What is the difference between numerology and science?

As far back as the Pythagoreans, and probably much further, people have noticed, or have believed that they have noticed, that numbers play a part in nature.  Indeed, many cultures have developed numerical skills which were adequate to predict the motions of the heavenly bodies, often for centuries into the future.

Many people will have noticed that many flowers have three, four, five, six or eight petals, but seldom seven, and that many other flowers appear to have bilateral symmetry.  Some people have been able to demonstrate convincingly that the occurrences of these numbers in plants are closely related, as well as the occurrence of certain other numbers as high as 144.  This is described in the page about Fibonacci numbers, where these numbers are related to the physical restrictions on growth.

The profound difference between scientists and numerologists is that scientists look for simple physical ideas that lead naturally to the occurrence of the observed numbers, while numerologists treat the numbers themselves as the important quantities.  That is not to say that numbers have not led to great scientific discoveries.

Even in science, the process of theory creation may have more than one stage.  Newton's Laws were very effective in describing the motion of objects, but a more general formalism, based on the Lagrangian function, has proved to be very powerful in explaining many phenomena.

An example of progression is found in the explanation of atomic spectra, beginning with the measured wavelengths, fitted by Balmer's formula, "explained" by Bohr' atomic model, with some violence to previously accepted principles.  After that came "old" quantum theory, followed by "new" quantum mechanics and quantum electrodynamics.

In the 19th century, Newlands put forward the law of octaves to fit the pattern of behaviour of the chemical elements.  This was not taken very seriously.  Later, when Mendeleev created the periodic table, he was much more systematic in the collection of supportive data, and he was able to predict the existence of at least three unknown elements and their properties.  These have all been found as predicted.

There is a deep idea here.  Mendeleev did not merely fit the current observations: he dared to suppose that something was missing.  Furthermore, some elements when ordered by atomic weight did not fit, and had to be interchanged. We now know that the correct variable is atomic number, which corresponds to an actual physical number, the number of protons in a nucleus.

If you try to fit only the current observations, you may be trying to account for a pattern that is incomplete, or even faulty.  And the numbers may have been measured poorly, especially at the birth of a new discipline.   Before the omega-minus particle was discovered, someone published a paper that explained why it did not exist.

The great scientist Sir Arthur Eddington "derived" the fine structure constant from an equation based on simple ideas.  At the time the measured value was about 1/136.  Eddington assumed that the denominator was a whole number.  When the value was measured more accurately and found to be close to 1/137, Eddington had  to change his theory.  In fact the actual number is 1/137.03604.  Nobody to this day knows the origin of this number, nor even whether it has meaning.

Arthur Eddington probably did not know the following fact.  If you write Arthur Stanley Eddington, and if you set 1 for A, 2 for B, etc, the total of all the numbers from the letters is 274, which is 2 X 137.  The 2 can be identified with the electron gyromagnetic ratio: 137 we have met already.  

Had Eddington been aware of this, he might have reflected on the wisdom of believing something just because the numbers agree with something else.  That his idea did not lead to anything does not detract from Eddington's great work in physics and astronomy: quite a few scientists have had sidelines which were not rated highly by other scientists.  

Another problem with certain types of numerology is the use of numbers which depend on units.  If you use the length of a certain building in metres, and the mass of the materials used in it in kilograms, any results you obtain will change the units to feet and pounds, for example.  Numbers can enter into a formula only in combinations that give every term in the formula the same dimensions in consistent units.  In this way, a change of units affects all the terms in the same way, and the formula either always works or always doesn't work.

The Balmer formula related a number of spectrum lines in the spectrum of hydrogen.  It was inexplicable by classical theories.  The existence of the formula did not lead anyone to an explanation, until more knowledge was obtained in other areas, such as nuclear physics.  The Bohr atom that gave the first explanation required several changes in thinking that simply would not have been made with only the formula, and no other hints from elsewhere.

We know the masses of the quarks to some accuracy, but it is extremely unlikely that there is a simple formula for them, because the dynamics are likely to be complicated, as the great range of the values suggests.

Suppose that you believe that one or more artists have used golden section.  How would you try to prove it?  You could mark "important points" on a set of reproductions, and measure their positions.  Then you could calculate the ratios of all the  "important" distances in a given picture.  If there were any truth in your idea, a histogram of all the ratios would show a sharp peak at about 0.618.  This would not prove that golden ratio is pleasing or meaningful: it would only suggest that the artist thought it was, and put it into his work.  If the results were more positive for the artists' most important work, that might add to the significance of the results.

The next diagram shows an array of fifty places.

The diagram above includes fifty spots, many of which have been joined by straight lines.  Some of the spots are clearly more important than the others; they have more lines connected to them than the others.

In order to investigate the significance of this diagram we could measure the following quantities -

Number of points that have 0, 1, 2, 3, 4 and 5 lines through them.

Number of lines that have 3,4 and 5 spots on them.

Number of lines that cross 0, 1, 2, 3, 4 and 5 other lines.

Distribution of angles of the lines, measured from the horizontal.

Distribution of angles of intersection of lines.

Number of unconnected islands of connected spots.

Distribution of spots within the islands.

Distribution of the proportions of the two parts of the lines that have three spots.

In order to decide where to draw the lines we have to decide how exactly collinear the spots have to be, for the probability that three are exactly in a line is zero.  We are here ignoring screen resolution and assuming that the diagram represents a smooth distribution of the original data.

Next we could use pseudo-random numbers created by a computer program to generate a large number of pictures containing fifty spots each, and we could find out the distribution of the variables listed above.  Then we could compare our special picture with the random ones to see whether it differed significantly, using suitable tests of statistical significance.  Of course, the more variables we examine, the more likely we are to find something "unusual", and this factor has to be included in the estimation of statistical significance.

In fact, the diagram above was generated using pseudo- random numbers.  Only eight of the fifty spots have no lines "through" them.  Six have two lines, and two have three lines.  One even has four lines.

Laws and Numbers

Consider the game of cricket.  There are several ways of describing it.  Firstly there are the laws, from which everything else flows.  Secondly we could attend a match, where twenty two players and some umpires would create an example of cricket.  Thirdly, we could hear a running commentary of this match.  Fourthly, we could read an analysis in a newspaper after the match, including a score card giving many details of the unfolding events.  Fifthly, we could read an evaluation of the match in the context of a series of matches, and in the context of the recent performance of the two teams.

Which of these is most real?  From one point of view, all that matters to the teams is who won, or how many points they scored.  Nobody would imagine that the exact numbers of runs scored, or wickets taken had any significance beyond their magnitude: they are but a reflection of what happened.

We need to be very careful, in any investigation, to find out what are the fundamental variables, and which are only dependent.

Rules

As Ian Stewart says in his book, Life's Other Secret by Ian Stewart - Penguin, ISBN 0 14 025876 0, simple rules can generate complex behaviour.  The behaviour of cars on the streets and roads can be quite complicated, yet it is generated by simple rules (not always obeyed) about giving way, maintaining gaps, and so on.  In fact there are two sets of rules, those given in the highway code, and those which govern the behaviour of people in cars.  Understanding of the basic behaviour of drivers can explain phenomena such as periodic bunching and even halting on motorways.

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