The idea that as a matter of principle there should be truths in mathematics that can only be reached by some form of inductive reasoning—like in natural science—was discussed by Kurt Gödel in the 1940s and by Gregory Chaitin in the 1970s. … With the release of Mathematica in 1988, mathematical experiments began to emerge as a standard element of practical mathematical pedagogy, and gradually also as an approach to be tried in at least some types of mathematical research, especially ones close to number theory.

Quantum field theory In standard approaches to quantum field theory one tends to think of particles as some kind of small perturbations in a field. … And indeed standard perturbation theory is based on starting from these and then looking at the expansion of Exp[  s/ ℏ ] in powers of the coupling constant. … At larger distances something like color flux tubes that act like elastic strings may form.

Some non-standard versions of quantum field theory involving discrete space did however continue to be investigated into the 1960s, and by then a few isolated other initiatives had arisen that involved discrete space. The idea that space might be defined by some sort of causal network of discrete elementary quantum events arose in various forms in work by Carl von Weizsäcker (ur-theory), John Wheeler (pregeometry), David Finkelstein (spacetime code), David Bohm (topochronology) and Roger Penrose (spin networks; see page 1055 ). … Models that involve discreteness have been proposed—most often based on spin networks—but there is usually still some form of continuous averaging present, leading for example to suggestions very different from mine that perhaps this could lead to the traditional continuum description through some analog of the wave-particle duality of elementary quantum mechanics.

What the theorem shows is that there are statements that can be formulated within the standard axiom system for arithmetic but which cannot be proved true or false within that system. … He suggested that his results could be avoided if some form of transfinite hierarchy of formalisms could be used, and appears to have thought that at some level humans and mathematics do this (compare page 1167 ).

Ornamental art Almost all major cultural periods are associated with certain characteristic forms of ornament. Often the forms of ornament used on particular kinds of objects probably arose as idealized imitations of earlier or more natural forms for such objects—so that, for example, imitations of weaving, bricks and various plant forms are common. … The pictures below illustrate its relation to standard cursive Arabic writing.

And by the end of the 1920s basic practical quantum mechanics was established in more or less the form it appears in textbooks today. … At first the path integral was viewed mostly as a curiosity, but by the late 1970s it was emerging as the standard way to define a quantum field theory. … And the result is that beyond perturbation theory there is still no real example of a definitive success from standard relativistic quantum field theory.

The standard big bang model assumes that the universe starts with matter at what is in effect an arbitrarily high temperature. … The actual process of inflation is usually assumed to reflect some form of phase transition associated with decreasing temperature of matter in the universe. … But above a critical temperature thermal fluctuations should prevent the background from forming—leading to at least some period in which the universe is dominated by a cosmological term which yields exponential expansion.

But when two gliders are present, their collision forms a wall, which prevents output of the spaceship. If one considers rules with more than two colors, it becomes straightforward to emulate standard logic circuits.

Einstein equations In the absence of matter, the standard statement of the Einstein equations is that all components of the Ricci tensor—and thus also the Ricci scalar—must be zero (or formally that R ij = 0 ). … 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.

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 . … But in general for arbitrary a there is no standard mathematical function that seems to satisfy the functional equation.