MS 464-465 (1903) - Lowell Lecture III - 3rd Draught

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formulate a similar routine. But the performance of it would be utterly impracticable owing to the stupendous complications that it would lead to. For example, here is a graph, not particularly intricate:

Yet from this graph as a premiss no less than ninety entirely independent conclusions can be drawn, showing the misconception involved in the popular expression, “the conclusion” from given premisses. Moreover, none of these 90 conclusions uses any part of the assertion of the graph more than once. By repeated making use of the same assertion, any

Last edit over 6 years ago by gnox
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number of conclusions could be drawn. In the alpha-part of logic to which the ordinary textbooks of logic are virtually confined, because although if the problem they consider were stated in graphical form, there would be ligatures, yet they are only such as can all be joined throughout the graph, so that the rules of operation become the same as if there were no ligature at all,— this is not a minutely accurate statement because the books are so unsystematic that no brief statement of what they contain could be quite accurate, but it is substantially so— in this alpha-part of logic a premiss can only be efficient once. But in the beta part of logic premisses can be applied efficient over and over and over again endlessly. A moment's reflexion on any simple branch of mathematics will show that it must be so. For instance, the whole theory of numbers

Last edit over 6 years ago by gnox
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depends upon five premisses represented in this graph

The red ligatures refer to numbers, the brown ligatures to any universe such that there is a relation that u may be understood to express, that will make the entire graph true.

X is to be understood to be replaceable by any monad graph whatever without altering the truth of the entire graph.

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We shall see, in due time, what those five premisses are. Suffice it for the present to say that there are only five. Now if each of them could be efficient and that in only one way, as one would have a right to infer from the account of reasoning given in the text-books, it would necessarily follow that there could not be but 32 theorems of the theory of numbers in all; whereas of highly interesting theorems already known there are hundreds. Euclid's Elements, which was never designed to be more than an introduction to geometry and algebra (or that theory which with the Greeks served the purpose of our algebra) has only 5 postulates, which are the main premisses, together with 9 axioms and 132 definitions. From these Euclid deduces 369 theorems, 96 problems,

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17 lemmas, and 27 corollaries. There are also 2 scholia. At the same time, I ought to remark that, while the possible conclusions are innumerable, yet after all premisses have been iterated so as to exhaust all the different ways of using them together with all simpler ways, there will be no more theorems of any particular interest, and the branch of mathematics in question may be said to be substantially exhausted. The theory of conics is an instance. The great geometer Chasles after he attained to a great age continued to grind out, by the bushel-basket full, such theorems as that in a plane, the number of conics that touch five given conics is 3264. I think that when a mathematical theory has nothing new to discover but such propositions as that it may be said to be

Last edit over 6 years ago by gnox
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