Нумерология

Дискуссия в журнале "Nature"

Nature 304, 7 July 1983, p.11 (N304p11.tif)

The temptations of numerology

Too much innocent energy is being spent on the search for numerical coincidences with physical quantities. Would that this Pythagorean energy were spent more profitably.

Ever since the Pythagoreans sought an ex-planation of their puzzling world in terms of simple numerical relationships, there has been a small but ingenious industry given over to the search for numerical relationships between supposedly fundamental quantities that specify the Universe. And the numerologists do have a few successes to their credit. Prout's hypothesis, early in the nineteenth century, that the masses of atoms relative to hydrogen would turn out to be integers when accurate measurements were carried out was for a long time discredited when accurate measurements of chlorine, for example, seemed to make such a simple rule untenable. But as things have turned out, atomic masses are indeed multiples of the nucleon mass with an appropriate allowance for what was called the"packing fraction" half a century ago. Much the same may be true of what is known as Bode's law, the rule that the distance in astronomical units from the Sun to the Solar System planets is given by a simple arithmetical series, at least as far as Neptune and if it is supposed that the asteroid belt takes the place of the planet between Mars and Jupiter supposed to be "missing". For although none of the attempts so far to account for Bode's law by the condensation of material from the solar nebula is wholly convincing, there is always a chance something may turn up.

These are not, however, the fields in which the numerology industry concentrates its efforts. With the proliferation of different particles of matter in the past several years, most of the practitioners have turned their attention to the discovery of numerical relationships between particle masses. Some have been impelled in that direction by the recognition that the reciprocal of the fine structure constant, a dimensionless quantity, is almost (but, significantly, not quite) the integer 137. More than a score of papers in this genre turn up in the Nature office each year, and then make their way back to their authors. The accompanying argument, from Mr Peter Stanbury, a factory worker from Tunbridge Wells in Kent, is a comparatively elegant piece of numerology which the author fears may be kept from the public by a stuffy establishment.

There are three kinds of reasons why numerology usually gets and more often deserves scant attention. First, if indeed there is some underlying numerical ideal, why are the relationships never exact? How can Pythagoreans be so tolerant of departures from their golden rules, in this case by several parts in a thousand? Second, sheer coincidence is by no means as unlikely as the numerologists like to think. Indeed, given that simple algebraic combinations of numbers such as π are literally infinite, the chance of being able to match an arbitrary set of numbers to within a fractionof a per cent must be high, tedious though the task may be. Third, there is the sceptical riposte "So what?". In spite of the hard work lavished on these numerical comparisons, nobody is any wiser about the way that matter is constituted even when they are successful. The pity is that such devotion is spent on such fruitless pursuits.

John Maddox

The alleged ubiquity of π

It has long been known that the proton to electron mass ratio is very nearly 6π⁵ - that is to say that m_p/m_e = 1.83615152 compared with 6π⁵ = 1.836118. What I have done is to look for and find a more general relationship between the value of π and the masses of the sub-atomic particles.

The particles of the basic octet are π⁰, π⁺, π^-, κ⁺, κ^-, κ⁰, ̅κ⁰ and η. The sum of their masses is 3.14006m_p. It can hardly escape one's attention that the multiplier of m_р is very nearly π or 3.1415926. When I first discovered this, my natural reaction was to see if anything similar occurred for baryon masses. The basic baryon octet contains the particles p, n, Λ, ε⁺, ε^-, ε⁰,Ξ⁰,Ξ^-, and the sum of their masses is 9.812 m_р, where the multiplier is significantly close to π² or 9.869604. Can one really say that both these results are coincidence?

Having observed that m_p/m_e = 6π⁵, let us do away with man-made electron volt units for mass and instead use a system in which m_p = π⁶ so that m_e = π/6. In these units, the masses of the particles become

m_p = 961.389 (= π⁶)

m_μ = 108.26

mπ⁰= 138.28

mπ^± = 143.006

mκ^± = 505.827

m_η = 562.3

m_e = 0.5235987 (π/6)

I note that

m_μ/π⁴= 1.111434 (1)

m_η/mκ^± = 1.11167 (2)

(m_η/mκ^±)/m_p= 1.1102 (3)

Relations (2) and (3) are both independent of the choice of units. Note also that 1 +π^-2 + π^-4= 1.111587.

Now we come to the really interesting part of my work. The fine-structure constant is a dimensionless quantity whose reciprocal is equal to 137.03604 and is very nearly equal to 4π³ + π² + π or 137.03630. Considering that this represents the ratio of the strength of the strong nuclear force (for which π⁰ is the main carrier) to the electromagnetic force, it must surely be of some significance that the value of mπ⁰ is π⁴+ π³ + π² or 138.286.

How does all this fit together? For a long time I could see no really simplified pattern until I found the following:

            x             y           z

A         π⁴           π²           1

В         π⁴           π³           π²

С         π⁴           π⁴           π⁴

The values in row В (column y) add up to the value of mπ⁰already quoted. The values in row A (column z) add up to 108.276, which is close to the value 108.2618 for mπ⁰. The values in row С (column x) do not add up to a particle mass, but the sum of rows В + C (columns x + y) is the same as the expression I have previously derived for πα^-1. Although row С does not produce a mass value, m_μ + mπ⁰+ mπ^± = 4π⁴.

Bearing this in mind, we note that

mπ^± / mπ⁰ = 1.03441   (A)

(mπ^±- π³)/ mπ⁰= 1.0345   (B)

mπ⁰ / (mπ^± - π³)= 1.1111²   (C)

The two values 1.0345 and 1.1115 turn up in so many of our results that I would state categorically that coincidence is ruled out. Indeed, there are many more of them, of which the following are two examples

π¹¹/m_η·mκ^± = 1.0343

[(m_μ / mπ) - mπ³] / mπ^± = 1.0346

В этой формуле (см. выше исходный скан в tif файле), похоже, есть какая-то неочевидная опечатка, если кто сообразит какая – сообщите.

Peter Stanbury

Nature 305, 1 september 1983, p.8 (N305p8.tif)

Playing games with numbers

Peter Stanbury (Nature 7 July, p. 11) presented an interesting account of the ubiquity of π in particle physics. Your accompanying comment led me to do some quick number crunching of my own. I soon discovered that, if the number of the Nature volume containing the article (304) is divided into the issue number (5291) and the resulting quotient is divided by the page number on which the article is printed (11), the resulting number, 1.771, is 99.9 per cent of the value π^½. Surely this is simply a coincidence, or is it?

R.B. Rosenberg, Department of Ophthalmology, University of Louisville, Louisville, Kentucky 40202, USA

In "The temptations of numerology" (Nature 7 July, p. 11) you ask "...why are the relationships never exact?" The answer is that sometimes they are and you said yourself that "the numerologists do have a few successes to their credit". Some famous examples, which perhaps led to modern physics and society, were:

(1)    Kirkhoff's observation in 1857 that the ratio of the electrical units was equal to the velocity of light. It was explained by Riemann in 1858 (see Max Mason and Warren Weaver, The Electromagnetic Field (1929), p.x).

(2)    In 1885 Balmer gave a formula that fitted the spectral lines of hydrogen. It was explained in 1913 by Niels Bohr, and better by Dirac and Pauli in 1926.

(3)    Kepler's third law was numerological, and was explained by Newton.

All three of these examples are of great historical and scientific interest. They were fairly accurate and also simple, and were right. Eddington's attempts, for elementary particles, were accurate at the time, forced, and wrong. There is of course such a thing as obviously bad numerology. The worst published one that I can recall appeared, believe it or not, in Nature! (185, 602; 1960).

I.J. Good, Department of Statistics, Virginia Polytechnic Instituteand State University, Blacksburg, Virginia 24061, USA

Your correspondent Peter Stanbury (Nature 7 July, p. 11) has applied considerable effort and ingenuity to numerical speculations about mass ratios of fundamental particles. However, like many amateurs in science, he has missed one important point: the significance of (inaccuracies. Specifically, he takes the sum of the masses of the lightest baryon octet. He then finds that the ratios of these mass sums to the proton mass are 3.14006 and 9.812 respectively, and suggests that these numbers are "very nearly π or 3.14159" and "close to π² or 9.86960".

Now, using the values given in the 1982 tables¹, I obtain 3.13935 ± 0.00066 and 9.81460 ± 0.00062 for these ratios; these are not as quoted by Mr Stanbury, presumably because he used older tables. However, the chief significance lies in the errors (standard deviations here). They exclude Mr Stanbury's assignments by 3.4 standard deviations for the first and no less than 89 for the second.

In general (and it is sad to have to state this in Nature) any experimental result has at least two parts: the measurement itself, and the error on it. If any theory purports to predict the result, we have an immediate indication of the worth of that theory. On this basis Mr Stanbury's theory is in error by some 90 standard deviations; it therefore fails.

To forestall one possible defence, I accept that Mr Stanbury's later relations (2) and (3) do pass this particular test; but two "hits" and two bad "misses" do not justify any theory. (Since his other results depend on the choice of mass units, I decline to discuss them here.) To forestall another, I am a professional physicist, and therefore perhaps one of that stuffy establishment so feared by Mr Stanbury. But surely it does not take years of training and experience to realize that (to illustrate) if a theory predicts x, and if experiment says 2x±0.1x, then that theory is wrong.

John F.Crawford, Schweizerisches Institut fürNuklearforschung, 5234 Villigen, Switzerland

1. Phys Lett 111B, April 1982.

Nature 306, 8 december 1983, p.530 (N306p530.tif)

Numerology

In the argument about numerology (Nature 7 July, p. 11 and 1 September, p.8), both the numerologists and their critics miss the point. Being impressed by the near match of numbers does not make science, nor is it scientific simply to stress the remaining inaccuracy and complain of the lack of sense in the whole effort. This is a simple scientific problem: is there a phenomenon to be explained, or not?

This question can only be answered by making use of a statistical model which allows us to calculate accurately the probability of matching a number by a certain type of combination of specified numbers (such as 1, e or π) with an accuracy as good as the one actually obtained. If this should turn out to be well below the usual critical limits of 5 x 10^-2 or 10^-2, the numerological result has to be taken seriously and cannot be dismissed by any non-mathematical reasoning. In cases not totally evident no numerologist can expect scientific credibility without first establishing such statistical evidence.

Marcus Gossler, Universitaetsbibliothek, A-8010 Graz, Austria

Nature 313, 14 february 1985, p.524 (N313p524.tif)

Numerology

In July 1983 (Nature 304, 11; 1983) I presented a series of numerological results concerning the masses of elementary particles. My article was accompanied by a sceptical commentary, in which you gave the view that the results are likely to be no more than coincidence and are in any case a "fruitless pursuit". Whether or not it is a fruitless pursuit is, in my view, still very much open to debate, although I doubt whether any such debate would reveal anything other than a wide variety of subjective opinion on the matter. But the real issue is, or should be, whether or not the results can be dismissed as no more than coincidence. As such, the crucial factor is the statistical evidence.

Among the results in my article were two relating to the combined mass of the particles in the basic meson octet, and the basic baryon octet, relative to the proton mass. The combined mass of the particles π⁰, π⁺, π^-, Κ⁺, Κ^-, Κ⁰, Κ⁰, and η, of the basic meson octet, is 3.13935 times the proton mass. The multiplier here is close to π, or 3.1415926. Such a close relationship clearly has a specific chance of having arisen. 3.13935 is π to the power of 0.999376, and a little elementary mathematics shows that only one random rational number in 801.2 will have such accuracy relative to any integral power of π.

The combined mass of the particles ρ, η, Λ, Σ⁺, Σ⁰, Σ^-, Ξ⁰,and Ξ^-, of the basicbaryon octet, is 9.8146 times the proton mass. The multiplier here is close to π², or 9.869604. Once again, the relationship had a specific chance of occurring. 9.8146 is π to the power of 1.995118, so only one random rational number in 102.4 will have such accuracy relative to any integral powerof π. This means that the odds are 82,000 to one against any pair of random rational numbers occurring with an overall accuracy, relative to integral powers of π, similar to that of the above two results.

Given the simple nature of the results, it seems clear that coincidence is unlikely (though not impossible). Here we have a phenomenon that deserves serious scientific attention, and not a casual dismissal in terms of coincidence. Furthermore, in view of the absence of any full physical theory of particle mass, there ought to be greater attention given to those observations that might provide useful clues pointing in the right direction. Can science really afford to ignore such results?

Peter Stanbury, Flat 2, 3 Ferndale, Tunbridge Wells, Kent TN2 3RL, UK