No heavenly power will give us a definite assurance to the contrary, and all of us may be just as false in our judgment of accepted celebrities as were the Byzantines eight hundred years ago in their estimate of Psellus.
Whereas the latter had obtained a value "less than 3," there are learned documents of about the same date that have been preserved, according to which the value of π comes out as exactly 4. Compared with this grandiose bungling, even the observations mentioned in the Old Testament are models of refinement. For, as early as three thousand years ago, it is stated of the mighty basin in the temple of Solomon (First Book of Kings, chapter VII.): "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits; and a line of thirty cubits did compass it round about." Thus π here appears as 3, an approximation which no longer satisfied later generations. The wise men of the Talmud went a step further, in saying 3 plus a little more; and this agrees roughly with the actual value.
The view became more and more deeply rooted that this π was a main pillar of mathematical thought and calculation. The more the problem of the quadrature of the circle seized on men's minds, the greater were the efforts made to find the exact value of this "little more" of the Talmud. Since 1770 we know that this is not possible, for π is not rational, that is, it can be represented only as an infinite and irregular (that is, non-repeating) decimal expression. It occupies, further, a special rank as a transcendental quantity; this fact was proved by Lindemann as late as 1882 for the first time. Yet, even nowadays, there are incorrigible devotees of quadrature, who are still hunting a solution because they cannot rid themselves of the hallucination that such a simple figure as the circle must submit ultimately to a constructive process.
The correct way was to carry out an even more accurate determination of the decimal figures. The above-mentioned Ludolf van Ceulen got as far as the 35th place of decimals; at the turn of the eighteenth century the 100th decimal place was reached. Since 1844, thanks to the lightning calculator Dase, we have its value to the 200th decimal place, and this should satisfy even the most extravagant demands. This number, associated with the circle, is a classical example of how an approximation that is expressible in figures of very small value gives an order of accuracy that can be described only by using fantastic illustrations.
If we take a circle of the size of the equator, and also multiply the value of the diameter of the earth by π, we know that the latter result will not be exactly equal to the former, and that there will always be a small remainder. If this discrepancy were less than a metre, the order of exactness would be extraordinarily high, for a metre is practically insignificant compared with a mighty circle of the dimensions of the earth's circumference.
Let us stipulate still greater accuracy. We demand that the error is to be less than the thickness of the thinnest human hair. We find, then, that we must take for π at most 15 places of decimals. Thus, if we use π = 3.14159265358973, we are applying a means of calculation that reduces the possible error in all measurements of circles on the earth to a degree beyond the limits of human perception.
If we pass beyond the world out into celestial space, and consider circles of the dimensions of a planetary orbit, nay, further, if we pass on to the Milky Way or even to the limit of visible stars, to find space for our circle, and if in this case we still reduce the discrepancy so as to be less than any length that is observable under a microscope, then the last given value of π still suffices. Yet we must not forget the proviso: semper aliquid haeret, something unsolved still clings to the problem.
Such numerical approximations, however instructive they may be, nevertheless retain a comparatively playful character, and furnish only a superficial analogy to the most important approximations that are contained in our natural laws themselves. It is these, above all, that manifest themselves so clearly in Einstein's life-work, and they bear the same relation to the former as truth bears to correctness. Truth comprises the greatest conceivable circle of ideas and passes far beyond the sphere of correctness, which deals only with measure-relations, and not with the things in themselves. If Einstein, as we learn, emphatically declares truth to be the only object of science, he means the strictly objective truth that is to be derived from Nature, the true relationship of phenomena and occurrences, independently of whether restless philosophy assigns a question mark to this ultimate objectivity. A great discoverer in the realm of Nature cannot and dare not proceed otherwise. For him there is behind the veil of Maya not a phantom that finally vanishes, but something knowable, that becomes ever clearer and more real as he detaches each successive veil in his process of approximation.
During this conversation, when we were talking of the "Future of the Sciences," Einstein gave his ideas free rein, shooting far ahead of the views and prognostications of the above-mentioned scientists:
"Hitherto we have regarded physical laws only from the point of view of Causality, inasmuch as we always start from a condition known at a definite cross-section of time, that is, by taking a time-section of phenomena in the universe, as, for example, a section corresponding to the present moment. But, I believe," he added, with earnest emphasis, "that the laws of Nature, the processes of Nature, exhibit a much higher degree of uniformity of connexion than is contained in our time-causality! This possibility suggests itself to me particularly as the result of certain reflections concerning Planck's Quantum Theory. The following may be conceived: What belongs to a definite cross-section of time may in itself be entirely devoid of structure, that is, it might contain everything that is physically conceivable, even such things (so I understood him to say) as, in our ordinary physical thought, we consider impossible of realization, for example, electrons of arbitrary size, and having an arbitrary charge, iron of any specific gravity, etc. By our causality we have adjusted our thought to a lower order of structural limitations than seems realized in Nature. Real Nature is much more limited than our laws imply. To use an allegory, if we regard Nature as a poem, we are like children who discover the rhyme but not the prosody and the rhythm." I interpret this as meaning that children do not suspect the restrictions to which the form of the poem is subject, and just as little do we, with our causality, divine the restrictions which Nature imposes on occurrences and conditions even when we regard them as governed by the natural laws we have found.