History is full of people that were absolutely certain to be right, until somebody else or some event proved them wrong. The border between a wrong theory, a non-substantiated claim and a superstitious belief can be quite shady, depending on the caller. Numerology falls in all the three categories. All ancient civilizations, from the Egyptians to the Sumerians to the Hebrews, strived to find correspondences between numbers and meanings, religious, divinatory, astrological, or combinations thereof. The Greek circle of Pythagoras was the most famous in the antiquity, because of the links they tried to make between integer numbers and signs in the universe, from the planetary motion to the musical scale. Since the Greeks did not use positional notation, but letters to indicate numbers, it is likely that their notions of mystical numbers could arise from aligning objects according to their cardinality. For example, by arranging 10 pebbles in triangular shape with four stones on each side and one at the center, it appeared that by summing rows of pebbles, the “mystical” triangular numbers (tetraktys) appeared: 1, 1+2=3, 1+2+3=6, 1+2+3+4=10. Such “mysteries” were part of the alchemic creed and percolated up to relatively recent times, from the jewish Gematria, to the Hermetic Kabbalah, to the Rosicrucians, to the followers of Aleister Crowley… The belief that numerical coincidences should hide some mysterious meanings is so widespread, that even famous scientists have been tempted to explore the question.
I read somewhere (am too old to remember where, exactly) that given enough integers and enough π’s, one can approximate any number. That is, writing as a series:
X = Σn kn πn,
for any X, with kn integers, and n={0,…,inf}. (In fact, this is true also if you replace π by any fixed real number, but playing with π always looks fancier.) For some special combinations, however, curious coincidences of the kn coefficients appear, for example take the very old coincidence:
e6=π4+π5=403.42879…
This can be approximately explained by noticing that 403=13×31=(3+10)x31, and that π≈3, π2≈10 and π3≈31, with the respective approximations bearing errors that compensate each other. In the above series, it is just k4=k5=1 and all the other k’s are 0. And that already looks rather “mysterious”. Then, add to this the fact that e3=20.085537… is slightly larger than 20, but just by the right amount to match the RHS within 45 parts in a billion. Amazing! But just a coincidence. However, as the famous detective Miss Marple used to say, “Any coincidence is always worth noticing: you can throw it away later, if it was only a coincidence.”
Other “strange” coincidences involve e, π and some prime numbers. Everybody knows that π3=31 (very nearly). But then, what about the well-known:
eπ – π1-e = 23 (actually, 23.00081238…)
and
(π+e+163)1/2 = 13 (actually, 12.994609…), and 1631/2 – π1/2 = 11 (or better, 10.99469…)
Or even the transcendental:
π/2e = tg π/6, at better than 1 part in 1000.
The recipe to obtain such “coincidences” (at least some of them) is not always so complicate as it may seem. Remember the opening statement, “enough integers and enough π’s…” Let’s say, for example, that I wish to surprise you with something including the figure 1/π = 0.3183098… Maybe I can get close to 1 by manipulating a little bit some more π’s and integers? Approximately, 1 – 1/π = 0.682. Could I make, e.g., the ratio of two numbers to look close to 0.682, then sum it to 1/π and get 1? Well, (π/4)2 = 0.617 is already quite close, but not so great. Then by playing a little, I stumble on (π2 – 4)=5.8696… and if (by chance, I was just lucky) I multiply this by 0.682, I get very close to the integer 4. Therefore, let’s try 4/(π2 – 4)? That’s 0.68147… Bingo! I just obtained that (1/π)+4/(π2 – 4) = 1, at better than 2 parts in 10.000. (After this, I checked on the internet if this result was already known… unfortunately it is! My Fields medal is slipping away!)
Sometimes (or maybe “always”?) these are not coincidences, but true (albeit approximate) mathematical results. The well-known coincidence:
eπ – π = 20
has been recently shown to be one term in an infinite sum for a special function (the Jacobian theta-sum). The first term of the sum is also the largest, and is equal to (8π – 2)e–π, which is already very close to 1. So, rewrite it as: eπ = 8π – 2 = π + (7π – 2). Then, by using one of the rational approximations to π≈22/7, the term in parenthesis gets equal to 20, and the thing is done. It may also be interesting to note that the bad approximation of 22/7 for π (wrong by 4 parts in 10,000) makes the overall error on (eπ – π) one order of magnitude smaller.
Numbers are just numbers, but when we look at physical constants, such numbers suddenly become something tangible and measurable. Arthur Eddington wrote the fine-structure constant as a riddle of 16:
1/α=(162–16)/2+16+1=137.
Better approximations to α, again involving only π’s and integers are for example, (16π4/9)(120/π5)1/4= 137.036082…, or 108π(8/1843)1/6 = 137.03591… (and note the 1843, close to the proton/electron mass ratio). A most “physically mysterious number” for me is 1/(30^πe) (that is, 30 to the power π to the power e), which is equal to h, the Planck’s constant. But that is true only in S.I. units. When I use calories instead of joules, it is h=1.583 x 10-34 cal.s … sorry for the wrong coincidence. (Actually, by replacing 30 by 32 as the base, you get 1.569 x 10-34 with just the same fractional error). The delicate connections among certain physical constants, and between those constants and life on our planet, are called anthropic coincidences, which some people wish to interpret as evidence for an “intelligent design”. Early in the twentieth century, Herman Weyl expressed his puzzlement that the ratio of the electromagnetic force to the gravitational force between two protons is such a huge number, e2/(Gm2) ~ 1040 (here e is the elementary charge, not Euler’s e…) Obviously, one thinks, electromagnetic repulsion must be much larger than gravity otherwise no stars could ever form. But how much is “much”?
To help students make sense of the enormous differences in distance and size, in my class of Introductory Astrophysics I often use the joke “In physics we only use three numbers: 0, 1 and infinity.” Meaning that only when we can get physical quantities in the proper range (which is neither the negligible zero, nor the out-of-scale infinity) we can draw some conclusions, make hypotheses, or create models. And this was the same thinking that puzzled Weyl in the year 1919: once properly adjusted, physical properties should most naturally occur around the scale of 1. All electromagnetism and all of chemistry are regulated by just two pure numbers of order-1: the fine-structure constant 1/137, and the proton/electron mass ratio, about 1840. But why 1040? Why not 1074, or 10–123 ? Something (or “Somebody”…?) must have selected this special 1040. John Barrow wrote a beautiful book about the Constants of Nature, in which he plays with numbers following Eddington’s example: the number of protons in the observable universe is about 1080… the total action of the universe in Planck units is 10120… the cosmological constant Λ fitting with the current acceleration in the expansion of the universe is 10–120, again in Planck units. And asks: why such powers of 1040 keep popping up? What’s so special about it?
Arthur Eddington tried to build a theory that made the appearance of “large numbers” understandable. He wanted to prove that just by pure thinking one could make sense of all the laws and fundamental constants of nature. He dedicated his last 25 years of work to his Fundamental Theory (published only posthumously, thanks to his mentor John Wheeler). Obsessed with the number 136 (in the ‘20s the measured value of α was rather close to 1/136), he calculated the number 136×2256~1080, approximately equal to the number of protons in the observable universe, which would be called the Eddington number; noting that 1+3+6=10 and 102+62=136, he stated the quadratic equation
10u2–136u+1=0,
whose two roots are in the ratio 1847:1, very close to the proton/electron mass ratio. Combined with α and the famous 1040, all his theory was a tremendous effort to deduce physics from these numbers. He was mostly ridiculed by the establishment, or pitifully ignored; Hans Bethe and George Gamow were especially insolent about his endeavors. Yes, he failed to convince the scientific community that he was right. Yet, he succeeded in persuading people that there was something that needed explaining.
It was Paul Dirac who took the challenge, with a short paper in the Proceedings of the Royal Society in November 1937 (allegedly written while on honeymoon with his freshly married Margit, the sister of Eugene Wigner). Although noting that Eddington’s “correspondences” were just approximated (1078 instead of 1080, 1039 instead of 1040…), he could not ignore that they were too big to be just a casual result of the nature of the universe. He proposed that the reason for such large numbers should lie in some yet-undiscovered formula that connected them somewhat. Indeed, the number of protons (or H atoms) in the visible universe is a measure of its radius, and given the constant speed of light, also of its age TU. He thus boldly proposed that the combination of constants e2/Gmp should be proportional to TU: the “constants” are not constant, but should change with the age of the universe. And in this proposal, Dirac view G as the main suspect for a saecular variation: gravity should have been much stronger in the past, and should decrease in the future. While Eddington continued to sell hundreds of copies of his book (which, however, he never saw in print), Dirac’s proposal did not last long: the required variations of G were so large that the Sun would have burnt away in a few million years, and the Earth would have been so hot to boil its oceans. However, both theories (or pseudo-theories) of Eddington and Dirac had in common the key observation that the “large numbers” are a feature of present-day universe, linking the structure of the whole universe as we see it, to our local conditions that notably include – and must be connected with – the possibility of life on Earth.
Robert Dicke is today largely forgotten by our community. The author of a successful, and yet unquestioned, alternative interpretation of general relativity (the Brans-Dicke model), both an excellent mathematician and accurate experimentalist, he invented the lock-in amplifier, patented a device similar to the laser two years before Townsend and Schawlow, and predicted the cosmic background radiation four years before the accidental detection by Penzias and Wilson (the instrument he was already building, detected the 3K signal just a few months later). When jumping in the heated discussion between Eddington and Dirac, he produced the interpretation that today is better known as the weak anthropic principle. He made the interesting remark that the “large numbers” have a biological aspect: the only number that is truly independent is a statistical quantity, the ~1080 total number of protons in the visible universe; the other large numbers are a consequence of the fact that present-day universe is in the right conditions to host “intelligent” observers. In a beautiful, one-page letter to Nature, Dicke argued that biochemical life-forms owe their existence to carbon, oxygen, phosphorus and a few other heavy elements. These could not have been produced before the explosion of enough supernovae, and it will be difficult for life to survive after their star will have burnt out (the “heat death”). Such a time, which he calls “the epoch of man” is controlled by fundamental constants, and can be grossly estimated as TM=h2/(Gmp3).
According to the weak anthropic principle, the age of the universe in this very moment is not a random number, but it is limited by the fact that human physicists must be present to measure it. In order for a Big-Bang universe to contain the basic biochemistry, it must have aged at least the time it takes for the nuclear reactions in stars to produce heavy elements. This means that the observable universe must be at least ten billion years old, and, since it is expanding, it must be also ten billion light-years in size. We could not exist in a universe that was significantly smaller. And since it is old and big, the universe must also be also dark and cold, enough to allow a planet to retain water and sustain life (as we know it). Such a response of the cosmologists to the ancient metaphysical and religious questions about our “unique” position in the universe, also stroke the chords of philosophers like Karl Jaspers: “The fundamental fact of our existence is that we appear to be isolated in the silence of the universe. In the history of the solar system there has arisen on the Earth, for a so far infinitesimally short period, a condition in which humans evolve and realize knowledge of themselves, and of being. This is the place, a mote in the immensity of the cosmos, at which being has awakened with man.”