In the discrete Regge calculus that I mention on page 1054 this variational principle turns out to have a rather simple form. The Einstein–Hilbert action—and the Einstein equations—can be viewed as having the simplest forms that do not ultimately depend on the choice of coordinates. … Below 4D the vanishing of the Ricci tensor immediately implies the vanishing of all components of the Riemann tensor—so that the vacuum Einstein equations force space at least locally to have its ordinary flat form.

In effect it expresses the idea that the integers form a single ordered sequence, and it provides a basis for the notion of recursion. … (It is known, however, that essentially nothing is lost even from full Peano arithmetic if for example one drops axioms of logic such as ( ¬ ¬ a)  a .) A form of arithmetic in which one allows induction but removes multiplication was considered by Mojzesz Presburger in 1929.

Around 350 BC Aristotle claimed that a full explanation of anything should include its purpose (or so-called final cause, or telos)—but said that for systems in nature this is often just to make the final forms of these systems (their so-called formal cause).

But the table below gives for example the actual algebraic formulas obtained in the case a = 4 after applying FullSimplify —and shows that these increase quite rapidly in complexity. In the specific case a = 4 , however, it turns out that by allowing more sophisticated mathematical functions one can get a complete formula: the result after any number of steps t can be written in any of the forms Sin[2 t ArcSin[ √ x ]] 2 (1 - Cos[2 t ArcCos[1 - 2 x]])/2 (1 - ChebyshevT[2 t , 1 - 2x])/2 where these follow from functional relations such as Sin[2x] 2  4 Sin[x] 2 (1 - Sin[x] 2 ) ChebyshevT[m n, x]  ChebyshevT[m, ChebyshevT[n, x]] For a = 2 it also turns out that there is a complete formula: (1 - (1 - 2 x) 2 t )/2 And the same is true for a = -2 : 1/2 - Cos[(1/3) ( π - (-2) t ( π - 3 ArcCos[1/2 - x]))] In all these examples t enters essentially only in a t .