210 IV. STRUCTURAL FACTORS IN
A LANGUAGEStaking away all odd numbers, which, itself, represents infinite numbers of numbers.
This example was used by Leibnitz to prove that infinite arrays cannot exist, a conclusion which is not correct, since he did not realize that both finite and infinite arrays depend on definitions. We should be careful not to approach
infinite processes, arrays., with prejudices, or silent doctrines and assumptions, or, in general s.r, taken over from finite processes, arrays,.Thus we see that the process of generating natural numbers is structurally an infinite process because its
results can be put in a one-to-one correspondence with the results of the process of generating even numbers., which is only a part of itself. Similarly, a line AB has infinitely many points, since its points can be put into a one-to-one correspondence with the points on a segment CD of AB. Another * see page xii example can be given in the Tristram Shandy paradox of Russell. Tristram Shandy was writing his autobiography, and was using one year to write the history of one day. The question is, would Shandy ever complete his biography? He would, provided he never died, or he lived infinite numbers of years. The hundredth day would be written in the hundredth year, the thousandth in the thousandth year, etc. No day of his life would' remain unwritten, again provided his process of living and writing would never stop.Such examples could be given endlessly. It is desirable to give one more example which throws some light on the problems of 'probability', 'chance',. The theory of probability originated through consideration of games of chance. Lately it has become an extremely important branch of mathematical knowledge, with fundamental structural application in physics, general semantics, and other branches of science. For instance. Boltzman based the second law of thermodynamics on considerations of probability. Boole's 'laws of thought', and the many-valued 'logic' of Lukasiewicz and Tarski are also closely related to probability; and the new quantum mechanics uses it constantly,.
The term 'probability' may be defined in the rough as follows: If an event can happen in
a different ways, and fails to happen in b different ways, and all these ways are equally likely to occur, the probability ofLet us assume that in a certain city a lecture is held each day, and that, though the listeners may change each day, the numbers of listeners |
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