[FOM] Simpson on Tymoczkoism

[P.T.M.Rood@ph.vu.nl]

On Thu Oct 02 23:36:03 2003,
Sandy Hodges <SandyHodges at alamedanet.net> wrote:
> [...]
>I take it that Rood declines to assert:
>
>    Given a line and a point outside it, there is exactly
>    one line through the given point which lies in the
>    plane of the given line and point,
>    so that the two lines do not meet.
>
>I wonder if he would be willing to assert:
>
>    IF, given a line and a point outside it there is exactly
>    one line through the given point which lies in the
>    plane of the given line and point,
>    so that the two lines do not meet,
>  THEN the angles of any triangle add up to two right angles.
>[...]

Hodges proceeds by wondering whether I am willing to assert
the second conditional, although (as Hodges takes it) I am not
willing to assert the Euclidean axiom of parallels (or, more
accurately, Playfair's axiom). If I am willing to assert this conditional,
I am willing to assert, as Hodges puts it, logical facts (what does it
mean to assert a fact?), and thus I would be committed to propositional
knowledge concerning such logical facts. Hodges' appears to suggest
that this conflicts with my claim that it is highly problematic to say
that there is propositional knowledge in mathematics.

Let me begin by noting that my point (in my FOM message from
Fri Sep 26 13:36:14 EDT 2003) was not about assertion; it was
about truth. Also, Hodges does not seem always to distinguish clearly
between truth and logical truth. The point I  tried to make was that it
eventually became dubious to say that the propositions of mathematics
are true. To which I still hold (see below). This holds for the axioms
in particular. Hodges' suggestion to put the axioms of geometry into
the antecedent of a conditional (and letting some theorem to be its
consequent), and to say that this conditional is logically true, evades
the whole point. I am willing to admit that those conditionals are truths
of logic (and hence logically true). But then: so what? My point was
about the propositions--theorems and axioms--of mathematics,
not about those of logic.

But perhaps I haven't been clear enough in my message to which Hodges
reacts.

Let us keep on considering the case of geometry--the points I make
should apply also to other branches of mathematics.

We may say that if a mathematical axiom or theorem is true, it is true
with reference to a (unique) antecedently given domain of objects.
Given such a point of view, if the axioms or theorems of geometry are
true, they are true with respect to some antecedently given domain of
objects. Traditionally, it seems that the objects of geometry were taken
to be "figures" in (physical) space. Euclid defined those objects the first
pages of the Elements (and also at the beginnings of other books of the
Elements). There, he defined what a point is, what a line is, etc. (indeed,
I take Euclid to have presented real definitions--definitions of the
essence of objects--and not lexical definitions).

Again, given such a point of view, the idea of the truth of the axioms and
theorems of geometry makes sense. Also, the idea of (propositional) knowledge
of the axioms and theorems of geometry makes sense. Finally, the idea
of demonstrating the truth of a theorem of geometry (not: a theorem
of logic) makes sense.

It was Hilbert who broke with this view. For Hilbert, the axioms and
theorems of geometry are not about some antecedently given domain
of objects. In particular, in Hilbert's view the axioms of geometry are,
in a way, about any system of objects that satisfies those axioms. Thus,
the axioms cannot be said to be true outright. The most we can say
is that those axioms are true with respect to some given class of models.
Thus, Hilbert clearly formulated a view, a methodology, that became
characteristic for modern mathematics.

I have always found it striking that, contrary to what Euclid did in his 
Elements,
Hilbert did not open his Grundlagen der Geometrie with a list of definitions
determining the ontology of geometry--the domain of objects that geometry is 
about.
Rather, basically  the first thing he did was simply to present a list of 
(five groups) of
axioms. Accordingly, this would seem to confirm that he indeed dropped the 
view that
geometry is about some antecedently given domain of objects. And thus, he
equally well dropped the idea that the propositions of geometry are true with
respect to such a domain of objects. And thus, he let go the idea of 
(propositional)
knowledge of the propositions of geometry. Indeed, for Hilbert, questions of
dependency and consistency were far more important for the foundations of
mathematics.

(The novelty of Hilbert's views in the above respects is addressed in: P. 
Bernays.
Hilbert's significance for the philosophy of mathematics. Transl. and repr. 
in:
>From Brouwer to Hilbert. The debate on the foundations of mathematics in the
1920s, P. Mancosu (ed.), New York: Oxford University Press, 189-97.)

Ron Rood
-- 

*****************************
Ron Rood
Department of Philosophy
Vrije Universiteit Amsterdam
De Boelelaan 1105
1081 HV Amsterdam
The Netherlands
e-mail: p.t.m.rood at ph.vu.nl
FAX: +31-20-4446620
phone: +31-20-4446614
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