Alexander to Broad (4 August 1920)
My dear Broad,
I will try to answer your questions at once, returning you your own letter for convenience of reference. (Please send it back. I keep a copy of my own letter). The point I am trying to establish is that Space and Time really are identical, the one thing Space-Time from two sides, and that in strictness Time is not an additional character added to Space, but they repeat each other. I do this by taking them separately and showing that each is the same as the other over again (see p. 69). Of course you must use the four coordinates x, y, z, t; but in using x, y, z you are metaphysically already using t. So that mere correlation of Space and Time is not enough to describe their intimacy.
(i) See the note 2 on p. 50. I should say that Time is a stuff, not a relation (see ch. 6) which has the 3 properties of successiveness, irreversibility and betweenness, and on these the logical properties of successiveness, asymmetry, and transitiveness are founded. And as each may occur without the others I should say they were all properties alike. But I should not hold out very strongly on the point.
(ii) No, I mean “to distinguish the instants”. The point-instants are given as distinct. But I am taking points and instants independently at first and then showing that there could be no unambiguous distinction of A and B as before and after with one-dimensional Space.
(iii) This is the real difficulty. Remember that Time is only represented by the Space axis (which in this case is the whole of Space). But in fact if Time were bare succession, and Space one-dimensional the two would be identical, the one is the other over again. Now I go on to say that since Time is irreversible the one-dimensional Space cannot support Time.
I think you are wrong in saying “of course if you do this etc.” Your statement (1) is not true; for I am not forced to represent the identical instant A by a single point. On the contrary it is represented by many points. Are you supposing that the instants corresponding to a and a1 are separate As? On the contrary it is the same A though at different places. I think I ought to have inserted a note to guard against this possible misreading of the diagram. However I am not sure you are so misunderstanding, though I a little suspect it.
(iv) This answers I think your statement that A can’t be repeated at all etc. There is nothing to prevent repetition, but you can’t have irreversibility if the Space is one-dimensional. (See the note on p. 52). The argument is: Time is repeated in Space – that is empirical; but it is also empirical that Time is irreversible; therefore Space cannot be one-dimensional. And the spirit of the subsequent proofs is the same.
All the time Time is being represented spatially. But with one-dimensional Space, if that were possible, T and the one-dimensional Space would be identical. If a two-dimensional Space were enough to secure the full characters of Time, Time would be identical not with any one of the Space dimensions, but with both together (I admit, a conception hard to express). In fact Time is identical with the three dimensions taken together. But in pp. 54-55 it is still legitimate to represent the one-dimensional Time by a line in Space.
I hope I have made the matter clearer; and that if I haven’t you will tell me so. The fact is I have an obstinate belief that the proposition is true, but I also think it more than possible I have slipped in the proof. I might have imitated Causs and the other mathematical swells, and stated the theorem and left it unproved. But not being swell enough to be taken on trust I thought it cowardly not to give my reasons, so far as I could.
pp. 58-60 show better what I am aiming at. Time is not an addendum to Space as if there could be a Space without Time in much the same way as we can think a Space of one or two dimensions. I refer also to the reference in note 2 of p. 59.