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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 wordusing 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 suchandsuches is known when
we know the names or numerals (in their proper order) that belong to the suchandsuches
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 numberwords 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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 Spring '13
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 Philosophy, Thought, ........., Abstraction, ·by Locke

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