Since I have been establishing the fuzzy multivalued logical basis of not only mathematics but physics and other sciences since 1980, it is not too surprising to find that chaos and fractals are based on fuzzy multivalued logic. See fairly detailed abstracts of 69 of my papers (publications, papers presented at conferences, technical reports, internet contributions) at http://www.logic.univie.ac.at, Institute for Logic of the University of Vienna (after accessing that site, select in this exact order: 1. ABSTRACT SERVER, then 2. BY AUTHOR, then 3. my name). Although I did not discover the fuzzy multivalued logics mentioned below, I and my wife Marleen J. Doctorow Ph.D. discovered most of their applications that you will read below. The 3 types of fuzzy multivalued logics (Lukaciewicz or its predicate extension Rational Pavelka, Product/Goguen, and Godel) correspond to the three types of events in the physical universe, respectively: Memory (M), Semi-Memory (S), and Non-Memory (N). Remarkably, they also correspond respectively to Logic-Based Probability (LBP - which subtracts probabilities), Bayesian Conditional Probability (BCP - which divides probabilities), and Independent Probability-Statistics (IPS - which multiples probabilities) which are the three types of probability concerned with either the influence or the dependence of events/processes on each other. Even more remarkably, they also correspond respectively to events/processes which are influenced by 2 or more previous times (human memory/consciousness/perception, quantum entanglement, spherical expansion-contraction as in radiation from a point source and the expansion aspect of the universe, viscoelastic material memory), exactly one previous time (Markov chains and Markov processes, games like chess in which the present move depends only on the state of the board on the previous move), or no previous time (tosses of two fair coins, two dice, etc.). To summarize very succinctly what the fuzzy multivalued logical basis of chaos and fractals is, it is sufficient to say that the basis is Lukaciewicz or its extension Rational Pavelka fuzzy multivalued logic. This is because LBP turns out to correspond also to rare or extremely rare events/processes, and such events/processes (events for short) in mathematics are respectively modeled, at least for continuous random variables which correspond to them, by events of probability epsilon (very small positive number) or 0. Fractals in general have probability and for bounded fractals Lebesgue measure 0. Contrary to erroneous opinions of many people, events of probability 0 are not impossible - in fact, they occur *all the time*. This goes back to Lebesgue measure, which is used in Lebesgue integration, which is 0 for all n-k dimensional subsets of Euclidean and Euclidean-like spaces of dimension n where k = 1, 2, ..., n. For example, a single point or a single line or curve or even a plane or surface (which are 2-dimensional automatically) in 3 dimensional space or 3 + 1 dimensional spacetime (3 spatial and one time dimensions) have Lebesgue measure and hence continuous probability 0 if a continuous random variable can be found on the region in question - or, to put it another way, if a continuous probability distribution (such as the normal/Gaussian) is found on a volume of space(-time) enclosing the events in question. Likewise a single point in time has probability 0 under the above assumptions, so that extremely rare events like catastrophes or a particular person winning a lottery have probability 0 with the above assumptions. If the space is 2 dimensional, then 0 and 1 dimensional bounded objects have probability and Lebesgue measure 0 under the above assumptions. It must be pointed out that Julia and Mandelbrot sets and hence fractals and chaos are generally boundaries of regions, and the boundary of a 3 or 2 dimensional region of Euclidean type has lower dimension than the space (even the complex plane can be regarded as of Euclidean type although its multiplication and conjugates are quite different from real Euclidean operations). It also must be pointed out that events of probability 0 in LBP have MAXIMUM INFLUENCE on other events, which is not necessarily true for BCP and IPS, but that other events in LBP may also have *maximum influence*. However, this has the unusual consequence that fractals and chaos are among the events/processes/objects which have maximum influence on other events. This means, in turn, that fractals and chaos should be applicable to physics and engineering not only occasionally as in NASA's use of controlled chaos in a recent space scenario, but is a large variety of circumstances. Of course, similar results should hold for life and behavior sciences, etc. Finally, and most remarkably of all perhaps, fractals and chaos belong to the Memory (M) part of the universe. This puts them alongside human memory/consciousness/perception, spherically expanding/contracting motion such as radiation from a point source and the universe with respect to its expansion, quantum entanglement, viscoelastic material memory, most biological processes, etc. This opens up several important areas for research, including fractal and chaotic applications to the other things like human memory mentioned above, since such events can be expected to either intersect or be contained in each other (or one contained in the other). What about the origins of fractals and chaos in iteration of functions on a computer? Iteration and similar methods with algorithms certainly produce fractals and chaos, but both mathematically and in view of the above paragraphs what is key to this process is the fact that with iteration the function involved usually approaches an EQUATION OF A CURVE/CURVES/ISOLATED POINT or something which is extremely irregular but which mathematically is like a curve/curves/isolated point. This has to do with the nature of limits in a real analysis/calculus sense and fixed points and so on. Whether there exists a generalization of fractals in which not the same function but different functions are used *iteratively* (each new function replaces the same-function iterate) is an interesting open question as far as I know. Aczel and Dhombres and their colleagues have done some work in this area, summarized in Cambridge University Press volumes of the last 15 years. Osher Doctorow