Prime Obsession 325
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics | |
author | John Derbyshire |
pages | 422 |
publisher | Plume |
rating | 9/10 |
reviewer | jkauzlar |
ISBN | 0452285259 |
summary | History of the attempt to prove the Riemann Hypothesis |
Bernhard Riemann came to the University of Goettingen in 1846 at the age of 19, originally to study theology. The University, however, was home to Carl Friedrich Gauss, "the greatest mathematician of his age and possibly of any age," and the impressionable young Riemann, succumbing to the privilege of Gauss's presence and following his already blossoming interest in mathematics, refocused his studies on the area in which he would soon attain distinct immortality. As early as 1851 he was impressing even Gauss with the results of his doctoral dissertation and in 1859 was appointed a corresponding member of the Berlin Academy. To this honor, Riemann responded with his most famous paper, entitled "On the number of prime numbers less than a given quantity," containing therein what became known as the Riemann Hypothesis.
At the heart of the RH is the Zeta function which, in its basic form, looks like this: Z(s)=1 + 1/2^s + 1/3^s + 1/4^s + ... and which, through some simple algebraic manipulation as demonstrated by the mathematically gifted journalist Derbyshire, can be given in the form (1 - 2^-s)^-1 * (1 - 3^-s)^-1 * (1 - 5^-s)^-1 * (1 - 7^-s)^-1 * ... And it is in this second form which Derbyshire calls "The Golden Key" where the non-mathematician gets the first glimpse of the Zeta function's relationship with prime numbers.
But where this Golden Key appears as this "novel's" turning point--its central conflict-- it is not until Prime Obsession's climax when the Key is at last turned and the Zeta function's true relationship to the prime counting function pi(x)--the number of primes less than a given x--is at last made clear. Along the way, from the introduction of the Zeta function to the final explanation of its relevance to prime numbers (the turning of the Key), Derbyshire enlightens us with clear, mostly English language descriptions of the mathematics involved, as well as plentiful anecdotes that give readers a sense of the life and work of the major figures in the history surrounding the RH from Euler, Gauss and Dedekind in the late 18th century through Riemann's 1859 paper, and from 1859 onward to recent advancements in the '80s and '90s.
The Riemann Hypothesis states that "all nontrivial zeros of the Zeta function have real part one-half." Understanding the statement of the hypothesis is Derbyshire's first mission for the reader. In short, most functions with a dependent variable, say f(x)=x^2-2x+1, have a value for which if you replace x with this value, the function returns zero. In the example given, it is at the value x=1 where f(x)=0. The Zeta function has an infinite number of these zeroes and an infinite number of these is "non-trivial." The non-trivial zeroes come from complex number values. Riemann's guess, his hypothesis, is that the real part of each of these non-trivial zeroes is equal to one-half. The imaginary part can be anything.
Derbyshire explains all of the mathematics in very readable language. It's unlikely that anyone who did well in high school mathematics will not be able to follow Derbyshire's mathematics (and it's unlikely that those who didn't do well will pick up a 400-page book on this topic). The Zeta function is explored from a number of angles--numerically, graphically, algebraically, statistically, and there's even a link between the non-trivial zeroes of the Zeta function and quantum physics! By a larger margin, however, Prime Obsession's intrigue lies in Derbyshire's expositions on Riemann, Hilbert, Turing, Gauss, et al, as well as those modern mathematicians he's interviewed personally. The line between the mathematical half of the book and the historical is clearly defined; the odd-numbered chapters are devoted to the former, the even to the latter.
Those fans and foes of Derbyshire's most public line of work as a journalist/editorial writer for National Review will be comforted to know all political polemics have been set aside. John Derbyshire gives a virtuoso performance as an informed journalist and maintains his stance as a personable and careful guide through a sometimes difficult terrain. Anyone with some interest in the topic will find it hard to put down Derbyshire's book once begun. If we are lucky (hint, hint, JD) perhaps Derbyshire's next book will cover the newly-proven Poincare Conjecture ...
You can purchase Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, carefully read the book review guidelines, then visit the submission page.
If you are interested in solving math puzzles (Score:4, Interesting)
Not quite related to primes, but close and can certainly create an obsession. Also, look behind the scenes for something simpler to solve.
My favorite. (Score:4, Interesting)
It's very readable, and has chapters on interesting stuff like knot theory, cellular automata and primes.
I highly recommend it. It isn't going to turn anyone into a math professor, but it is very interesting reading.
Of Course (Score:2, Interesting)
I've heard a lot of criticism of [Hofstadter's "Godel, Escher and Bach"] from mathematicians and musicians..
Of course mathematicians and musicians will criticize the book. It challenges the very logical foundations upon which their theories are based. Perhaps the most dogmatic disciplines outside of Christian fundamentalism are the sciences. It's the age old case of man believing his logic is impenetrable, where in reality it amounts to nothing more than the finger pointing to the moon. The sciences may have theory-this and theory-that, but they will never in an infinite amount of lifetimes be able to run the full course of reality with their tools. And that's a fact that mathematics itself asserts.
I once ran into an old friend of mine I knew back in middle school. He has a twin brother that, over ten years since he left highschool is still in university plugging away at some mathematics doctorate. I silently asked myself. What are his aims, his purpose? To solve the universe? It was clear he was always a brilliant student; I'm sure his I.Q. is off the chart, but ambitious mathematicians have to learn to let go. All their combined knowledge amounts to one drop in the Pacific ocean of reality.
If you take a liking to esoterics and esoteric knowledge, you will notice there's a smooth transition between scientists and esoterics; that is, there is the complete scientist who deems it worthless to search for truth in the unseen and the non-constant -- that the only universe worth pursuing is the visible and measurable universe. Then you have the transition scientists (Godel, Heisenberg) who through experiments of their own come to the realization that the sciences are not adequately equipped to be able to completely ascertain truth and that there must be more -- another form of reasoning perhaps outside the realm of postulation and thought where paradox becomes perfectly logical, but they may at the same time reserve making any definite statement about one or the other, effectively taking up the agnostic position.
Finally you have the esoteric, who acknowledges science as a method for ascertaining some degree of truth, though a limited portion of it, but through experience is assured that complete truth is to be found outside the dualistic disciplines of science and philosophy. Zen masters, enlightened sufis or Christian mystics might fall into that category. Due to their highly honed awareness, they are able to acertain more in a ten minute period about the laws of life than ten scientists could over the course of a hundred years. These, quite rightfully are higher order human beings. I imagine it's the same sort of higher order, perhaps to a somewhat lesser degree, that allows the idiot savant to blast through hundreds of years of perpetual calendars or calculate ridiculously large numbers in their heads almost instantaneously. Savants appear to have a firm, instinctual understanding of computational causality. They may very well be solving our mathematics from some other conscious plane the rest of humanity haven't yet achieved, a plane that allows them to blaze logical trails in parallel and from a figurative bird's eye view, through our "world." The same thing goes for enlightened men. Though we may plug along attempting to understand the unverse with 4-bit effectiveness, they do it from a conscious vantage point that may exceed a figurative 1024-bits or more. They simply know.
- IP
Why I believe this book to be of interest. (Score:5, Interesting)
I agree with the reviewer's sentiment that the book is well written, and it is very enjoyable. The author writes in a very audience-centric fashion, even going as far to discuss the "scaffolding" of the book itself (all of the "hard math" stuff is found in odd chapters, the author had debated putting this information in only the "prime" chapters, but then said "there is such a thing as being too cute.")
Anywho, if you have a math friend you need to buy a gift for, definitely consider this book.
Re:Majored in math, away from it for a year (Score:3, Interesting)
This wouldn't be a book to get someone that works in a heavily mathematical field...
Why not? I have a Ph.D. in math from one of the top scientific institutions in the world and I think the book sounds interesting. Quoting from the above review:
This book is not a "dip in the math kiddie pool" like you suggest; it's an examination of the personalities behind the equations. I know quite a bit of math but very little about the people who developed the theories and the trials they had to go through. I suspect that most people who work in "heavily mathematical field" are in the same boat. And these stories can be very interesting. Ever read the account (or see the PBS special) about Andrew Wiles's proof of Fermat's Last Theorem? Fascinating stuff!
I think this idea that popular math books appeal only to those who know something about math is unfortunate. It prevent the public at large from understanding and appreciating mathematics. Carl Sagan did wonders for astronomy with his Cosmos series (although the isn't too tough to do because, as my astronomer friend Nicole says, "astronomy is a 'sexy' science") but we don't have any real champion for other fields of science. I think books like this should appeal to a large audience: mathematicians and the math-phobic alike.
GMD
Re:If you are interested in solving math puzzles (Score:3, Interesting)
Actually, it is possible that the problem is unsolvable. You have 2^160 (not 128) inputs, 2^160 outputs. If there is a strict one-to-one mapping, it might not hold true for any pair. Personally, I think I'd be willing to give $50 for a proof that a solution exists at all. A lot more for a proof that a solution didn't exist.
To demonstrate, I'll use a silly 2 bit "hash" function.
00->01
00->10
10->11
11->00
There is no pair a,b so that hash(a) == b && a == hash(b). There's a strict one-to-one map. The same could be true of MD5 sums.
Kjella
Re:lay person? (Score:4, Interesting)
Consider yourself "shown".
I am not speaking of general math - rather I am speaking of the esoteric stuff such as "new math" stuff that has no "purpose" other than to be a neat trick.
I was deeply impressed by Richard Feynmans chapter on his reviewing high school math books. He was livid that a number of things being taught were useless. He wanted the books to teach the students not only what they were learning, but why. One example has him in an uproar because there was a question about taking the average tempurature of a number of stars. This made Feynman angry because there is no reason to get an average star tempurature for a number of stars, it is just not something that you do. Feynman called it "a trick to get the students to add".
Furthermore, he was furious at a physics problem in one book, that had wrong answers, and in fact, Feynman actually performed the experiment listed in the book, and found out the "observed" results were wrong. The author did not even take the time to DO the experiment listed.
Again, this made him furious because he felt that teaching students math in a deceptive manner would never give them a feeling as to where the math can take you in fields in the sciences. I agree.
So, I don't want to learn fluff. I was at a disadvantage because I was just told "learn this" and in answer to the question of "why?" I was only given "so you can pass the exams."
In high school I deeply wanted the answers to some questions in Physics, that were available with mathematics, but I was not shown these, and I developed an unfortunate disgust with mathematics because of this.
So many people here on slashdot can take me to task for being bad at math - and I know I am. I don't know if you would have been so interested in it either if it was drilled into you in a dull manner, and a feeling that it lacked a purpose.
Am I learning math now? Yes, but then I understand much more about the why, the how, and the history now than I did then. I don't know about the rest of you but I detest rote learning. So take me to task on my math skills if you wish (or my typing