George Walford: Of Apples and Arithmetic

When, in the course of argument, one asks for an example of universal truth, the proposition that one and one make two usually comes up. A little thought reveals exceptions. One heap of sand added to another heap of sand does not make two heaps of sand, the same applies to puddles of water and, more dramatically, one critical mass of U235 added to one critical mass of U235 does not make two critical masses of U235; it makes one big bang.

The response to this tends to be that although these exceptions have to be granted, and doubtless some others also, yet by and large the rule holds; one and one do, nearly if not quite always, make two. But closer examination shows that one and one make two only in certain specific, carefully selected, highly artificial and very uncommon circumstances, namely when dealing with abstractions.

In ordinary daily living, if we have one apple and somebody gives us another, we say we have two apples. This is good enough for most purposes, but it falls a long way short of complete exactitude. Close examination of those apples shows each of them to have its own unique features. In calling them simply apples we overlook these and take notice only of what they have in common. That is to say, we abstract their common features and ignore all others. Only by doing this can we legitimately treat them both as apples and not as two unique entitities. The ‘apples’ which, when one is added to another, make two, are not apples we have ever seen, much less tasted. They are purely conceptual apples, consisting only of their ‘appleness’ and exhibiting none of those unique features that distinguish every real apple from every other real apple.

The same applies, with the appropriate modifications, to all concrete objects. Every one of them, examined closely enough, is a unique individual, different from all others, and they can be treated together, one added to another to make two, only by selecting for consideration exclusively that which they have in common, which means ignoring most of what attracted our attention to them in the first place.

This does not mean arithmetic has no use in practical life; we do add one real apple to another real apple and say we have two apples. But in doing this we depart from the rigour of mathematics, which holds good only for abstractions. This appears if we hold on to our two apples for more than a few days. We find ourselves with no apples at all, only two lumps of corruption, and if they had been the apples with which arithmetic is concerned this would not have happened, since these are subject to no time limit. If there be any universal truth it must be something other than ‘one and one make two.’

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“HISTORICAL WRITING should not – cannot – begin with ‘the facts.’ Assumptions and questions plainly come first: they construct the net that culls what seems relevant in the sources.” (Michael Bentley, The Climax of Liberal Politics, London, Edward Arnold, 1987, p.1.

IS SYNTAX imposed only on sinners?

from Ideological Commentary 35, September 1988.