It is definitely not a standard or mainstream view, but it could be a flavor of https://en.wikipedia.org/wiki/Finitism which has been defended by a very small but not infinitesimal :-) number of professional mathematicians and which isn't a logically inconsistent position.
By that I mean that I believe that physical systems can be completely described by constructive mathematics based on intuitionistic logic[2] operating on computable reals[3]. I believe that any other kind of mathematics, e.g. classical logic with axiom of choice can create unphysical models.
That being said, I don't object to classical logic as a purely abstract concept. Everything proved in ZFC is certainly true in ZFC! And I don't think any finitist will contest that.
Indeed, there are an almost infinite number of crank theories because they don't require logic, evidence, or proof - just belief. One crank can churn out a hundred nonsense theories in the time it takes someone to validate one scientific idea or mathematical proof.
This has a corollary in the startup world: everyone has an idea, what matters is execution.
Professor Norman Wildberger's homepage:
http://web.maths.unsw.edu.au/~norman/
...a paper by Wildberger tackling some of the issues in this thread:
Set Theory: Should You Believe?
http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf
Some interesting works by Gregory Chaitin
Meta Math! The Quest for Omega
https://arxiv.org/abs/math/0404335
Exploring Randomness
https://www.cs.auckland.ac.nz/~chaitin/ait/
How Real are Real Numbers?
https://arxiv.org/abs/math/0411418
People who are interested in ultrafinitism would also want to check out Doron Zeilberger
http://www.math.rutgers.edu/~zeilberg/
and
Edward Nelson
https://web.math.princeton.edu/~nelson/