In general one tends to talk of purpose only when doing so allows one to give a simpler description of some aspect of behavior than just describing the behavior directly.

History [of continued fractions] Euclid's algorithm states that starting from integers {a,b} iterating {a_, b_}  If[a > b, {a - b, b}, {a, b - a}] eventually leads to {GCD[a, b], 0} . … It was discovered in antiquity that Euclid's algorithm starting with {x, 1} terminates only when x is rational.

2D fluids The cellular automaton shown in the main text is purely two-dimensional. … The basic argument for Kolmogorov's k -5/3 result for the spectrum of turbulence is independent of dimension, but there are reasons to believe that in 2D eddies will tend to combine, so that after sufficiently long times only a small number of large eddies will be left. … Formally, there is evidence that the Navier–Stokes equations in 2D might have a ∇ 2 Log[∇ 2 ] viscosity term, rather than a ∇ 2 one.

(Typical models involve the notion of stigmergy: that elements are added at a particular point based only on features immediately around them; see also page 1184 .)

The standard mathematical framework of relativity theory implies that any massless particle must propagate at c in a vacuum—so that not only light but also gravitational waves presumably go at this speed (and the same is at least approximately true of neutrinos). The effective mass for massive particles increases by a factor 1/Sqrt[1 - v 2 /c 2 ] at speed v , making it take progressively more energy to increase v .

Monadic pure predicate logic—in which predicates always take only a single argument—reduces in effect to basic logic and is not universal. … And indeed this is the case even if one considers only statements with quantifiers ∀ ∃ ∀ .

Set theory [and axioms] Basic notions of finite set theory have been used since antiquity—though became widespread only after their introduction into elementary mathematics education in the 1960s. … Note that by using more complicated axioms the only construct beyond predicate logic needed to formulate set theory is ∈ .

In organisms with a total of only a few thousand nerve cells, each individual cell typically has definite connections and a definite function.

But by the 1980s hidden Markov models were being studied, in which only some of the states or transitions could be distinguished by outside observations. … He argued that the best scientific model is one that minimizes this complexity—which with probabilities 0 and 1 is equivalent to minimizing the number of nodes in the network.

If there is only one inequivalent model the axiom system is said to be categorical—a notion discussed for example by Richard Dedekind in 1887.