Re: Name the thesis: "Formal sentences capture informal ones"

tchow_at_lsa.umich.edu
Date: 01/30/05 

Date: 30 Jan 2005 01:26:31 GMT

In article <ctgt5c$34t$1@phys-news1.kolumbus.fi>,
Aatu Koskensilta <aatu.koskensilta@xortec.fi> wrote:
>The problem with the thesis under consideration is that, unlike the
>Church-Turing thesis, it doesn't equate a mathematically defined concept
>with an informal one, it equates two equally informal and vague
>concepts.

O.K., let me try another version.

   (*) Intension-preserving formalization of informal mathematical
       statements is always possible.

Maybe this should be thought of not as a thesis but as a "thesis schema"?
Instances of the schema would be things like:

   (+) Con("PA") is an intension-preserving formalization of "PA is
       consistent."

Yeah, I know I'm abusing the term "schema" here, but I think you know
what I mean. Con("PA") is formal; "PA is consistent" is informal, so
like the Church-Turing thesis I'm equating---or at least making a tight
correspondence between---something formal and something informal.

Something like (+) is tacitly assumed by most people. Whenever we
draw philosophical conclusions from Goedel's 2nd theorem such as "The
consistency of PA cannot be proved using methods formalizable in PA"
we are tacitly assuming (+) and a whole host of statements like it.
This degree of acceptance seems to me to be very parallel to the
widespread acceptance of the Church-Turing thesis.

Failure to recognize that (+) is being assumed leads to all kinds of
confusions and misunderstandings. It seems to me that this warrants
the attempt to be more explicit about what is happening. 

-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences