propositions that can perish and their role in the foundations of mathematics
José Manuel Rodríguez Caballero
josephcmac at gmail.com
Fri Jun 17 05:42:06 EDT 2022
Harvey Friedman asked:
> Are there foundational advances that cannot be greatly improved by some
> supporting philosophically coherent advance?
Before proceeding, I recall the definition of "philosophically coherent":
> In short, a "philosophically coherent" development of f.o.m. (a la
> Friedman) is supposed to be a development that builds up the entire
> conceptual structure of mathematics from the most basic concepts possible.
> It is not enough to formally mimic existing mathematical practice while
> punting on the hard work of analyzing compound concepts in terms of atomic
The rather subjective notion of "most basic concepts possible" doesn't
guarantee that when they are put together they will not generate a
contradiction in the long run, e.g., Gottlob Frege's foundations of
mathematics after Bertrand Russell's paradox (the formal version of the
barber paradox). After a contradiction is found, the candidates for the
"most basic concepts possible" are updated, and a new attempt of "
philosophically coherent" development is pursued (just like in the myth of
Sisyphus and the rock, which represent the foundations of science, and
foundations of mathematics in particular). Therefore, we may conclude that
foundational advances happen precisely when philosophically coherent
advances lead to a contradiction. In other words, foundational advances may
be defined as the realization of the limit of our intuition concerning the
development of mathematics in a philosophically coherent way. This is like
the space-filling curve that was a realization of the limit of our
intuition in Analysis during a concrete historical period.
Using the language of Diodorus Cronus' Master Argument, we can rephrase the
above conclusion as identifying the foundational advances to the discovery
that something we thought possible is in fact impossible. This was
precisely Chrysippus's solution to this paradox. Thus, the solution to the
problem of future contingents when applied to f.o.m., at least from this
point of view, is that:
(i) the past of the foundations of mathematics is fixed;
(ii) the future of the foundations of mathematics is open;
(iii) philosophically coherent developments that are possible in the
present can become impossible in the future.
Notice the role of undecidability in point (iii): in general, this
knowledge is obtained by chance, without any algorithmic guarantee. Claim
(i) corresponds to the list of failed attempts to develop mathematics. For
claim (ii), we can imagine the possibility of post-ZFC mathematics, after a
hypothetical contradiction in ZFC is found several millennia in the future.
The transition from the possibility to the impossibility of the same
proposition in (ii) can't be formulated in traditional modal logic.
Chrysippus defended its validity by introducing the concept that
propositions can perish. Here is a reference:
Ide, Harry A. "Chrysippus's response to Diodorus's Master Argument."
History and Philosophy of Logic 13.2 (1992): 133-148.
Therefore, trying to use a contemporary version of Chrysippus's language,
we may define foundational advances as the progressive perishment of the
possibilities of philosophically coherent foundations.
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