FOM: Transfinite Logic

[Dean Buckner]

On Heck's posting, there is little I have to disagree with (even the bits
where we seem to disagree).

Martin Davis, by contrast says, more worryingly  "Why are you posting (and
repeatedly) issues about the use of English on a  forum devoted to
foundations of mathematics".

Davis, who is on the FOM editorial board, clearly finds these postings
irrelevant and irritating (as do some others who have commented offline),
and consequently I will discontinue them.  But I would like the opportunity
to comment on why I placed them at all.

My background is in philosophy.  I had the good fortune to study briefly
with John Mayberry, and to discuss many issues around philosophy and the
philosophy  of mathematics, in the seminars.  But my main teachers were
Michael Welbourne and Christopher Williams, whose thinking was ingrained
with the "ordinary language" style of philosophy that came from Oxford, and
which dominated philosophical and logical thinking in the period of 1960's
and early 1970's (and after).  Students were encouraged to express ideas in
clear, plain English, and the use of symbolism was generally discouraged
(not that there wasn't any).  There was, too, the idea that plain English
was in order exactly as it was, and its structure embodied logic and
thought, and formal formal systems suspicious precisely because of they were
formal, & therefore invented.

This is in contrast to the style of logic that dominated in the US, highly
formalised, systems of natural deduction that people of course studied in
England, but which had far less impact in the philosophy of language.

Why, given this, have I been posting (repeatedly) issues about the use of
English on a  forum devoted to foundations of mathematics, dominated by
mainly US mathematicians?

(1) Because I am interested in issues that have *historically* been regarded
as foundational.  From offline correspondence I learnt to my surprise that a
number of the mathematicians had little idea of these.  They regard
Zermelo-Fraenkel as the one and only foundation, the idea that there was a
historical background to this came as a surprise to them.  I discontinued my
postings on Frege's arguments, for example, when I found out that many
correspondents did not really know, and were uninterested in, the arguments
of the Grundlagen.

(2) I continue to think that there is close connection between the logic of
natural language, and the number system, that has not yet been explored in
detail, and from which we could learn things of great importance.  In
particular, the idea that our language for numbers consist of two things (a)
The ordinals (the 1st, the 2nd ...) as a set of proper names, together with
a method of generating new ones, that ensures we do not run out of names ( a
symbolic concept) (b) a system of predication, i.e. "x is different from y
and z and y different from z" that can also be ordered, and whose use is
intimately bound up with the system of ordinals (a semantic concept).  Most
work on the concept of number has focused on counting.  this "sematnic"
concept of number eschews counting in favour of ideas tractable to a very
small number of logical concepts like reference, predication and identity.

I really think that something very powerful could come out of this, but I'm
aware these are just ideas at the moment, not very well connected or
explained.  Also they have no obvious connection with the
mathematically-orientated postings (such as Harvey's) that appear frequently
on these pages.  So, regarding with horror the idea of boring or irritating
people in a group which I am a guest, I will discontinue for now.

I'll close with the comment that neither Heck (nor Davis) has really
understood my point about the use of English.  Which was that the (obviously
true) formalised statements made by Heck are true in a quite banal and not
really very interesting sort of way.  Do they not become interesting only
when we impose the "folk" ideas of ordinary English upon them?  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.

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.



I'll recommend again Richard Arthur's paper, here:

http://community.middlebury.edu/~arthur/LeibCant.pdf



Dean Buckner
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