FOM: The meaning of truth

[Kanovei]

Goedel misinterpretation: conclusion

This discussion was started with the thesis 

THESIS (by Kanovei)
There is no any mathematical statement, 
(ontologically) true but not provable mathematically. 
That the known Goedel's theorem implies the existence of 
such a statement is a misinterpretation of the theorem. 
(end of THESIS)

Comment 1. By definition, any computation (manual or by 
a computer), any geometrical demonstration, any simple 
combinatorical reference are considered as kinds of 
mathematical proof. 

Comment 2. By (ontological) truth we understand any scientifically 
defined notion that can be seen as "to be factually true in the 
universe of things independently of any human ideology". 
This can be said sharper: to be physically true, in some 
physically valid sense. 

FACT 1. 
No any single counterexample for the THESIS above was 
presented, just no any one. 
By counterexample I mean a concrete mathematical sentence, 
like CH, or con ZFC, whatever, which is unprovable mathematically 
but can qualify as (ontologically) true in some well defined sense. 

Remark. To well-define the required sense is a duty of those who 
may want to present a counterexample, not my duty.

FACT 2. 
The "true but not provable" (TBNP below) party claimed the THESIS 
to be wrong by different reasons, which can be classified as follows. 

*Kanovei's THESIS is wrong because*:

(1) Kanovei is a member of a wild philosophical sect of fanatics, 
defined under titles like "superfinitists", 
perhaps, a Russian nigilist of sort (Mr. Black, Nov.2, was shy of this), 
which should intend that anything written by him is wrong by definition

(2) because Goedel proved otherwise

(3) (detalization of 2) because there is a ZFC truth predicate 

(4) because a sentence, say con ZFC, can be qualified as true by 
just an opinion of a particular mathematician or by a reference 
to an oracle previously known to be truthworthy (Mr. Shipman) 

(5) because there can be a device which produces, in some 
mathematically well defined manner, a dyadic sequence which is 
not recursive (Mr. Shipman)

(6) because there is (ontologically, not as a mathematical abstraction) 
a structure <N,+,x,...> determined enough to make any arithmetical 
statement "already" true or false by fact (a hypothesis of Mr. Shipman, 
from which, to his credit, he distanced himself).

Now my comments follow. 

(1) I am not a philosopher, nor a member of any philosophical 
group or sect, I have no published works in philosophy, nor   
philosophical parts in my mathematical papers, and basically 
I have not been a member of any societies, formal or not 
(except for purely mathematical like ASL) since my quit from 
"comsomol" due to age in end-70s. 
(Remark: membership in "comsomol", i.e., young communist league, 
was almost absolutely necessary 
for a non-Jew for any mathematical career in Moscow.) 
I can barely guess what "superfinitist" mean. At least it is not me.

That (2) and (3) demonstrate only that some members of 
the TBNP party just do not understand the 
point of discussion and/or the difference between mathematics 
and reality and/or mathematics in general, is more than clear.  
To make (2) and (3) ridiculous, it is enough to ask the 
TBNP party to apply their favorite version of "truth predicate"  
to CH or any their favorite unprovable sentence and figure out 
is it true or false, and then try to define, in clear, 
scientifically valid terms, WHY it has been qualified true or 
resp. WHY it has been qualified false, and WHAT does it mean. 

(4) can be discussed sociologically, that is, if CH was 
acknowledged as TRUE by the Den Haag tribunal, is it enough to 
agree that it is ontologically true ? Interesting question.

(5) needs explanations. Surely the existence of such a device 
would be the most unlikely event in mathematics perhaps after a   
proof that a major theory like ZFC is inconsistent. 

(6) I guess that this is just what some of the TBNP party really 
think but were enable to put clear, due to different reasons, 
including obligations to share (1) above or perhaps personal 
incompatibility with scientifically rigorous way of thinking and 
writing. 

To me, (6) is (unlike the rest) a discutable point, 
but far not obvious, and the recent post of Mr. Hazen 
(after paying his tall to (1), of course) 
shows how approximately it can be critisized. 
I could add more to that, but I am waiting that somebody 
of the TBNP party finally clarifies THEIR OWN vision of the 
topic of discussion, in scientifically wellfounded manner 
and without irrelevant arguments like (1,2,3) above. 

V.Kanovei