>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.




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