[Ben]>> I don't know a whole lot about math, and I tend to be fallibilist, so I
wonder whether anybody really does "know," like Penrose claims, that those
maths are in fact really and truly are consistent, which are consistent
_provably_ only on the unprovable assumption of arithmetic's consistency. I
think of seeming inconsistencies that get patched up, 0 divided by 0 equals
"any number" you want -- so, more carefully define equality to exclude that
problem. Denominators seemingly turning to 0 in calculus got remedied. And so
[Bruno]> I think that all mathematicians (99,999...%) believes correctly in the
consistency of Peano Arithmetic (PA). Note that PA is a "simple" example of a
lobian theory or machine).
Then I think the main reason not to call this sort of thing "knowledge" is the
context of epistemology as a discipline which has focused often on hyperbolic
doubts and an idea of knowledge or episteme rather like that of Aristotle's
idea, where knowledge is reached deductively from firm principles. I wish that
it were not so, accustomed as I am to the English word "knowledge" with its
vagueness on such distinctions as between _scire_ and _cognovisse_, and between
_savoir_ and _connai^tre, etc.
If mathematicians "believe correctly" or know that Peano Arithmetic is
consistent, then I think that the simplest explanation is that they have -- not
formally, but nevertheless -- drawn this as an ampliative inductive conclusion
from general corroboratory experience with Peano Arithmetic and with
mathematics generally. Generalizations and surmises, the latter of which
arguably include perceptual judgments, can be quite compelling though
A problem here may be an inevitable reliance on small number of words for a
less small number of ideas. Something sufficiently corroborated or confirmed
for a given theoretical or practical purpose can be called "knowledge" in one
sense, but, according to another standard, that, for instance, of Aristotle, is
[Bruno]> Few mathematician doubt about the consistency of Zermelo-Fraenkel Set
Theory (ZF), although George Boolos makes a case that ZF could be inconsistent.
[Bruno]> Famous results by Godel and Cohen have shown the relative consistency
of many "doubtful" math assertion: precisely it has been shown that IF ZF is
consistency THEN ZF + the axiom of choice is consistent. The same for the
continuum hypothesis, etc.
[Bruno]> Note that (a theorem prover for) ZF is also a Lobian machine.
[Bruno]> Much more difficult is the question of the consistency of (rather
exotic) set theories like Quine's New Foundation (NF).
[Bruno]> There is no sense to ask about the consistency of the whole math,
because the whole math cannot be presented in a formal theory or theorem
proving machine (if only by Godel incompleteness result). It is a reason to
doubt about Penrose' use of the notion of consistency in his "godelian"
argument against Mechanism/Comp. All what such types of reasoning show is that:
IF I am a sound Lobian Machine THEN I cannot know which machine I am (and then
I cannot know which computational history extends me, and that is what I use
eventually for solving the OM measure problem).
You've outline a whole range of degrees of cognitive assurance from firm to
uncertain, and I continue to doubt that it can all be fitted under the notion
"faith" or "belief" at all. Now, if it is agreed (and maybe it isn't) that some
sort of ampliative induction is involved in all these cases, then it's worth
pointing out that the Greek word is _epagögé_ (in case the extended characters
don't survive the server, that's epago"ge'), and the field of interest is a
kind of "epagogics." But this seems too general, because we use this kind of
inference in order to infer about many more things than the consistency of an
arithmetic or the overall sanity of oneself. I dislike pointing to all these
problems without offering a solution. If only I had formally studied Greek!
Besides, I'm not aware (and am not the kind of person who would be likely to
know) that anybody has done work outlining the character of "informal inductive
inferences about arithmetic consistency" etc., and I'm kin!
d "winging it" with regard to the character of G*, I don't have the background
to understand it well.
[Bruno]> PS I will answer your long "naming-issue" post, Ben, and some others
Thanks, I'll be quite interested in it.
Best, Ben Udell