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*To*: <categories@mta.ca>*Subject*: categories: Fixed points of cross-category relationship - Doctorow*From*: "Osher Doctorow" <osher@ix.netcom.com>*Date*: Mon, 11 Sep 2000 22:22:01 -0700*Sender*: cat-dist@mta.ca

From: Osher Doctorow, Ph.D. osher@ix.netcom.com, Monday Sept. 11, 2000, 10:02PM Consider the fixed point g(x,y) = (x,y) for g(x,y) = (x-->y, (x-->y)^) where x-->y is 1 - x + y and its real conjugate, labeled (x-->y)^, is 1 + x - y. This is equivalent to 1 - x + y = x, 1 - y + x = y, and subtracting the second equation from the first results in -x + y + y - x = x - y or 3(y - x) = 0 so y = x and substituting for either x or y in the first or second equation yields the unique solution x = 1, y = 1, so the fixed point is (1, 1), and since x-->y = x and y-->x = y, we also have x-->y = 1 and y-->x = 1. It turns out that even if the problem had been g(x,y) = (y,x) for the above g, the same fixed point would have been obtained. This is remarkable for those interested in fixed points. Fixed points have turned out to be of considerable importance across categories, and similarly for symmetry groups and invariance properties, both in mathematics and physics. From the above paragraph, the multiplicative unit of the category has turned out to be its unique fixed point. Also it suggests that categories for which the multiplicative unit is very important are those in which x-->y = 1 - x + y plays an important role. This holds for at least two categories, which I will refer to for short as probabilities and (fuzzy) multivalued logic - especially Product/Goguen logic and Lukaciewicz logic, although both are related to Godel logic. The multiplicative unit is important in probability because probabilities by definition are between 0 and 1 - their multiplicative unit is also their maximum. In (fuzzy) multivalued logics, the unit 1 is similarly an extreme value, although it is usually considered to be the "trivial" case of either tautology or contradiction (usually tautology). This would at least preserve the intuitive idea of tautology expressing "complete logical truth", however trivial we may consider that to be. The above results also suggest that a generalization of fixed point results to the general case where x-->y and/or y-->x = 1 might be quite useful, and this is precisely what logic-based probability (LBP) has found to be the case (recall my previous contributions to categories@mta.ca). There is an interesting application to history that I might cite as an amusing aside. History is usually taught in terms of stories - the story of a war, a civilization, an era, etc. The above results would suggest, if they are applicable to history, that it is best to teach history as the study of rare discrete events and how they causally influence each other. For example, what caused a war, what ended a war, or in different language, why did the war occur, why did the war end, etc. The discrete events would correspond to fixed points, and we would be especially interested in them when they are repeated in history - the repetition would correspond to fixed points or invariance in time or symmetry in time. Then history would become the study of why errors repeat rather than merely the study of what happened, where it happened, who it happened to, how it happened, etc. I find this somewhat amusing because of arguments that one can get into when discussing matters with the more detail-obsessed historians to whom any suggestion of causation or relationships across categories results sometimes in remarkable responses. Osher Doctorow

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