258 V. MATHEMATICS A A LANGUAGE |
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occupied and there are no empty seats and no persons standing; then we would say that the number of persons is equal to the number of seats, and a
symmetrical relation of equality would be established. If all seats were occupied and there were some persons standing in the hall, or if we found that no one was standing, yet not all seats were occupied, we would establish the asymmetrical relation of greater or less.In the above processes, we were using an important principle; namely, that of
one-to-one correspondence. In our search for relations, we assigned to each seat one person, and reached our conclusions without any counting. This process, based on the one-to-one correspondence, establishes what is called the cardinal number. It gives us specific relational data about this world; yet it is not enough for counting and for mathematics. To produce the latter, we must, first of all, establish a definite system of symbolism, based on a definite relation for generating numbers; for instance, which establishes adefinite
order. Without this ordinal notion, neither counting nor mathematics would be possible; and, as we have already seen, order can be used for defining relations, as the notions of relation and order are interdependent. Order, also, involves asymmetrical relations.unique and specific symmetrical relations and all other numbers also unique and specific asymmetrical relations. Thus, if we have a result '5', we can always say that the number 5 is five times as many as one. Similarly, if we introduce apples. Five apples are five-times as many as one apple. Thus, a number in any form, 'pure' or 'applied', can always be represented as a relation, unique and specific in a given case; and this is the foundation of the exactness of dealing with numbers. For instance, to say that a is greater than b also establishes an asymmetrical relation, but it is not unique and specific; but when we say that a is five |
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