On Jul 3, 2014, at 8:06 AM, Benjamin Udell <[email protected]> wrote:
> ...but I remembered it because I had already wondered what Peirce thought
> about mathematical undecidables of the kind that mathematical platonism would
> hold to have a true answer (i.e., not whether parallel lines meet, which is
> postulational, but whether, say, a given sequence of a trillion digits
> appears an infinite number of times in the decimal expansion of pi - well
> actually I'm unsure whether that's a good example but I hope my idea is
> clear).
I still had that discussion in my email archives. Way back from 2006 when we
were discussing New Elements.
Personally I think Peirce would be open to any of the main foundations of
mathematics in terms of what counts as proof. i.e. logicism, platonism,
intuitionalism etc. That’s not to say he doesn’t have a particular view on that
due to where he places mathematics and proof. Just that I’m not sure his type
of scholastic realism logically entails a particular choice between those.
I say that mainly because of his famous example of the phoenix.
"Every phoenix, in rising from its ashes, sings 'Yankee Doodle,’" will be, we
may be confident, not in conflict with any experience. If so, it is perfectly
true.
I can’t seem to find the reference but I also have this as attributed to Peirce
"Every four-sided triangle is deep blue," is necessarily true, since it is
impossible that any experience should conflict with it.
Now clearly there’s a strong realist tendency in that (without getting into the
nuances of different sorts of realisms). But for various constructivist
approaches to mathematics we could say that if we can’t find an experience
(construction of a proof) that conflicts then it is true. It’s an interesting
approach to such matters. For more full throated platonism we can ask what
would count as an experience in conflict.
Again, I’m not making a claim about Peirce’s overall beliefs here. Just noting
an application of certain elements of his thought that I find interesting.
Clark Goble
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