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This source was then processed by an elaborate automated procedure much like a standard software build. … Then within Mathematica various transformations and tests were done on this expression—with for example every program in these notes being formatted and broken into lines using rules similar to Mathematica StandardForm .

All of these curves have fairly simple, essentially repetitive forms. … Plots of some standard mathematical functions. … In all cases the curves shown have fairly simple repetitive forms.

Higher Forms of Perception and Analysis…Higher Forms of Perception and Analysis
In the course of this chapter we have discussed in turn each of the major methods of perception and analysis that we in practice use. … So the point is that these regularities are just not ones that can be detected by our standard methods of perception and analysis.
Yet the fact that there are in the end regularities means that at least in principle there could exist higher forms of perception and analysis that would succeed in recognizing them.

Note that in several cases axiom systems are given here in much shorter forms than in standard mathematics textbooks.

The pictures below show a standard very simple example of how this can happen. … A standard simple example of a continuous system in which there is a discrete change in behavior as a consequence of a continuous change in initial conditions. … There are many systems in nature that follow the same general form of mathematical equations as those that describe the energy and motion of the ball.

Picture (a) is essentially the standard representation of mobile automaton evolution that I have used in this book. … Pictures (e) through (g) show how a network can be formed with nodes corresponding to updating events.

Model theory
In model theory each form of operator that satisfies the constraints of a given axiom system is called a model of that axiom system. … The model intended when the axiom system was originally set up is usually called the standard model; others are called non-standard. In arithmetic non-standard models are obscure, as discussed on page 1169 .

Proofs of axiom systems
One way to prove that an axiom system can reproduce all equivalences for a given operator is to show that its axioms can be used to transform any expression to and from a unique standard form. For then one can start with an expression, convert it to standard form, then convert back to any expression that is equivalent. … A standard form in terms of Nand can be constructed essentially by direct translation of DNF; other methods can be used for the various other operators shown.

Various models of snowflake growth exist in the standard scientific literature, typically focusing on one or two of these effects. … In nature a variety of forms are seen. … But in other cases there are needle-like forms, tree-like or dendritic forms, as well as rounded forms, and forms that seem in many respects random.

And this is why it has been reasonable to think of the standard axiom system of arithmetic as being basically just about ordinary integers.
But if instead of this standard axiom system one uses the reduced axiom system from page 773 —in which the usual axiom for induction has been weakened—then the story is quite different. … At this juncture it should perhaps be mentioned that in their raw form quite a few well-known axiom systems from mathematics are actually also far from complete.