Berkeley, Principles of Human Knowledge

Afterwards they discovered the more compact ways of

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Unformatted text preview: e most 45 trifling numerical theorems that in practice are of no use, and serve only to pass the time; and it has infected the minds of some people so much that they have dreamed of mighty mysteries involved in numbers, and tried to explain natural things by means of them. But if we look into our own thoughts, and consider the doctrines I have laid down, we may come to have a low opinion of those high flights and abstractions, and to look on all enquiries about numbers as merely trivial difficulties insofar as they are not practically useful in making our lives better. 120. Unity in the abstract I have considered in section 13. From that discussion and from what I said in the Introduction, it plainly follows there is no such idea. But number being defined as a collection of units, we can conclude that if there is no such thing as unity or unit in the abstract, there are no ideas of number in the abstract denoted by names and numerals. Therefore, if theories in arithmetic are abstracted Ÿfrom the names and numerals, and Ÿalso from all use and practical application as well as Ÿfrom particular things that are numbered, they have no subject matter at all. This shows us how entirely the science of numbers is subordinate to practical application, and how empty and trifling it becomes when considered as a matter of mere theory. 121. There may be some people who, deluded by the empty show of discovering abstracted truths, waste their time on useless arithmetical theorems and problems. So it will be worthwhile to consider that pretence more fully, and expose its emptiness. We can do this clearly by looking first at arithmetic in its infancy, observing what originally set men going on the study of that science, and what scope they gave it. It is natural to think that at first men, for ease of memory and help in calculations, made use of counters, or in writing made use of single strokes, points, or the like, each of which was made to stand for a unit - that is, some one thing of whatever kind they were dealing with at that time. Afterwards they discovered the more compact ways of making one symbol stand in place of several strokes or points. ·For example, the Romans used V instead of five points, X instead of ten points, and so on·. And lastly, the notation of the Arabians or Indians - ·the system using 1, 2, 3, etc.· - came into use, in which, by the repetition of a few characters or figures, and varying the meaning of each figure according to its place in the whole expression, all numbers can be conveniently expressed. This seems to have been done in imitation of language, so that the notation in numerals runs exactly parallel to the naming of numbers in words: the nine simple numerals correspond to the first nine names of numbers, and the position of a simple numeral in a longer one corresponds to the place of the corresponding word in a longer word-using name for a number. ·Thus, for example, ‘7’ corresponds to ‘seven’; and the significance of ‘7’ in ‘734’ - namely, as standing for seven hundreds - corresponds to the significance of ‘seven’ in ‘seven hundred and thirtyfour’·. And agreeably to those conditions of the simple and local value of figures, were contrived methods of finding from the given figures or marks of the parts, what figures, and how placed, are proper to denote the whole, or vice versa. [The preceding sentence is exactly as Berkeley wrote it.] Having found the numerals one seeks, keeping to the same rule or parallelism throughout, one can easily read them into words; and so the number becomes perfectly known. For we say that the number of such-and-suches is known when we know the names or numerals (in their proper order) that belong to the such-and-suches according to the standard system, For when we know these signs, we can through the 46 operations of arithmetic know the signs of any part of the particular sums signified by them; and by thus computing in signs (because of the connection established between them and the distinct numbers of things each of which is taken for a unit), we can correctly add up, divide, and proportion the things themselves that we intend to number. 122. In arithmetic therefore we have to do not with the things but with the signs, though these concern us not for their own sake but because they direct us how to act in relation to things, and how to manage them correctly. Just as I have remarked concerning language in general (section 19 (intro.)), so here too abstract ideas are thought to be signified by numerals or number-words at times when they don’t suggest ideas of particular things to our minds. I shall not go further into this subject now, but shall only remark that what I have said makes it clear that those things that are taken to be abstract truths and theorems concerning numbers are not really about anything except Ÿparticular countable things - or Ÿnames and numerals, which were first attended to only because they are signs that can represent aptly whatever particular things men needed to calculate...
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