number theory and time

"I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in regard to time, which like space may be in essence poly-dimensional and that this and other such sort sort of truths would become self-evident to a being whose mode of perception is according to superficially as opposed to our own limitation to linearly extended time."

J.J. Sylvester, from "On certain inequalities relating to prime numbers", Nature 38 (1888) 259-262, and reproduced in Collected Mathematical Papers, Volume 4, page 600 (Chelsea, New York, 1973)
Translated into contemporary English, Sylvester is saying more-or-less this:

"I have sometimes thought that if we were able to perceive time in some multi-dimensional way, more like a surface than like a line, then perhaps the distribution of prime numbers would be entirely self-evident, and would not seem at all mysterious to us."    [a curious aside]

This is the earliest printed statement I am aware of which attempts to link the nature of time (or our perception of it) with the mysteries surrounding the distribution of primes.

However, there are a few other items of interest in this general direction:

The work of Michel Planat of the Laboratoire de Physique et Métrologie des Oscillateurs du CNRS concerns the measurement of time and of frequencies, and has led to some fascinating connections with number theory in general and the Riemann hypothesis in particular:

M. Planat, "1/f noise, the measurement of time and number theory", Fluctuation and Noise Letters 0 (2001).

[abstract:] "Time and frequency measurements of a high frequency oscillator need the comparison to a reference oscillator: the physical units of the measurement are the integers and the relevant approach is analytical number theory. We show this in the context of the moon-sun calendar discovered in ancient Greece and in the context of a communication receiver. It is shown that the resets in time measurements are governed by continued fraction expansions and that their low frequency statistics connects to prime number theory. A link between Riemann hypothesis and 1/f noise arises in this context."

The following is to be published in the proceedings of the recent conference "The Nature of Time: Geometry, Physics & Perception" (Tatranská Lomnica, Slovak Republic, May 2002):

M. Planat, "Time measurements, 1/f noise of the oscillators and algebraic numbers"

"Many complex systems from physics, biology, society...exhibit a 1/f power spectrum in their time variability so that it is tempting to regard 1/f noise as a unifying principle in the study of time. The principle may be useful in reconciling two opposite views of time, the cyclic and the linear one, the philosophic view of eternity as opposed to that of time and death. The temporal experience of such complex systems may only be obtained thanks to clocks which are continuously or occasionally slaved. Here time is discrete with a unit equal to the averaging time of each experience. Its structure is reflected into the measured arithmetical sequence. They are resets in the frequencies and couplings of the clocks, like in any human made calendar. The statistics of the resets shows about constant variability whatever the averaging time: this is characteristic of the flicker (1/f) noise. In a number of electronic experiments we related the variability in the oscillators to number theory, and time to prime numbers. In such a context, time (and 1/f noise) has to do with Riemann's hypothesis that all zeros of the Riemann zeta function are located on the critical line, a mathematical conjecture still open after 150 years."

M. Planat, "On the cyclotomic quantum algebra of time perception" (preprint 03/04)

[abstract:] "I develop the idea that time perception is the quantum counterpart to time measurement. Phase-locking and prime number theory were proposed as the unifying concepts for understanding the optimal synchronization of clocks and their 1/f frequency noise. Time perception is shown to depend on the thermodynamics of a quantum algebra of number and phase operators already proposed for quantum computational tasks, and to evolve according to a Hamiltonian mimicking Fechner's law. The mathematics is Bost and Connes quantum model for prime numbers. The picture that emerges is a unique perception state above a critical temperature and plenty of them allowed below, which are parametrized by the symmetry group for the primitive roots of unity. Squeezing of phase fluctuations close to the phase transition temperature may play a role in memory encoding and conscious activity."

1/f noise plays a significant role in the work of Planat and his various collaborators, and appears to be a crucial ingredient in an emerging understanding of time.

some more publications by Planat,

This is an excerpt from I.V. Volovich, "Number theory as the ultimate physical theory" which hints that time might have a p-adic or adelic character:

"If K is a field, it either contains the field of rational numbers Q as a subfield (in this case it is said to have characteristic 0) or the field Fp for some prime number p (in this case K is said to have characteristic p). So, there are two simplest fields: Q and Fp.

Some fields have an additional property of being 'ordered' according to a relationship of 'greater than' or 'less than'. For example the field of rational numbers and the field of real numbers are ordered. This order is customarily expressed in terms of the relation a < b (a is less than b). The relation a < b means that the difference b-a is a positive number. Consequently, every property of the relation a < b can be derived from properties of the class of positive numbers. A field K is said to be ordered if it contains a set P of 'positive' elements with the additive, multiplicative and trichotomic properties. The real numbers can be described as a complete ordered field. But their is no possible definition of a 'positive complex number' that would make the field of complex numbers an ordered field. Moreover, the field of p-adic numbers and the field of characteristic p are not ordered. Nevertheless it is sometimes useful to introduce a partial order for the field of p-adic numbers (see [27]).

Note that the notion of the arrow of time is closely connected with the order of the field of real numbers. Usually, one considers time as the real axis. but in such a case it is difficult to understand the appearance of the arrow of time (see an interesting discussion of this problem in [28] and [29]), because the field of real numbers has a natural order. It seems that only if we try to represent time as a non-ordered field do we get a possibility to understand the appearance of the arrow of time."

[27] W.H. Schikhof, "Non-archimedean monotone functions", Report 7916, Mathematisch Instituut, Nijmegen, The Netherlands (1979)

[28] R. Penrose in Quantum Gravity 2, eds. C.J. Isham, R. Penrose and D.W. Sciama (Clarendon, 1981)

[29] S.W. Hawking, "Quantum cosmology", in 300 Years of Gravity (Cambridge U.P., 1981)

There are also these:

O. Sotolongo-Costa and J. San-Martin, "p-adic properties of time in the Bernoulli map", Apeiron 10 No. 3 (2003)

[abstract:] "The Bernoulli Map is analyzed with an ultrametric approach, showing the adequacy of the non-Archimedean metric to describe in a simple and direct way the chaotic properties of this map. Lyapunov exponent and Kolmogorov entropy appear to yield a better understanding. In this way, a p-adic time emerges as a natural consequence of the ultrametric properties of the map."

Y. Meurice, "Quantum mechanics with p-adic time", Preprint CINVESTAV-FIS-12-89 (1989) 1-14

B. Dragovich presented a talk at the Russian Interdisciplinary Temporology Seminar in the spring of 2000 entitled "Real, p-adic and adelic time":

"It is well known that result of any measurement, including measurement of time, is represented by rational numbers. Starting from the set of rational numbers one can obtain real and p-adic numbers, which can unify with the help of adeles. Since 1987, many physical models have been constructed, whose space and time can be described not only by real numbers but also with p-adic numbers and adeles. One can expect that p-adic and adelic time play the essential role at the Planck scale (t ~ 10-44 sec), and also can play a significant role in description of variation of some very complex natural systems. Study of p-adic and adelic time gives possibility to get some knowledge not only on the new forms of time realization, but also on better understanding of the concept of standard (real) time itself. Nowadays, it seems that time, as well as space and matter, are in fact adelic, but under some conditions may be described by real or p-adic numbers only."

It has been observed that the Riemann Hypothesis is ultimately about the relationship between addition and multiplication. Our inability to prove it could perhaps be seen as the result of some failure to fully understand this relationship:

"The Riemann Hypothesis is the most basic connection between addition and multiplication that there is, so I think of it in the simplest terms as something really basic that we don't understand about the link between addition and multiplication."

B. Conrey, quoted in K. Sabbagh, Dr. Riemann's Zeros (Atlantic, 2003) p.160

"[The Riemann Hypothesis] is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication. It's a gaping hole in our understanding, because until we really understand it we cannot say that we understand the line. Even the line is still extraordinarily mysterious."

A. Connes, quoted in K. Sabbagh, Dr. Riemann's Zeros (Atlantic, 2003) p.208

In his novel Uncle Petros and Goldbach's Conjecture (Faber, 2000), the author A. Doxiadix, makes the following point through his fictional mathematician character Petros (p.184):

"'Multiplication is unnatural in the same sense as addition is natural. It is a contrived, second-order concept, no more really than a series of additions of equal elements. 3 x 5, for example, is nothing more than 5 + 5 + 5'...'If multiplication is unnatural,' he continued, 'more so is the concept of 'prime number' that springs directly from it. The extreme difficulty of the basic problems related to the primes is in fact a direct outcome of this. The reason there is no visible pattern in their distribution is that the very notion of multiplication - and thus of primes - is unnecessarily complex. This is the basic premise...'"

Finally, Louis Kauffman writes in "Virtual Logic - Formal Arithmetic" (from Cybernetics & Human Knowing 7 no. 4 (2000) p. 93):

"Multiplication is more complex [than addition]. When we muliply 2 x 3 we either take two threes and add them together, or we take 3 twos and add these together. In either case we make an operator out of one number and use this operator to reproduce copies of the other number."

This all seems to be suggesting that the primes, and the difficulty of the Riemann Hypothesis (which concerns their 'deep' behaviour) are ultimately the result of the uneasy relationship between the operations of addition and multiplication. We shall consider briefly the nature of this relationship. I'm sure someone has written something far more profound and erudite on this matter than what is to follow (please let me know!), but I think it gets the general direction of thought across. I am grasping at something rather elusive here.

Imagine a number line marked out on the ground, with consecutive integers separated by about a pace. You are standing at 0. We are going imagine 'acting out' the operations of addition and multiplication.

First consider "4 + 7". To act this out, you walk four steps in the positive direction (bringing you to 4), then seven more steps in the same direction, bringing you to 11. This is a simple demonstration of the fact that 4 + 7 = 11.

Now consider "4 x 7". To act this out, you must walk four steps in the positive direction seven times

The immediate reality is that multiplication isn't as simple as addition. In fact it's a sort of iteration or compounding of additions. 4 x 7 = 7 + 7 + 7 + 7 so the 4 and 7 are playing different roles: the 7 is the thing being added, the 4 is the number of times it is being added. Despite the fact the symmetry of "4 x 7 = 7 x 4" there's an asymmetry in there somewhere, and it's something to do with time. When we consider 4 + 7, we find complete symmetry in that 4 and 7 are both numbers being added to something. Their roles are identical.

People still generally prefer to say "seven times twelve" (rather than "seven multiplied by twelve"). This makes sense if we remind ourselves that

84 = 7 x 12 = 12 + 12 + 12 + 12 + 12 + 12 + 12

12 added to itself 7 times. In slightly more outdated language, this is "7 times 12".


84 = 12 x 7 = 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7

or twelve times seven.

English schoolchildren informally call the multiplication tables they are encouraged to memorise "times tables".

Another way we can make sense of 7 x 12 is to make 7 copies of the number line, mark dots on the first 12 dots in each, and count the dots in the rectangular array that this produces. This gives an impression of the way multiplication relates to the mathematics concerning areas. Again, time/repetition is involved here - You must mark out the 12 dots 7 times.

. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .

The key here is that multiplication involves the concept of repeating something a certain number of times, whether it's the act of addition, or the laying out of a row of dots. Obviously time is required in order to 'act out' both multiplication and addition as described above. In some sense, though, the relationship with time differs in a fundamental way.

Our inability to completely understand the primes is evident in our inability to prove the Riemann Hypothesis, which is described above by two eminent mathematicians as an inablility to properly understand the relationship between addition and multiplication. I am beginning to think that this inability has its roots in something akin to what Sylvester described above - an incomplete understanding/experience of time.

Recall what Connes said above:

"[The Riemann Hypothesis] is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication. It's a gaping hole in our understanding, because until we really understand it we cannot say that we understand the line. Even the line is still extraordinarily mysterious."

Western civilisation, which has developed the study of number to the point where the Riemann zeta function could be detected and the Riemann Hypothesis hypothesised, is also unique in its characterisation of time as a line - a featureless, homogeneous continuum. Other cultures tend to conceptualise and experience time as something more multi-faceted, fluid, 'organic', cyclic. Western science's increasing tendency to measure and quantify time (and everything else) led to a conceptual link being made between time and the real number line. But since the discovery of p-adic numbers and adeles, which is what Connes is alluding to here, we cannot claim to understand the line.

In other words, if time is in some sense a line, as children in history lessons have been led to believe for many years - everyone is familiar with the 'timeline' - then we cannot claim to fully understand the nature of time. The children encounter the same line in their mathematics lessons as the 'number line'. Those who end up studying mathematics at University may learn about the conceptual foundations behind this 'number line', i.e. the axioms underlying the continuum of real numbers. Despite the informed assurances of Professor Connes, they are very unlikely to be shown anything about the number line to suggest that it could be "extraordinarily mysterious". Nor would those who happened to study physics suspect this of the time continuum that their variable t is meant to be varying in.

The rise of "p-adic physics" may change this, and may ultimately change the way we understand and think about time.

Our understanding of time and our understanding of number have both shifted considerably over the centuries, and we have every reason to believe that this will continue, and perhaps even accelerate.

Because of the extent to which our concept of time has become based on the real number model, I suspect that the next big shifts in our understandings of time and of number are likely to occur simultaneously.

The appearance of type III factors in Alain Connes' groundbreaking number theory work (see for example:

J.-B. Bost and A. Connes, "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. (New Series), 1 (1995) 411-457)

suggests an interesting possibility in terms of relating number theory to concepts of time. Type III factors are noncommutative algebras with an intrinsically dynamical quality (a 'canonical flow'). As Connes himself has written, such algebras "...evolve with time!".

One curious thing struck me when learning about type III factors: Whereas a 4-dimensional hypercube may be said to 'exist' in the 'mental hyperspace' of a mathematician (no physical representation being possible), a type III factor would have to be said to exist in the 'mental hyperspacetime' of the mathematician. For it is a mathematical object which undeniably 'evolves with time'. But unlike the mathematical model of a spinning top or pendulum, this mathematical model can't ever be realised in the physical time continuum for humans to observe in the form of a physical model. The only kind of 'time' it can be evolving with is a kind of abstract 'mental time'.

Psychologists and cognitive scientists have undoubtedly carried out detailed research into 'subjective' or 'psychological time' (the extent to which it seems to speed up or slow down in certain situations, relative to 'objective' clock time, for example). However, I sense that there is more to be explored here. A type III factor could be said to be "flowing". "Where is it flowing?" we might ask. "When is it flowing?". If we consider the role which type III factors are playing in Connes' very promising approach to unravelling the mysteries of the primes (as well as in Lapidus' recent research), and bearing this in mind, go back to Sylvester's quote at the top of this page, it again seems not unreasonable to suspect that humanity will not be able to clear up the various confounding issues associated with the primes (typified by the Riemann hypothesis) until our understanding of time has been appropriately expanded or adjusted.

The sages and priests of many ancient civilisations were concerned to an extraordinary degree with time. The development of calendars for practical, agricultural purposes is understandable, but if we look at, say, the complex set of interlocking Mayan calendric systems, or the philosophical infrastructure behind the Chinese I Ching, there appears to be a much deeper, less mundane concern - the nature, the 'quality' or even the 'structure' of time as opposed to just its measurement.

Gauss described mathematics as "the queen of the sciences" and number theory as the "queen of mathematics". Could it be that number theorists - arguably the inner priesthood of today's 'scientistic' civilisation - are unconsciously laying the groundwork for a similar quest - to explore the ultimate nature/structure/meaning of time?

Incidentally, in more recent work, Connes has related the Riemann Hypothesis to a flow on a space of adele classes.

J.-F. Burnol, "The Explicit Formula and a propagator" (preprint, 09/98)

[excerpt:] "To express his positivity criterion Weil uses conventions slightly distinct from ours. He moves the local terms to be together with the poles, and makes a shift of 1/2 in the Mellin transform. In this way he gets a distribution C and translates the Riemann Hypothesis into a positivity criterion:

$C(F \star F^{\tau}) \geq 0$

for an arbitrary test-function F on C...In the function field case $C(F \star F^{\tau})$ can be given a geometric interpretation as an intersection number of cycles on an algebraic surface, and the positivity follows from the Hodge Index Theorem...

But another interpretation is possible that does not seem to have been pushed forward so far. To prove that a number is non-negative it is enough to exhibit it as the variance of a random variable. In our case this means that there should be a generalized, stationary, zero mean, stochastic process with C as "time" whose covariance would be C. That is we have a probability measure $mu$ on the distribgutions on the classes of ideles...

If such a probability measure $mu$ could be constructed, corresponding to an 'arithmetic stochastic process', then the Riemann Hypothesis would follow of occurs.

But such a generalized process could not be obtained and I now believe that this "temporal" interpretation of the idele classes and the accompanying purely probabilistic formulation of the Riemann Hypothesis are slightly misguided. We should keep the poles and the zeros together and not make the shift by 1/2."

J.-F. Burnol, "An adelic causality problem related to abelian L-functions" J. Number Theory 87 Série I (2001) 423-428

In this paper, Burnol uses a Lax-Phillips scattering framework to reveal "a natural formulation of the Riemann Hypothesis, simultaneously for all L-functions, as a property of causality."

P. Riot and A. Le Méhauté, "From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis" (preprint 10/2017)

[abstract:] "The novelty of the Jean Pierre Badiali last scientific works stems to a quantum approach based on both (i) a return to the notion of trajectories (Feynman paths) and (ii) an irreversibility of the quantum transitions. These iconoclastic choices find again the Hilbertian and the von Neumann algebraic point of view by dealing statistics over loops. This approach confers an external thermodynamic origin to the notion of a quantum unit of time (Rovelli Connes' thermal time). This notion, basis for quantization, appears herein as a mere criterion of parting between the quantum regime and the thermodynamic regime. The purpose of this note is to unfold the content of the last five years of scientific exchanges aiming to link in a coherent scheme the Jean Pierre's choices and works, and the works of the authors of this note based on hyperbolic geodesics and the associated role of Riemann zeta functions. While these options do not unveil any contradictions, nevertheless they give birth to an intrinsic arrow of time different from the thermal time. The question of the physical meaning of Riemann hypothesis as the basis of quantum mechanics, which was at the heart of our last exchanges, is the backbone of this note."

T.H. Ray, "Self organization in real and complex analysis" (preprint, 2006)

[abstract:] "We identify specific properties of the complex plane that allow functions of a continuous n-dimensional (Hilbert) measure space to be transformed into a well ordered counting sequence. We discuss proof strategies for problems in number theory (Goldbach Conjecture) and topology (Poincaré Conjecture) that suggest correspondence between the physical principle of least action and the mathematical concept of well ordering. The result implies a deeply organic connection between physics and mathematics."

K. Scharff's has created an extensive bibliography of literature concerning alternative models of time and causality. As regards the issue of time and number, he suggests the following:

T.L. Hanking, "Algebra as pure time", from Motion and Time, Space and Matter (ed. Machamer, 1976)

D. Lohmer, "On the relation of mathematical objects to time", Journal of Indian Philosophical Reserach 10 (3) 1993

W.R. Hamilton, "Theory of Conjugate Functions or Algebraic Couples with a Preliminary and elementary Essay on Algebra as the Science of Pure Time", Transactions of the Royal Irish Academy 17 (1837)

Here is an excerpt from pp.3-5 of the introduction. Note that while Hamilton is talking about algebra and not number theory, the essay which follows this intoduction considers the basic arithmetic operations, ratios, reciprocals, roots, but not the more recent ideas we now associate with the word 'algebra' (groups, rings, fields, cosets, modules, ideals, etc.). Remember that this was 1837. It is particularly interesting to read this bearing in mind the discoveries of Connes, discussed above (von Neumann algebras, type III factors, etc. with canonical flows).

"...The History of Algebraic Science shows that the most remarkable discoveries in it have been made, either expressly through the medium of that notion of Time, or through the closely connected (and in some sort coincident) notion of Continuous Progression. It is the genius of Algebra to consider what it reasons on as flowing, as it was the genius of Geometry to consider what it reasoned on as fixed. EUCLID defined a tangent to a circle, APOLLONIUS conceived a tangent to an ellipse, as an indefinite straight line which had only one point in common with the curve; they looked upon the line and curve not as nascent or growing, but as already constructed and existing in space; they studied them as formed and fixed, they compared the one with the other, and the proved exclusion of any second common point was to them the essential property, the constitutive character of the tangent. The Newtonian Method of Tangents rests on another principle; it regards the curve and line not as already formed and fixed, but rather as nascent, or in process of generation: and employs, as its primary conception, the thought of a flowing point. And, generally, the revolution which NEWTON made in the higher parts of both pure and applied Algebra was founded on the notion of fluxion, which involves the notion of Time.

Before the age of NEWTON, another great revolution, in Algebra as well as in Arithmetic, had been made by the invention of Logarithms; and the "Canon Mirificus" attests that NAPIER deduced that invention, not (as it is commonly said) from the arithmetical properties of powers of numbers, but from the contemplation of a Continuous Progression; in describing which, he speaks expressly of Fluxions, Velocities and Times.

In a more modern age, LAGRANGE, in the Philological spirit, sought to reduce the Theory of Fluxions to a system of operations upon symbols, analogous to the earliest symbolic operations of Algebra, and professed to reject the notion of time as foreign to such a system; yet admitted that fluxions might be considered only as the velocities with which magnitudes vary, and that in so considering them, abstraction might be made of every mechanical idea. And in one of his own most important reseraches in pure Algebra, (the investigation of limits between which the sum of any number of terms in TAYLOR's Series is comprised,) LAGRANGE employs the coneption of continuous progression to show that a certain variable quantity may be made as small as can be desired. And not to dwell on the beautiful discoveries made by the same great mathematician, in the theory of singular primitives of equations, and in the algebraical dynamics of the heavens, through an extension of the conception of variability, (that is, in fact of flowingness,) to quantities which had before been viewed as fixed or constant, it may suffice for the present to observe that LAGRANGE considered Algebra to be the Science of Functions, and that it is not easy to conceive a clearer or juster idea of a Function in this Science, than by regarding its essence as consisting in a Law connecting Change with Change. But where Change and Progression are, there is TIME.


That a moment of time respecting which we inquire, as compared with a moment which we know, must either coincide with or precede or follow it, is an intuitive truth, as certain, as clear, and as unempirical as this, that no two straight lines can comprehend an area. The notion or intuition of ORDER IN TIME is not less but more deep-seated in the human mind, than the notion of intuition of ORDER IN SPACE; and a mathematical Science may be founded on the former, as pure and as demonstrative as the science founded on the latter. There is something mysterious and transcendent involved in the idea of Time; but there is also something definite and clear..."

Conference "The Nature of Time: Geometry, Physics & Perception" (Tatranská Lomnica, Slovak Republic, May 2002):

The Jungian scholar Marie-Louise von Franz wrote most interestingly on the interrelation of time- and number-based concepts. If she had been able to study the essentials of number theory, I suspect this particular line of her research might have led in some very exciting directions. The best starting point for exploring this work is her book Number and Time.

"As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to such an extent that it can legitimately be asked whether the subject of study of arithmetic is not the human mind itself. From this a strange fascination arises: how can it be that these numbers, which lie so deeply within ourselves, also give rise to such formidable enigmas? Among all these mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant."

G. Tenenbaum and M. Mendès France, from The Prime Numbers and Their Distribution (AMS, 2000) p.1

number theory, time and ancient Chinese aesthetics

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