Tuesday, 8 March 2016

Great Discovery: But Most People Seem To Miss What This Actually Says About Mathematics

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.