Page:Eddington A. Space Time and Gravitation. 1920.djvu/30

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14

PROLOGUE

had four dimensions because four mathematical variables were used to describe it. Your use of the term "dimensions" is probably more restricted than mine.

Phys. I know that it is often a help to represent pressure and volume as height and width on paper; and so geometry may have applications to the theory of gases. But is it not going rather far to say that geometry can deal directly with these things and is not necessarily concerned with lengths in space?

Math. No. Geometry is nowadays largely analytical, so that in form as well as in effect, it deals with variables of an unknown nature. It is true that I can often see results more easily by taking my $x$ and $y$ as lengths on a sheet of paper. Perhaps it would be helpful in seeing other results if I took them as pressure and density in a steam-engine; but a steam-engine is not so handy as a pencil. It is literally true that I do not want to know the significance of the variables $x$ , $y$ , $z$ , $t$ that I am discussing. That is lucky for the Relativist, because although he has defined carefully how they are to be measured, he has certainly not conveyed to me any notion of how I am to picture them, if my picture of absolute space is an illusion.

Phys. Yours is a strange subject. You told us at the beginning that you are not concerned as to whether your propositions are true, and now you tell us you do not even care to know what you are talking about.

Math. That is an excellent description of Pure Mathematics, which has already been given by an eminent mathematician.^[1]

Rel. I think there is a real sense in which time is a fourth dimension—as distinct from a fourth variable. The term dimension seems to be associated with relations of order. I believe that the order of events in nature is one indissoluble four-dimensional order. We may split it arbitrarily into space and time, just as we can split the order of space into length,

↑ "Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such a proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true…Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

Bertrand Russell.

[1] "Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such a proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true…Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

Bertrand Russell.

[1]