Hello,
Just a brief follow-up on the discussion above about factorization, which, in terms of mathematical beauty at least, looks like an ugly duck.
On the other hand, factorization is the mathematical equivalent of splitting matter: prime numbers are the "atoms" out of which any integer number is made of. Not being able to factorize efficiently numbers seems to be a symptom
of the more general idea that it is not always possible to break down complex, large-scale entities into distinct, "elementary" units. That is, it hints to a failure of reductionism already at the
mathematical level. P.W. Anderson wrote a famous essay in 1972 titled "More is different" - and indeed so: although things are made of parts,
given the large numbers involved, it could be
principially impossible to express the higher-order, emergent laws, in terms of elementary components - and, while many of us accept that this is indeed the case for example with biology versus
physics, it is quite surprising that such a limitation could be embedded already at the mathematical level.
... Then also the fact that
gravity is notoriously difficult to reconcile with quantum mechanics becomes by this a bit more understandable:
the general theory of relativity is an "intensely classical" theory, requiring concepts such as
clocks, measuring rods, and so on, which have intrinsic properties (unlike quantum-mechanical entities).
Many years ago, Wigner wrote an
excellent review on the conceptual tension between quantum theory and relativity
(Rev. Mod. Phys. 29,255 (1957)).
In this sense,
if, in general, reducing higher-level laws to lower-level laws would require exponentially-increasing
(in the number of components) resources, then it could be that there exists a true epistemological gap
between "elementary" and "emergent" laws - and not only that we will never find a theory of everything
but the concept itself is inconsistent. In other words, our ability of completely classifying
reality into elementary entities is impaired when
complex objects - which have no underlying symmetry - are considered.
Quantum computing
can be then regarded as a particular attempt to jump-pass this gap, by assuming full control over
the emergent properties of a complex system through local manipulations of the individual components.