FOM: Goedel: truth and misinterpretations

[V. Sazonov]

Torkel Franzen wrote:
> 
> Vladimir Sazonov says:
> 
>   >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.
> 
>   They don't usually work in any formal systems at all. 

Look more attentively, and you will see this. I do not know 
well school education system outside of Russia, but our, even 
not the best beginner University students know and even 
can use or just follow after the teacher, for example, the 
formal rule reductio ad absurdum. They learned this from some 
proofs in elementary geometry. The formal system where 
mathematician is working may be not realized explicitly and 
its precise formulation may be not very important. Does it 
matter, where you prove a theorem, in PA or ZFC if actually 
the proof may use only a rather small fragment of PA? Also, 
which precisely basic language you use, with + and * as 
function or relation symbols, etc.? If we do not recognize 
(implicit) using of some axiom or rule, we usually recognize 
this later. Semiformal proofs became formal (formalizable) 
later. Otherwise, it is anything, but not mathematics. It is 
simply the form of existence of mathematics. 

Of course, if your prejudiced view is that there is some 
(mysterious!) mathematical TRUTH which we should find (which 
way, if not by using rigorous proofs?) then you will not see 
the evident fact that mathematics is (first of all) a formal 
science. 

We learn, explain,
> talk about and apply mathematical concepts informally in innumerable
> contexts. I'm sure I haven't grasped what you regard as the proper way
> of thinking about mathematics, but it seems to me extremely esoteric.
> It's not clear what relevance it has to mathematics as it actually
> exists and as it is pursued (or can be pursued) by human beings.

Nothing esoteric. I wrote about this in FOM many times. 
Mathematics is just a mechanizing (formalizing) of thought 
(or of that part of thought which could be mechanized; 
same for other kinds of engineering helping in human physical 
activity). E.g., how would we multiply big decimal numbers 
without a formal rule which we learned at school? 
In mathematics we act formally (actually, sufficiently 
formally) and simultaneously think about what we are doing 
informally (as informally as we want, as it is useful for 
the result). We may think informally in terms of truth or 
in any other terms we prefer. Essentially, mathematical informal 
thinking is just a way of oral or written presentation of their 
formal activity in terms of a natural human language. No problem! 
Until a philosophy begins. But it should be rather careful. 

Say, outside the ordinary (FORMAL, with formally defined 
"|-" and "|=") mathematical context I have no idea 
WHAT DOES IT MEAN that 

"a mathematical statement is true, but unprovable". 

In which sense true? In which sense unprovable? 
It is just a big nonsense, nothing else. Mathematicians usually 
do not assert such statements. Even if they do, it has NO 
relation to their real mathematical activity. They would rather 
to formalize any semantically suspicious assertion and after that 
it becomes something strongly different from the first, actually 
meaningless version. Different, but subject to mathematical methods. 

Truth may be present in the real life. In mathematics this 
word is just a helper for our intuition in finding formal 
proofs. In mathematical logic corresponding formally defined 
notion |= is something like a chess piece. We are playing with 
truth, and the play may be very reasonable, meaningful 
and useful. A good philosophy probably should explain this, 
but not create meaningless concepts. 

> 
>    A clarification:
> 
>    >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 ...
> 
>   "In the ordinary mathematical sense, of course" was in response to
> your parenthetical "(IN WHICH SENSE, PLEASE?)".


We cannot consider that "something is true but unprovable" 
*in the ordinary mathematical sense* if this statement is not 
put in a formal context, as it ALWAYS required in mathematics. 
Forget temporary your philosophical inclinations and start to do 
something mathematical where you should formulate some new 
notion (let not so revolutionary as e.g. it was the continuity 
concept) and investigate it. After some preliminary very informal 
considerations you will become an acting formalist rather quickly
(e.g. by writing some differential equations and manipulating 
with them). 

On the other hand, *from philosophical point of view*, e.g. 
"unprovable" should be, I think, considered something like 
as "unprovable by human beings". Do you agree that we cannot 
uncritically replace this by mathematical 
\not\exists x Proof_{PA}(x,y)? 
We may try to consider relations between these statements 
(only one of which is mathematical). 
Say, if the latter is proved in ZFC then the former is 
most probably true. Here the word "true" (considered 
non mathematically) seems to me quite appropriate. It has 
a sufficiently REAL meaning. What is (formally) proved in 
mathematics may have SOME relation to the reality and by 
this reason we can sometimes use mathematics in practice. 

> 
>   The "philosophical point of view" of the observation that Goldbach's
> conjecture, even if true, need not be provable in PA, is that of
> ordinary informal mathematics. This point of view has indeed been
> criticized by philosophers (I think most forcefully by Wittgenstein)
> as resting on an illusion or misconception ("false picture") of
> arithmetic as dealing with arithmetical facts analogous to physical
> facts ("the mineralogy of numbers", in Wittgenstein's phrase).
> As in other similar cases (such as Hume's criticism of everyday
> assumptions) it is a weakness of the criticism that it doesn't really
> offer any workable alternative to our ordinary ways of thinking.

Which ordinary ways of thinking? Those using meaningless phrases 
as above or may be as the description 

"the least natural number which cannot be denoted by 
an English phrase containing less than 200 symbols"

of analogous sort? 

It is NOT from the everyday activity of mathematicians. 
Their thinking has formal nature (with involving informal 
intuition and natural language). 


Vladimir Sazonov