ll wait.Then he challenged the mathematical community to create a mathematical proof demonstrating this must be true While tantalizing them with the knowledge he d already created a proof for this with a scribbled note in the margin of his notebook I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain Asshole So sparked centuries of people trying to write this proof, to no avail Until Professor Andrew Wiles who has the most apropos name ever he has wiles indeed of Cambridge, in 1993 For clarity, this is Fermat s Equation which I ll refer to as FE an bn cn where n 2And THIS is Fermat s Last Theorum which I ll refer to as FLT There are no solutions to Fermat s Equation TANGENT GODEL At this point, it s been assumed that math is logically perfect that if you correctly build a proof using axioms like m n n m , it must be true and if it s proven to be true, nothing can prove it to be false This is known as axiomatic set theory Along came Godel who is analyzed to death by my beloved, Doug Hofstadter see here and here His ideas are as follows 1 If axiomatic set theory is consistent, then there have to be theorems that can t be proved or disproved Why Because of paradoxes Godel translated the following statement into mathematical notation This statement has no proof If that s false, then you COULD create a proof for that statement however, that would make the statement false, so how could you have a proof for that So it has to be true But if it s true, it can t be proved because that s what it literally says It s a mathematical statement that is true, but could never be proved to be true an undecidable statement 2 There s no way to prove that axiomatic set theory is consistent in a way, it s one of those undecidable statement that s true but can t be proved to be true Interestingly, this parallels the physicist Heisenberg s discovery of the uncertainty principle, but we won t get into that Now, there aren t very many of those undecidable statements Godel couldn t really point to any other undecidable statements besides the one above, so people assumed they were found only in the most extreme math and would probably never even be encountered Welp A young student named Paul Cohen at Stanford discovered a way to test whether a question is undecidable, and in doing so, discovered several.Which sparked some fear in mathematicians What if Fermat s Last Theorum was undecidable What if they were wasting their time trying to prove the unprovable Interestingly, if it were an undecidable statement, it couldn t be proved yet it would have to be true The theorum says there are no whole numbers to the equation an bn cn where n 2 If this were false, then it would be possible to prove this by offering a solution to this by finding a whole number N that s greater than 2 that allows the equation to be solved Which would make it a decidable statement, which is a contradiction So it can t be false and also be an undecidable statement In other words, Fermat s Last Theorum might be totally true but there might be no way to prove it STEP TWO TANIYAMA SHIMURA CONJECTURE Modular forms are a mathematical tool, sort of like impossible forms or shapes, that reveal a lot about how numbers are related The Taniyama Shimura Conjecture TSC , created by two Japanese mathematicians one of whom tragically and abruptly killed himself quite young says every modular form is related to a specific elliptic equation elliptic equations were Andrew Wiles s main area of study they re a type of equation, not super important that you understand them The fact that these were unified meant there s a kind of Rosetta stone that s been discoveredSimple intuitions in the modular world translate into deep truths in the elliptic world, and vice versa Very profound problems in the elliptic world can get solved sometimes by translating them using this Rosetta stone into the modular world, and discovering that we have the insights and tools in the modular world to treat the translated problem Back in the elliptic world we would have been at a loss Beyond that awesomeness, the TSC suggests something eveninteresting that possibly all of mathematics, all the different worlds of mathematics, might have parallels in other worlds, as with the elliptic world and the modular world All of mathematics might be unified arguably the absolute ultimate goal of abstract mathematics, because this would give us the most complete picture, and the biggest arsenal of tools to solve mathematical problems STEP THREE RIBET S THEORUM This is all important for FLT because of something called Ribet s Theorum Ribet was a colleague of Wiles Ribet s Theorum goes like this the imagined solution to FE can be translated into an elliptic equation And that elliptic equation doesn t seem to have a modular world equivalent But the TSC claims that every elliptic equation must be related to a modular form So if you can PROVE that the elliptic form of the solution to FE has no modular form which we can The proof was done in 1986 , the following is true if the TSC is true i.e all elliptic equations have modular forms , and the elliptic form of the imagined solution to FE has no modular form, then the imagined solution cannot exist, proving FLT So now, all we have to do is prove that the TSC is true, and FLT is automatically proven to be true And this THIS is what Andrew Wiles focused on Proving the TSC Taniyama Shimura Conjecture STEP FOUR PROVING THE THEORUM Wiles finally succeeded when he applied a new method called the Kolyvagin Flach method, which groups elliptic equations into families and then proves that an elliptic equation in that family has a modular form if that elliptic equation has a modular form, then all other elliptic equations in that family also has a modular form However, the Kolyvagin Flach method has to be adapted for each family of elliptic equations Wiles successfully adapted the method for all families of elliptic equations, thereby proving that all elliptic equations have modular forms which, as a reminder, is basis of the Taniyama Shimura Conjecture, and proving the TSC proves FLT Two months before I was born, in 1993, he proved FLT in a series of 3 lectures Kinda Actually, there was a minor flaw in his proof essentially, he might have failed to properly adapt the Kolyvagin Flach method for some of the families of equations but it doesn t really matter because a year later, Wiles published a work around to that flaw, so he proved FLT once and for all CONCLUSION All I can think about is maybe if math looked like this, if THIS had been our material in high school maybe then I wouldn t have disliked it Maybe other people wouldn t, either What a shoddy job we do teaching our children the wonder of learning