[FOM] RE: FOM digest, Vol 1 #94 - 2 msgs

[Matt Insall]

Buckner:

My view is that set theory is wrong, in the sense it does not capture our
ordinary intuitions about sets and things.


Davis:

This makes about as much sense as saying that physics is wrong because it
doesn't capture our ordinary intuitions about work and energy. A scientific
discipline is not
intended to capture our ordinary (fallible) intuitions, but rather to
improve on them. Particularly on FOM, set theory is of special interest in
providing a foundation for mathematics where "our ordinary intuitions" have
proved utterly misleading.


Insall:

The axioms of set theory are intended to capture a ``minimal amoount'' of
``ordinary intuition'' about sets.  The intuition helps us formulate
conjectures, which may be true or false, but are worth testing for truth or
falsity (or undecidability).  Set theory appears to be consistent, but
undecidably so.  Thus Buckner's claim that it is ``wrong'' is really a claim
about the axioms of set theory, which, as I said are ``right'', because they
actually do capture a reasonable portion of our intuition about sets.  Thus
Buckner's claim makes sense for the axioms, but is not reflective of the
current mathematical reality.  On the other hand, the reason physics is
wrong is not that it does not capture our ordinary intuitions about work and
energy.  Physics is a science that allows an unsound rule of inference,
namely an inductive inference scheme that allows one to draw sweeping
infinitary conclusions from a finite amount of data.  This is not, in and of
itself a bad thing, because it is part of what makes physics so interesting.
The currently popular theme of finding a ``theory of everything'' in the
jumble of stuff that is put out by a multitude of theorists is a red herring
that fails to be well-formulated.  For this reason, if for no other, physics
is currently ``ill''.  One who learns a bit of logic will realize that what
is being referred to popularly is not just a ``theory''.  It is, as some
have put it to me, a single equation.  If they really want a ``theory'' of
everything, and if, as I have seen and heard before, they merely mean to
``unify'' theories of quantum and relativistic mechanics, they need only
learn to think mathematically, and put some hypotheses in front of their
equations, and conjunct together the various theories (each of which, modulo
real analysis, appears to be essentially finitely axiomatizable equational
theories themselves).  Then one has a foundation for all of modern physics
in a single (very long) well-formed formula of the form

(p --> q) & (r --> s) & ...

You see, they seem to want to write

q & r & ...,

leaving out the hypotheses to the laws, and then wrap everything into a big
equation, Q:

(q & r & ...) <---> Q,

(where Q is an equation).  But the conjunct on the left side here is, by its
very nature, inconsistent, whereas the one that includes hypotheees may
actually be consistent.  Moreover, translating into set theory the whole
mess - hypotheses and conclusions - and using characteristic functions for
domain switching, one should be able to convert the formula

(p --> q) & (r --> s) & ...

into one large equation.  An example of this in mathematics is available in
group theory.  Consider the statements p, q, r and s below:

p :  The group G is Abelian.

q :  The group G has only normal subgroups.

r :  The group G is SO_3.

s :  The group G has a nonnormal subgroup.

In the theory of groups (formulated as a fragment of set theory, but for a
single set, together with axioms that deal with the group properties, such
as ``there is an associative binary operation such that ...''), which is
consistent and correct, by the way, the assertion

(p --> q) & (r --> s)

is not only consistent, but is provable, and hence is true, by the
appropriately formulated version of the soundness theorem.  However, the
assertion

q & s

is provably false, in a trivial way.  I consider this to be a simple
illustration in mathematics of how one can produce false statements from tru
e ones by ignoring the hypotheses, in the way I contend several physicists
do.  (If no physicist really does perform this fallacious act, please let me
know.  The quest for a TOE seems to me, as I indicated above, to be the most
amazing form of this fallacy in modern science.)

Of course, there are likely to be some mathematical physicists out there who
will disagree with my analysis of the situation, but so far, I get no
responses I can understand, or that clearly shows me that my analysis is
incorrect.


Matt Insall