A Treatise Concerning the Principles of Human Knowledge by George Berkeley. - HTML preview

sensible things, to which nevertheless in their own nature they bear no

relation at all.

111. As for Time, as it is there taken in an absolute or abstracted

sense, for the duration or perseverance of the existence of things, I

have nothing more to add concerning it after what has been already said

on that subject. Sect. 97 and 98. For the rest, this celebrated author

holds there is an absolute Space, which, being unperceivable to sense,

remains in itself similar and immovable; and relative space to be the

measure thereof, which, being movable and defined by its situation in

respect of sensible bodies, is vulgarly taken for immovable space. Place

he defines to be that part of space which is occupied by any body; and

according as the space is absolute or relative so also is the place.

Absolute Motion is said to be the translation of a body from absolute

place to absolute place, as relative motion is from one relative place to

another. And, because the parts of absolute space do not fall under our

senses, instead of them we are obliged to use their sensible measures,

and so define both place and motion with respect to bodies which we

regard as immovable. But, it is said in philosophical matters we must

abstract from our senses, since it may be that none of those bodies which

seem to be quiescent are truly so, and the same thing which is moved

relatively may be really at rest; as likewise one and the same body may

be in relative rest and motion, or even moved with contrary relative

motions at the same time, according as its place is variously defined.

All which ambiguity is to be found in the apparent motions, but not at

all in the true or absolute, which should therefore be alone regarded in

philosophy. And the true as we are told are distinguished from apparent

or relative motions by the following properties.--First, in true or

absolute motion all parts which preserve the same position with respect

of the whole, partake of the motions of the whole.

Secondly, the place

being moved, that which is placed therein is also moved; so that a body

moving in a place which is in motion doth participate the motion of its

place. Thirdly, true motion is never generated or changed otherwise than

by force impressed on the body itself. Fourthly, true motion is always

changed by force impressed on the body moved. Fifthly, in circular motion

barely relative there is no centrifugal force, which, nevertheless, in

that which is true or absolute, is proportional to the quantity of

motion.

112. MOTION, WHETHER REAL OR APPARENT, RELATIVE.--But, notwithstanding

what has been said, I must confess it does not appear to me that

there can be any motion other than relative; so that to conceive

motion there must be at least conceived two bodies, whereof the

distance or position in regard to each other is varied.

Hence, if there

was one only body in being it could not possibly be moved. This seems

evident, in that the idea I have of motion doth necessarily include

relation.

113. APPARENT MOTION DENIED.--But, though in every motion it be

necessary to conceive more bodies than one, yet it may be that one

only is moved, namely, that on which the force causing the change

in the distance or situation of the bodies, is impressed. For, however

some may define relative motion, so as to term that body moved

which changes its distance from some other body, whether the force

or action causing that change were impressed on it or no, yet as

relative motion is that which is perceived by sense, and regarded in

the ordinary affairs of life, it should seem that every man of common

sense knows what it is as well as the best philosopher.

Now, I ask any

one whether, in his sense of motion as he walks along the streets, the

stones he passes over may be said to move, because they change distance

with his feet? To me it appears that though motion includes a relation of

one thing to another, yet it is not necessary that each term of the

relation be denominated from it. As a man may think of somewhat which

does not think, so a body may be moved to or from another body which is

not therefore itself in motion.

114. As the place happens to be variously defined, the motion which is

related to it varies. A man in a ship may be said to be quiescent with

relation to the sides of the vessel, and yet move with relation to the

land. Or he may move eastward in respect of the one, and westward in

respect of the other. In the common affairs of life men never go beyond

the earth to define the place of any body; and what is quiescent in

respect of that is accounted absolutely to be so. But philosophers, who

have a greater extent of thought, and juster notions of the system of

things, discover even the earth itself to be moved. In order therefore to

fix their notions they seem to conceive the corporeal world as finite,

and the utmost unmoved walls or shell thereof to be the place whereby

they estimate true motions. If we sound our own conceptions, I believe we

may find all the absolute motion we can frame an idea of to be at bottom

no other than relative motion thus defined. For, as has been already

observed, absolute motion, exclusive of all external relation, is

incomprehensible; and to this kind of relative motion all the

above-mentioned properties, causes, and effects ascribed to absolute

motion will, if I mistake not, be found to agree. As to what is said of

the centrifugal force, that it does not at all belong to circular

relative motion, I do not see how this follows from the experiment which

is brought to prove it. See Philosophiae Naturalis Principia Mathematica,

in Schol. Def. VIII. For the water in the vessel at that time wherein it

is said to have the greatest relative circular motion, has, I think, no

motion at all; as is plain from the foregoing section.

115. For, to denominate a body moved it is requisite, first, that it

change its distance or situation with regard to some other body; and

secondly, that the force occasioning that change be applied to it. If

either of these be wanting, I do not think that, agreeably to the sense

of mankind, or the propriety of language, a body can be said to be in

motion. I grant indeed that it is possible for us to think a body which

we see change its distance from some other to be moved, though it have no

force applied to it (in which sense there may be apparent motion), but

then it is because the force causing the change of distance is imagined

by us to be applied or impressed on that body thought to move; which

indeed shows we are capable of mistaking a thing to be in motion which is

not, and that is all.

116. ANY IDEA OF PURE SPACE RELATIVE.--From what has been said it follows

that the philosophic consideration of motion does not imply the

being of an absolute Space, distinct from that which is perceived

by sense and related bodies; which that it cannot exist without the

mind is clear upon the same principles that demonstrate the like

of all other objects of sense. And perhaps, if we inquire narrowly,

we shall find we cannot even frame an idea of pure Space exclusive

of all body. This I must confess seems impossible, as being a most

abstract idea. When I excite a motion in some part of my body,

if it be free or without resistance, I say there is Space; but if I

find a resistance, then I say there is Body; and in proportion as the

resistance to motion is lesser or greater, I say the space is more or

less pure. So that when I speak of pure or empty space, it is not to be

supposed that the word "space" stands for an idea distinct from or

conceivable without body and motion--though indeed we are apt to think

every noun substantive stands for a distinct idea that may be separated

from all others; which has occasioned infinite mistakes.

When, therefore,

supposing all the world to be annihilated besides my own body, I say

there still remains pure Space, thereby nothing else is meant but only

that I conceive it possible for the limbs of my body to be moved on all

sides without the least resistance, but if that, too, were annihilated

then there could be no motion, and consequently no Space. Some, perhaps,

may think the sense of seeing doth furnish them with the idea of pure

space; but it is plain from what we have elsewhere shown, that the ideas

of space and distance are not obtained by that sense.

See the Essay

concerning Vision.

117. What is here laid down seems to put an end to all those disputes and

difficulties that have sprung up amongst the learned concerning the

nature of pure Space. But the chief advantage arising from it is that we

are freed from that dangerous dilemma, to which several who have employed

their thoughts on that subject imagine themselves reduced, to wit, of

thinking either that Real Space is God, or else that there is something

beside God which is eternal, uncreated, infinite, indivisible, immutable.

Both which may justly be thought pernicious and absurd notions. It is

certain that not a few divines, as well as philosophers of great note,

have, from the difficulty they found in conceiving either limits or

annihilation of space, concluded it must be divine. And some of late have

set themselves particularly to show the incommunicable attributes of God

agree to it. Which doctrine, how unworthy soever it may seem of the

Divine Nature, yet I do not see how we can get clear of it, so long as we

adhere to the received opinions.

118. THE ERRORS ARISING FROM THE DOCTRINES OF

ABSTRACTION AND EXTERNAL

MATERIAL EXISTENCES, INFLUENCE MATHEMATICAL REASONINGS.-

-Hitherto of

Natural Philosophy: we come now to make some inquiry concerning

that other great branch of speculative knowledge, to wit, Mathematics.

These, how celebrated soever they may be for their clearness and

certainty of demonstration, which is hardly anywhere else to be

found, cannot nevertheless be supposed altogether free from mistakes, if

in their principles there lurks some secret error which is common to the

professors of those sciences with the rest of mankind.

Mathematicians,

though they deduce their theorems from a great height of evidence, yet

their first principles are limited by the consideration of quantity: and

they do not ascend into any inquiry concerning those transcendental

maxims which influence all the particular sciences, each part whereof,

Mathematics not excepted, does consequently participate of the errors

involved in them. That the principles laid down by mathematicians are

true, and their way of deduction from those principles clear and

incontestible, we do not deny; but, we hold there may be certain

erroneous maxims of greater extent than the object of Mathematics, and

for that reason not expressly mentioned, though tacitly supposed

throughout the whole progress of that science; and that the ill effects

of those secret unexamined errors are diffused through all the branches

thereof. To be plain, we suspect the mathematicians are as well as other

men concerned in the errors arising from the doctrine of abstract general

ideas, and the existence of objects without the mind.

119. Arithmetic has been thought to have for its object abstract ideas of

Number; of which to understand the properties and mutual habitudes, is

supposed no mean part of speculative knowledge. The opinion of the pure

and intellectual nature of numbers in abstract has made them in esteem

with those philosophers who seem to have affected an uncommon fineness

and elevation of thought. It has set a price on the most trifling

numerical speculations which in practice are of no use, but serve only

for amusement; and has therefore so far infected the minds of some, that

they have dreamed of mighty mysteries involved in numbers, and attempted

the explication of natural things by them. But, if we inquire into our

own thoughts, and consider what has been premised, we may perhaps

entertain a low opinion of those high flights and abstractions, and look

on all inquiries, about numbers only as so many difficiles nugae, so far

as they are not subservient to practice, and promote the benefit of life.

120. Unity in abstract we have before considered in sect. 13, from which

and what has been said in the Introduction, it plainly follows there is

not any such idea. But, number being defined a

"collection of units," we may conclude that, if there be no such thing as unity or unit in

abstract, there are no ideas of number in abstract denoted by the numeral

names and figures. The theories therefore in Arithmetic, if they are

abstracted from the names and figures, as likewise from all use and

practice, as well as from the particular things numbered, can be supposed

to have nothing at all for their object; hence we may see how entirely

the science of numbers is subordinate to practice, and how jejune and

trifling it becomes when considered as a matter of mere speculation.

121. However, since there may be some who, deluded by the specious show

of discovering abstracted verities, waste their time in arithmetical

theorems and problems which have not any use, it will not be amiss if we

more fully consider and expose the vanity of that pretence; and this will

plainly appear by taking a view of Arithmetic in its infancy, and

observing what it was that originally put men on the study of that

science, and to what scope they directed it. It is natural to think that

at first, men, for ease of memory and help of computation, made use of

counters, or in writing of single strokes, points, or the like, each

whereof was made to signify an unit, i.e., some one thing of whatever

kind they had occasion to reckon. Afterwards they found out the more

compendious ways of making one character stand in place of several

strokes or points. And, lastly, the notation of the Arabians or Indians

came into use, wherein, by the repetition of a few characters or figures,

and varying the signification of each figure according to the place it

obtains, all numbers may be most aptly expressed; which seems to have

been done in imitation of language, so that an exact analogy is observed

betwixt the notation by figures and names, the nine simple figures

answering the nine first numeral names and places in the former,

corresponding to denominations in the latter. 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. And

having found the sought figures, the same rule or analogy being observed

throughout, it is easy to read them into words; and so the number becomes

perfectly known. For then the number of any particular things is said to

be known, when we know the name of figures (with their due arrangement)

that according to the standing analogy belong to them.

For, these signs

being known, we can by the operations of arithmetic know the signs of any

part of the particular sums signified by them; and, thus computing in

signs (because of the connexion established betwixt them and the distinct

multitudes of things whereof one is taken for an unit), we may be able

rightly to sum up, divide, and proportion the things themselves that we

intend to number.

122. In Arithmetic, therefore, we regard not the things, but the signs,

which nevertheless are not regarded for their own sake, but because they

direct us how to act with relation to things, and dispose rightly of

them. Now, agreeably to what we have before observed of words in general

(sect. 19, Introd.) it happens here likewise that abstract ideas are

thought to be signified by numeral names or characters, while they do not

suggest ideas of particular things to our minds. I shall not at present

enter into a more particular dissertation on this subject, but only

observe that it is evident from what has been said, those things which

pass for abstract truths and theorems concerning numbers, are in reality

conversant about no object distinct from particular numeral things,

except only names and characters, which originally came to be considered

on no other account but their being signs, or capable to represent aptly

whatever particular things men had need to compute.

Whence it follows

that to study them for their own sake would be just as wise, and to as

good purpose as if a man, neglecting the true use or original intention

and subserviency of language, should spend his time in impertinent

criticisms upon words, or reasonings and controversies purely verbal.

123. From numbers we proceed to speak of Extension, which, considered as

relative, is the object of Geometry. The infinite divisibility of finite

extension, though it is not expressly laid down either as an axiom or

theorem in the elements of that science, yet is throughout the same

everywhere supposed and thought to have so inseparable and essential a

connexion with the principles and demonstrations in Geometry, that

mathematicians never admit it into doubt, or make the least question of

it. And, as this notion is the source from whence do spring all those

amusing geometrical paradoxes which have such a direct repugnancy to the

plain common sense of mankind, and are admitted with so much reluctance

into a mind not yet debauched by learning; so it is the principal

occasion of all that nice and extreme subtilty which renders the study of

Mathematics so difficult and tedious. Hence, if we can make it appear

that no finite extension contains innumerable parts, or is infinitely

divisible, it follows that we shall at once clear the science of Geometry

from a great number of difficulties and contradictions which have ever

been esteemed a reproach to human reason, and withal make the attainment

thereof a business of much less time and pains than it hitherto has been.

124. Every particular finite extension which may possibly be the object

of our thought is an idea existing only in the mind, and consequently

each part thereof must be perceived. If, therefore, I cannot perceive

innumerable parts in any finite extension that I consider, it is certain

they are not contained in it; but, it is evident that I cannot

distinguish innumerable parts in any particular line, surface, or solid,

which I either perceive by sense, or figure to myself in my mind:

wherefore I conclude they are not contained in it.

Nothing can be plainer

to me than that the extensions I have in view are no other than my own

ideas; and it is no less plain that I cannot resolve any one of my ideas

into an infinite number of other ideas, that is, that they are not

infinitely divisible. If by finite extension be meant something distinct

from a finite idea, I declare I do not know what that is, and so cannot

affirm or deny anything of it. But if the terms

"extension," "parts,"

&c., are taken in any sense conceivable, that is, for ideas, then to say

a finite quantity or extension consists of parts infinite in number is so

manifest a contradiction, that every one at first sight acknowledges it

to be so; and it is impossible it should ever gain the assent of any

reasonable creature who is not brought to it by gentle and slow degrees,

as a converted Gentile to the belief of transubstantiation. Ancient and

rooted prejudices do often pass into principles; and those propositions

which once obtain the force and credit of a principle, are not only

themselves, but likewise whatever is deducible from them, thought

privileged from all examination. And there is no absurdity so gross,

which, by this means, the mind of man may not be prepared to swallow.

125. He whose understanding is possessed with the doctrine of abstract

general ideas may be persuaded that (whatever be thought of the ideas of

sense) extension in abstract is infinitely divisible.

And one who thinks

the objects of sense exist without the mind will perhaps in virtue

thereof be brought to admit that a line but an inch long may contain

innumerable parts--really existing, though too small to be discerned.

These errors are grafted as well in the minds of geometricians as of

other men, and have a like influence on their reasonings; and it were no

difficult thing to show how the arguments from Geometry made use of to

support the infinite divisibility of extension are bottomed on them. At

present we shall only observe in general whence it is the mathematicians

are all so fond and tenacious of that doctrine.

126. It has been observed in another place that the theorems and

demonstrations in Geometry are conversant about universal ideas (sect.

15, Introd.); where it is explained in what sense this ought to be

understood, to wit, the particular lines and figures included in the

diagram are supposed to stand for innumerable others of different sizes;

or, in other words, the geometer considers them abstracting from their

magnitude--which does not imply that he forms an abstract idea, but only

that he cares not what the particular magnitude is, whether great or

small, but looks on that as a thing different to the demonstration. Hence

it follows that a line in the scheme but an inch long must be spoken of

as though it contained ten thousand parts, since it is regarded not in

itself, but as it is universal; and it is universal only in its

signification, whereby it represents innumerable lines greater than

itself, in which may be distinguished ten thousand parts or more, though

there may not be above an inch in it. After this manner, the properties

of the lines signified are (by a very usual figure) transferred to the

sign, and thence, through mistake, though to appertain to it considered

in its own nature.

127. Because there is no number of parts so great but it is possible

there may be a line containing more, the inch-line is said to contain

parts more than any assignable number; which is true, not of the inch

taken absolutely, but only for the things signified by it. But men, not

retaining that distinction in their thoughts, slide into a belief that

the small particular line described on paper contains in itself parts

innumerable. There is no such thing as the ten--

thousandth part of an

inch; but there is of a mile or diameter of the earth, which may be

signified by that inch. When therefore I delineate a triangle on paper,

and take one side not above an inch, for example, in length to be the

radius, this I consider as divided into 10,000 or 100,000 parts or more;

for, though the ten-thousandth part of that line considered in itself is

nothing at all, and consequently may be neglected without an error or

inconveniency, yet these described lines, being only marks standing for

greater quantities, whereof it may be the ten--

thousandth part is very

considerable, it follows that, to prevent notable errors in practice, the

radius must be taken of 10,000 parts or more.

128. LINES WHICH ARE INFINITELY DIVISIBLE.--From what has been said

the reason is plain why, to the end any theorem become universal in

its use, it is necessary we speak of the lines described on paper

as though they contained parts which really they do not.

In doing

of which, if we examine the matter thoroughly, we shall perhaps

discover that we cannot conceive an inch itself as consisting of,

or being divisible into, a thousand parts, but only some other line which

is far greater than an inch, and repres