These two numbers can't be explained by an ultimate theory of physics. Instead, we must find another constant of Nature which has the dimensions of a velocity. The ratio of this quantity to the speed of light will then be a pure number, with no dimensions. There is then the possibility that it might be a number that could be calculated in terms of quantities like π or any of the other numbers of mathematics.

Rosenthal-Schneider replies9 and mentions the ideas of Planck, with whom she studied as a student, about the three special constants that he used to create his ‘natural’ units:

‘However, I am still worrying – and that is why I pester you again with my questions – about what are the universal constants as Planck used to enumerate them: gravitational constant, velocity of light, quantum of action, … which are not dependent on external conditions like pressure, temperature, … and which therefore are pleasantly distinct from the constants of irreversible processes? If all these were entirely non-existent, the consequences would be catastrophic,

If I understood Planck correctly he regarded such universal constants as “absolute quantities.” If now you were to state that they are all non-existent, what at all would be left for us in the natural sciences? It is much more worrying for an ordinary mortal than you can imagine.’

Einstein's penfriend is worried about the consequences of there being no true constants of Nature. If they are all illusory, what bedrock is there for physical reality; why does the Universe seem to be the same from one day to the next? She misunderstands Einstein's statement that there are no free constants of Nature, thinking that he means that they are not constant when he means only that he believed they are not free.

A deeper theory will eventually determine them. Sensing that he has misled his correspondent, he responds in greater detail10 on 13 October 1945, with a complete analysis of the situation. First, he notes that there are just quantities like 2, π or e (a numerical constant equal to about 2.718) which appear in physical formulae. In a later chapter we shall discuss them further. Einstein notices that they tend to appear in physical formulae but their values are neither very large nor very small:11 they are never very different from the number 1. They might be ten times greater or smaller but not millions of times greater or smaller. This is something he cannot explain. It just seems like a piece of good luck for physicists.12

‘I see from your letter that you did not grasp my hint about the universal constants of physics. I will therefore try to make the matter clearer.

1. Basic numbers. These are those which, in the logi-cal development of mathematics, appear by a certain necessity as unique individual formations.

e.g., e = l + l + l/2! + l/3! + …

It is the same with π, which is closely connected with e. In contrast to such basic numbers are the remainingnumbers which are not derived from 1 by means of a perspicuous construction.

It would seem to lie in the nature of things that such basic numbers do not differ from the number l in respect of the order of magnitude, at least as long as consideration is confined to “simple” or, as the case may be, “natural” formations. This proposition, however, is not fundamental and not sharply definable.’

But Einstein knows that these basic numbers are not the most interesting constants of Nature. Einstein explains that the usual constants, like the speed of light, Planck's constant, or the gravitation constant, have dimensions of different powers of mass, length and time. From them we can create combinations which are pure numbers but we might need to introduce other quantities to do it. He says,

‘Now let there be a complete theory of physics in whose fundamental equations the “universal” constants c1, … cn occur. The quantities may somehow be reduced to gm. cm. sec. The choice of these three units is obviously quite conventional. Each of these c1, … cn has a dimension in these units. We now will choose conditions in such a way that c1, c2, c3 have such dimensions that it is not possible to construct from them a dimensionless product cα 1cβ 2cγ 3. Then one can multiply c4c5, etc., in such a way by factors built from powers of c1, c2, c3 that these new symbols, c*4, c*5,c*6 are pure numbers. These are the genuine universal constants of the theoretical system which have nothing to do with conventional units.

Suppose his c1, c2, c3 are Planck's c, h and G, then there is no way to combine them in powers so that you can get a pure number with no dimensions.13 To do that you need to multiply by some other dimensional constants of Nature. For example, by multiplying G/hc by the square of some mass, for example the mass of a proton, we get the pure number Gmpr2/hc, say c * 4, which is approximately equal14 to 10–38. The ‘starred’ number we have just created is made by measuring some constant of nature with units of a mass by Planck's mass.