Mathematical fictionalism vs. physical fictionalism

[Thomas Klimpel]

Vaughan Pratt wrote:
> While I'm sympathetic to the point of view of mathematical fictionalism, I'd be interested in hearing from any FOM participants about physics fictionalism.

Like many other mathematicians, I believe in a principle of
'conservation of difficulty'. This allows me to believe that
mathematics stays useful, even if it would be fictional. I believe
that often the main difficulties of a real world problem will still be
present in a fictional mathematical model. Therefore analyzing that
model and understanding the difficulties in that context will provide
useful insight into the real world problem.

>From my experience with physicists (being an applied mathematician
working in semiconductor manufacturing), their trust in 'conservation
of difficulty' is often less pronounced. As a consequence, physics
fictionalism has a hard time (at least that is my impression). So
instead of accepting Bohmian mechanics as a useful fictional model
with huge potential for analyzing various difficulties of quantum
mechanics (and extracting insights about the real world from it), it
was initially dismissed for being too obviously fictional.


> Whereas Bohmian mechanics maintains that particles such as electrons move continuously,
> the Copenhagen Interpretation maintains that electrons in orbitals only
> exist as distributions and not as actual continuously moving particles.
>
> These being contradictory points of view, must we infer that at least one of them is fictional?

The Copenhagen Interpretation is a bit more subtle than that,
especially since it is not intended as a fictional mathematical model.
The starting point is not that particles such as electrons don't exist
as actual continuously moving particles, but that the macroscopic
classical physical world exists (or at least the results of actually
performed measurements exist). The reality of the quantum world only
needs to be denied in case of conflicts, especially with respect to
counterfactual reasoning (i.e. the typical "results of measurements
that have not been performed don't exist" slogan).

But back to fictionalism (and the principle of 'conservation of
difficulty'). In "Collapse. What else?"
(https://arxiv.org/abs/1701.08300), Nicolas Gisin says the following
about Bohmian mechanics:
"Let’s return to Bohmian quantum mechanics. As said, it is a
remarkable existence proof of non-local hidden variables. But does it
answer the deep question of the quantum measurement problem? I don’t
think so. As with the many-worlds, it assumes hyper-determinism and
relies on infinitesimal digits for its predictions."
In a previous section on many-worlds, Gisin had already attacked real
numbers as fictions:
"Do these infinitesimal digits have a real impact on the real world?
Is this still proper physics? For sure, such assumptions can’t be
tested. Hence, for me, hyper-determinism is a non-sense"

But Gisin didn't stop at attacking real numbers as fictions. He later
tried to use intuitionistic mathematics to overcome the issues he saw
with those fictions:
https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/
"Indeterminism in Physics and Intuitionistic Mathematics"
(https://arxiv.org/abs/2011.02348)

This provides a nice link to FOM, but I still want to come back to
'conservation of difficulty' in the context of Bohmian mechanics. John
Stewart Bell appreciated Bohmian mechanics as a mathematical model,
and observed the role of nonlocality for it. And then he made the
crucial next step and extracted an important insight about the real
world, namely his inequalities that allowed to verify the nonlocal
aspects of quantum mechanics experimentally.

Which leads me back to Nicolas Gisin, because he also performed
important experiments in this context. And in his short (popular) book
"Quantum Chance," he tries to go one step further. He somehow goes
beyond the notion of true randomness or “bit strings with proven
randomness” and tries to capture the experimentally observable nature
of quantum chance itself, independent of any interpretation of quantum
mechanics (or even the validity of quantum mechanics). Something like
that it is a nonlocal randomness, and because the nonlocal
correlations of quantum physics are nonsignalling, it has to be
random, because otherwise it would allow faster than light
communication.