Using the original discussion as a template I have attempted here to provide detailed answers to many of the issues raised though a novel mathematical approach.
For the original discussion (without insertions) see Prime Evolution.
Matthew Watkins’ Web-based Number Theory and Physics Archive and its speculative twin Inexplicable Secrets of Creation – hosted by the mathematics department of Exeter University in the UK where Watkins is an honorary research fellow – have grown into a unique resource. The archive brings together work from the plurality of disciplines contributing to an as yet unnamed field of research concerned with the startling connections between number theory – particularly the Riemann Hypothesis on the distribution of the prime numbers – and the physical sciences. Watkins talks to COLLAPSE about his role in, and motivations for, catalysing and disseminating the field, about the latest developments in the search for the hypothetical ‘Riemann dynamics’, about the nature of discovery in mathematics and its academic and cultural status.
At http://www.maths.ex.ac.uk/~mwatkins/. Dr. Watkins has kindly assembled a ‘primer’ for the mathematical concepts discussed in this interview: at http://www.maths.ex.ac.uk/~mwatkins/zeta/collapseglossary.htm
COLLAPSE: The primes have perennially been hailed as ‘mysterious’. In modern mathematics this mystery has condensed around the problem which Riemann's Hypothesis concerns. We can find primes as we count along the number line, but we have no way of predicting in general where and how densely they will occur. A lack of determinable global order, then.
MATTHEW WATKINS: But first there’s a major question concerning what is meant by ‘order’. I’m often asked, is there a pattern in the primes, is there an order, but what does that mean? If you try to reformulate these questions very precisely, you’re forced to consider what it would it mean for there to be order, or a pattern. I mean, there are patterns like wallpaper patterns, where you have a block of something repeating. Well, almost by definition the primes can’t do that. But what sort of pattern could there be, what sort of order could there be? The idea that there might be a pattern, the importance of there being a pattern in the primes – these aren’t things you can rigorously pin down.
C: Couldn’t you use an information-theoretical definition of pattern?
MW: You could come up with a definition, one of an endless number of possible definitions, from information theory or some related discipline, of what a pattern is, and then apply it. But I think there’s still the basic fact that when people who aren’t familiar with any of those definitions are asking ‘is there a pattern?’ they don’t mean anything that such a definition could capture, they mean something else – they don’t really know quite what, but it seems important to them that there should be; whether or not there is a pattern in the primes they see as an important question. And it struck me when I was thinking about this that it’s more feeling-based, it’s not a rational question they’re posing. You can try to construct rational questions around it. People have, and such questions have given rise to a large part of that body of work we call number theory.
C: But initially it’s more like the expression of an instinct for pattern recognition?
MW: Perhaps. As Jung said – almost as the culmination of his work on archetypes – the set of positive integers, taken as a whole, corresponds to the archetype of order. So, in a sense, all notions of order, of something coming before something else, of things being in a sequence, all of that ultimately can be linked back to our instinctive grasping of there being a number system underlying our experience. Now that number system turns out to have embedded within it an enigma, a problem bordering on the paradoxical: is there order in the way this thing’s put together or not? We feel there should be, but we aren’t entirely sure how to ask the question – basically, we don’t really know. So we start by asking whether there’s order in the number system, and the unintended result of our probing into this matter is that what we ultimately mean by order in any sense gets indirectly thrown into question. People also frequently ask about the existence of a formula – is there a formula for the prime numbers? Well, again, that’s difficult because, yes there is, there’s the Riemann-von Mangoldt explicit formula, which effectively generates exactly the distribution of prime numbers as ‘output’ – but you need the complete set of Riemann zeros as input. This is an infinite set, and to produce it you effectively need the complete set of prime numbers, so there’s a circularity. So it’s a formula, but not the kind of formula which people who ask this question have in mind. There are also algorithms – rigorous procedures – which can systematically generate the primes. One could arguably call these ‘formulas’ , but they’re basically methods of computation, and the computations quickly become intractably huge…so we’re not talking about anything that can systematically spit out primes one after another in the sense that people might have in mind when they ask about the existence of a formula.
C: As the years have gone on, mathematicians’ ingenuity and the employment of new technologies have seen an acceleration in the conquest of the critical line of Riemann zeta zeros. But does the fact we’ve got, say, one billion of the zeros make it any less mysterious than when we had a hundred? Does the apparent success of the Riemann Hypothesis (RH) militate against the conception of the primes as mysterious?
MW: First of all, you can’t really talk about RH being ‘successful’, it’s still a hypothesis. RH doesn’t predict the primes as such, but the theory of Riemann’s zeta function, from which it emerges, allows us to understand the distribution of primes much more deeply. At the heart of this theory is the peculiar sequence of ‘zeros’ now known as ‘Riemann zeros’, ‘Riemann zeta zeros’ or sometimes just ‘zeta zeros’ – these are what RH directly concerns. What’s happened really is that RH has displaced the mystery. The primes are no longer mysterious, you could argue – we now know that they are exactly governed. Initially, it was found that they’re governed by a logarithmic distribution, a sort of gradual thinning out, in an almost statistical sense – that provides reliable but approximate information about the primes. Riemann later found that the logarithmic distribution is ‘modulated’ by an infinite set of waves, where each wavelength corresponds to one of the Riemann zeros. We’re in the realm of proven mathematical results here, and these precisely pin down the primes, so in that sense, all mystery is gone; but in actuality the mystery has been pushed back, or displaced. The mystery now is, where the hell did these Riemann zeros come from? We can calculate hundreds of billions of them, we’ve got a vast, intricate body of precise mathematical results concerning them which ultimately brings us to a big, important, question about whether they’ll all lie on the ‘critical line’ – that question is RH. But ultimately, what are they? Since the seventies, this idea that they might be vibrations of something has taken root and has now been more-or-less universally accepted, on the basis of a lot of computational evidence together with a mysterious, suggestive mathematical ‘coincidence’ involving something called the Selberg Trace Formula – and that ties in with certain unexpected connections with physics. So if we’ve got vibrations of a mysterious ‘something’ underlying the number system, in a sense the primes are no longer the mystery, the primes have been taken care of, the mystery has been displaced. The primes are our obvious way into the mystery, but ultimately it’s a mystery about the system of positive integers, about ‘order’, and arguably even about time.
Peter Collins: Firstly, I congratulate you on your Number Theory and Physics Archive, which I have found most useful and have recommended in turn to other interested correspondents.
However it is important to state at the outset that because physical and psychological reality are in dynamic terms complementary in nature, number – especially with respect to the famed zeta zeros – has an equally important connection with psychological understanding. And this aspect has been all but ignored in recent research. Indeed – because of their inherent complementarity – without addressing the psychological, it is not possible to provide a proper context for understanding the fascinating physical findings that have emerged. So hopefully I will have more to say on this matter as we proceed.
Your comments on the meaning of order with respect to the number system are well taken. However to appreciate better why this is the case we need to make the massive step of accepting that – rather than representing some abstract fixed entity frozen as it were in time and space – that number is inherently of a dynamic nature comprising the interaction of opposite polarities in experience.
So, for example, one cannot envisage the notion of a number – say 2 – representing an external object without the corresponding internal perception of that number.
Thus the actual experience of number necessarily involves a two-way dialogue involving both physical (external) and psychological (internal) aspects.
Likewise one cannot appreciate any particular number without reference to the general category of number. So from an external physical perspective we have the general number object in relation to specific numbers; then from the internal psychological perspective we have the general concept of number in relation to specific perceptions.
So inevitably our actual experience of number also involves a two-way dialogue involving the relationship of part to whole (and whole to part). And this in turn entails a two-way dialogue involving both the quantitative and qualitative aspects of number.
However, though the experience of number is dynamically interactive with respect to these two polar sets (external/internal and part/whole), in conventional mathematical interpretation the notion of number is dramatically reduced in a manner that is quite distorted.
So in rational terms, numbers are given an abstract objective validity (independent of our interaction with them). Likewise the whole in every context is reduced to the quantitative parts.
This is most obvious with the interpretation of addition (e.g. 1 + 1 = 2) where the (whole) sum is represented in a merely quantitative manner as the sum of the constituent part members.
I refer to this quantitative understanding – which defines the accepted approach – as the Type 1 (analytic) aspect of mathematics.
However, when it is explicitly accepted that the necessary interaction as between opposite poles implies their qualitative interdependence, this entails a unique holistic aspect to all mathematical symbols (with complementary physical and psychological attributes).
In direct terms this entails a refined intuitive capacity to literally “see” the interdependent relationships involved. These are then, indirectly, rationally interpreted in a (circular) paradoxical manner.
So quite remarkably, though the present Type 1 completely defines what is formally accepted as valid mathematics, an equally important Type 2 (holistic) aspect – all but unrecognised – potentially exists, applying to every symbol and relationship, which operates in an utterly distinctive manner.
Then when both Type 1 and Type 2 aspects are coherently combined, we obtain the truly comprehensive Type 3 approach that I term radial mathematics.
It has been my firm contention for some time that the Riemann zeta function (with its key related issues of the non-trivial zeros and the Riemann Hypothesis) needs at a minimum a preliminary Type 3 mathematical approach for proper interpretation.
This thereby requires showing how both the analytic (quantitative) and holistic (qualitative) aspects of number interact in a dynamic complementary manner in physical and psychological terms.
And with this more comprehensive understanding, many of the seeming intractable issues with respect to the function simply dissolve while greatly enhancing true appreciation of the inherent mystery involved.
So once again from a Type 3 perspective, the primes are no longer understood as fixed entities in an abstract manner. Rather they are viewed as truly dynamic in nature combining both analytic and holistic aspects.
The primes now therefore possess a relative identity, where quantitative notions of independence and qualitative notions of interdependence ceaselessly interact with each other.
This same dynamic approach likewise characterises the randomness of the primes, which can only be successfully viewed in a relative approximate manner.
So notions of randomness and order are now seen as complementary (where one implies the other).
We can say that the individual primes are randomly distributed. However appreciation of this fact implicitly requires the order of the natural numbers with which it is diametrically opposed.
Therefore they are as two sides to the same coin with no absolute meaning independent of each other.
Perhaps the best we can say is that the individual primes are distributed as randomly as possible that is consistent with preservation of the greatest degree of order with respect to the natural numbers. But as you rightly suggest, trying to pin these notions down in a totally precise manner is not actually possible.
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