I can say, with some modesty, that I am familiar with the subject of mathematics more than an average person is. Despite that I hadn’t ever read a non-technical book on mathematics, so this book ended up being a very engrossing and a pleasant surprise. Marcus has done a tremendous job detailing history of search of the proof of Reimann Hypothes: interspersed with personal stories and personalities of mathematicians involved – one person at a time.
What is Reimann Hypothesis? I am not smart enough to
explain, but I regurgitate what I gleaned from the book. As mathematicians are
wont to do, they wanted to understand what and why certain numbers are prime and
what is the distribution of prime numbers across number line. When that proved
to be tad difficult of an endeavor, one of them decided to find pattern in number
of prime numbers less than a certain number, say primes less than 100, prime
less than 1000, and so on. There was some success in generating a formula for
such counting (see later), but it wasn’t exact. Meanwhile when a mathematician B.
Riemann decided to pass on complex numbers to simple looking Zeta function (1+1/2^n+1/3^n+…) in 1850s, he observed that values of “n” for
which function takes value of zero relate to counting of primes (I don’t understand
how, though!). Moreover, he guessed that all such “n” have the real part of 1/2.
This guess is the Riemann Hypothesis. Line passing through this point is called
Reimann’s critical line.
Book gave me a fascinating glimpse into how theoretical
mathematicians work and how does mathematics advance. They often work by
posing ad-hoc puzzles (“what happens when kick a rotating sphere of liquid”),
trying new mathematical operations on existing formulas or statistics on existing numbers (“taking
moments of function”, “passing imaginary numbers to known equations”), looking
for patterns and trying to find equation and explanation of those patterns, translating
numbers and equations from one branch of math to another (“from numbers to
graph”), trying to prove intuitions, and finally aiming for elegant, closed form
solutions and formulas to problems. To an application oriented engineer, this
was delightfully ridiculous – trying anything to anything and spending decades
trying to make sense of it toward no immediately discernible end!
In doing so, mathematical properties sprout starting with educated guesses
on observed patterns called ‘conjectures’. When supported by available data
become, conjectures become ‘hypotheses’ which make a novel yet untested prediction. Hypotheses
graduate to ‘theory’ when the non-trivial novel prediction comes true and proven
mathematically, and eventually to ‘law’ when validated in multiple domains
across time to multiple level of rigour with no scope for any further
enhancement or exception.
Growth of mathematics is also, surprisingly, driven by friendly
bets between prominent mathematicians. Bets, often of little material value,
but taken seriously enough, are reason for many important advancements in the
field. For instance, 100 million Reiman’s zeros were calculated by costing CPU
processing in 1970s just to (dis)prove a bet! I wonder if this is true of other
disciplines of science.
Till very recently before advent of internet, exchange of
ideas through letters was usual form of brainstorming and communication among
mathematicians. Yet, many also worked in isolation through their rough sketches
never sharing ideas till they were comfortable of elegance of proof, or of not
being challenged or ridiculed, or not wanting to share credit of their
discovery too early! Consequently, the amount of knowledge, which is lost to
humanity, because their family couldn’t comprehend the importance of scribbles
in their papers when they died (and just threw away) is heart wrenchingly disappointing.
Ramanujan’s and Reimann’s notes on having figured out proofs but couldn’t be
bothered to write them out because they thought them trivial have costed
mathematicians efforts of centuries.
Book underscores a unique manner of working of theoretical
mathematicians – their rigorous reliance on proof. Unlike other spheres of
science, where theoretical predictions are useless unless observed under
experimentation, mathematicians are content with solid proof despite lack of observations,
and as corollary, they still do want solid proof despite what would be
overwhelming experimental observations in other sciences. And why is it so?
Because math being math, incongruence with theory may occur even after
trillions of observations – 10^10^10^34 is theoretical number when Gauss’sPrime Conjecture is proven false. Does it matter in practice, though? This reliance of solid theoretical proofs
also means that proofs once established don’t change forever. While theory of
gravity has been refined every century, mathematical proofs established from 1600s
have stood test of time.
Role of computers is hinted on later parts of books, but it
is aptly concluded that computer can only help in disproving a hypothesis by
providing counter examples but cannot prove a hypothesis simply because any
number of matching examples (evidence) don’t count and cannot constitute a
proof.
Now let’s come to the personalities of mathematicians narrated in the book. If you imagine mathematicians to be hard working geniuses & nerds,
you will not be wrong, on average. Like all geniuses, most of them show their
signs before they turn ten. For instance, Gauss (of the Gaussian distribution)
was correcting his father’s math at age of three where normal children do not know
beyond counting. Most mathematicians are pacifists reveling in their own cocoon
with little care or awareness of the rest of the world. Some were eccentric – for
instance, G H Hardy, lived his life fighting with, tempting, and challenging
God, with often comical outcome where he ended up believing in God to prove
that he didn’t believe in Him.
Ramanujan’s level of genius is, of course, unfathomable. If
partition number is defined as number of ways to partition N things (partition
number of 3 is 3, of 4 is 5) then he literally dreamt up formula to calculate
partition number of any N just in flashing insight without intermediate steps
which included square root, pi, sum, mod, natural logarithm, integration,
differentiation, imaginary number, and trigonometry! If not Namagiri Goddess,
then how?!
Of course, there are sprinkling of revolutionaries,
womanizers, and partygoers. While the book does reference tumults in the rest
of world – world wars, famines, Nazi politics – and their impact on
mathematicians, and hence on mathematics, they are not dwelled into more than
mere mentions.
Incentives for theoretical mathematics are few and far (are they?) –
most of them do this simply for the joy and, occasionally, for the fame within
mathematician community. There is no Nobel Prize in math. The highest award, the Field Medal is given
only once in four years for living mathematics under forty years of age and
accompanies just CAD 15,000. Opportunities for commercialization are rare –
though of course, cryptography and encryptions are important exceptions. Makes
you wonder how cheap and easy to sponsor a prize in math and yet why hasn’t anyone
done so?
While reading the book, I also realized that my level of
thinking as commoner, despite my admissions in the opening line of this review,
is closer to what mathematicians were doing 400 years ago. Hence, what mathematicians
are doing today, reflected in the later chapters of the book, went completely
over my head! Marcus has done excellent job in talking about mathematics
without using mathematics, but I do feel that book can be slightly better by
exposing readers to real math – say equations or screenshots of seminal papers –
even with clear expectations that reader won’t understand. Else there is risk
of getting too lost in analogies!
Search for Reimann Hypothesis has led to multiple discoveries
and developments on the way, and in recent years even found a connection to Quantum
Physics, where distance between consecutive Reimann Zeros is found to mimic
distance between energy levels in atoms! Was this coincidence or some
universal pattern? Being able to predict next number in moments of Reimann
Zeta function by observed energy level of atoms confirmed this as a tangible
connection. Not only is the universe stranger than we imagine, but it is also
stranger than we can imagine!
Longevity of Reimann Hypothesis has given it an enigmatic status. Some mathematicians take this as challenge of ultimate proportion. Others
shy away knowing that it may as well be the end of their careers. Many proofs in
other areas of mathematics depend on Reimann Hypothesis being true. Will we ever
prove this almost 200 hundred year old hypothesis? There were 300 million zeros
discovered which follow Reimann Hypothesis – as of book’s writing in 2003 – and
10 trillion as of writing of this review, giving overwhelming evidence about its
truth, but alas no solid conceptual proof exists yet (how Reimann Zeros look?). It’s even proved that
there are many more zeros within an arbitrarily small band around critical line
than outside, but that doesn’t prove that all zeros are on that line. It’s even
proved that there are infinite zeros on the Reimann’s critical line, but that still
doesn’t prove that all zeros are on that line.
Godel’s incompleteness theorem says that we cannot prove
that same set of math axioms won’t lead to two different answers (contradictions),
and there are going to be some true statements which cannot be proven. It’s
amazing how mathematical logic can even prove such abstract generalized
statements! A possible consequence of this is that Reimann’s Hypothesis may be
one of those statements which cannot be proven. But mathematicians believe in
elegance of solution and haven’t yet surrendered that such important hypothesis
won’t have a proof.
A person like me who doesn't understand Reimaan Hypothesis, has no comprehension of its importance or role in the world, cannot even being to understand even after trying - and I did try, during and after this book, and still ended the book feeling the sadness one feels when one departs from a lovable character at the end of the book: feeling of parting the long journey and deep melancholic awareness that next chapter of that story may never come within my lifetime. This is the power of well told story written by Marcus du Sautoy.
***
Below are some noteworthy facts about Prime numbers and Reimann hypotheses which have nothing to do with the book review but elicited aha for me and documented just for fun.
- Number of primes up to number N is approximately 1/ln(2) + 1/ln(3) + … + 1/ln(N)
- Harmonic series, 1/n, is called so because pot filled with water up to 1, 1/2, 1/3, … levels produce harmonious sound but not at other levels!
- Mathematicians have created a 26 variable equation to generate all primes. This works when results are positive numbers, but most often, they are not.
- AT&T played key role in mathematical research, even as telephone company they had little applications for mathematics, but it was not purely altruistic behaviour towards science & humanity, but a commercial and public relation consideration since they had restrictions on how much profit to absorb because they were a monopoly.
- How to test if a number is Prime, without necessarily trying out all divisors? X^p=X(modulo p) only if p is Prime for any X. This is important to find keys for Public Key Cryptography.