An Essay Concerning Humane Understanding · John Locke
1. Both capable of greater and less.
Though we have in the precedent chapters dwelt pretty long on the considerations of space and duration, yet, they being ideas of general concernment, that have something very abstruse and peculiar in their nature, the comparing them one with another may perhaps be of use for their illustration; and we may have the more clear and distinct conception of them by taking a view of them together. Distance or space, in its simple abstract conception, to avoid confusion, I call EXPANSION, to distinguish it from extension, which by some is used to express this distance only as it is in the solid parts of matter, and so includes, or at least intimates, the idea of body: whereas the idea of pure distance includes no such thing. I prefer also the word expansion to space, because space is often applied to distance of fleeting successive parts, which never exist together, as well as to those which are permanent. In both these (viz. expansion and duration) the mind has this common idea of continued lengths, capable of greater or less quantities. For a man has as clear an idea of the difference of the length of an hour and a day, as of an inch and a foot.
2. Expansion not bounded by Matter.
The mind, having got the idea of the length of any part of expansion, let it be a span, or a pace, or what length you will, CAN, as has been said, repeat that idea, and so, adding it to the former, enlarge its idea of length, and make it equal to two spans, or two paces; and so, as often as it will, till it equals the distance of any parts of the earth one from another, and increase thus till it amounts to the distance of the sun or remotest star. By such a progression as this, setting out from the place where it is, or any other place, it can proceed and pass beyond all those lengths, and find nothing to stop its going on, either in or without body. It is true, we can easily in our thoughts come to the end of SOLID extension; the extremity and bounds of all body we have no difficulty to arrive at: but when the mind is there, it finds nothing to hinder its progress into this endless expansion; of that it can neither find nor conceive any end. Nor let any one say, that beyond the bounds of body, there is nothing at all; unless he will confine God within the limits of matter. Solomon, whose understanding was filled and enlarged with wisdom, seems to have other thoughts when he says, ‘Heaven, and the heaven of heavens, cannot contain thee.’ And he, I think, very much magnifies to himself the capacity of his own understanding, who persuades himself that he can extend his thoughts further than God exists, or imagine any expansion where He is not.
3. Nor Duration by Motion.
Just so is it in duration. The mind having got the idea of any length of duration, can double, multiply, and enlarge it, not only beyond its own, but beyond the existence of all corporeal beings, and all the measures of time, taken from the great bodies of all the world and their motions. But yet every one easily admits, that, though we make duration boundless, as certainly it is, we cannot yet extend it beyond all being. God, every one easily allows, fills eternity; and it is hard to find a reason why any one should doubt that He likewise fills immensity. His infinite being is certainly as boundless one way as another; and methinks it ascribes a little too much to matter to say, where there is no body, there is nothing.
4. Why Men more easily admit infinite Duration than infinite Expansion.
Hence I think we may learn the reason why every one familiarly and without the least hesitation speaks of and supposes Eternity, and sticks not to ascribe INFINITY to DURATION; but it is with more doubting and reserve that many admit or suppose the INFINITY OF SPACE. The reason whereof seems to me to be this,--That duration and extension being used as names of affections belonging to other beings, we easily conceive in God infinite duration, and we cannot avoid doing so: but, not attributing to him extension, but only to matter, which is finite, we are apter to doubt of the existence of expansion without matter; of which alone we commonly suppose it an attribute. And, therefore, when men pursue their thoughts of space, they are apt to stop at the confines of body: as if space were there at an end too, and reached no further. Or if their ideas, upon consideration, carry them further, yet they term what is beyond the limits of the universe, imaginary space: as if IT were nothing, because there is no body existing in it. Whereas duration, antecedent to all body, and to the motions which it is measured by, they never term imaginary: because it is never supposed void of some other real existence. And if the names of things may at all direct our thoughts towards the original of men’s ideas, (as I am apt to think they may very much,) one may have occasion to think by the name DURATION, that the continuation of existence, with a kind of resistance to any destructive force, and the continuation of solidity (which is apt to be confounded with, and if we will look into the minute anatomical parts of matter, is little different from, hardness) were thought to have some analogy, and gave occasion to words so near of kin as durare and durum esse. And that durare is applied to the idea of hardness, as well as that of existence, we see in Horace, Epod. xvi. ferro duravit secula. But, be that as it will, this is certain, that whoever pursues his own thoughts, will find them sometimes launch out beyond the extent of body, into the infinity of space or expansion; the idea whereof is distinct and separate from body and all other things: which may, (to those who please,) be a subject of further meditation.
5. Time to Duration is as Place to Expansion.
Time in general is to duration as place to expansion. They are so much of those boundless oceans of eternity and immensity as is set out and distinguished from the rest, as it were by landmarks; and so are made use of to denote the position of FINITE real beings, in respect one to another, in those uniform infinite oceans of duration and space. These, rightly considered, are only ideas of determinate distances from certain known points, fixed in distinguishable sensible things, and supposed to keep the same distance one from another. From such points fixed in sensible beings we reckon, and from them we measure our portions of those infinite quantities; which, so considered, are that which we call TIME and PLACE. For duration and space being in themselves uniform and boundless, the order and position of things, without such known settled points, would be lost in them; and all things would lie jumbled in an incurable confusion.
6. Time and Place are taken for so much of either as are set out by the Existence and Motion of Bodies.
Time and place, taken thus for determinate distinguishable portions of those infinite abysses of space and duration, set out or supposed to be distinguished from the rest, by marks and known boundaries, have each of them a twofold acceptation.
FIRST, Time in general is commonly taken for so much of infinite duration as is measured by, and co-existent with, the existence and motions of the great bodies of the universe, as far as we know anything of them: and in this sense time begins and ends with the frame of this sensible world, as in these phrases before mentioned, ‘Before all time,’ or, ‘When time shall be no more.’ Place likewise is taken sometimes for that portion of infinite space which is possessed by and comprehended within the material world; and is thereby distinguished from the rest of expansion; though this may be more properly called extension than place. Within these two are confined, and by the observable parts of them are measured and determined, the particular time or duration, and the particular extension and place, of all corporeal beings.
7. Sometimes for so much of either as we design by Measures taken from the Bulk or Motion of Bodies.
SECONDLY, sometimes the word time is used in a larger sense, and is applied to parts of that infinite duration, not that were really distinguished and measured out by this real existence, and periodical motions of bodies, that were appointed from the beginning to be for signs and for seasons and for days and years, and are accordingly our measures of time; but such other portions too of that infinite uniform duration, which we upon any occasion do suppose equal to certain lengths of measured time; and so consider them as bounded and determined. For, if we should suppose the creation, or fall of the angels, was at the beginning of the Julian period, we should speak properly enough, and should be understood if we said, it is a longer time since the creation of angels than the creation of the world, by 7640 years: whereby we would mark out so much of that undistinguished duration as we suppose equal to, and would have admitted, 7640 annual revolutions of the sun, moving at the rate it now does. And thus likewise we sometimes speak of place, distance, or bulk, in the great INANE, beyond the confines of the world, when we consider so much of that space as is equal to, or capable to receive, a body of any assigned dimensions, as a cubic foot; or do suppose a point in it, at such a certain distance from any part of the universe.
8. They belong to all finite beings.
WHERE and WHEN are questions belonging to all finite existences, and are by us always reckoned from some known parts of this sensible world, and from some certain epochs marked out to us by the motions observable in it. Without some such fixed parts or periods, the order of things would be lost, to our finite understandings, in the boundless invariable oceans of duration and expansion, which comprehend in them all finite beings, and in their full extent belong only to the Deity. And therefore we are not to wonder that we comprehend them not, and do so often find our thoughts at a loss, when we would consider them, either abstractly in themselves, or as any way attributed to the first incomprehensible Being. But when applied to any particular finite beings, the extension of any body is so much of that infinite space as the bulk of the body takes up. And place is the position of any body, when considered at a certain distance from some other. As the idea of the particular duration of anything is, an idea of that portion of infinite duration which passes during the existence of that thing; so the time when the thing existed is, the idea of that space of duration which passed between some known and fixed period of duration, and the being of that thing. One shows the distance of the extremities of the bulk or existence of the same thing, as that it is a foot square, or lasted two years; the other shows the distance of it in place, or existence from other fixed points of space or duration, as that it was in the middle of Lincoln’s Inn Fields, or the first degree of Taurus, and in the year of our Lord 1671, or the 1000th year of the Julian period. All which distances we measure by preconceived ideas of certain lengths of space and duration,--as inches, feet, miles, and degrees, and in the other, minutes, days, and years, &c.
9. All the Parts of Extension are Extension, and all the Parts of Duration are Duration.
There is one thing more wherein space and duration have a great conformity, and that is, though they are justly reckoned amongst our SIMPLE IDEAS, yet none of the distinct ideas we have of either is without all manner of composition: it is the very nature of both of them to consist of parts: but their parts being all of the same kind, and without the mixture of any other idea, hinder them not from having a place amongst simple ideas. Could the mind, as in number, come to so small a part of extension or duration as excluded divisibility, THAT would be, as it were, the indivisible unit or idea; by repetition of which, it would make its more enlarged ideas of extension and duration. But, since the mind is not able to frame an idea of ANY space without parts, instead thereof it makes use of the common measures, which, by familiar use in each country, have imprinted themselves on the memory (as inches and feet; or cubits and parasangs; and so seconds, minutes, hours, days, and years in duration);--the mind makes use, I say, of such ideas as these, as simple ones: and these are the component parts of larger ideas, which the mind upon occasion makes by the addition of such known lengths which it is acquainted with. On the other side, the ordinary smallest measure we have of either is looked on as an unit in number, when the mind by division would reduce them into less fractions. Though on both sides, both in addition and division, either of space or duration, when the idea under consideration becomes very big or very small, its precise bulk becomes very obscure and confused; and it is the NUMBER of its repeated additions or divisions that alone remains clear and distinct; as will easily appear to any one who will let his thoughts loose in the vast expansion of space, or divisibility of matter. Every part of duration is duration too; and every part of extension is extension, both of them capable of addition or division in infinitum. But THE LEAST PORTIONS OF EITHER OF THEM, WHEREOF WE HAVE CLEAR AND DISTINCT IDEAS, may perhaps be fittest to be considered by us, as the simple ideas of that kind out of which our complex modes of space, extension, and duration are made up, and into which they can again be distinctly resolved. Such a small part in duration may be called a MOMENT, and is the time of one idea in our minds, in the train of their ordinary succession there. The other, wanting a proper name, I know not whether I may be allowed to call a SENSIBLE POINT, meaning thereby the least particle of matter or space we can discern, which is ordinarily about a minute, and to the sharpest eyes seldom less than thirty seconds of a circle, whereof the eye is the centre.
10. Their Parts inseparable.
Expansion and duration have this further agreement, that, though they are both considered by us as having parts, yet their parts are not separable one from another, no not even in thought: though the parts of bodies from whence we take our MEASURE of the one; and the parts of motion, or rather the succession of ideas in our minds, from whence we take the MEASURE of the other, may be interrupted and separated; as the one is often by rest, and the other is by sleep, which we call rest too.
11. Duration is as a Line, Expansion as a Solid.
But there is this manifest difference between them,--That the ideas of length which we have of expansion are turned every way, and so make figure, and breadth, and thickness; but duration is but as it were the length of one straight line, extended in infinitum, not capable of multiplicity, variation, or figure; but is one common measure of all existence whatsoever, wherein all things, whilst they exist, equally partake. For this present moment is common to all things that are now in being, and equally comprehends that part of their existence, as much as if they were all but one single being; and we may truly say, they all exist in the SAME moment of time. Whether angels and spirits have any analogy to this, in respect to expansion, is beyond my comprehension: and perhaps for us, who have understandings and comprehensions suited to our own preservation, and the ends of our own being, but not to the reality and extent of all other beings, it is near as hard to conceive any existence, or to have an idea of any real being, with a perfect negation of all manner of expansion, as it is to have the idea of any real existence with a perfect negation of all manner of duration. And therefore, what spirits have to do with space, or how they communicate in it, we know not. All that we know is, that bodies do each singly possess its proper portion of it, according to the extent of solid parts; and thereby exclude all other bodies from having any share in that particular portion of space, whilst it remains there.
12. Duration has never two Parts together, Expansion altogether.
DURATION, and TIME which is a part of it, is the idea we have of PERISHING distance, of which no two parts exist together, but follow each other in succession; an EXPANSION is the idea of LASTING distance, all whose parts exist together and are not capable of succession. And therefore, though we cannot conceive any duration without succession, nor can put it together in our thoughts that any being does NOW exist to-morrow, or possess at once more than the present moment of duration; yet we can conceive the eternal duration of the Almighty far different from that of man, or any other finite being. Because man comprehends not in his knowledge or power all past and future things: his thoughts are but of yesterday, and he knows not what to-morrow will bring forth. What is once past he can never recall; and what is yet to come he cannot make present. What I say of man, I say of all finite beings; who, though they may far exceed man in knowledge and power, yet are no more than the meanest creature, in comparison with God himself. Finite or any magnitude holds not any proportion to infinite. God’s infinite duration, being accompanied with infinite knowledge and infinite power, he sees all things, past and to come; and they are no more distant from his knowledge, no further removed from his sight, than the present: they all lie under the same view: and there is nothing which he cannot make exist each moment he pleases. For the existence of all things, depending upon his good pleasure, all things exist every moment that he thinks fit to have them exist. To conclude: expansion and duration do mutually embrace and comprehend each other; every part of space being in every part of duration, and every part of duration in every part of expansion. Such a combination of two distinct ideas is, I suppose, scarce to be found in all that great variety we do or can conceive, and may afford matter to further speculation.
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.
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End of part 11 of 41
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