Re: Name the thesis: "Formal sentences capture informal ones"
Helene.Boucher_at_wanadoo.fr
Date: 02/01/05
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Date: 1 Feb 2005 08:54:01 -0800
tchow@lsa.umich.edu wrote:
> 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
I agree with your division.
Because you put ZFC in (*), I just presumed that you would have said
that (s union s) = d - where s is your favorite way of representing 1
in ZFC and d is your favorite way of representing 2 - adequately
expresses its informal counterpart, which is 1 + 1 = 2. Am I wrong in
that?
Otherwise how can the formal expression of "ZFC is consistent"
adequately express the informal assertion that ZFC is consistent??
In short, it seems to me - or at least this is where my confusion lies
- that (*) does not itself make the distinction between the two steps
(mathematical informalism to mathematical formalism, mathematical
formalism to logical formalism) but conflates them.
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