[Universality of] predicate logic The universality of predicate logic with a single two-argument function follows immediately from the result on page 1156 that it can be used to emulate any two-way multiway system.

The concept of orthogonal bases was historically worked out first in the considerably more difficult case of continuous functions. … As discovered by Joseph Fourier around 1810, this is satisfied for basis functions such as Sin[2 n π x]/ √ 2 .

Universality in Mathematica As an example of how different primitive operations can be used to do the same computation, the following are a few ways that the factorial function can be defined in Mathematica: f[n_] := n! f[n_] := n f[n - 1]; f[1] = 1 f[n_] := Product[i, {i, n}] f[n_] := Module[{t = 1}, Do[t = t i, {i, n}]; t] f[n_] := Module[{t = 1, i}, For[i = 1, i ≤ n, i++, t ⋆ = i]; t] f[n_] := Apply[Times, Range[n]] f[n_] := Fold[Times, 1, Range[n]] f[n_] := If[n  1, 1, n f[n - 1]] f[n_] := Fold[#2[#1] &, 1, Array[Function[t, # t] &, n]] f = If[#1  1, 1, #1 #0[#1 - 1]] &

Mathematical impossibilities It is sometimes said that in the 1800s problems such as trisecting angles, squaring the circle, solving quintics, and integrating functions like Exp[x 2 ] were proved mathematically impossible. But what was actually done was just to show that these problems could not be solved in terms of particular levels of mathematical constructs—say square roots (as in ruler and compass constructions discussed on page 1129 ), arbitrary roots, or elementary transcendental functions.

Any sound can be specified by giving its amplitude or waveform as a function of time. … Other simple mathematical functions can also yield distinctive sounds. FM synthesis functions such as Sin[ ω (t+ a Sin[b t])] can be made to sound somewhat like various musical instruments, and indeed were widely used in early synthesizers.

And the crucial point that turns out to be the basis for much of the success of traditional theoretical science is that in fact most standard mathematical functions can be evaluated in a number of steps that is far smaller than the numerical value of their input, and that instead normally grows only slowly with the length of the digit sequence of their input. … And indeed, as we saw in Chapter 10 , if one uses just standard mathematical functions then it is rather difficult even to reproduce many simple examples of nesting.

General topology [and axioms] The axioms given define properties of open sets of points in spaces—and in effect allow issues like connectivity and continuity to be discussed in terms of set theory without introducing any explicit distance function.

The picture below compares their periods as a function of n .

Continuous mathematics [and networks] Even though networks are discrete, it is conceivable that network-based models can also be formulated in terms of continuous mathematics, with a network-like structure emerging for example from the pattern of singularities or topology of continuous surfaces or functions.

Other integer functions IntegerExponent[n, k] gives nested behavior as for decimation systems on page 909 , while MultiplicativeOrder[k, n] and EulerPhi[n] yield more complicated behavior, as shown on pages 257 and 1093 .