Re: Isomorphism between Mereology and Boolean algebra without least element
----- Original Message -----
From: "John F. Sowa" <sowa@bestweb.net>
To: "Jay Halcomb" <jhalcomb8@comcast.net>
Cc: "Michel Eytan" <eytan@umb.u-strasbg.fr>; "Avril Styrman"
<Avril.Styrman@helsinki.fi>; <standard-upper-ontology@listserv.ieee.org>;
<cg@CS.UAH.EDU>; <alexander.heussner@gmx.net>; <aapo.halko@helsinki.fi>
Sent: Thursday, July 07, 2005 21:24
Subject: Re: Isomorphism between Mereology and Boolean algebra without least
element
> Jay,
>
> Those aren't foundational crises. Those are just
> new theories that extend, but do not invalidate
> any of the old ones:
Well, there is still theoretical debate about mathematical validity, and its
nature. It's true enough that core engineering and physics maths aren't
practically disputed. But there remain plenty of areas where theoretical
issues impinge on the possibilities of practical results. Even in Peano
Arithmetic there are still simply stated theorems to be sought, and
questions about undecidability, and essential undecidability.
Even though there's a broadly accepted practical 'core' for engineers, it's
not always just a case of simple extensions ('additions') of FOL theories.
The picture is only somewhat rosy; the millenium hasn't quite arrived, even
yet. Life is difficult, but not impossible. QED.
>
> JH> Mathematics has been enjoying foundational crises
> > since at least the invention of zero.
>
> The word "enjoying" is a good choice, because new
> additions make math more interesting. But they
> don't create crises -- except for people who don't
> like to be bothered with newfangled ideas.
>
> > What Peano and the others did was to recognize
> > the need for formal axioms of that sort.
>
> The 19th century did a better job of stating axioms
> more precisely, but that can still be considered
> normal math. Even Peano's axioms are about integers
> -- they don't reduce integers to set-theoretic
> construction.
>
> Dedekind cuts for defining real numbers, however,
> are different. They do replace a real number with
> something else -- the set of all rational numbers
> less than or equal to the cut.
>
> The difference is that Peano's axioms are actually
> used by working mathematicians while doing number
> theory, etc. Dedekind cuts, however, are *never*
> used by working mathematicians. Nobody uses a
> Dedekind cut to calculate a square root, a cosine,
> or a logarithm or to prove a theorem about them.
>
> So Peano's axioms are part of "object-level" math,
> but Dedekind cuts are part of metamathematics.
>
> The following is true, but working mathematicians are
> content to worry at the object level, not the metalevel:
Perhaps so, but computer scientists are also working mathematicians in some
sense, and at least some of them worry about consistency issues and about
the distinction between meta- and object-.
>
> > 'Working' mathematicians should worry, though, about
> > consistency in some way or another, if they are to
> > have any notion of proof (and of communication) at
> > all. And, in fact, they do worry about definitions,
> > principles, and so forth. Some worry more precisely
> > than others.
>
> First of all, consistency is generally accomplished by
> finding a model
...in the metalanguage or object language, as might be.
> -- and the two most important sources
> of models are the integers and Euclidean geometry --
> the two subjects that are far more secure (i.e., free
> from contradiction) than any work on metamathematics.
That "any" is a source of some dispute, I think -- requiring further
clarification, at least, since some metamathematical work makes just
such minimal/secure assumptions. And there are also ways in which
consideration of non-standard models impinges even here.
There are also 'non-standard' logics to be considered in the *soup*.
>
> Remember that Goedel used integer arithmetic to model his
> famous theorem about logic. That's because mathematicians
> have more far faith in the consistency of arithmetic than
> they have in the consistency of higher-order logic (and
> with very good reason -- despite the fact that nobody has
> proved or could ever prove the consistency of arithmetic).
Have more faith "with very good reason" or have faith with very nearly
or exactly the same reason? That is what foundational work is about.
What is this 'faith'?
>
> Precision is important at any level, not just the
> metalevel. Defining a real number as a Dedekind cut
> does *not* make any object-level math more precise.
> It does not improve the qualify a proof about logarithms,
> cosines, or Bessel functions.
>
> Bottom line: Arithmetic and Euclidean geometry got along
> quite well for millennia without any need for "foundations".
> And whenever anybody tries to prove that their notion of
> foundation is consistent, they use arithmetic or geometry
> as the standard of security.
Or not, as the case may be. 'Anybody' is too broad. Debate and research
about the nature of consistency and satisfiability continues in many
different forms -- as also about the nature of identity. There are also the
categorists and the intuitionists, as we've mentioned already. Etc.
http://www.cs.nyu.edu/mailman/listinfo/fom
"The FOM list is intended to provide a venue for discussing the
provocative, sometimes controversial, ideas which drive
contemporary research in foundations of mathematics..."
Jay
>
> John