Page:Mind (New Series) Volume 15.djvu/525

 SYMBOLIC REASONING. 511 no way clash with, but, on the contrary, render more precise, the rather vague significations usually attached to the words in ordinary language and even in mathematics ; and, lastly, that they are very convenient and, if accepted, would ma- terially increase the formal accuracy of our reasoning whether the questions discussed be metaphysical, logical, or math- ematical. Clear working definitions of the words finite, infinite and infinitesimal are imperatively needed ; and if those I here propose be not found suitable, others should be substituted. By " working definitions " I mean definitions which, by their formal precision, would prevent ambiguities and staggering paradoxes paradoxes that may be true or false according to the meanings attached to the words in which they are expressed. In ordinary informal speech variability of meaning is, of course, permissible, as the con- text generally prevents all ambiguity. No one, for instance, would misunderstand the meaning of such a statement as " He took infinite pains, yet his gains were infinitesimal " ; but in logic and mathematics the case is different. Here also, it is true, the context as a rule prevents ambiguity, but not always ; and when it does not, the errors into which we fall are often serious. We should especially remember that such expressions as - - -, like their reciprocals-, -, - 1 2 oo , etc., are not real ratios at all, but pure unrealities, though they have their utility as symbols. 11. Paradoxes also arise from the fact that our unit of reference is not always constant. A pound of tea is lighter at the equator than in London or Paris or Melbourne ; yet a pound of tea always weighs a pound, neither more nor less, wherever we may weigh it, provided the scales we use be correct. And if we took our pound and scales and weights to the moon or to the planet Mars, the result, in spite of the greatly diminished attraction, would be the same ; the pound of tea would be much lighter, but so would our unit of com- parison, the metallic pound with which we weighed it. It is the same with all spatial dimensions. We can never be sure that our units of comparison remain constant ; or, rather, we may be quite sure that they do not. The actual length of the standard yard or standard metre varies with the tem- perature. True, so far as our experience goes, the variation owing to this cause is slight ; but then our experience of the possibilities and actualities of nature is literally infinitesimal. For aught we know to the contrary, there may be other and far more powerful causes at work, causes which (unlike heat) act equally on all the perceptible substances of our universe,