If, like
me, you're a bit of a numbers nerd, you'll have been excited by the recent news
that a new largest
ever prime number has been discovered - one that exceeds 22 million digits.
The operative word here is 'discovered', and the article explains how:
"The search for new and bigger
prime numbers is conducted using software developed by the GIMPS team, called
prime95—it grinds away, day after day, until a new prime number is found."
It may
be a subtle point that's so often missed, but the fact that mathematics has
unknown properties that we need to discover empirically ought to tell us an
awful lot about the primary nature of mathematics - a primacy, as I argued last
July in
this blog post, that indicates
that even the entire physics of our universe is only a tiny representation of
mathematics as a whole. Or to put it in terms your grandma would understand, not
only is mathematics more than just a human invention, it actually gives every
indication that its truths exist over and above anything that exists in the
physical universe.
As
well as being an exciting finding in the field of mathematics, the discovery of
this new prime number also, to me, serves as a subtle reminder that mathematics is both
a creation and a discovery, and that it's in David Hume that we can see the astounding way that this plays out philosophically (as
I write about here in this blog post - Why
We Never Have, & Never Will, Predict Anything New). Some of you may be
inclined to read it later if you have time. Some of you may not. For the latter
group, to illustrate the point in that blog, I said that if you find one day
that a fundamental law of nature gets changed (such as we no longer can have a
magnetic field through the application of electricity), the only way to discover
this would be through experience.
The
reason being, we are learning something new about nature by the process of
discovery. As I will expound below, it is with prime numbers we can see how we
are doing the same thing with numbers - a fact which enables us to understand
just how complex and other-worldly mathematics actually is. It is here that we
can see the important difference between physical discoveries and mathematical
discoveries. With physical properties there comes a point when there is no
longer anything new to discover about them. Suppose you have a plank of wood in
front of you. Once you try to discover new things about the wood you find that
it is made up of smaller constituents: atoms, electrons, protons, and perhaps
even small elements as yet undiscovered. But there will come a point when there
is nothing more we can discover about the wood - we will be left with the bare
bones of mathematics.
Things
are very different, though, with numbers - we never reach the point at which we
have discovered everything about them - and prime numbers are a wonderful
illustration of this. Given that almost every fact about numbers is not known
by humans, it’s pretty obvious that they are far more than mere human
inventions (think about it: by its very nature, mathematics is so infinitely complex
that however much we know about numbers we will still only know a tiny fraction
of all the possible things there is to know). How can anyone expect us to
believe that we invented something that we don’t know most things about? Prime numbers give us a great indication that
our construction of integer symbols is the construction of a map that relates
to a much wider territory of undiscovered mathematics.
In
1989 mathematicians discovered that between the 2 consecutive prime numbers
90874329411493 and 90874329412297 there is an astounding 803 composite numbers.
We didn't know that fact back in 1889 or 1789, because in 1889 and in 1789 the
fact “there exists 803 composite numbers between prime numbers 90874329411493
and 90874329412297” was a fact still awaiting discovery. There would have been
a time when no one knew that wood was made up of atoms, electrons, protons, and
so forth, but the difference between wood and prime numbers is that there is guaranteed
to always be a time when no one knows about most prime numbers.
The
reason being; we discover facts about numbers in the same way that we discover
facts about nature – through experience (as per the Hume blog I linked earlier),
and many numbers are just too vast to be experienced. Prime numbers is a perfect
example of the consistency of mathematics combined with our having to deal with
the subject probabilistically as we increase in complexity.
One
famous modelling of the primes is the Riemann map, which consists of the
distribution of the primes in the shape of a staircase (showing the steps as
each prime is higher than the last). Then running a Gaussian curve through it
Riemann composed it into a sum of simple waves, which are graphed with positive
and negative discrepancies.
The complexity of the natural numbers lies not in generating the
numbers per se, but in generating true statements about those
numbers. The complexity of a set as a measure of how hard it would be to
generate that set is a good indicator. On complexity alone, generating all the
natural numbers isn't too difficult, although it is infinitely time
consuming. You could spend the rest of your life counting integers if you
like …1..2..3..4..and so on, and not much will stand in your way. But
generating all the prime numbers is harder to do because you need
more than simple bit by bit addition - you need vast division. The
checking of whether a very large natural number is actually a prime
before including it in the set is a measure of complexity that increases as the
task increases in size. If it wasn't then the discovery of a 22 million digit
prime number wouldn't make the news at all.
Let’s look further at how this relates to knowledge and
probability. We can plot the primes as a binary sequence; that is as
11101010001. In this sequence I have
plotted a "1" wherever a position in the sequence is a prime number
(you'll notice that a '1' appears in positions 1, 2, 3, 5, 7 and 11 - denoting
the primes). The Riemann hypothesis about primes states that an infinite number
of frequencies is needed to define this sequence of primes in their entirety. In
other words, he is telling us that it is not possible to use short-cutting
compression to reduce the sequence of primes to a finite form. This translates
as; an infinite amount of data is needed to specify the primes in terms of
frequencies - but this is not to be confused with the term 'infinite' which
necessitates that primes will go on and on infinitely just like the natural
numbers will. In actual fact, just as we can specify all the natural numbers
with a very short counting algorithm, we can also define all the primes using a
very short prime number generation algorithm.
The infinite number of frequencies Riemann conjectures to be
needed to specify the primes is not because they go on infinitely - it is
because the only known prime number generation algorithm that generates primes
with certainty involves the vast and lengthy task of factorising. Factorising
is a very long winded task which means working through all the possible ways a
number might be confirmed as a prime number - which basically means that after
a very large prime number in a sequence one has to check each large number that
follows it in the sequence and work out whether it can be divided by a number
other than 1 and itself, and carry on doing that until another prime number is
reached in the sequence. That is how a prime number is identified. If we are
only interested in the length of the data string then the infinite prime number
sequence can be defined with a finite amount of data. What it cannot be is specified with a finite amount of data
because no periodicities exist in this sequence that we can exploit to help us
calculate primes with certainty using any quicker method than factorising.
The way it relates to knowledge and probability is because of this
need to factorise. To predict the next prime one has to keep factorising the
numbers ahead until one strikes a number that won't factorise - but this comes
at a huge computational cost because the numbers one is targeting are very
unrepresentative compared with the numbers one has to compute and discard. Riemann
tried to stumble upon a more succinct method of calculating (predicting) primes
using the perceived organisation of their layout in the integer sequence. But
he found that this couldn't be done because purely from the patterned layout
point of view it is not possible to predict primes with certainly - one can
only predict them probabilistically (which is better than chance predictions).
As interesting as this is, more generally, I think this is
interesting as a template for knowledge in general, as least by way of analogy.
Just as prime number sequences in binary form (11101010001 ...etc) sit between
the spectrum of being a maximally disordered sequence yet retaining enough
order predictable with a range of outcomes, so too is knowledge much like that.
In fact, if you can paint yourself a fairly clear picture of what factorising
for primes is like, you’ll see that it is a beautiful illustration for
knowledge of the complex world in more generalised terms. The main difference
between numbers and information about the physical world is that with the
latter we are discovering knowledge of what we can add to our map, whereas with
the former we are discovering more about the territory to which those maps
relate.
