ON THE DOGMA OF THE NATURAL NUMBERS1

P. K. Rashevskii

God gave us the integers, all else is

man's handiwork

L. Kronecker

Of course, no one nowadays takes Kronecker's words literally, indeed,

hardly anyone ever meant them literally, including Kronecker himself. But

if we read them in an appropriate transcription, then they do, in some

sense, express the prevailing frame of mind of mathematicians right up to

the present time.

On this point I should like to say that even today the natural numbers

are the unique mathematical idealization of the processes of real calcula-

tion.2 This monopoly position illuminates them with the halo of some

truth in the highest instance, absolute, the only one possible, to which the

mathematician must have recourse in every case when he is working on the

counting of his objects. Moreover, since the physicist uses only the appar-

atus proffered to him by the mathematician, the absolute power of the

natural numbers spreads to physics as well, and by means of the real axis

predetermines to a considerable extent the possibilities of physical theories.

It may be meaningful to compare the present position regarding the

natural numbers with the position of Euclidean geometry in the eighteenth

century, when it was the only geometrical theory, and hence was considered

as some absolute truth, as binding for the mathematician as for the physicist.

It was considered quite reasonable that physical space must ideally obey

Euclidean geometry exactly (what else?). Similarly, today we consider that

the counting of sets, as large as we please, of material objects, the measure-

ment of distances as large as we please in physical space, etc. must obey

existing schemes of the natural numbers and the real axis (what else?)