FOM: Goedel: truth and misinterpretations

[Kanovei]

> From: "Jeffrey Ketland" <ketland at ketland.fsnet.co.uk>
> Date: Sun, 22 Oct 2000 22:09:08 +0100
> 
> Semantics and the Problem of Reference

We must thank Ketland for a substantial post, including the 
opportunity to focus once again on typical philosophical 
mistakes and misinterpretations of mathematics. 
  
Basically, what does it mean: A is true ? 
I will not pretend to any depth in saying that there 
should be a Structure the truth in which is studied 
and an Observer who studies and evaluates. 
And the most problems in this issue appears of a mess 
with the latter.

(1)
> CH is true just in case any infinite subset of |R is either denumerable
> or equipollent to |R

Who is here the Observer~? 
Either this is a ZFC-daimon, an imaginary ZFC-dweller, 
or it is a Human mathematician to whom the Mathematics 
as a whole is an external structure. 

For a ZFC-dweller, (1) of Kerkland is just the definition 
(see Cantor, Hausdorff, etc.), then CH is either true or false, 
and to see which the ZFC-dweller must simply thumb through all 
infinite subsets and to see if there is a counterexample. 
 
(2)
> The following T-sentence can be *proved* from the definition
> (v) and axioms (i)-(iv):
> 
>  CH is true iff CH

If (i)-(v) are literally as in Kerkland's post then ZFC-dweller 
will characterize (2) as meaningless because (i)-(v) appeal to 
quantification over proper classes, nonexisting in his world, 
and by no means can be soundly converted to a ZFC 
sentence, see Tarski, "undefinability of truth definition". 
If (i)-(v) are modified so that to define truth in a 
mathematical structure then (2) becomes a tautology 
(for the structure that 
consists of reals, sets of reals, and everything else   
involved in the definition of CH). 

Consider the other case, when Observer is Human and the 
Structure is Mathematics as a whole. 
Now, how can it be claimed that any given mathematical 
statement is either true or false? 
Not, and exactly by a very 
elementary reason: the Structure is not fully determined. 

To make this more precise, let us ask: is the number of stars 
in the Milky way even or odd ? 
To answer this at least we have to specify 
where the boundaries of the Milky Way are, to claim that it is 
really either even or odd. 

The same for the Mathematical universe: its "parameters" are 
not fully specified to answer ALL questions, even  
CH in particular, and it is what Goedel demonstrated that 
(possibly unlike the Milky Way boundaries) 
the Mathematical universe just CANNOT be fully specified and 
determined as to allow to answer ALL mathematical questions. 
 
(3)
> 5. Does mathematical truth mean "having a mathematical proof"? (Kanovei)
> 
> Obviously NOT. The truth set for (|N, 0, s, +, x) is not axiomatizable. This
> was discovered by Kurt Goedel in 1930. N.B. The definition (v) of truth (for
> sentences of the lang of set theory) above makes no mention of *proof*.

This is probably the most splendid misinterpretation of any 
mathematical result, that persists 70 years in discussions 
between clones of philosophers (and mathematicians, and, shamefully, 
logicians, too). 

Goedel proved (in a very weak fragment of ZFC, essentially, in PA) 
that for any theory T of certain kind, if Cons T then there is an 
arithmetical statement A such that neither Prov_T A nor Prov_T (not-A), 
where Cons and Prov express the consistency and provability (in T). 
Then Ketland notes: gee, either A or not-A is true, hence, there is 
something true but not provable. 

Yes, ZFC-dweller may argue this way.  

But what does this mean for a real, Human Observer of the Mathematical 
universe (we are not ZFC-dwellers so far...) ? 
Take, for instance, Con ZFC as A, a statement neither provable 
nor refutable mathematically (in ZFC), by Goedel (unless ZFC happens 
to be inconsistent). 
Kertand says, by (2) above: go ask ZFC-dweller if A is true or false. 
But whom to ask ? 
Steel may say that ZFC-dweller must obey a class of Woodin cardinals, 
Harvey Friedman will say that Con(some large cardinal) must be respected, 
Shipman will require a certain measure on the reals, 
which yield even partially incompatible concrete consequences. 

Thus, the only meaningful consequence that we can draw from 
Goedel's result mentioned in (3) is just that there is no 
axiomatization of Mathematics, of certain kind (recursive etc.) 
which is both consistent and complete. 
Any further reference to "true but not provable" is a misconception, 
whose philosophical nature is the misidentification of the 
Human and ZFC-dweller as Observers. 

> 6. How do we *know* that a mathematical statement/axiom is true?
> 
> Question postponed!
> (Intuition? Self-evident? Follows from logic alone? Indispensable to
> empirical science? Most pragmatically coherent explanation of our basic
> intuitions?).

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. 
There is no other way to figure out is, e.g., Riemann hypothesis 
true or false, other than to prove or disprove it, and no intuition, 
empirical observation, "reference theories", etc. can help to achive  
the goal surpassing mathematical proof. 
There is no exceptions. 
The real philosophical question here is the ratio 
of empirical/social/individual in the status of 
"mathematically sound proof", including both the choice of axioms 
(why everything is so focused on ZFC ?)
and the methods allowed in proofs.

V.Kanovei