The Eddington’s number: A case of scientific arrogance?
Still speaking in a low voice, the stranger said, "It can't be, but it is. The number of pages in this book is no more or less than infinite. None is the first page, none is the last."
Jorge Luis Borges
The Book of Sand.
In some textbooks of science, and in the popular literature, there are some unusual numbers that grasp our attention in a special way. An example of one of those numbers is the Avogadro’s number:
NA= 6.022 141 99 * 1023.
This number is said to be the amount of atoms or molecules needed to make up a mass equal to the substance's atomic or molecular mass, in grams. For example, 55.847 g of iron contains NA iron atoms.
This extremely big number is also approximately equal to the number of protons in a gram of pure protons. But do you imagine somebody telling us the exact quantity of protons in a gram of pure protons (pure in the sense of nothing else but protons)? Note that no matter how big this number is, the last 15 digits of this number are all zeros; it is not an exact quantity up to the last digit. So the Avogadro's number is also a big approximation to an ideal number.
But if computing exactly how many protons are there in a gram of protons is a chimerical task, how about of computing all the protons in the Universe? This was exactly what Eddington did. Barrow & Tipler2 in their book the Anthropic Cosmological Principle write:Eddington was the first to suggest that the total number of particles in the Universe, N, might play a part in determining other fundamental constants of Nature. He evaluated this number to high precision and it is often termed the ‘Eddington number’
NEdd = 2.136 x 2^256 ~ 1079.
One of the attractions of this quantity for Eddington was the necessity that its value be integral. This meant that it could, in principle be calculated exactly.
In its expanded form, the Eddington number becomes:
NEdd = 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296
This number is related to what is known as the 'fine structure constant', a pure number (hence a dimensionless number), that by 1930, was estimated to be near 1/136. This concept was introduced by Arnold Sommerfeld in 1916 from the predictions of the Bohr’s atom model. This constant is in relation with other physical constants, like the electron charge, the Planck constant, the velocity of light, etc. However, the role of the physical constants are under continuous debate by physicists and philosophers of science.
Can we really count the pieces of the Universe?
The fantasy and speculation is not exclusively of the literary-inclined minds. I’m no saying with this that literature is all about fantasy, but that this is an intellectual activity with permits and licenses to write about anything that come to our minds.
On the contrary, the scientific literature tries to be auto-regulated and peer-revised with projections and intentions verifiable in any epoch and geographical situation.
But science as an objective intellectual activity does not exist; what does exist are people called scientists, that being part of the same community of all other terrestrials, are also subjected to the same forces and influences as other writers.
To exceed in projections is a natural temptation, and the scientists —in all the fields— seem to have adopted it without remorse in their consciousness. This seems to be happening since humankind began to count one, two, three, ...
Archimedes tried to calculate the grains of sand needed to fill the known universe at his time, but he was careful to establish upper bounds by not giving an exact count, the opposite of what Eddington did for the amount of protons in our known Universe.
Writing about those two similar attempts –although very distant in time-- Harrison4, in his essay The Cosmic Numbers, says:
Eddington attached great importance to his number. Because the Eddington universe is closed he was able to argue that N2, equal rough to 1080, is the actual number of nucleons in a finite universe. But, in general, this number is no more a rough estimate of the nucleons in the observable universe of radius equal to the Hubble length. Archimedes’ 1063 grains of sand contain, by pure chance, 1080 nucleons, and Archimedes’ number and Eddington’s number are therefore equivalent to each other.
Newton, so careful in his mathematical expositions never abandoned his investigations in alchemy, ‘a science’ no so regulated and structured as other practiced at his times.
Evocating Archimedes, Jorge Luis Borges, with all the possible literary licenses, in his short story The Book of Sand, makes a trade of and old English Bible for an old book that a salesman brought to him.
He told me his book was called the Book of Sand, because neither the book nor the sand has any beginning or end. ...
The number of pages in this book is no more or less than infinite. None is the first page, none the last. I don’t know why they’re numbered in this arbitrary way. Perhaps to suggest that the terms of an infinite series admit any number. ...
I realized that the book was monstrous. What good did it for me to think that I, who looked upon the volume with my eyes, who held it in my hands, was any less monstrous?
The Eddington number, this figure so perfectly computed that came to the mind of this gentleman for the exact quantity of protons in the Universe, is at the same time an act of pride and arrogance that doesn’t seem to have no philosophical foundation, but at best, an exercise in numerology and arithmetic.
For that reason, in mathematics and physics this number has no transcendence; it is not a respectable and transcendental number like pi, or the Avogadro's number, or other of the fundamental "constants" of nature. This number is just a mere curiosity and maybe, unnecessary.
Barrow cites that Beck, Bethe, and Riezler, in 1931, and Max Born, in 1944, designed ‘spoofs’ to parody the Eddington methodology. Born called Eddington’s number 'apocalyptical' and even rewrote one of St. John’s Revelation verse using Eddington’s algebra.
However, despite the frequent credibility attacks toward Eddington’s number, Barrow & Tipler, have some encouraging words for the later implication Eddington’s work:
Although Eddington’s Fundamental Theory is very easy to criticize, it is still interesting for the vision of an underlying unity in Nature which it displays. A vision that has since materialized in an entirely different form. Through his work in this area Eddington directed the attention of many of many other workers to the ubiquity of large dimensionless numbers. This, in turn, stimulated other approaches to cosmological theory that have borne more fruit than their progenitor.
Why this number has to be so exact? Why not an additional proton or a proton less? If the universe is so immense and the protons are so small why is it that there is no space for one more infinitesimal body? Why the number has to end in 296 and not 297 or 715, or 000?
At the end, it is so difficult to digest the relevance of this number that I prefer to accept that Borges had in his hands and infinite book rather than to accept that Eddington computed the protons of the Universe.
E. Pérez
jul-10
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