Berkeley, Principles of Human Knowledge

Nothing can be more obvious to me than that the

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Unformatted text preview: about. To study them for their own sake, therefore, would be just as wise and pointful as to neglect the true use or original intention and purpose of language, and to spend one’s time on irrelevant criticisms of words, or on purely verbal reasonings and controversies. 123. From numbers we move on to discuss extension, which (considered as relative) is the object of geometry. The infinite divisibility of finite extension, though it is not explicitly asserted either as an axiom or as a theorem in the elements of geometry, is assumed throughout it, and is thought to have so inseparable and essential a connection with the principles and proofs in geometry that mathematicians never call it into question. This notion is the source of all those deceitful geometrical paradoxes that so directly contradict the plain common sense of mankind, and are found hard to swallow by anyone whose mind is not yet perverted by learning. It is also the principal source of all the fine-grained and exaggerated subtlety that makes the study of mathematics so difficult and tedious. So if I can make it appear that nothing whose extent is finite contains innumerable parts, or is infinitely divisible, that will immediately Ÿfree the science of geometry from a great number of difficulties and contradictions that have always been thought a reproach to human reason, and also Ÿmake the learning of geometry a much less lengthy and difficult business than it has been until now. ·My discussion of infinite divisibility will run to the end of section 132·. 124. Every particular finite extension [= ‘finitely extended thing’] that could possibly be the object of our thought is an idea existing only in the mind, and consequently each part of it must be perceived. If I cannot perceive innumerable parts in any finite extension that I consider, it is certain that they are not contained in it: and it is evident that indeed I cannot distinguish innumerable parts in any particular line, surface, or solid that I either perceive by sense or picture to myself in my mind; and so I conclude that it does not contain innumerable parts. Nothing can be more obvious to me than that the extended things I have in view are nothing but my own ideas, and it is equally obvious that I cannot break any one of my ideas down into an infinite number of other ideas - which is to say that none of them is infinitely divisible. If ‘finite extension’ means something distinct from a finite idea, I declare that I do not know what it means, and so cannot affirm or deny anything regarding it. But if the terms ‘extension’, ‘parts’, and the like are given any meaning that 47 we can conceive, that is, are taken to stand for ideas, then to say ‘a finite quantity or extension consists of infinitely many parts’ is so obvious a contradiction that everyone sees at a glance that it is so. And it could never gain the assent of any reasonable creature who is not brought to it by gentle and slow degrees, like bringing a converted pagan to believe that in the Communion service the bread and wine are turned into the body and blood of Jesus Christ. Ancient and rooted prejudices do often turn into principles; and once a proposition has acquired the force and credit of a principle, it is given the privilege of being excused from all examination, as is anything that is deducible from it. There is no absurdity so gross that the mind of man can’t be prepared in this way to swallow it! 125. Someone whose understanding is prejudiced by the doctrine of abstract general ideas may be persuaded that extension in the abstract is infinitely divisible, whether or not the ideas of sense are. And someone who thinks the objects of sense exist outside the mind may be brought by that to think that a line an inch long may contain innumerable parts really existing, though they are too small to be discerned. These errors - ·abstract ideas, and existence outside the mind· - are ingrained in geometricians’ minds as in other men’s, and have a similar influence on their reasonings; and it would not be hard to show how they serve as the basis for the arguments that are employed in geometry to support the infinite divisibility of extension. At present I shall only make some general remarks about why the mathematicians cling to this doctrine so fondly. 126. I have pointed out that the theorems and demonstrations of geometry are about universal ideas (section 15 intro). And I explained in what sense this ought to be understood, namely that the particular lines and figures included in the diagram are supposed to stand for innumerable others of different sizes. In other words, when the geometer thinks about them he abstracts from their size; this doesn’t imply that he forms an abstract idea, only that he doesn’t care what the particular size is, regarding that as irrelevant to the demonstration. Thus, an inch-long line in the diagram must be spoken of as though it contained ten thousand parts, since it is regarded not in its particular nature but as something universal, and it is universal only in it...
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