Broad to Alexander (3 August 1920)
In reading your book I have been completely puzzled by the arguments in Vol. 1 p. 52-56, in which you attempt to correlate the characteristics of temporal order with the number of dimensions in space. If it will not bore you I should be very much obliged by your answering the following questions. (I think it very likely that what you mean to say may be true; but I certainly have failed altogether to grasp your argument at this point, after repeated attempts).
(i) On p. 51 you talk of three independent characteristics of temporal order – successiveness, transitivity, and asymmetry. Is successiveness a characteristic in the same state as the other two? Would one not rather say that the relation is the relation of successiveness, and that it has the two logical properties of being transitive and asymmetrical?
(ii) [You write] ‘… aA and bB are two point-instants. The points a and b … suffice to distinguish the instants …’ (Space, Time and Deity, book 1, ch. 1, p. 52). You mean, I suppose, ‘… to distinguish the point-instants’, not the instants A and B themselves. For, although on your theory it is necessary that A and B should occupy various points of space, no particular point of such points are either necessary or sufficient to distinguish the instants A and B. Not sufficient, therefore the same instant can also occupy different points. Not necessary because the same point can occur at different instants.
(iii) [You write] ‘… if there were only one dimension of space and we take the line ab, as we may, to represent the time-dimension as well …’ (Space, Time and Deity, book 1, ch. 1, p. 52). Why may we do this? Surely the fact that there were only one dimension of space would not entitle us to identify this one spatial dimension with the time-dimension? Of course if you do this the repetition of A at two points a and a1 is simply unrepresentable; for, by identifying your time and space axes you are forced (1) to choose a single point to represent the identical instant A and (2) two points to represent the different points a and a1. And you have yet to identify both a and a1 with A to represent the points-instants aA and a1A.
(iv) [You write] ‘… A cannot be repeated at two points a and a2, both on the same side of b, because in that case their dates would be different’ (Space, Time and Deity, book 1, ch. 1, pp. 52-3). A can’t be repeated at all if you insist on identifying the time dimension with the one space-dimension. In fact it seems to me that your whole argument then rests on that identification and I can’t see that you have given the slightest justification for it.
Of course I must be making some stupid mistake here; but I should be very much obliged if you would write and point it out to me.
With kind regards, I am