FOM: precursors of Cantor (&Transfinite Logic)

[Everdell@aol.com]

On June 10, 2002, Dean Buckner wrote:

<<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.>>

I've hesitated to comment, but I feel bound to express my regret.  It has 
always seemed to me that symbolism was suspiciously simple and somewhat 
overrated as an aid to foundational thinking.  My reason for this heresy is 
similar to Prof. Buckner's.  I find that every symbol we use must be shared, 
and this requires each symbol to be defined, and this in turn inevitably 
requires clear philosophical thinking and a translation into a current 
language in which, as Frege might say, people may communicate, agree and thus 
be "objective" about the objects of thought.  One might even argue, in 
Wittgensteinian fashion, that the meaning of any symbol is no more than the 
resultant of (or response to) all uses made of it, which means symbols are no 
more to be trusted than words or phrases--and indeed may turn out to be not 
even philosophically *distinguishable* from words and phrases in our joint 
pursuit of rigor.  Unable, therefore, to profess the Leibnizian faith in 
symbols, I am tempted to return to outer darkness to listen to the weeping 
and gnashing of teeth.

But on June 13, Martin Davis wrote:

<<Leibniz was among those who observed the one-one correspondence between the 
natural numbers and one of its infinite subsets - in his case, the even 
numbers. He concluded that it makes no sense to speak of the number of 
natural numbers. See my "The Universal Computer" aka "Engines of Logic" p. 
65>>

What is a "subset"?  I once searched my Cantor for a definition and found 
none.  Usually when the mathematical writer feels a certain slipperiness in 
the concept, he or she adds the word "proper,"  as Cantor did (in German), 
although "proper" is not defined--and neither have I seen any referent for 
"improper subset."  It seems to me, therefore, that the proof that a set is 
"equinumerous" (that's not the same as "equipollent," right?) with one of the 
set's subsets, especially an infinite subset, assumes one of the premises 
necessary to the proof, viz., that the "subset" in question is somehow 
"included" in the set, despite they're being the same size.  

I hope that if "subset" is indeed defined in any of these earlier works, 
someone on FOM will point it out to me (so I can ask about the terms used to 
define it).  I mean only to suggest to colleagues here that mathematical 
symbolism is never really complete and that crucial ideas continue to be 
assumed through language, which is almost always used to define and 
communicate foundational concepts.  This is why mathematics has a history, 
and why there is such a thing as the philosophy of mathematics.  I like the 
argument made by Poincaré and quoted by Lucas Wiman (which must of course be 
about French, not English).  "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)"

And by Dean Buckner:


<<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 suspic
ious precisely because of they were formal, & therefore invented. [...]  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. >>

So I shall not join the apostates like Bertrand Russell in outer darkness, 
and I expect other historians like me will continue to lurk here with writers 
and philosophers.  For the education we get, much thanks.
 

-Bill Everdell
History
St. Ann's School
Brooklyn, NY