contingency or necessity for philosophically coherent developments of f.o.m.

[José Manuel Rodríguez Caballero]

Tim wrote:

> 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
> ones.


An immediate question may be: Is there a unique way to make the development
of f.o.m. philosophically consistent?

I will use Wolfram's example from the Arrival movie to illustrate this
question:

Wolfram: One thing I thought about for the "Arrival" movie was what do you
> use to communicate? Do you use math-like stuff or do you use more
> computation/programming language type stuff? I think the
> computation/programming language type stuff is actually better than the
> math-type stuff, but people don't know the words for that so it's not very
> useful for dialogue in the movie.


https://www.space.com/34783-stephen-wolfram-arrival-interview.html

For example, if an extraterrestrial civilization contacts Earth and has
developed its own practice similar to what we call mathematics, e.g.,
Wolfram's "computation/programming language type stuff", should it need to
adapt its work to what humans did in order to be philosophically consistent
in a universal sense? Or are they philosophically consistent with respect
to themselves, but philosophically inconsistent with respect to humans?

Wolfram's solution to this problem is to consider mathematics as a
manifold, where different developments are like coordinate systems. No
coordinate system is better than another. In his own words:

But in the abstract, I think we can expect that there are “views of
> mathematics” that are incoherently different from our own, and that while
> in a sense they are “still mathematics”, they don’t have any of the
> familiar features of our typical view of mathematics, like numbers.


https://writings.stephenwolfram.com/2021/05/how-inevitable-is-the-concept-of-numbers/

Using this geometric language, my question really is whether the property
of being philosophically consistent is coordinate-independent, i.e.,
independent of the contingent history of mathematics of a given
civilization and the authority of a community.

Kind regards,
Jose M.
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