Central Limit Theorem Averages of large collections of random numbers tend to follow a Gaussian or normal distribution in which the probability of getting value x is Exp[-(x - μ ) 2 /(2 σ 2 )] / (Sqrt[2 π ] σ ) The mean μ and standard deviation σ are determined by properties of the random numbers, but the form of the distribution is always the same. … The Central Limit Theorem leads to a self-similarity property for the Gaussian distribution: if one takes n numbers that follow Gaussian distributions, then their average should also follow a Gaussian distribution, though with a standard deviation that is 1/ √ n times smaller.

The theory of types used in Principia Mathematica introduced some distinction, and following the proof of Gödel's Completeness Theorem for first-order logic in 1930 (see page 1152 ) standard axiom systems for mathematics (as given on pages 773 and 774 ) began to be reformulated in first-order form, with set theory taking over many of the roles of second-order logic. In current mathematics, second-order logic is sometimes used at the level of notation, but almost never in its full form beyond. And in fact with any standard computational system it can never be implemented in any explicit way.

My guess is that unless one asks about very specific details there is really not—and that standard logic is in a sense distinguished in the end only by its historical context. … Indeed, given many forms of operator there are always axiom systems that can be found to describe it. Axiom systems that reproduce equivalence results for the forms of operators shown.

Nested structure of attractors Associating with each sequence of length n (and k possible colors for each element) a number Sum[a[i] k -i , {i, n}] , the set of sequences that occur in the limit n  ∞ forms a Cantor set. For k = 3 , the set of sequences where the second color never occurs corresponds to the standard middle-thirds Cantor set. … Note that if the possible sequences cannot be described by a network, then the Cantor set obtained will inevitably not have a strictly nested form.

And while traditional mathematical notation suffers from some inconsistencies and ambiguities, it was possible in developing Mathematica StandardForm to set up something very close that can be interpreted uniquely in all cases.

Non-standard diffusion To get ordinary diffusion behavior of the kind that occurs in gases—and is described by the diffusion equation—it is in effect necessary to have perfect uncorrelated randomness, with no structure that persists too long. … The result is that on page 464 the limiting form of the average behavior does not end up being an ordinary Gaussian.

Standard mathematical functions There are an infinite number of possible functions with integer or continuous arguments. … And if one modifies the usual hypergeometric equation y''[x]  f[y[x], y'[x]] by making f nonlinear then solutions typically become hard to find, and vary greatly in character with the form of f . … A few functions that involve manipulation of digits have also become standard since the use of computers became widespread.

Theism and the standard Western religions generally attribute thinking to a person-like God who governs the universe but is separate from it. … In its typical religious form in Eastern metaphysics—as well as in philosophical idealism—the contents of the universe are identified quite directly with the thoughts of God.

And with this definition, a region that approximates a cylinder can be formed just by setting up spheres with centers at every point on the path. But there is now another issue to address: at least in its standard formulation general relativity is set up in terms of properties not of three-dimensional space but rather of four-dimensional spacetime. … The idea is to start at a particular event in the causal network, then to form what is in effect a cone of events that can be reached from there.

Picture (a) is a version of the standard representation that I have used for mobile automaton evolution elsewhere in the book—in which successive lines give the colors of cells on successive steps, and the position of the active cell is indicated at each step by a gray dot. … For rather than having a picture based on successive individual steps of evolution, one can instead form a network of the various causal relationships between updating events, with each updating event being a node in this network, and each stripe being a connection from one node to another.