FOM: Transfinite Logic

[wiman lucas raymond]

Dean Buckner writes:
>The idea embodied  by Heck's statement  "what an incredible conceptual
leap Cantor
>made when he extended the idea that equinumerosity is one-one
correspondence
>to the infinite"   This is only incredible from the "folk" perspective.
The
>formal statement that really underlies it - that the word
"equinumerosity"
>will from now on mean plain one-one correspondence - is I suspect not
very
>interesting, and not much of a conceptual leap.

Yes, it's not a great conceptual leap now.  I don't know enough about
history to posit whether it was then or not.  However, it's a conceptual
leap forward in the sense that it distills a great deal about what has
shown to be mathematically important about infinite sets.  It also
formalizes the fairly confused ideas of intuitive infinity.  Most people
I've talked to have found it difficult to believe both the ideas that
there are different "sizes" of infinities, and that there are the same
"number" of evens as whole numbers.  Yet if one of these is false, the
other is surely true!

>By analogy, the idea that the earth is round is only incredible, if we
>continue to think of it as flat, so Australians walk upside down.  
>Our ordinary "folk" way of thinking dominates the world of theory, and
makes it
>fantastic, and incredible (and perhaps, interesting).  I remember the
first
>impact that the transfinite arguments made on me (a very long time ago
now)
>and a recall a deep sense of awe and mystery.  Except that was all
wrong.
>What was awesome was the remnant of my old finitist way of thinking.

The idea of the round earth is deeply ingrained in much of our
collective culture and thought processes.  The idea of different sized
infinities is deeply ingrained in my thought processes, but *not* in my
culture.  As you say, it's not "really" surprising--we just defined it
that way.  Yet mathematicians often look for a surprising theorem.  If
we were totally logical creatures (and I've yet to meet one who wasn't
an insect or a computer), then a logically correct proof of a theorem
would cause us to say "of course, it's an obvious tautology."  

This, of course, does not happen.  Why?  Because we have intuition about
what should and should not be true.  If a logically correct proof is
presented of an unbelievable thing, then we search for another framework
in which to look at it.  If such a framework is found (for example, the
notion of a group of transformations on a space), then mathematicians
are able to "see" the theorem in its correct obvious light--but only
after the right mode of thought is introduced.

In the end, it's the same "trivial" tautology, but we've expended a
great deal of effort to make it "evident", or to obtain an elucidative
or constructive proof.  It is for this reason that the "logicist" school
of mathematics has always annoyed me.  As I understand it (I haven't
read his writings), Frege found it ridiculous that mathematicians
couldn't define the concept of a number to themselves.

This is an idiotic view!  I doubt that there were any mathematicians who
didn't fully understand the notion of a number, but Frege (or perhaps
Peano) gave them arithmetic axioms to fully elucidate their imagined
failings.  The definitions of any given number are quite circular:  3 is
defined as s(s(s(0)))--the third successor of zero.  n is defined as the
n-th successor of zero, and so on.  You need to understand the notion of
a number in order to "get" the logical definition of a number.

The logical definition is secondary to the intuitive one.  Useful to
study the logical structure of PA, but never (or hardly ever) useful to
practicing number theorists.  Furthermore (though neither Frege nor
Peano could have known this), one cannot even completely describe
arithmetic by logic alone.  

Bruno Poizat said the following on a highly related topic: "As for the
idea that a malicious God put us in a nonstandard model of arithemetic,
which could not even be the real arithmetic, and which we could
naturally be aware of, it can only be the product of a foggy brain.  It
is to lose sight of the fact that the language familiar to model
theorists is principally used by them for technical purposes, ... and
that it is only by a deformation of the mind that we end up believing it
to be the natural frame for mathematical thought:  In fact,
mathematicians easily break loose from it."  ("A Course in Model
Theory," Springer, 2000.  pp. 157)

The obviousness of claims like those about elementary arithmetic is one
thing which caused much annoyance to  many FOMers (myself included) with
regards to Buckner's postings.  Buckner's anti-Fregean arguments (among
other things) were obviously wrong.  Don't take me the wrong way.  This
doesn't mean they were in fact wrong (many "common sense" arguments are
really wrong, though they seem right & vice versa), but he didn't
explain them very well.  This caused everyone to look to the obvious to
refute them, which I'm sure angered him greatly (apparently having some
rationale for them).  In the end, to quote the rock group Guns 'n Roses:
"what we have here is a failure to communicate."

In the end, I think Buckner has some deep misunderstandings of
mathematics (not factual ones necessarily, but perhaps cultural?).
However, I think his discussion was fairly appropriate for this list.
While I'm certainly not the one to make such a determination, this list
is devoted to the foundations of math, and much of what Buckner wrote on
was definitely foundational in nature.  He didn't deal with the much
more highly technical research areas of set theory, second order
arithmetic, etc., but he did deal with very foundational issues indeed:
the definition of numbers, counting, & infinity.  I leave you with the
following quote of Henri Poincar\'e:

"...we cannot reduce mathematical thought to an empty form without
mutilating it.  Even admitting it has been established that all theorems
can be deduced by purely analytical processes, by simple logical
combinations of a finite number of axioms, and that these axioms are
nothing but conventions, the philosopher would still retain the right to
seek the origin of these conventions, and to ask why they were judged
preferable to the contrary conventions."  ("The Value of Science,"
Modern library, 2001.  pp. 464)

-Lucas Wiman