Observations on the Distribution of the Prime Numbers

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the first question to ask is: Could the Distribution of the Primes be other than what it is?    Seemingly not.  Could a ‘universe’ exist where the basic constants of physics, such as c, the speed of light in a vacuum,  and g, the gravitational constant, were completely different? Most physicists I have talked to say yes. In any physical universe there would have to be an upper limit to the transmission of information, but there is no reason why it should be anywhere near the value of c. As for g, it has been suggested by some physicists that it actually has changed during the evolution of the universe we live in.  Could there be a universe where Hook’s Law, or Boyle’s Law or any of the other basic laws of physics was not valid? Conceivably.
The distribution of the primes is thus, in some sense, more ‘necessary’ or more fundamental than even the most basic physical constants and principles. (It has apparently been shown that not only is there no formula which will give us the distribution of the primes exactly, but that no such formula can possibly exist.)  But the distribution of the primes is not a logical law nor even a mathematical one : it is a physical law. Consider a hen laying an egg —  I assume a hen that can only lay one egg at a time. This hen carries on laying eggs indefinitely. We make copies of the egg situation at successive moments thus deriving all the natural numbers (in concrete form). They are already ordered, firstly temporally and secondly quantitatively, so there is no need for any ‘Axiom of Order’, let alone for the Axiom of the Least Upper Bound or the Axiom of Choice. We then test as to whether we can make each ordered collection of eggs into so many smaller, non-unitary, numerically equal collections. If we can,  we put such collections on the right, if we can’t we leave them on the left.   Thus   000000 goes on the right, since we can break it down into 
00 00 00, while 0000000 goes  on the left because we can only break it down into 0 0 0 0 0 0 0 . This procedure can be continued indefinitely and requires absolutely no knowledge of mathematics whatsoever. There is no ‘intelligence’ involved as such, no need to posit the existence of a supreme Mind behind the scenes. As soon as you have a ‘world’ where there are ‘little bits floating around’ you have primality and non-primality and you are landed with the distribution of the primes whether you know it or not, and whether you like it or not.

            As far as I am concerned I am very pleased about this : it is the victory of Nature, ignorant, witless Nature over human intelligence. “Pull down thy vanity, mathematician, pull down”. To judge from the writings of certain people, you would imagine that the actual distribution  of the primes in reality was a crude and misguided attempt to approach the Li function or the Riemann distribution.   And yet at the same time there is nothing special about the distribution of the primes — at any rate not to my eyes. I am sceptical about the so-called ‘beauty’ of this distribution and am convinced that no one would for a moment pay any attention to it were it not for the extraordinarily complicated mathematics that is (indirectly) involved. The curve on the graph is no different from hundreds of others, and the coloured 3-D pictures no different from thousands of other vaguely psychedelic computer simulations. Show patterns based on the distribution of the primes to an assortment of people who don’t know their origin and see whether they pick them out from the rest. I wager no one would. And, incidentally, this is not true of all shapes, all curves : there are plenty of mathematical shapes which really are inherently beautiful and fascinating, the parabola, the equiangular spiral spiral mirabilis, that Bernoulli was so enamoured of he had it inscribed on his tombstone. Even the graph of log x  is, to me, aesthetically more satisfying than the Prime Distribution.
So in a sense the prime distribution is ‘nothing special’ : it has the supreme Zen quality of being as it is and not otherwise — but then so does everything else.  We should, I think, consider what exactly we are looking for if we want a ‘reason’ for the distribution of the primes. An explanation should involve facts or principles that are much more fundamental than the fact or behaviour we want to explain. A vast amount of so-called ‘elementary’ Number Theory is based on such basic truths as A number cannot be at once odd and even or pragmatic procedures such as  the Euclidian Algorithm which is so simple (yet so powerful) that it could be carried out by a caveman using collections of twigs or pebbles.
But on what great truths is the Prime Distribution based? I see none. Only that Some numbers are prime and others are  not and they come in a certain order  which is not so much a truth as a fact of experience. It is for this reason that I have always derided any attempt at finding any  great significance in the distribution — until I stumbled across Matthew Watkins’ website.

             Seemingly, we have to recognize that there are certain mathematical assertions that may will never be ‘proved’, not for any highfaluting Gödelian reason, but simply because they are about as basic as one can go. Quite possibly Goldbach’s Conjecture (“Every even number > 4 is the sum of two odd primes”) is a case in point. Wiles’ much vaunted proof of Fermat’s Last Theorem  is only valid if one considers that the assertion on which it is based, namely the Taniyama-Shimura Conjecture, is in some sense more basic than the (apparent) fact that there are no cubic or higher order Pythagorean Triples. (Actually, it would seem to me there must be some strictly physical rather than mathematical reason for the truth of Fermat’s Last Theorem, something involving dimensionality of the real, rather than the mathematical, world.)   Also, the Distribution of the Primes is almost completely useless — if we except its recent use in codes.  No great  discoveries in physics depend on it, or seem likely to. (I am aware of the ‘chaos’ interpretation of the Prime Distribution and find this interesting but it was not observation of the Prime Distribution that gave rise to Chaos Theory.) 

            The uselessness of the Prime Distribution highly significant. As I see it, the Distribution of the Primes in itself gives us negligible knowledge about the physical world (I would say none at all) : if such knowledge really were embedded in it surely we would have got the pearl out of the oyster by now. Of course, the problem of the Prime Distribution has been a most stimulating and entertaining intellectual exercise for generations of mathematicians but that is not the point. But the  mathematics enveloping  the Prime Distribution is no more revealing of the structure of the world we live in than Mozart’s symphonies, and we don’t go to Mozart for knowledge but pleasure.

            At the same time, the interest the Prime Distribution is currently arousing (of which I was not aware until scanning Dr. Watkins’ website) is not just intellectual and aesthetic. There are articles on the distribution of the primes which view it as a ‘chaotic’ phenomenon, there is the claim that the Riemann zeta function is a generator of a vast class of functions, and, most significant of all, we have the interpretation of the zeta function as “a thermodynamic partition function defining an abstract numerical gas”. What this amounts to is that the Distribution of the Primes has a quasi-physical nature : so maybe it does have something  to tell us about reality after all.  

So what to conclude?  The only possible way forward is to suppose that the Distribution of the Primes tells us something about a deeper level of reality from which the visible and intellectual universe we know once emerged, and is still emerging.  Is the universe self-sufficient?  Self-explanatory?   It would seem not. All societies and practically all thinkers have at some stage found it necessary to appeal to some being or principle which is outside the physical universe. Newton and Kepler still believed in a supremely intelligent Creator God  and the rationalist thinkers of the Enlightenment, despite their hostility to organised religion, still needed a Prime Mover or a vague impersonal Deity. Mathematicians found themselves in a quandary when the nineteenth century brought about the death of God : they were left with a handful of equations and formulae without a supreme intelligence that produced them. And curiously, the twentieth has taken us right back to the idea of a beginning in time and a Space-Time singularity beyond which is….?  

            Most pure mathematicians, closely followed by theoretical physicists, are secret — sometimes overt —  Platonists and do indeed posit a reality beyond the material. However, they are unanimous about this ‘higher reality’ being mathematical in nature. They do not ever think for a moment that it may be professional blindness that impels  them to this conclusion. Musicians would doubtless be more attracted to the Vedic doctrine that “In the beginning was the sound”  and lovers to the idea that the universe came about through amorous play.

           I do believe that there is “an order of things of an entirely different kind lying at the foundation of the physical order”, as Schopenhauer put it, but I am equally convinced that this order is utterly unmathematical. “The Tao that can be named is not the original Tao”. Lao Tze in the fifth century BC was living in a mainly verbal, not numerical,  culture : today he would almost certainly  write “The Tao that can be numbered is not the original Tao”. Within the reality beyond this one — let us call it K0 as opposed to K1 — there would, as far as I can see, be no separability and no discreteness, no shape and no form. It would be  a domain beyond,  and prior to, plurality : the only number appropriate to it would  thus be 1 (or equivalently 0). Is there anything at all we can say about it?  A little. It is presumably ‘continuous’ which nothing in this universe actually is. The current universe must in some sense be contained within it, since otherwise nothing would come of nothing and manifestly something has. Also, surprisingly in a way, it would seem that K0 is in motion, in perpetual motion.  In the more speculative part of Dr Watkins’ website he speaks of the ancient Chinese concept of li  and views li as “essentially dynamic formations”, perhaps analogous to Newton’s mysterious ‘fluxions’ which have been completely rejected by modern mathematicians. (Newton was incidentally much interested in alchemy and mystical literature.).

            The only way I can take on board the idea that the Prime Distribution is significant and meaningful is by interpreting it as a sort of ‘frozen wave’ on the ocean that is K0. The physical world, K1, is not primary, but is a residue, an offshoot, of K1  in much the same way as David Bohm’s Explicate  Order is aan offshoot from the original Implicate Order.  However, it may be a ‘first order’ residue or offshoot and thus hold precious information about the reality that is beyond this one.

“The child is father to the man” (Wordsworth) —  a most paradoxical statement. Wordsworth presumably  meant that the child was closer to the source and therefore had a more vivid memory of what existed before birth. It may be, then, that we see in the Distribution of the Primes a relatively pure trace of what is almost (but not quite) unknowable and from which the entire physical universe has emerged. Whether true or not, this is certainly a beautiful thought and I am most grateful to Dr Watkins for introducing me to it. For what is striking is that the natural numbers, which by their discreteness and separateness, are entirely of this world and can say nothing about the beyond, but nonetheless by their distribution  perhaps point towards a reality that is the very opposite of all this since it is single, unitary, continuous and in perpetual motion.       

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