FOM: Goedel: truth and misinterpretations

V. Sazonov V.Sazonov at
Thu Oct 26 15:09:44 EDT 2000

Torkel Franzen wrote:
> Vladimir Sazonov says:
>   >It is
>   >greatest mistake to identify the formal statement Con(PA)
>   >with "Peano Arithmetic is consistent". But things like this
>   >are permanently happening here in FOM!
>   Not only in FOM, but in mathematics. As you know, mathematicians in
> general neither know nor care about the distinctions you emphasize,
> but speak freely about the truth or falsity of the Riemann hypothesis,
> the properties of algorithms, and so on, as do people in computer
> science and other formal fields, 

Yes, I explained this in terms of their illusions. 
There is (practically) nothing dangerous in this way 
of thinking of working mathematicians because all they 
do is under a strongest control of formal systems in 
which they are working. 

It absolutely does not matter whether they realize this fact 
or not! (However, mathematical logicians should realize this 
professionally.) They DO THIS AUTOMATICALLY, (with the help 
of some mechanisms of their brains, I think, and by making 
notes on a paper) as YOU DO when turn bicycle pedals without 
even thinking about this. It is a very good idea to be able 
to do something automatically! Mathematics allows to think 
mechanically, at least to check correctness of reasoning.  
Instead of turning bicycle pedals we turn our thoughts. 
This is the main idea of mathematics, I think. 

> without reference to any formal
> theories. 

See above. Especially computer scientists are so closely 
related with *explicit* formalisms (like programming languages), 
even more than mathematicians or logicians. 

> I would say that we have no grounds for thinking it is
> possible to uphold in practice the distinction between mathematics and
> "philosophy" that you urge.

If we will not even try (as I, believe, a GOOD philosophy of 
mathematics should do), we will always mix "black" with "white" 
and infinitely twaddle. 

A good rule: do not use the word TRUTH (or GOD) without a 
very serious need or reason. I believe, there should be a 
good corresponding dictum in English. (Ne pominaj Boga v sue 
- in Russian - kazhetsa tak). 

>   >It is completely unclear which way we could conclude (without
>   >making all the necessary distinctions!) that from some
>   >philosophical point of view there are arithmetical truths
>   >(IN WHICH SENSE, PLEASE?) which are unprovable (IN WHICH SENSE,
>   >PLEASE?).
>   Why, in the ordinary mathematical sense, of course. 

PROVABLE BY HUMAN BEING or in the sense of 
\exists x Proof(x,y) with Goedel formalization of 
Proof predicate? 

Be careful and precise, please! 

When we speak of
> e.g. the possibility that Goldbach's conjecture is true but unprovable
> in PA, we are speaking of the possibility that even if every even
> number greater than two is the sum of two primes, there may be no
> formal deduction in PA of the canonical formalization of this
> statement.

I understand this phrase ONLY as a mathematical sentence 
(i.e. FORMAL one, using the formal quantifier symbols, 
the formally defined predicate Proof and the entailment 
notion |=, etc.; see the above comments on formal character 
of mathematics). Then there is nothing philosophical to 
discuss. Just prove or disprove the corresponding formal 
sentence in a formalism where it was formalized/formulated 
(say, in ZFC). 

But returning again to yours 

>   Why, in the ordinary mathematical sense, of course. 

I recall that this was said concerning 

>   >from some
>   >philosophical point of view ...

I have no idea how mathematically describe that 
"philosophical point of view" (see above). That view 
was (quasi)philosophical and therefore should be grounded 
philosophically. By a GOOD PHILOSOPHY, PLEASE, making 
all the necessary distinctions and having nothing to do 
with (theological) beliefs (in truth of axioms or in anything 
else). It should be a philosophy of SCIENCE. 

What about the following principle: 

Our illusions, even mathematical, cannot be formalized 
in principle. (Who doubts?)

It seems Goedel's incompleteness theorems may play a role of 
a rigorous mathematical counterpart (not a proof or formalization) 
of this self-evident principle. 

Vladimir Sazonov

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