The Concept of Number

8 min

1. Number the simplest and most universal Idea.

1. Number is 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.

Of all our ideas, none comes to mind in more ways, and none is simpler, than UNITY, or one. This idea contains not the slightest variety or combination of parts. Every object we sense, every idea we understand, and every thought we have brings with it the idea of one. So this idea is the most closely connected to our thoughts. It is also our most universal idea, since it applies to all things. For number applies to people, angels, actions, and thoughts, to everything that exists or can be imagined.

2. Its Modes made by Addition.

2. Its modes are formed 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.

By repeating this idea in our minds and adding the repetitions together, we form COMPLEX ideas of its MODES. Thus, by adding one to one, we form the complex idea of a pair. By putting twelve units together, we form the complex idea of a dozen. The same applies to twenty, a million, or any other number.

3. Each Mode distinct.

3. Each mode is 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?

The SIMPLE MODES of NUMBER are the most distinct of all simple modes. Even the smallest change, a single unit, makes each combination clearly different from every other combination. The nearest combination is as clearly different as the most distant. Two is as distinct from one as it is from two hundred. And the idea of two is as distinct from the idea of three as the earth’s size is from a mite’s. Other simple modes are not like this. We cannot so easily, or perhaps at all, distinguish between two nearly identical ideas that are nevertheless truly different. For who would claim to distinguish the whiteness of this paper from the very next shade of white? Or who can form distinct ideas of every tiny increase in size?

4. Therefore Demonstrations in Numbers the most precise.

4. Therefore, proofs involving numbers are 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.

Each number is clearly distinct from every other number, even those closest to it. This leads me to think that numerical proofs have broader uses and more definite applications than proofs about spatial dimensions. This is so even if numerical proofs are no more evident or exact. For ideas of numbers are more precise and easier to distinguish than ideas of spatial dimensions. In space, equalities and differences in size are not so easy to observe or measure. This is because our thoughts cannot reach a fixed smallest size in space that serves as a unit. We can always imagine something smaller. So we cannot discover the size or proportion of the smallest difference. Numbers, by contrast, make such differences clear. As I have said, 91 is as distinct from 90 as from 9000, though 91 is only one above 90. But spatial dimensions are different. Something longer than exactly a foot or an inch cannot always be distinguished from that standard length. Two lines may appear equal, yet one may exceed the other by countless small parts. Nor can anyone specify an angle that is the next largest after a right angle.

5. Names necessary to Numbers.

5. Numbers need names.

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.

As already explained, we repeat the idea of one unit and join that unit to another. We form a single combined idea and give it the name two. Anyone who can do this can continue by adding one more unit to the last named collection. They may count, or form ideas of, distinct collections of units. They can go as far as they have names for successive numbers and can remember the sequence and its names. For all counting consists of adding one more unit and giving the resulting whole a new name or sign. We understand that whole as a single idea. Its name or sign distinguishes it from the numbers before and after it, and from every smaller or larger collection. So anyone who can add one to one, then one to two, can continue counting. They must keep track of the distinct name belonging to each step. They can also work backward, subtracting one unit from each collection and making the collections smaller. Such a person can form all the ideas of numbers their language covers, or for which they have names. But perhaps they cannot form ideas of numbers beyond those. For the different simple forms of numbers in our minds are just combinations of units. These units have no variety, and their combinations can differ only in having more or fewer units. Names or marks for each distinct combination therefore seem more necessary than for any other kind of idea. Without such names or marks, we can hardly use numbers properly in calculation, especially when a collection contains many units. Without a name or mark identifying that exact collection, we can hardly keep those units from becoming a confused heap.

6. Another reason for the necessity of names to numbers.

6. Another reason why numbers need names.

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.

I think this explains why some Americans I have spoken with could not count to 1000 as we do. In other respects, they were quick-minded and quite capable of reasoning. Yet they had no clear idea of that number, though they could count very well to 20. This was because their language had few words, suited only to the few necessities of a poor, simple life. They knew neither trade nor mathematics, and their language had no words for 1000. So when people spoke to them about these larger numbers, they pointed to the hairs on their heads. In this way, they expressed a great multitude that they could not count. I suppose their inability to count such numbers came from their lack of names for them. The Tououpinambos had no names for numbers above 5. They expressed any number beyond that by showing their fingers and the fingers of others present. And I have no doubt that we ourselves could clearly count in words much further than we usually do. We would only need to find suitable names for those larger numbers. Instead, we now name them by saying millions of millions of millions, and so on. With this method, it is hard to go beyond eighteen, or at most twenty-four, decimal places without confusion. But distinct names greatly help us count accurately and form useful ideas of numbers. To show how much they help, let us look at all the following digits in one continuous line. Read them as representing a single number, for example:

Nonillions. 857324

Octillions. 162486

Septillions. 345896

Sextillions. 437918

Quintrillions. 423147

Quintillions. 423147

Quartrillions. 248106

Quadrillions. 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.

The usual way to name this number in English is to repeat millions, of millions, of millions, of millions, of millions, of millions, of millions, of millions. Millions is the name given to the second group of six digits. Naming the number this way makes it very hard to form a clear idea of it. But suppose we gave each group of six digits a new name in an orderly sequence. Might we then easily count these digits distinctly, and perhaps many more in the same progression? Might we also form ideas of these numbers more easily and communicate those ideas more clearly to others? I leave these questions for consideration. I mention this possibility only to show how necessary distinct names are for counting. I do not mean to introduce new names of my own invention.

7. Why Children number not earlier.

7. Why children do not learn to count 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.

Children may lack names for the successive steps in counting. Or they may not yet be able to combine separate ideas into complex ones and arrange them in order. They must also hold these ideas in memory, as counting requires. So children do not begin counting very early. Nor do they count very far or reliably until long after they have acquired many other ideas. We can often observe children talking and reasoning quite well, with very clear ideas about several other things, before they can count to twenty. Some people, because of weaknesses in memory, cannot count or correctly work through even a moderately long series of numbers throughout their lives. They cannot remember the different combinations of numbers, together with their names, in the proper order. Nor can they remember how the steps in a long numerical sequence depend on and relate to one another. For anyone who counts to twenty, or has any idea of that number, must know that nineteen comes before it. They must also know the distinct name or sign for every preceding number, in its proper order. Wherever this knowledge fails, a gap appears, the chain breaks, and counting can go no further. So counting correctly requires two things: (1) The mind must carefully distinguish two ideas that differ only by adding or subtracting ONE unit. (2) The mind must remember the names or signs of the different combinations, from one unit up to that number. These must be remembered not in confusion or at random, but in the exact order in which the numbers follow one another. If the mind fails at either task, the whole process of counting is disrupted. Only a confused idea of a large quantity remains, without the ideas needed to count distinctly.

8. Number measures all Measurables.

8. Numbers measure everything that can be measured.

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.

Another thing to notice about number is that the mind uses it to measure everything we can measure. The main things we measure are EXPANSION and DURATION, or space and time. Even when applied to space and time, our idea of infinity seems to be nothing but the infinity of number. For what are our ideas of Eternity and Immensity but repeated additions of imagined portions of time and space? We add these portions using the infinity of number, so we can never reach an end to the additions. Of all our ideas, number most clearly provides us with such an inexhaustible supply, as everyone can plainly see. For suppose someone adds together as large a number as he pleases. However great the total, it does not reduce his ability to add to it in the slightest. Nor does it bring him any closer to exhausting the supply of numbers. There is still as much left to add as if none had been taken away. This ENDLESS ADDITION of numbers, so clear to the mind, gives us, I think, our clearest and most distinct idea of infinity. We could instead call it ADDIBILITY, if anyone prefers that word. I will discuss infinity further in the next chapter.