[FOM] quantum foundations of mathematics
José Manuel Rodriguez Caballero
josephcmac at gmail.com
Wed Sep 12 00:46:48 EDT 2018
Miroslav Dobsicek wrote:
> while it is true that an *absolute* certainty that a physical computer
> operates 100% correctly all the time is out of reach, there are theorems
> for fault tolerant computing which allow you to get the error rate to
> *arbitrary* low levels, epsilon > 0.
> So just pick your level of absoluteness and there you go.
Identifying mathematics with quantum mechanics (I will call it qMath for
short), we have that epsilon is zero by definition. Which kind of
mathematics can be naturally expressed in qMath? It seems that
noncommutative geometry is a good candidate.
How could noncommutative geometry be formalized in qMath, if we need the
separable infinite-dimensional Hilbert space and this concept require the
complex numbers from ZFC to be defined? Well, in qMath we obtain such a
Hilbert space as a primitive object (the quantum reality) and many of the
foundational issues from ZFC are resolved for free.
This is not my personal proposal for foundations of mathematics, but just a
way to try to formalize the relationships between mathematics and reality
in Connes' worldview. If I am not mistaken, the point of Connes' criticism
of hyperreal numbers is that they determine, in a canonical way,
non-measurable sets (in the sense of Lebesgue) and these sets do not exist
in qMath, which is what Connes calls the primordial mathematical reality.
Reference about math and quantum reality:
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