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This is a highly stimulating novel about the
mathematician as artist, following the trend laid down by Simon Singh's
'Fermat's Last Theorem'. And it is attractive to watch artists suffering,
as Petros Anargyros does here. For Petros dares solve Goldbach's
Conjecture...
This Conjecture states that "EVERY EVEN
NUMBER GREATER THAN TWO IS THE SUM OF TWO PRIMES". This is an
apparently simple mathematical problem, yet it has remained unsolved for two
hundred and fifty years. The delight of this novel is that it seems to go
over all the options in layman's terms. For a student of Cultural
Studies, this conjecture is especially attractive, since one of our founding
principles is articulation - making bridges between two separate entities,
crossing barriers, making links. So, to me, it seemed a mistake that
Petros should write all prime numbers as horizontal lines, since this
representation denies the dynamic in the formation of any odd number - that of
two even numbers being bridged by the integer of 1, so that it could form an
upward arrow head ^ (it could also form a downward arrow head, but that looks
too negative, and I haven't got that symbol on my keyboard). Doxiadis has
chosen an excellent problem, in my view, for a mathematical novel, since Goldbach's
Conjecture is a literary quandary as well as a numerical one.
I
do have some literary complaint about Doxiadis though: he makes Petros more
romantic than his successful peers, and the narrator writes his account in the
style of a math paper.
It may as well be a cryptic crossword clue (very
apt in the case of one of the mathematicians Doxiadis mentions), with the
answer lying in the body of the question. This novel certainly makes you
want go out and try and prove Goldbach's Conjecture - you'll wake up in the
middle of the night, thinking about it.
At first glance, it seems very appropriate that 2
(the only even prime), is mentioned in the conjecture. After all, it's
common sense that there can only even be one even prime. If there was an
even prime number larger than 2, then it could be divided by 2, and therefore
it could not have been prime in the first place. The fact that there is
no even prime larger than 2 goes very much in favour of Goldbach's Conjecture,
since this discounts a possible exception.
Whenever I do a review of a novel, I'm also
reviewing the novelist, trying to find out what makes them tick. I think
that it's suitable to apply the same methods to Goldbach. Who was
he? Why did he use these particular words? My task is slightly more
difficult here, since Goldbach didn't write his conjecture in English, and
adapted it from the time that he first mentioned it in a letter to Euler.
But the grammar of the conjecture still gives us clues, and insights into his
mind. Part of the conjecture is that "EVERY EVEN NUMBER IS THE SUM
OF TWO PRIMES" with 2 being an exception, since 1 isn't a prime.
This is true for 2+2=4, 3+3=6, 5+3=8, 7+3=10, 7+5=12. However, for even
number 14, you can have both 11+3=14, or 7+7=14. This trend for having
more than a single set of primes possible of creating an even number
continues beyond 14. This is probably what lies at the heart of
Goldbach's Conjecture: he speculated that the bigger the even number, the
more likely it would be that he was correct, since the number of chances
would multiply. In this way, it seems, Goldbach's conjecture is very much
dependent on the 'unnatural' operation of multiplication. He and Euler
both knew and wrote about Fermat's work on chance. For Goldbach, this
meant that he could quite happily make his conjecture, because the chances of
his being proved wrong were so small, (probably smaller than the chance of
someone winning the lottery and being killed by a meteorite on the same
night!).
However, this is too simplistic. It is not
true to say that the higher the even number, the higher the number of pairs of
primes. Look at these:
10,0028=has 627 pairs of prime numbers under Goldbach
10,0030=972 pairs
10,0032=121 pairs
10,0034=670 pairs
10,0036=594 pairs
10,0038=119 pairs
10,0040=815 pairs
So, this is where some doubt comes in to play, since it
does not follow that the biggest even number ever created will have the largest
number of prime number pairs in its construction. However, there does
seem to be a pattern in the allocation of pairs.
This is important, because it shows that prime
numbers are not just random occurrences. The unwritten principle behind
the Eratosthenes' Sieve which Petros uses is that prime numbers are
predictable. Petros also mentions that Euclid proved that prime numbers
are infinite. I want to go further and say that ALL PRIME NUMBERS ARE
INFINITELY PREDICTABLE - which is probably why Doug Lenat's AM programme found
Goldbach's Conjecture so boring. The one big problem with Goldbach's Conjecture
is that there seems to be no magic formula with which to prove it, with the
only way forward seeming to be to check every even number in existence, to see
if Goldbach is right. But there's a never-ending amount of even
numbers...
This is where Kurt Godel comes into Petros'
story. His 'Incompleteness Theorems' state that some current mathematical
problems can never be resolved, which is linked to his 'undecidability
theorem'. Much of the drama in Doxiadis' novel comes from this disclosure.
So, many mathematicians believe Goldbach to be unsolvable, because we are never
going to be able to check every even number. But these mathematicians
could take a lesson from literature and Doxiadis. There is left a certain
element of doubt at the end of 'Uncle Petros', but there is also definite
closure. What I'm saying is that it is not the numbers which are
uncertain: it is we who have doubts. This is where the human factor
comes into mathematics. If we want certainty in mathematics, then it is
up to us to create that certainty. Could we not create new axioms?
Do we want to be positive or negative?
So, I say that Goldbach's Conjecture is true
because it has been proved on the finite stage many times. I want to move
the burden of proof away from 'we don't know whether is is true' to 'we know
this to be true because it has not been proved otherwise'. Therefore,
Goldbach's Conjecture is more likely to be true than most conjectures made
today, because it has survived two hundred and fifty years without being
disproved. Fermat's Last Theorem was a similarly famous problem which
Andrew Wiles proved positively after an even longer period of time.
It is somewhat appropriate to include a time factor here, since Goldbach was an
historian, as well as a mathematician. Geometry may well tell us all we
need to know about a three dimensional sphere, but can it tell us how long that
sphere may last? (Is there a mathematics of entropy?). Time has
been an important factor in the development of maths, with some ideas lasting
thousands of years, whilst others are instantly dismissed. Time, like
maths and language, is an artificial construct which we have created in order
for us to make some sophisticated sense of the world. We start from
the finite, and head for infinity when we choose.
Now, as I've said before, Goldbach's Conjecture is
provable on the finite level. For every new even number we create, there
is a finite amount of prime number pairs with which to construct it. After
all, all we're really interested is in the last two digits of this new even
number at most. If we go back to the very beginning, we can see the bare
bones of how the smallest even numbers are made, e.g. 11+11=22, 7+7=14,
3+3=6, 3+5=8, 13+7=20. From this, we can see that it is possible to
narrow down the amount of prime number pairs which will go into the
creation of an even number, since some pairs are more essential than
others. For instance, any even number ending in '0', must have a pair of
primes whose final digits are '3' and '7'. There always seems to be a
balancing act in this system, e.g. 17+3=20, 27+3=30, 37+3=40, 47+3=50, 57+3=60,
67+3=70. However, once we get to 77, we have a problem, since 77 is not a
prime number. Here, the two primes compensate for this, and become
67+13=80, with each still having '3' and '7' as their final digits. The
relationship between the two primes is very flexible.
Goldbach's Conjecture has already tried to deceive
us by writing of 'THE SUM' (singular), whereas we know that there are usually
more than one set of prime numbers involved in the creation of an even number
(another possible reason for Goldbach starting off at 2, since the lower even
numbers can only be created by a single pair). I think it deceives us
further by making us pool all our efforts into discerning how even numbers are
created, without even thinking about how prime numbers are constituted
also. So, here's another one of my conjectures: EVERY TIME AN EVEN
NUMBER IS CREATED, A NEW PRIME NUMBER IS CREATED ALSO. Basically, what
I'm saying here is that even numbers are crucial in the creation of prime
numbers, and that EVERY CONSECUTIVE EVEN NUMBER FORMS PART OF A PRIME NUMBER -
you can't have one without t'other. They're at it like rabbits, are prime
and even numbers, the perfect reproduction system, unhampered by acts of God -
which is just as well, if you want multiplication to work. So, if the new
even number is 'E' then a previous prime is 'p', and the sum of this pair is
the new Prime 'P':
E+p=P.
However, I suspect that the reproductive system of numbers has the same potential to be controlled and regulated by humans, as that of rabbits is. Or, in other words, the destiny of numbers is in our hands. We can be as certain, or as uncertain of numbers as we collectively decide to be. There is a choice for mathematicians as there is for novelists: shall we create a closure for our narrative, or shall we leave the end open? Which would be more thrilling for our audience?
Okay, let's go back a long way. Euclid
proved that prime numbers are infinite. It follows to reason then, that
even numbers are infinite also. Therefore, THERE IS AN EQUAL AMOUNT
OF PRIME AND EVEN NUMBERS. "What's that?" you say.
"An equal number of prime and even numbers? Don't you mean odd,
rather than prime?" At first glance, to say that there is an equal
number of prime and even numbers is ridiculous, because if you look at the
number 100, there are 50 evens within it, and 25 primes. The point
is, I'm using my imagination here to take a leap in the dark - if every time a
new even number is created, a new prime is created also, then there must be the
same number of primes as evens, stretching off into eternity,
never-ending. Therefore, if we want to create meaning, we must create
closure, we must erect an arbitrary barrier - 'thus far, and no farther'.
Within a closed system of whole numbers, much more productive work can be
done. There can still be great fun in trying to transgress the border, to
see if we can go beyond this a new 'zero', if we want to.
So, what I'm suggesting is that we try to
replicate the effect of infinity on a finite scale. Our last integer ('F'
for 'le fin', 'the finish') will be a prime number which can grammatically
be divided into 2 - the equal numbers of primes ('p') and the same
numbers of evens ('e'), where 'F' is in two places at once, the final even
number ('e') and the final prime number ('p'): e+p=F. Note that 'F' does
not equal the sum of 'e' and 'p' - what we're recognising here is that 'e' and
'p' are of equal significance.
32+32=131, where e=64, and p=131
33+33=137, where e=66, and p=137
34+34=139, where e=68, and p=139
35+35=149, where e=70, and p=149
36+36=151, where e=72, and p=151
72+151=151
72+x=151
x+151=151
Here, if 'F' equals 131 (that is, the 32nd prime),
then 'e' equals 64 (the 32nd even number). 'F' always has to be a prime
number, because if 'F' was an even number, then we couldn't arrive an
equal number of primes and evens. So, what we are saying is that if we
arbitrarily assume that 151 is the last number ever ('F'), then Goldbach's
conjecture is correct, because it had already been proved up to the even number
72. We can now create the positive statement, "where F=151,
Goldbach's Conjecture is true". So, what mathematicians can now do
is to conclusively prove conjectures and theorems such as Goldbach's by trying
to replicate 'Infinity' on a smaller scale, to create 'F', to create certainty,
to create closure.
Kevin Patrick Mahoney
|
Visit
our Apostolos
Doxiadis page, for Apostolos Doxiadis biography, Apostolos Doxiadis
bibliography, Apostolos Doxiadis articles, and a “Uncle Petros and Goldbach’s
Conjecture” reading guide |
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