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

tchow_at_lsa.umich.edu
Date: 02/01/05 

Date: 01 Feb 2005 15:27:50 GMT

In article <1107246412.251830.121830@z14g2000cwz.googlegroups.com>,
 <Helene.Boucher@wanadoo.fr> wrote:
>In any case, *if* that is the logicist thesis, then indeed it would
>seem to depend on your (*). That is, informal mathematics cannot
>reduce to formal logic unless informal mathematical assertions can be
>captured by assertions in formal logic.

I agree with this. However, I would describe the situation as follows.
There are two steps involved: first, we translate informal mathematical
statements into formal ones. Second, the formal mathematical statements
are reduced to purely logical ones.

The possibility of performing the first step is what I was focusing on.
The second step is, I think, the heart of logicism. If someone were to
propose a slightly different philosophical position from what you're calling
logicism, namely that informal mathematics reduces to informal logic, I
would still be inclined to call that a variant of logicism. On the other
hand, someone who only accepts the first step but rejects the second doesn't
sound at all like a logicist to me. So I wouldn't call the first step any
kind of "logicist thesis."

Something like "1+1=2" prima facie speaks of natural numbers. It is rather
controversial whether natural numbers are purely *logical* entities. Simply
formalizing the statement "1+1=2" without explicating how numbers reduce to
logic might be the *first* step to demonstrating how logicism "works," but
it is really the subsequent step (reduction of numbers to logic) that is
crucial for the logicist. 

-- 
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
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