IDEA OF NUMBER.
1. Number the simplest and most universal Idea.
Amongst all the ideas we have, as there is none suggested to the mind
by more ways, so there is none more simple, than that of UNITY, or one:
it has no shadow of variety or composition in it: every object our
senses are employed about; every idea in our understandings; every
thought of our minds, brings this idea along with it. And therefore it
is the most intimate to our thoughts, as well as it is, in its
agreement to all other things, the most universal idea we have. For
number applies itself to men, angels, actions, thoughts; everything
that either doth exist or can be imagined.
2. Its Modes made by Addition.
By repeating this idea in our minds, and adding the repetitions
together, we come by the COMPLEX ideas of the MODES of it. Thus, by
adding one to one, we have the complex idea of a couple; by putting
twelve units together we have the complex idea of a dozen; and so of a
score or a million, or any other number.
3. Each Mode distinct.
The SIMPLE MODES of NUMBER are of all other the most distinct; every
the least variation, which is an unit, making each combination as
clearly different from that which approacheth nearest to it, as the
most remote; two being as distinct from one, as two hundred; and the
idea of two as distinct from the idea of three, as the magnitude of the
whole earth is from that of a mite. This is not so in other simple
modes, in which it is not so easy, nor perhaps possible for us to
distinguish betwixt two approaching ideas, which yet are really
different. For who will undertake to find a difference between the
white of this paper and that of the next degree to it: or can form
distinct ideas of every the least excess in extension?
4. Therefore Demonstrations in Numbers the most precise.
The clearness and distinctness of each mode of number from all others,
even those that approach nearest, makes me apt to think that
demonstrations in numbers, if they are not more evident and exact than
in extension, yet they are more general in their use, and more
determinate in their application. Because the ideas of numbers are more
precise and distinguishable than in extension; where every equality and
excess are not so easy to be observed or measured; because our thoughts
cannot in space arrive at any determined smallness beyond which it
cannot go, as an unit; and therefore the quantity or proportion of any
the least excess cannot be discovered; which is clear otherwise in
number, where, as has been said, 91 is as distinguishable from 90 as
from 9000, though 91 be the next immediate excess to 90. But it is not
so in extension, where, whatsoever is more than just a foot or an inch,
is not distinguishable from the standard of a foot or an inch; and in
lines which appear of an equal length, one may be longer than the other
by innumerable parts: nor can any one assign an angle, which shall be
the next biggest to a right one.
5. Names necessary to Numbers.
By the repeating, as has been said, the idea of an unit, and joining it
to another unit, we make thereof one collective idea, marked by the
name two. And whosoever can do this, and proceed on, still adding one
more to the last collective idea which he had of any number, and gave a
name to it, may count, or have ideas, for several collections of units,
distinguished one from another, as far as he hath a series of names for
following numbers, and a memory to retain that series, with their
several names: all numeration being but still the adding of one unit
more, and giving to the whole together, as comprehended in one idea, a
new or distinct name or sign, whereby to know it from those before and
after, and distinguish it from every smaller or greater multitude of
units. So that he that can add one to one, and so to two, and so go on
with his tale, taking still with him the distinct names belonging to
every progression; and so again, by subtracting an unit from each
collection, retreat and lessen them, is capable of all the ideas of
numbers within the compass of his language, or for which he hath names,
though not perhaps of more. For, the several simple modes of numbers
being in our minds but so many combinations of units, which have no
variety, nor are capable of any other difference but more or less,
names or marks for each distinct combination seem more necessary than
in any other sort of ideas. For, without such names or marks, we can
hardly well make use of numbers in reckoning, especially where the
combination is made up of any great multitude of units; which put
together, without a name or mark to distinguish that precise
collection, will hardly be kept from being a heap in confusion.
6. Another reason for the necessity of names to numbers.
This I think to be the reason why some Americans I have spoken with,
(who were otherwise of quick and rational parts enough,) could not, as
we do, by any means count to 1000; nor had any distinct idea of that
number, though they could reckon very well to 20. Because their
language being scanty, and accommodated only to the few necessaries of
a needy, simple life, unacquainted either with trade or mathematics,
had no words in it to stand for 1000; so that when they were discoursed
with of those greater numbers, they would show the hairs of their head,
to express a great multitude, which they could not number; which
inability, I suppose, proceeded from their want of names. The
Tououpinambos had no names for numbers above 5; any number beyond that
they made out by showing their fingers, and the fingers of others who
were present. And I doubt not but we ourselves might distinctly number
in words a great deal further than we usually do, would we find out but
some fit denominations to signify them by; whereas, in the way we take
now to name them, by millions of millions of millions, &c., it is hard
to go beyond eighteen, or at most, four and twenty, decimal
progressions, without confusion. But to show how much distinct names
conduce to our well reckoning, or having useful ideas of numbers, let
us see all these following figures in one continued line, as the marks
of one number: v. g.
Nonillions. 857324
Octillions. 162486
Septillions. 345896
Sextillions. 437918
Quintrillions. 423147
Quartrillions. 248106
Trillions. 235421
Billions. 261734
Millions. 368149
Units. 623137
The ordinary way of naming this number in English, will be the often
repeating of millions, of millions, of millions, of millions, of
millions, of millions, of millions, of millions, (which is the
denomination of the second six figures). In which way, it will be very
hard to have any distinguishing notions of this number. But whether, by
giving every six figures a new and orderly denomination, these, and
perhaps a great many more figures in progression, might not easily be
counted distinctly, and ideas of them both got more easily to
ourselves, and more plainly signified to others, I leave it to be
considered. This I mention only to show how necessary distinct names
are to numbering, without pretending to introduce new ones of my
invention.
7. Why Children number not earlier.
Thus children, either for want of names to mark the several
progressions of numbers, or not having yet the faculty to collect
scattered ideas into complex ones, and range them in a regular order,
and so retain them in their memories, as is necessary to reckoning, do
not begin to number very early, nor proceed in it very far or steadily,
till a good while after they are well furnished with good store of
other ideas: and one may often observe them discourse and reason pretty
well, and have very clear conceptions of several other things, before
they can tell twenty. And some, through the default of their memories,
who cannot retain the several combinations of numbers, with their
names, annexed in their distinct orders, and the dependence of so long
a train of numeral progressions, and their relation one to another, are
not able all their lifetime to reckon, or regularly go over any
moderate series of numbers. For he that will count twenty, or have any
idea of that number, must know that nineteen went before, with the
distinct name or sign of every one of them, as they stand marked in
their order; for wherever this fails, a gap is made, the chain breaks,
and the progress in numbering can go no further. So that to reckon
right, it is required, (1) That the mind distinguish carefully two
ideas, which are different one from another only by the addition or
subtraction of ONE unit: (2) That it retain in memory the names or
marks of the several combinations, from an unit to that number; and
that not confusedly, and at random, but in that exact order that the
numbers follow one another. In either of which, if it trips, the whole
business of numbering will be disturbed, and there will remain only the
confused idea of multitude, but the ideas necessary to distinct
numeration will not be attained to.
8. Number measures all Measurables.
This further is observable in number, that it is that which the mind
makes use of in measuring all things that by us are measurable, which
principally are EXPANSION and DURATION; and our idea of infinity, even
when applied to those, seems to be nothing but the infinity of number.
For what else are our ideas of Eternity and Immensity, but the repeated
additions of certain ideas of imagined parts of duration and expansion,
with the infinity of number; in which we can come to no end of
addition? For such an inexhaustible stock, number (of all other our
ideas) most clearly furnishes us with, as is obvious to every one. For
let a man collect into one sum as great a number as he pleases, this
multitude how great soever, lessens not one jot the power of adding to
it, or brings him any nearer the end of the inexhaustible stock of
number; where still there remains as much to be added, as if none were
taken out. And this ENDLESS ADDITION or ADDIBILITY (if any one like the
word better) of numbers, so apparent to the mind, is that, I think,
which gives us the clearest and most distinct idea of infinity: of
which more in the following chapter.