# FOM: geometrical reasoning

Stephen G Simpson simpson at math.psu.edu
Wed Feb 24 20:14:43 EST 1999

```Jacques Dubucs 23 Feb 1999 14:53:32 writes:
> I assume here that higher-order geometrical notions as continuity
> are really required by our scientific representation of the world;

This is not clear to me.  Doesn't the atomic theory point to some sort
of discreteness?  I think there is a lot of work to be done toward
integrating f.o.m. (= foundations of mathematics) with foundations of
physics.

> 1) The euclidean structure of space is described without intuitive
> residue and with a maximal precision (i.e.: categorically) by
> Hilbert's system H of the Grundlagen der Geometrie

Why do you identify `maximal precision' with categoricity?  I dispute
this.

To me, `maximal precision' should mean that all assumptions are stated
explicitly.  H is categorical, but in order to make it maximally
precise, we need to embed it, one way or another, into (first-order!)
predicate calculus.

Let me elaborate on this point.

It seems to me that your identification of `maximal precision' with
categoricity does not take adequate account of the work of G"odel and
Cohen regarding the set-theoretic independence phenomenon.  Yes, it's
true and a theorem of ZFC that H is categorical, i.e. it has exactly
one model.  However, by Cohen's work, this `unique' model has
strikingly different properties when understood within different
models of ZFC!  Thus there is an opportunity to go beyond categoricity
to obtain a greater degree of precision.  One way to do so is to
augment the axioms of ZFC with additional assumptions, e.g. the
continuum hypothesis.  That is the thrust of much recent work in set
theory.

The general point here is: Given any informal scientific theory, one
way to force all the hidden assumptions of the theory out into the
open is to formalize the theory in (first-order!) predicate calculus.

> Maximal precision in the description of geometrical structure, or
> completeness of a system of formal rules for deriving geometrical
> truths, one has to choose.

This is a false dichotomy.  We don't have to choose between these two
alternatives.  Categoricity can coexist with explicit formal rules.
Hilbert, G"odel, Tarski, Cohen, ... have shown the way.

> Henkin-style strategies to extend first-order deductive
> completeness to the so-called "so-called "higher-order logics""
> basically rest on the interpretation of "higher-order" as
> "many-sorted".  ... as far as geometry is concerned, this
> interpretative decision implies that we have to renounce to
> consider lines, polygons, polyedres, a.s.o. as sets of points, and
> it leaves us with the archaic conception of them as heterogens
> kinds of geometrical beings, each sui generis.

I don't understand this remark.  If we embed Hilbert's axioms into
Henkin-style higher-order logic in the obvious way, then lines,
polygons, circles, etc. are conceived as sets of points, not
self-sufficient geometrical entities.

Actually, the predicate calculus is flexible enough to accommodate
either approach to geometrical figures: sets of points, or
self-sufficient geometrical entities.

The predicate calculus is a very flexible instrument.  Post-modernists
and continental philosophers do not adequately appreciate this
flexibility.

> Prof. Simpson, are you sure you were happy to live in such a
> vicinity ?

I'm not afraid of it.  There is merit in viewing geometrical figures
as self-sufficient entities.  There is also merit in viewing them as
sets of points.  Each approach has its advantages.  We can integrate
these two approaches by means of suitable interpretations between
theories in the (first-order!) predicate calculus.

-- Steve

```