# [FOM] Mathematics ***is*** formalising of our thought and intuition

Thu May 27 17:11:51 EDT 2010

```I again change the subject to not mix things with Timothy Y. Chow's
Formalization Thesis which makes a different accent.

See the discussion below.

-- VS

----- Original Message ----
> From: "hendrik at topoi.pooq.com" <hendrik at topoi.pooq.com>
> To: fom at cs.nyu.edu
> Sent: Wed, May 26, 2010 2:22:40 PM
> Subject: Re: [FOM] Formalization Thesis: A second attempt

> On Tue, May 25, 2010 at 08:57:14AM -0700, Paul Budnik wrote:
> > On 05/21/2010 02:11 PM, Timothy Y. Chow wrote:
> > > Vladimir Sazonov<vladimir.sazonov at yahoo.com>  wrote:
> > >>
> > >> Mathematics ***is*** formalising of our thought and intuition.
> > >>
> > >
> > That seems too general.
>
> Perhaps step back from formalisiation a bit.  Try
>
> Mathematics is the art and practice of devising and using conceptual
> spaces where long sequences of reasoning steps can be carried out
> without error.
>
> Formalism is an important tool, but it isn't the only one.

First, I like your formulation and I read it as intuitively
very close to my.

Evidently, "conceptual spaces" belong to our thought and intuition.

As I wrote in a recent posting, I also understand "formal" in a wide
sense as "semi-formal" including here e.g. also Euclid's geometrical
proofs which were not absolutely formal in the contemporary sense.
Mathematics since the time at least of Euclid clearly separates
the form of our thought/reasoning from its content/intuition,
and correctness of mathematical reasoning is determined exclusively
by its form rather that by content.

I hope that these comments should make it clear that we really
have very close formulations because "long sequences of reasoning
steps" are just derivation steps which, if are not completely formal,
are completely (and potentially, as I always emphasise) formalisable.

If a derivation is and remains doubtful to be (potentially) formalisable
it will never be accepted as mathematical one by the mathematical community.