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But as their names suggest Simplify and FullSimplify were intended to be less predictable—and just to do what they can and then return a result. And in many cases these functions end up trying to prove theorems; so for example FullSimplify[(a + b)/2 ≥ Sqrt[a b], a > 0 && b > 0] must in effect prove a theorem to get the result True .
This idea was used in the 1980s as a way of simplifying the Feynman diagrams to consider in QCD and other quantum field theories.
But inevitably functions like FixedPoint , ReplaceRepeated and FullSimplify can run into undecidability—so that ultimately they have to be limited by constructs such as $IterationLimit and TimeConstraint .
But at least at first, I suspect that huge simplifications will be made, with the result that all sorts of misleading conclusions will probably be reached, perhaps in some cases even seemingly contradicting the principle.
But the kinds of systems that I discuss in this book typically show much more complex behavior, and have no such simplifying properties.
Whenever natural selection is an important determining factor, I suspect that one will inevitably see many of the same simplifying features as in systems created through engineering.
So what all this means is that much of what we see in biology should correspond quite closely to the typical behavior of simple programs as we have studied them in this book—with the main caveat being just that certain aspects will be smoothed and simplified by the effects of natural selection.
But the whole point of a model is to have a simplified representation of a system, from which those features in which one is interested can readily be deduced or understood.
If one puts enough constraints on the axioms one uses, one can eventually prevent universality—and in fact this happens for commutative group theory, and for the simplified versions of both real algebra and geometry on pages 773 and 774 .
This is now in practice done by Simplify and other functions in Mathematica using methods of cylindrical algebraic decomposition invented in the 1970s—which work roughly by finding a succession of points of change using Resultant .