FOM: Goedel: truth and misinterpretations

[Kanovei]

> Date: Tue, 24 Oct 2000 09:31:04 +0200
> From: Torkel Franzen <torkel at sm.luth.se>
  
> ... after all it is mathematically
> provable that if ZFC is consistent then there is a true arithmetical
> sentence A such that the canonical translation A* of A into the
> language of ZFC is not a theorem of ZFC. Apparently you don't regard
> this particular result as even meaningful. Why not? 

I wrote that the result is widely misinterpreted by philosophers 
(and philosophying mathematicians). The nature of misinterpretation 
is that the wording of true mathematical result is interpreted in 
terms of relations between a human-mathematician (or the mathematical 
community) and the mathematical universe. 

It seems that you do not buy my conceptual explanation of the 
yesterday post, so let me explain my point in simpler, but 
more practical fascion. 

You write: 
> true arithmetical sentence A such that ... 
True -- where ? in which sense ? 

Let's forget Goedel's mystical "sentences" and consider CH 
which everybody knows what it means (at least formally). 
neither CH nor not-CH is a theorem of ZFC -- and this is 
a mathematical fact (I let aside the assumption of Con ZFC). 

Now, the misinterpretation which I am talking about reduces 
to the following: 
BUT either CH or not-CH is TRUE ,
which is here nothing but an example of the excludedmiddle. 

Now let me ask again: True -- where ? in which sense ? 

Are you going to answer anything more meaningful here than 
just claiming CH or not-CH as a ridiculous successor of the 
Danish prince ? No, you are more reasonable

> that's no obstacle to statements that refer only to a
> determinate part of that indeterminate structure being determinately
> true or false.

For instance, astronomers know that there is a mid-size black hole 
in the center of the Milky Way -- despite possibly there is no 
rigorous and fully determined concept of the Milky Way (its boundaries, 
for example). This is because there is factual information enough 
by the standards of astronomy to make the above conclusion. 
But we have reasonable doubts can the concept of the Milky Way 
ever be reasonably specified to make the statement that 
"it has either even or odd number of stars" to have any real 
meaning (I even do not say any verifiable meaning). 

And this is about the empirically existing structure, what it remains 
to say about the mathematical universe whose empirical core consists 
of things like finite combinatorics and observable shapes and the 
rest are just "ideas", i.e., axioms and their logical consequences~?   
Is there anything more than just a dogma, in saying that either CH 
is true or not-CH is true (for a mathematician, in the mathematical 
universe) ? 
And if yes then -- again, in which sense other than 
the meaningless in this context "CH-or-not-CH"? Any answer ? 
 
>   >Since Euclid, "we *know*" well that a mathematical statement is true  
>   >if there is a (mathematically sound) PROOF of the statement -- this 
>   >is by the way why Mathematics has been called EXACT science.
> 
>   Right, but what is your view of the axioms used in this proof?

This is the point on which there sometimes has been 
substantial disagreement between even "peers" of 
mathematics (the "Sinq lettres" between Borel, lebesgue, and 
Hadamard can be mentioned, let alone the Berkeley critics of 
the use of infinitesimals). 
My view is that ZFC concentrates, in the form of a few 
simply formulated principles, everything which has been 
practiced in mathematics, although, perhaps, in more 
generality that 99.9/100 of mathematics really needs. 
Perhaps we can trace ZFC to finite combinatorics of 
empiric collections of pebbles, that would give it some 
real substance.
There are writings by those who fully know the subject 
e.g. Shoelfield's article in the "Handbook of Math. Logic".

V.Kanovei