User Reviews
Rating: really liked it
Simon Singh has the ability to present a story about a mathematics problem, and tell it like a detective story. He makes the subject exciting, even though the outcome is well known. Singh intersperses history with discussions about the mathematics, and makes it quite understandable.
Singh starts with the roots of the famous Fermat's Last Theorem, by recounting the stories and mathematics of Pythagoras, Euclid, and Euler. Other, less well-known mathematicians are also given credit, for example Sophie Germain, Daniel Bernoulli, Augustin Cauchy, and Evariste Galois.
Three hundred fifty years ago, Fermat wrote the following theorem in the margin of a mathematics book:
And, Fermat wrote that he had a marvelous proof, but no room in the margin for it. For centuries, mathematicians have attempted to prove the theorem, without success. It had been sort of a "holy grail" of mathematicians to prove the theorem, and many brilliant minds spent years on it. Perhaps in was unprovable, and worst of all, Kurt Godel showed that some theorems are actually
undecidable--that is to say, it is impossible even to decide whether or not a theorem is true.
Singh recounts a fascinating story of the gifted mathematician, Paul Wolfskehl. He was very depressed, and decided to commit suicide on a particular night, at midnight. While waiting for that time to arrive, he started to read about the failed attempts to prove Fermat's Last Theorem. He became so engrossed in the subject, that he worked well past midnight. He found a gap in the logic of a predecessor, and was so proud of himself that he gained a new desire for life. And, in his will he established a fund of 100,000 marks to be given to the mathematician who first completes the proof of the theorem!
Much of the book describes how Andrew Wiles developed a growing interest in the theorem. He worked in almost total isolation for seven years, in order not to be distracted. He occasionally published little tidbits unrelated to his real endeavor, in order to dispel suspicions about what his real work entailed.
The central piece of the proof entailed proving the Taniyama-Shimura conjecture, that linked modular forms with elliptic equations. This was a linkage between two far-flung branches of mathematics that seemed to be totally unrelated. To prove the conjecture would allow incredible advances to be made. And then, Ken Ribet showed that a proof of the Taniyama-Shimura conjecture would, in effect, be a direct proof of Fermat's Last Theorem. But many people tried and failed to develop the proof. But that is exactly what Andrew Wiles worked on for so many years.
I had previously read that during Andrew Wiles' famous lecture, he just casually let the unsuspecting audience know, "and that is a proof of Fermat's Last Theorem." Well, this book tells a somewhat different story. Most of the audience had heard rumors that the third of Wiles' lectures would be of historical significance. They came prepared with cameras, and took photographs during the lecture. So, it was a surprise, but not a total surprise.
After Wiles' manuscript of the proof was sent to a publisher, six mathematicians reviewed it, and a crucial gap was found in it. Wiles worked furiously for a nightmarish year, and with the help of Richard Taylor, finally closed the gap. Today, Wiles is recognized as the one who developed the proof. But it is clear, that Wiles "stood on the shoulders of giants"; he used--and developed--mathematical techniques that had not existed just a few decades previously.
Simon Singh writes with a wonderful style. It is clear, not too jargon-heavy but contains enough mathematical "meat" to seem satisfying. The book is followed by ten appendixes that contain more details about some of the mathematics; they are not overly technical, and give the reader a better understanding of some of the issues. I highly recommend this book to everyone interested in math.
Rating: really liked it
This book is endowed with all elements of a perfect story, which are curiosity, surprising events, tragedy, ambition, alternating defeat / triumph and finally an astonishing success.
The “Fermat’s enigma” is a tremendously inspiring narrative of one of the most important and most difficult conundrums of mathematics. It was devised by the reverend Pierre de Fermat who was called “The prince of amateurs” because he had neither had any formal education in mathematics nor he had any known tutor. He learned maths by some copies of ancient books written by Euclid’s disciple named Diophantus.
When mathematicians scrutinized margins of his books they saw a short note which baffled the brightest minds for more than 350 years. Fermat claimed a groundbreaking conjecture in the margin of Diophantus book without any proof, but the line next to the proof made the case much worse for other mathematicians. he wrote beside the conjecture:
“I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain” .This showed the fact that the formula had indeed a proof but it was lost. If you want to know what that theory is you can search on google because it can’t be properly explained in this review.
Since the last 350 years the formula had been attacked by the most eminent mathematicians including Leonhard Euler but none of them could prove it thoroughly.
After so many years of fruitless endeavor Andrew wiles managed to prove it. For Andrew the Fermat’s last theorem was personally and emotionally significant because he had been in love with the theorem since he first encountered it when he was just ten years old. After thirty years he used so many new mathematical techniques and finally he put more than 8 years on proving this conjecture in order to firmly prove it and finally he did.
This book is only recommended for those who appreciate the true value of mathematics.
Rating: really liked it
Before delving into the book itself, I thought I’d start things off by introducing the problem it’s concerned with, just in case you aren’t already familiar with it.
So, what exactly is Fermat’s Last Theorem? Well, basically, this is it:
As you can see, the conjecture is quite easy to understand, and yet, believe it or not, it was so remarkably difficult to prove that it took over 350 years to accomplish! The fact that Fermat (teasingly?) scribbled this rather infuriating note in the margin only added to the frustration felt by the scores of mathematicians who did battle with it over the centuries:
“I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.”No doubt the cheeky bastard would’ve enjoyed Twitter:
Okay, now that that’s been taken care of, it’s time to look at the actual book.
What Kind of Book is Fermat’s Enigma?Simply put,
Fermat’s Enigma is a history book. It is most definitely NOT a math book, so don’t expect to find any degree of mathematical rigor or complexity here. There are a handful of fairly simple proofs included in the appendices, but overall, the concepts under discussion are glossed over in a superficial manner, never examined in any kind of detail. If you want something that fleshes out how the proof actually works, I’m afraid you’ll have to look elsewhere. (Singh was kind enough to include quite a few “further reading suggestions” at the end of this book. While I’ve not looked into any of the titles he recommended, I assume many of those might prove more to your liking if you prefer a “math book” on the subject.)
In any case, while Singh did not pursue the actual mathematics in any real sense, he did positively
excel at telling the story of an utterly fascinating struggle, one which spanned hundreds of years and ensnared countless brilliant, talented minds. Readers make the acquaintance of such notable mathematicians as Pythagoras (whose work paved the way for Fermat), Leonhard Euler, Paul Wolfskehl, Sophie Germain, Daniel Bernoulli, Augustin-Louis Cauchy, Evariste Galois, Yutaka Taniyama, Goro Shimura, and of course Andrew Wiles, the man, the myth, the legend who finally proved the damn thing.
Overall, I was surprised and delighted by just how compelling the story actually was. For me, this book quickly became a veritable page-turner, one I was loathe to put aside. Some may argue that in order to accomplish this, he omitted too much relevant information, that he sacrificed depth for readability. Perhaps this is true to an extent, but in my opinion, while it was admittedly easy to read and follow, it still managed to include a fair amount of pertinent, interesting material. More importantly, it never got bogged down with unnecessary details or lost in minutiae, and never meandered down exasperating tangents, as many otherwise outstanding history books are wont to do.
And ultimately, what made this book so very stimulating was that the manner in which the story was told really made it come
alive. Singh bestowed a truly suspenseful, exciting quest upon the reader, one full of twists and turns, and even *gasp* its fair share of drama. He enthusiastically demonstrated just how action-packed and exhilarating the life of the mind can be. And for accomplishing this tremendous feat, I heartily recommend the book, warts and all.
And Now, For a Few Words on the Star of the ShowAndrew Wiles is an extraordinary human being. Fascinated by Fermat’s Last Theorem since he was ten years old, he vowed to conquer that most impossible of proofs. This was
at the age of ten, mind you. I seriously can’t get past that. And then, true to his word, the little rascal grew up to become an eminent mathematician, one who finally went into seclusion for seven years in order to hack away at this tremendous proof. While a not insignificant error marred the first release of said proof, he didn’t let that deter him, but persevered and managed to rectify the error, and, within a couple of years, came out with THE proof. Holy goddamn shit.
To me, Wiles’ story was completely and utterly inspiring. I was frankly amazed by what the human mind can achieve; I think I will always be in awe of Wiles’ fierce determination and incredible tenacity. Mad respect.
Anyway, as you can probably tell, Andrew Wiles is a personal hero of mine. He is an undeniable, ultimate badass. Wayne and Garth said it best:
Rating: really liked it
Being a scientist of long standing and loving all aspects of science and maths, Fermat's Last Theorem in itself was a wonderful mystery, what I would give to see Fermat's note book with a note in the margin about cubic numbers as opposed to squares. A very trite remark, too lengthy to write in the margin so it is elsewhere, and no one has ever found it or managed to prove his statement, until - - - this book is a brilliant read, you would think it would be as dry as dust, but no! It is a superb account of the proof of the last theorem from Fermat's notebook to be proven. The only thing that still niggles at me, although the mathematical proof is fabulous, it uses modern techniques not available to Fermat, so it is proven but how the hell did Fermat do it??????
A brilliant book, beautifully written a tremendous historical question answered in a very modern way, fabulous, well done for readability.
Rating: really liked it
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 lot more though 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...
Rating: really liked it
That's an excellent history of mathematics from antiquity to the present day. The author skillfully captures the pedagogical trick of the Last Theorem. This
"mathematical siren, ..., which attracted geniuses to destroy their hopes" better is a common thread that compellingly guides us through anecdotes, concepts very well explained in the maze of mathematics. The author manages to give both a technical and societal dimension to his story and, thereby, maintain the reader's attention from start to finish.
Very successful.
Good reading!
Rating: really liked it
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 any more digestible 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).
Rating: really liked it
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.
Rating: really liked it
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 or more. What 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 even more tantalizing 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?
Rating: really liked it
This book is as interesting as a detective story while being about quite advanced mathematics - as such it is quite a book showing the remarkable skill of its writer to explain complex ideas in ways that are always readable and enjoyable.
A mathematician finds a simple proof to what seems like a deceptively simple problem of mathematics - that pythagoras's theorem only works if the terms are squared, and not if they are any other power up to infinity. Sounds dull. Except that the mathematician jots down that he has found this proof, but not what the proof is. And for hundredsd of years the greatest minds in mathematics have tried to find this simple proof and been beaten by the problem time and again.
This really is a delightful book and one that gives an insight into how mathematicians think about the world. The proof of Pythagoras's theorem given in this book is so simple that the beauty of mathematical proof is made plain to everyone. Just a little knowledge of algebra is needed for this part of the book - the rest requires no maths at all.
Rating: really liked it
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, a² + b² = c², 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 > 2
And 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 more.
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 discovered:
“Simple 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 even more interesting: 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.
Rating: really liked it
From my reading journal:
May 31, 2009.
Yesterday I finished reading Fermat's Last Theorem. I plan to write a glowing book review but this space is too limited to contain it.
Rating: really liked it
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. Even more intrigued than before, I started the second one.
Fermat's Last Theorem was much more relatable - 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
Rating: really liked it
This is a very interesting book. Don't be scared about the contents, it's an enjoyable read even if you don't know anything about mathematics.
Rating: really liked it
My brain doesn't do numbers good but I enjoyed this book.
