FOM: Goedel: truth and misinterpretations

[V. Sazonov]

Dear Torkel Franzen, 

You wrote:
> 
> Vladimir Sazonov says, with reference to
> 
>     (2) Even if every even number greater than 2 is the sum of two
>         primes, this is not necessarily provable in ZFC.
> 
>   >Very good! Eventually, instead of "A is true" it is written simply A.
>   >The ill-stared "truth" is banished!
> 
>   I'm aware that you're of the opinion that we should not "use the word
> TRUTH (or GOD) without a very serious need or reason". However, I don't
> believe we need to approach the foundational problems of mathematical truth
> in any such spirit of awe and mystery. 

Is it me who introduces the spirit of awe and mystery? My words 
cited above were addressed to those who use the term "truth" 
in a way non-explainable in rational (scientific) terms which 
could be rather characterized as theological or the like. 
I believe (and I am not original in this) that we should put 
any our faith aside when doing science, even if it is a philosophy 
(of science). 

Probably I am insufficiently convincing for some participants, 
but it proves that "the other side" is also not very adequately 
(from scientific point of view) replies to the arguments touching 
somebody's faith. It is too principal question whether we can use 
faith (in Platonic mathematical world and in corresponding 
absolute truth or in anything analogous). It is here where awe 
and mystery can be found. 

Looking through the last postings to FOM I see again and again 
that there is a real problem of mutual (mis)understanding 
between the two sides (if there are only two). 

By the way, I should note, that I agree with almost everything 
said by Vladimir Kanovey who often (very witty and clearly) 
wrote some comments just before I intended to do that. This 
is the reason that sometimes I referred to him. Actually, it 
was somewhat unexpected and a good surprise for me that our 
views during this discussion almost coincided. There were no 
other relation or cooperation with him. However, attributing 
to him ultrafinitism because I wrote during this discussion 
some corresponding notes is also incorrect. (This is concerning 
the whole discussion.) 


Why not just soberly consider
> the concepts we use and the use we make of them? 

I think that I soberly distinguish the following correct ways of 
using the term "truth": (i) in the real world, as in everyday life 
or in (experimental) physics and other natural sciences and (ii) in 
mathematics (ii-a) as synonym of "provable" and (ii-b) as an 
analogue of (i), but in an illusory mathematical world always 
relativized to a given mathematical formalism. This illusory truth 
is just helper for our intuition and cannot play the same role as 
in (i), even to have some scientific or philosophical analogy with 
(i). It is illusory truth about illusory world and should be soberly 
always considered as such. It is rather the subject for psychology 
and I strongly believe it is philosophical mistake to ignore these 
distinctions. 

Is not this reply to your question what I consider wrong in your 
position? Didn't I say this some time ago and now essentially only 
repeat this position in slightly different terms? 

Now, I would be very grateful to anybody to say what is wrong 
in this schema and which kind of truth is omitted there. 

(I omitted relation of mathematics to reality and to corresponding 
truth via applications in physics, etc. But it was Hilbert
who 
already explained this by distinguishing and relating 
Geometry-as-physics and Geometry-as-mathematics. 
Analogously, 
we should clearly distinguish 4CC as sentence about real maps 
and as a mathematical sentence, what is unfortunately mixed in 
some recent postings to FOM when it was essentially said that 
a scientific proof of 4CC (as a physical fact about maps) may 
be considered as mathematical proof of 4CC. Anyway, even if 
it could be said that this is my mistaken impression, the 
distinctions between mathematics and physics, etc., should be 
made extremely precise in such discussions as in FOM.) 

I have no idea how to characterize "true, but not provable" 
according to the above classification, if this sentence is not 
understood in the well known technical (meta)mathematical sense. 
Moreover, during the current discussion I have not seen any 
reasonable explanation how to understand this phrase. This was 
the main request, not formalization of this sentence. However, 
I believe that formalization is probably the best way to find 
out anything rational in any informal idea related with FOM. 


In the present context,
> I have explained that "true" in "Goldbach's conjecture is true" is to
> be understood in the mathematically defined sense, whereby "A is true"
> is a statement mathematically equivalent to A itself.

I am not completely sure what did youn mean by making this step. 
It seems that this was a step to formalization where any mystery
usually disappears. From my point of view such a step (not necessary 
exactly this one) was inevitable to make any sense of this sentence. 
Therefore my note above about ill-stared truth which seems affected 
you so strongly. I am very sorry if I was somewhat venomous! 


>   Your idea that (2) should be made mathematically precise is an odd
> one, since (2) isn't even a mathematical statement. 

I suggested, seemingly at the very beginning, to be sufficiently 
precise and to make all necessary distinctions, but you replied, 
as I remember, that it is impossible. For example, taking into 
account that your intended meaning of (1) was not mathematical, 
I proposed corresponding non-mathematical version of reading 
the term "provable" to make at least this part meaningful. 
I did not insist on mathematically precise meaning of the whole 
phrase (1), but as it was seemingly impossible for you to make 
it precise in any reasonable sense (or may be to understand what 
I ever mean; it seems that we both have somewhat different culture 
or scientific roots), it was natural to try some mathematical 
version. And this seems to me not a bad idea. 

Let us recall (independently on rigorous character of mathematics), 
what a mathematician do if somebody asks to give a definition of a 
term? Just gives a definition and the problem of misunderstanding 
disappears. I wrote that I do not understand what is "truth" in 
(1) and added I have no problems with understanding of Tarski's 
(mathematical!) definition of truth. I will try to "play" with 
(1) and (2) below, but it probably will not help because the 
question about "truth" in (1) will inevitably appear again. What 
does it mean "truth" concerning mathematical statements in that 
(external to mathematics) context? 

If you will allow
> me to be a bit repetitive, the general tenor of your remarks isn't
> difficult to understand. (2) is a statement of the same form as a
> number of everyday observations, for example "Even if John is at home,
> he will not necessarily answer the phone". Inspired, I believe, by
> certain very natural metaphysical inclinations, you take the view that
> a corresponding observation involving mathematical statements, like
> (2), does not really make good sense. 

It seems what this comment is not applicable to me. I just do 
not understand how is related this (quite meaningful for me) 
phrase on John to (2), except some syntactic analogy. 
I also cannot guess which my metaphysical inclinations 
do you mean here. 


I would at first rewrite (2) in slightly different simpler 
way which seems should be equivalent for you (and it is 
unclear for me why you did not use a simpler language): 

(2') It is possible that GC (holds? is true?) and GC is not 
provable in ZFC.

or, what probably is not your intention, as purely mathematical 
sentence (except "It is possible"), actually in the language of PA  

(2'') It is possible that GC and ~Prov_{ZFC}(`GC'). 


I guess that you would not use here the formally defined 
predicate Prov, but then I would suggest a third, 
non mathematical version: 

(2''') It is possible that GC and GC is not provable in ZFC 
by human beings (by proofs of feasible length, even with allowed 
abbreviations of any kind).

Again I guess that you would not mention here human beings by 
using simply (2'), but then it would be mysterious for me. 
I cannot understand what is "not provable" in (2) if it is not 
interpreted as in (2'') (where ~Prov_{ZFC}(`GC') is just formula 
which understanding is my personal problem related with my personal 
intuitions and illusions; if anybody wants, I can tell him on these 
my illusions) or as in (2''') (where "not provable" has a real, 
however, vague and not mathematical meaning). 

Thus, I have only two readings (2'') and (2''') of (2) to consider 
it further. I still have problems with understanding. Can I read 
"possible" as "consistent", i.e. as 

nobody (no human being) will deduce a contradiction from 
GC & ~Prov_{ZFC}(`GC') in ZFC (or in PA?)? 

or as formal, but of course not identical version: 

~~Prov_{PA}(GC & ~Prov_{ZFC}(`GC') => 0=1)

or, equivalently (say, in ZFC), by completeness of logic 

\exists M(M |= PA + GC + ~Prov_{ZFC}(`GC')). 


I guess all of this is inappropriate to you because your 
(in my rewording) "it is possible" assumes a so called 
"standard model" for PA understood not in that ordinary 
formalized way in terms of definability (as the least infinite 
ordinal) and provability in ZFC, but as existing "objectively", 
independently of ZFC in which "it is possible" that GC is 
TRUE (YOU CANNOT AVOID THIS TERM ACCORDING TO YOUR VIEWS!), 
but GC is NOT PROVABLE IN THE SENSE THAT IN THIS (MYSTERIOUSLY 
KNOWN ONLY TO YOU AND SOME OTHER ELITE PEOPLES) STANDARD 
MODEL THERE EXISTS NO GOEDEL NUMBER OF A CONTRADICTION 
PROOF. 

Sorry, I am a simple sinful and I am unable to understand 
these highest matters. I only can go step-by-step, following 
explanations, definitions, deductions, calculations, etc. 
If nobody explain me what is STANDARD MODEL and TRUTH I 
will never know what it is. 


However, it is pointless merely
> to *assert* that (2) does not make good sense, unless you have no
> interest at all in putting your point of view across to other people
> or in bringing them around to your way of thinking. 


It seems I made this in FOM so many times and during this 
discussion, too, so that I can only repeat. 


You need to engage
> in argument, which admittedly is always a prolonged and difficult
> matter, and often frustrating. Nevertheless, as we are aware, it is
> not impossible to influence both the thinking and the practice of
> mathematicians and others by philosophical argument, as long as you
> can make a connection with their actual intellectual concerns, in a
> way that respects their actual mathematical or intellectual
> experience.

I would be glad to, but sometimes there are so strong 
intellectual barriers in understanding! Just recall that it 
was explained here to Prof. Kanovei, a known specialist in 
logic, what is Tarski truth! Does it witness any mutual 
understanding? One thing is clear to me. We discussed very 
principal questions of foundation of mathematics - what is 
the nature of mathematical thinking and of so called 
mathematical objects. We are not first, and, surely, not 
the last. 

>   With this, I suspect that our particular exchange on this topic cannot
> profitably be taken further. 

It is really very pity for me if no profit was gained. 

I'd like to mention here that I included
> your essay "On Feasible Numbers" in the literature for a graduate course
> in the philosophy of science at the cs department here, where it caused
> many a furrowed brow as the students pondered the existence of 2^1000.

If you read that paper not just a curious, you would probably 
began to doubt that TRUTH is a healthy concept. 

> ---
> 
>   Torkel Franzen


With the hope that understanding is, nevertheless, reachable, 
and with the best wishes. 


Vladimir Sazonov