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Prime Obsession
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.
Who isn't obsessed with the leader of the Autobots, Optimus Prime?
Ummm...what would its peers be? Just how many "classic" math books does the lay-person have now?
Could it be that the lay-person wouldn't be interested in any book about math, no matter how well written?
I dunnnoooo...almost sounds completely probable.
Superior mathematician.
The answer? 42.
The question? What is 6 times 9.
The part he didn't tell you is that the question/answer machine was devised by a group of aliens that had 13 fingers. They wouldn't count in base 10, they would count in base 13, naturally.
6 x 9 does in fact equal 42. In base 13.
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Why is it, that if you have studied math that people get you these books for Christmas, etc. People say, "Wow, he's into math, I'm sure he'd like that", when books like this are written for the lay person, as a fun introduction to the subject. People don't get Literature majors "Shakespeare for Dummies".
Rhymes that keep their secrets will unfold behind the clouds.There upon the rainbow is the answer to a neverending story
One man's propaganda is another man's editorial opinion.
Why must we use such slanted terms to describe the views of people we disagree with?
perhaps I just answered my own question.
There's a Mercedes gap too. I want one and can't afford one, but it's not government's job to do anything about it.
You might check out my current MD5hash Challenge. Some people have told me that it is impossible to solve, some have said that mathematically it is solveable.
Not quite related to primes, but close and can certainly create an obsession. Also, look behind the scenes for something simpler to solve.
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.
It's been a long time since I read Douglas Hofstadter's "Godel, Escher, Bach", but didn't it use the same kind of formula, alternating between dialogs and discussion chapters? I really loved that book. I've heard a lot of criticism of it from mathematicians and musicians, but that noise always sounded like so much professional nitpicking to me.
ISBN 0452285259 = 3 * 1009 * 149417
The author must be sad.
Try Mathematics for the Million by Hogben - it's fantastic, and the most coherent Calculus explanation I've ever encountered.
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Let serve as motivation the fact that anyone who can actually proof (but not disproof) the Riemann Hipothesis will won a prize of US$ 1E6 (i.e, US$ 1000000.00)!
*gasp* He's a conservative! Run for the hills!
It's entirely people's prerogative to mix politics and pleasure, but my God, what a silly prerogative to exercise.
Mathematics And Sex (2004)
Pi: A Biography of the World's Most Mysterious Number (2004)
Chance: A Guide to Gambling, Love, the Stock Market and Just About Everything Else (2004)
Entanglement: The Unlikely Story of How Scientists, Mathematicians, and Philosphers Proved Einstein's Spookiest Theory (2003)
The Mathematical Century : The 30 Greatest Problems of the Last 100 Years (2003)
The Golden Ratio : The Story of PHI, the World's Most Astonishing Number (2003)
When Least Is Best : How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible (2003)
The Honors Class: Hilbert's Problems and Their Solvers (2001)
An Imaginary Tale (1998)
e: The Story of a Number (1998)
Just to pick some recent examples (i.e. not including the masterpieces of Martin Gardner and other recreational mathematicians in the 1960s and 70s, and apologies if I left off your favorite). I would agree, however, that good pop-math books are a great deal more rare.
It's called white space. Look into it. Humans parse on it much faster then they parse on operators.
I currently have no clever signature witicism to add here.
After working at Initech for a year and not using any of my math skills, this was a welcome dip in the math kiddie pool.
I would probably need to do a few laps before I could go playing around near the high-dive again or anything. I don't think this speaks to my grasp of the subject or my intelligence, but to my complete abandonment of study for a long period.
This wouldn't be a book to get someone that works in a heavily mathematical field, but its a great choice for the coder in your live that likes math but has to write boring code all day to pay the bills.
My favorite book on math is The Mathematical Tourist by Ivars Peterson.
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.
Could be worse -- he might've been *gasp* a liberal!
;)
Besides, in the field of mathematics, being conservative is not a bad thing as long as the Earth is still round.
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
I know very few mathematicians and math students who aren't familiar with the Riemann Hypothesis (largely due to the million dollar prize associated with its proof), so a book exclusively on such a topic probably wouldn't interest too many people. What makes this book interesting, at least to me, is the Math History covered in it. In particular, the author goes into great depth into the personality and character of each of the principle figures in this book: the anecdote regarding Hilbert's torn pants, Gauss's (perhaps justified) arrogance, and Riemann's quiet nature. All of these aspects of the book add a lot more depth to the people behind this problem, and I find that to be far more valuable, as a mathematician, than yet another essay on the Riemann Hypothesis.
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.
I have discovered a truly remarkable sig which this margin is too small to contain.
Those interested in his other writings should check out John Derbyshire's homepage.
These comments do express the opinions of my employers, and, personally, I think they're complete rubbish.
I'm in charge in Moria!!
Don't blame Durga. I voted for Centauri.
People get those gifts because they try. They don't understand math at all, but they know that you do "something mathy".
Exactly right. They are trying to get you something that they think you might like even though they don't know very much about math. Instead of the grandparent getting all hot under the collar that his family and friends dare insult his grant intellect by purchasing him a "Math for Dummies" book (as he seems to think this historical work is), he should feel gratified to know that he family cares enough about him to actually put forth some effort to getting something that attempts to match his interests. There are lots of people who simply buy generic gifts for family like socks or shit like that. Isn't this book a lot better than a gift like that?
Reading the grandparents rant, I was reminded of an article in The Onion awhile back about some film snob getting all upset because his family -- damn their incompetance! -- dared buy him the widescreen edition of one of the Matrix sequels when he actually wanted the letterboxed edition (opportunity for karma whoring here if someone can link to it). For chrissake, your family and friends are trying their goddamned best and you get your panties in a bunch over details? That's so incredibly childlike, I can hardly believe this above "rant". Christmas isn't really about getting exactly what you want -- at least once you're an adult it's not. Christmas is just an opportunity to get together with loved ones and exchange gifts as a token of affection. It doesn't have to be the "perfect gift"; as long as it's somewhere in the ballpark you should feel happy that your family is at least aware of your interests.
GMD
watch this
He had a little bitchy slap-fight with the student body of the University of Michigan a while ago that resulted in this pretty good guide though his profound love of the word "buggery". Seriously, he writes articles where he'll use it like 16 times in a single paragraph.
Why not cite his own explaination of his homophobia?
These comments do express the opinions of my employers, and, personally, I think they're complete rubbish.
Can't believe nobody has mentioned this yet (maybe they have?), but this book (I think it is the same book) can be read for free online at the National Academies Press
I've started it and it is very good so far. Haven't had time to get past the first few chapters unfortunately.
Nothing disturbs me more than blind loyalism towards some unrealistic and over-idealistic notion of one's nationality.