Tipler has interesting ideas, but I think his cosmological model in this paper is wrong. He thinks the entropy of the universe increases without limit. That is wrong. He has a future Omega Point - literally a point in a collapsing universe rather than a hologram future event horizon with a finite asymptotic future area/entropy.

*Dirac derived his equation from the purely mathematical requirements of linearity, correspondence with Schr"odinger’s equation and sameness between space and time in the order of the highest derivative. But more often, the Platonic idea misleads physicists. ... The Platonic ideal was a prime motivation of the development of string theory. The hope was that there would be a unique mathematically consistent string equation, which would have a single unique solution. Alas, this hope has been dashed. String perturbation theory is termby- term finite, but the entire perturbation series is as divergent as the corresponding quantum field theory perturbation theory. And as string theory was extended to M-theory, the number of solutions was also realized to be infinite. ...*

*But in spite of these failures, there have been some notable successes in inferring the natureof reality from the structure of mathematics, or more simply, from the simple requirementof mathematical consistency. A classic example is found in Einstein’s 1905 paper ‘Onthe Electrodynamics of Moving Bodies’. Einstein referenced no experiments in this paper.Instead, he pointed out that the two fundamental theories of the day, Maxwell’s equationsfor the electromagnetic field and Newton’s equations for the motion of charged particles inan electromagnetic field, were mutually inconsistent: the former were invariant under theLorentz group, whereas the latter were invariant under the Galilean group. Einstein, inhis autobiography, gave a simple way to see that the Galilean group was inconsistent withMaxwell’s equations. Imagine, wrote Einstein, a plane electromagnetic wave. Use a Galilean transformation to move to the rest frame of this wave. In this frame, the wave should appear as a stationary sinusoidal field. But there are no such solutions to Maxwell’s equations.*

Such solutions do exist off mass shell as coherent states of virtual photons not real photons.For example the static Coulomb electric near field of a point charge in its rest frame is a superposition of Fourier components of different wave vectors k each of zero frequency f. These virtual quantum solutions do not, of course, obey the classical free far field Maxwell wave equations.

*Einstein realized that this inconsistency could not be removed by modifying Maxwell’sequations to make them Galilean invariant, without the resulting theory being inconsistent with experiment. But he realized that the same was not true of Newtonian mechanics, because a modification would involve terms of the order v/c, which would be tiny in the experiments conducted up to 1905. The rest, as they say, is history. Imposing mutual mathematical consistency on the theories of the day was to use mathematics to understand reality.*

INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 68 (2005) 897–964 doi:10.1088/0034-4885/68/4/R04

The structure of the world from pure numbers

F J Tipler

Department of Mathematics and Department of Physics, Tulane University, New Orleans,