Re: Paper+Exercises+Naming Issue-faith

From: Benjamin Udell <budell.domain.name.hidden>
Date: Mon, 9 Jan 2006 11:43:10 -0500

Bruno, list

[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 on.
 
[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 non-deductive.
 
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 merely belief.
 
[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 to, asap.
 
Thanks, I'll be quite interested in it.
 
Best, Ben Udell
Received on Mon Jan 09 2006 - 11:46:44 PST