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Berkeley, Principles of Human Knowledge

The procedure i have described here seems to be the

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Unformatted text preview: s signification, through which it represents innumerable lines longer than it is, in which ten thousand parts or more may be distinguished, even though it is itself a mere inch in length. In this manner the properties of the lines signified are (by a very usual figure of speech) transferred to the sign, and from that are mistakenly thought to belong to the sign - ·the inch-long line· - considered in its own nature. 127. Because there is no number of parts so great that there could not be a line containing more, the inch-line is said to contain parts more than any assignable number; which is not true of the inch itself but is true for the things it signifies. But men lose sight of that distinction, and slide into a belief that the small particular line drawn on paper has in itself innumerable parts. There is no such thing as the ten-thousandth part of an inch; but there is a ten-thousandth part of a mile or of the diameter of the earth, which may be signified by that inch. When therefore I delineate a triangle on paper, and take one inch-long side (for example) to be the radius ·of a circle·, I consider this as divided into ten thousand or a hundred thousand 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 any error or inconvenience, yet these drawn lines are only marks standing for greater lengths of 48 which a ten-thousandth part may be very considerable; and that is why, to prevent significant errors in practice, the radius must be taken to have ten thousand parts or more. 128. What I have said makes plain why, if a theorem is to become universal in its use, we have to speak of the lines drawn on the page as though they had parts that really they do not. When we speak in this way, if we think hard about what we are doing we shall discover that we cannot conceive an inch itself as consisting of (or being divisible into) a thousand parts, but only some other line that is far longer than an inch and is represented by it. And ·we shall discover· that when we say that a line is infinitely divisible, we must mean a line that is infinitely long. The procedure I have described here seems to be the chief reason why the infinite divisibility of finite extension has been thought necessary for geometry. 129. The various absurdities and contradictions that flowed from this false principle might have been expected to count as so many arguments against it. But ·this did not happen, because· it is maintained - I know not by what logic - that propositions relating to infinity are not to be challenged on grounds of what follows from them. As though contradictory propositions could be reconciled with one another within an infinite mind! Or as though something absurd and inconsistent could have a necessary connection with truth, or flow from it! But whoever considers the weakness of this pretence will think that it was contrived on purpose to humour the laziness of the mind, which would rather slump into an indolent scepticism than take the trouble to carry through a severe examination of the principles it has always embraced as true. 130. Recently the speculations about infinites have run so high and led to such strange notions that large worries and disputes have grown up among contemporary geometers. Some notable mathematicians, not content with holding that finite lines can be divided into an infinite number of parts, also maintain that each of those infinitesimals is itself subdivisible into an infinity of other parts, or infinitesimals of a second order, and so on ad infinitum. I repeat: these people assert that there are infinitesimals of infinitesimals of infinitesimals, without ever coming to an end! According to them, therefore, an inch does not merely contain an infinite number of parts, but an infinity of an infinity of an infinity . . . ad infinitum of parts. Others hold that all orders of infinitesimals below the first are nothing at all, because they reasonably think it absurd to imagine that there is any positive quantity or part of extension which though multiplied infinitely can never equal the smallest given extension. And yet on the other hand it seems no less absurd to think that the square-root, cube-root etc. of a genuine positive number should itself be nothing at all; which they who hold infinitesimals of the first order, denying all of the subsequent orders, are obliged to maintain. 131. Doesn’t this, then, give us reason to conclude that both parties are in the wrong, and that there are really no such things as infinitely small parts, or an infinite number of parts contained in any finite quantity? You may say that this will destroy the very foundations of geometry, and imply that those great men who have raised that science to such an astonishing height have all along been building a castle in the air. To this I reply that whatever is useful in geometry and promotes the benefit of human life still remains firm and unshaken on my principles. That science, considered as practical, will be helped rather 49 than harmed by what I have said; though to show this clearly fully might require a separate book. For the rest, even if my doctrines imply that some of the more intricate and subtle parts of theoretical mathematics may be peeled off without prejudice to the truth, I don’t see what damage this will bring to mankind. On the contrary, it is highly desirable that men of great...
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