>From Tipler's March 2005 paper "The Structure of the World From Pure Numbers":
Can the structure of physical reality be inferred by a pure mathematician? As Einstein posed it, "Did God have any choice when he created the universe?" Or is mathematics a mere handmaiden to the Queen of the Sciences, physics? Many Greeks, for instance Plato, believed that the world we see around us was a mere shadow, a defective reflection of the true reality, geometry. But the medieval universitywas based on the primacy of the physics of Aristotle over mere mathematics. Galileo, a poor mathematician, had to live on a salary of 520 ducats at the University of Padua, while Cesare Cremonini, the university's natural philosopher (physicist), had a salary of 2000 ducats (Tipler 1994, pp 372.3). Recently, mathematics has regained some of its primacy as theoretical physicists and mathematicians have struggled to determine if there is a Brane theory picked out by mathematical consistency. I shall investigate the idea that physical reality is pure number in the second section of this paper. I shall point out that quantum mechanics.more precisely the Bekenstein Bound, a relativistic version of the Heisenberg uncertainty principle.implies that the complexity of the universe at the present time is finite, and hence the entire universe can be emulated down to the quantum state on a computer. Thus, it would seem that indeed the universe is a mere expression of mathematical reality, more specifically an expression of number theory, and of integers to boot. I shall challenge this conclusion in the third section of this paper. I shall point out that even though quantum mechanics yields integers in certain cases (e.g. discrete eigenstates), the underlying equations are nevertheless differential equations based on the continuum. Thus, if we consider the differential equations of physics as mirroring fundamental reality, we must take the continuum as basic, not the integers. I review the field of mathematical logic, and point out the implications for pure mathematics of taking the continuum as fundamental. But if we take the continuum as fundamental, we are faced with the infinities of quantum field theory, and the curvature singularities of general relativity. I shall argue in the fourth section of this paper that taking proper account of the latter allows us to avoid the former. In particular, I shall argue that the mathematical difficulties of the most successful quantum field theory, the Standard Model (SM) of particle physics, all experiments carried out to date confirm the SM, naturally disappear if one requires that the SM be consistent with quantum gravity. ... http://www.iop.org/EJ/abstract/0034-4885/68/4/R04 Lee
