We could make others by dividing some time by Planck's time or some length by Planck's length. It is these pure ‘starred’ numbers that Einstein regards as the most fundamental. It does not matter what units are employed to measure them or to express them, they will always have the same value. Where do they come from? What fixes them? Why is Gmpr2/hc about equal to 10–38, rather than to 103 or 10–68? Einstein doesn't know, but he has a strong belief that they are fixed absolutely.15 There is no latitude for them to be different:

‘My expectation now is that these constants c* 4 etc., must be basic numbers whose values are established through the logical foundation of the whole theory.

Or one could put it like this: In a reasonable theory there are no dimensionless numbers whose values are only empirically determinable.

Of course, I cannot prove this. But I cannot imagine a unified and reasonable theory which explicitly contains a number which the whim of the Creator might just as well have chosen differently, whereby a qualitatively different lawfulness of the world would have resulted.

Or one could put it like this: A theory which in its fundamental equations explicitly contains a non-basic constant would have to be somehow constructed from bits and pieces which are logically independent of each other; but I am confident that this world is not such that so ugly a construction is needed for its theoretical comprehension.’

Elsewhere, Einstein, is famously quoted as saying what really interests him is whether ‘God had any choice in making the world’. What he meant by that cryptic statement is made clear by the extract from his letter to Rosenthal-Schneider. He wants to know whether the dimensionless constants of Nature could have been given different numerical values without changing the laws of physics or whether there is only one possible choice for them. Going further he might wonder whether different choices in their values are possible for different laws of Nature. We still don't know.16

The illuminating exchange of letters with Rosenthal-Schneider on constants ends on 24 March 1950 with Einstein reiterating his ‘religious’ view that God did not have any choice when it came to the fundamental constants and their values:

‘Dimensionless constants in the laws of nature, which from the purely logical point of view can just as well have different values, should not exist. To me, with my “trust in God” this appears to be evident, but there will be few who are of the same opinion.’

As we leave Einstein's thoughts about the inevitability of the constants of Nature it is interesting to pick up on the view of other great physicists who have been drawn to speculate about the significance and attainability of a final understanding of their values. Take George Gamow, the eccentric Russian physicist who risked his life escaping from the Soviet Union to live and work in America, where he became one of the founders of modern cosmology and even contributed to the early understanding of the DNA molecule and the genetic code. Gamow, like all his contemporaries, could see that there were four distinct forces of Nature (gravity, electromagnetism, weak and strong forces). The strength of each would create one of Einstein's pure numbers that define the world. Gamow was not drawn especially into the issue of whether they could have only one quartet of possible values. But for him a full understanding of those values – an ability to calculate or predict them precisely – would be like the waving of the chequered flag to a physicist. They would have attained a complete understanding of the forces of Nature when that day happened. Gamow is a little depressed at the prospect, like reaching the end of a great story, or sitting at the summit of a mountain one has striven to scale, for

‘If and when all the laws governing physical phenomena are finally discovered and all the empirical constants occurring in these laws are finally expressed through the four independent basic constants, we will be able to say that physical science has reached its end, that no excitement is left in further explorations, and that all that remains to a physicist is either tedious work on minor details of the self-educational study and adoration of the magnificence of the completed system. At that stage physical science will enter from the epoch of Columbus and Magellan into the epoch of National Geographic Magazine.’17

THE DEEPER SIGNIFICANCE OF STONEY-PLANCK UNITS: THE NEW MAPPA MUNDI

‘One Ring to rule them all, One Ring to find them.

One Ring to bring them all and in the darkness bind them.’

J. R. R. Tolkien18

The interpretation of the natural units of Stoney and Planck was not at all obvious to physicists. Aside from occasional passing remarks it was not until the late 1960s that the renewed study of cosmology led to a full appreciation of these strange standards. One of the curious problems of physics is that it has two beautifully effective theories – quantum mechanics and general relativity – but they govern different realms of Nature.

Quantum mechanics holds sway in the microworld of atoms and elementary particles. It teaches us that every mass in Nature, however solid or pointlike it may appear, has a wavelike aspect. This wave is not like a water wave. It is more analogous to a crime wave or a wave of hysteria: it is a wave of information. It tells you the probability that you will detect a particle. If an electron wave passes through your detector you will be more likely to make a detection, just as you are more likely to be robbed if a crime wave hits your neighbourhood. The quantum wavelength of a particle gets smaller the more massive the particle. Situations are dominated by quantum waviness when the quantum wavelength of their participants exceeds their physical size.