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Show 108 Utah Academy [Vol. XIII, of Sciences) Arts and Letters the single idea that a relation is a definiton and a defintion is a relation. The most perfect definition one can obtain is a statement of the way a thing is acted upon, and how it reacts to all the rest of nature. In other words, if any term used in any discussion is used to infer that it might have an absolute definition, independent of any other term. then that term vitiates the entire discussion to the degree that it is necessary to the discussion. I have time here. only to mention how two classes of paradox and mystery can and should be abolished. . First, the pure mathematical. This type, as all others, arises from the of terms which are meaningless. or at least ambiguous in their definitions. Ask a mathematician for the meaning of a variable, .and according to custom he will seek to define it in absolute terms. This custom is our re tained heritage of the chaotic, as knowledge has to do only with relations. To define a variable. I need to construct a sentence, then I have to show that the nature of the use of a variable agrees with the implications of the sentence. You must bear in mind that the language one uses implies a phi losophy, and that the language we now use implies its ambiguous philosophy. I have to of necessity, use the terms which are defined in one of the modern dictionaries. In one respect they are no better than those of the Dark Ages. Each term is defined in such a way as to show that the compilers desired the impossible,-knowledge in and of one thing by itself only. My philosophy is, therefore not the philosophy of the language of my listeners and hence I am sure not to be understood by all of you, but if some of you will read over the mimeographed sheet and try to define the terms of any paradox you may hear, you will be surprised to find that knowledge is but a play on words; and the words are created to express a relationship. A variable, then, is one term in a relationship of two, or more, used to express a coordination which has a virtual new position with at least one other term for every value of the term. That is to say, if x is a variable in y f(x) then y must be a variable also, and dy/dx must exist. The equation y f(x) has but one solution, the curve y equals f of x. This same curve can be written y-l-k f (x-l-k), but here we have four quantities to express something for which two are sufficient; this condition permits us to define x and y as dependent variables, and k and h as another set of dependent variables, or for which k and h or x and y individually have no existence except as a result of the ratio of the other two, This fixes the two equations as being the expressions of the same curve, or their only solution is but a single curve but a different point is reached for a given 'x, y. The mathematicians present are, undoubtedly, thinking of the Weierstrass example of a continuous function for which there is no derivative. By defi nition 14, as I have it listed, if it were possible to use a set of terms con sistently and obtain a paradox then of necessity my philosophy is no better than that in present use. Let us, therefore, look at this so-called example of Weierstrass. We find not one but many inconsistencies. First of all if we differentiate his function we find that for an infinity of points within any interval we can evaluate and although he tells us we cannot differentiate term by term, our result has all the properties of being what a derivative is, giving points of maxima and minima, points of inflection, etc. This is like the story of the master of ceremonies in making an announcement, said, "For about six reasons we will have to excuse Mil'. John Jones from our program this evening, first he died last night, and the others do not matter now." But let me give another of Weierstrass' inconsistencies,-his increment Llx that he chooses he defines to lie within the interval 1I2am and 3/Zam. This Llx is therefore not capable of having zero a-s its limit. See definition No. 16 for use . = = = |
