[FOM] On Euclidean and Hilbertian geometry and predicativeness

Aatu Koskensilta aatu.koskensilta at xortec.fi
Wed May 21 08:52:01 EDT 2003

Luigi Borzacchini wrote:
> As I told before, comparison is not easy, and I do not agree completely with
> Aatu's opinions.
> For example:
>>In Euclid's Element's the axioms and postulates are stated in terms of
> This is not true for axioms, which concern with equality, and also the V
> postulate concerns with parallels and does not seem a construction.

This is true. I should have spoken only of postulates. The fifth 
postulate is not an existence postulate; it says that if three given 
lines are in a specific relation to each other, then something will also 
be true of them (namely that two of the lines, if extended indefinitely, 
  meet on one side of the third line, the side being determined by the 
interior angle being less than two right angles). There seems to be 
nothing predicative nor unconstructivistic here.

> There has been a interesting discussion about how far the construction is
> actual or simply reveals implicit properties of the diagram (Mugler,
> Cambiano). Maybe in the IV century B.C. there was a keen discussion between
> a more theoretical (Plato-Speusippos) and a more problematic
> (Eudoxus-Menaechmus) approach.

Perhaps. I faintly recall reading a book on the development of the 
deductive method in greek mathematics which argued that in fact the 
diagrams are part of the semiformal system (other part contain rigid 
syntactic proof-elements expressed by repeated and recursive use of a 
very limited set of expressions), and that some of the propositions in 
Euclid which have been taken to contain tacit premises don't in fact 
cotnain anything hidden provided we assume that the diagram is also to 
be considered part of the set up for the theorem and not as a mere 
"visualisation aid". I'm not sure but perhaps the book was
Reviel Netz's The shaping of deduction in Greek mathematics : a study in 
cognitive history.

>>Euclid's geometry is clearly predicative; in order to define a thing and
>>be secure that the definition is valid we need only to ascertain that it
>>can be constructed from things we already know can be constructed from
>>constructible things.
> In Euclid many definitions are given before their construction.

Perhaps I should have been more explicit about what I mean here. I'm not 
  speaking about definitions of concepts (predicates, relations, &c.) 
that apply to geometric objects, e.g. isoscele, but of "naming" of 
geometric objects themselves. In Euclid (it seems to me that) all of 
these objects are either given (in the diagram or in the conditions of 
the theorem) or constructed from these given objects. In a sense, to 
refer to a object we must show a construction leading to it from the 
given objects. Also, I know of no definition in Euclid presupposing 
quantification over a domain which would include the deifned (named) 
thing. This clearly shows that there cannot be (or perhaps simply 
aren't?) impredicatively named or defined objects in Euclid at all.

This could be paraphrased by saying that Euclid's geometry is 
constructivistic (in the present sense of the term).

 > In addition
> we must remember that Aristotle distinguish neatly between definition and
> proof of existence. 

This may be. It's not, however, what I'm interested in here. From the 
fact that a definition is predicative, it does not follow that there 
must exist something that falls under the definition anymore than it 
does for impredicative definitions.

 > It is not sure that Euclid shared this opinion. I
> believe he didn't for a deeper reason.
> The problem is that in Euclid, in my opinion, the mathematical concepts had
> a meaning also before the definition, which played, together with the
> postulates, the role of starting points (i.e. elements). This is the crucial
> difference with Hilbert. In Euclid before the axiomatization everything had
> its meaning, in Hilbert before the axiomatization nothing has a meaning.

I don't really think this characterises Hilbert's approach. With 
Hilbert, surely the concepts *do* have a meaning prior to 
axiomatisation; axiomatisation is part of being "conscious" of ones 
intellectual undertaking and being precise about what one's concepts 
mean. Of course, in order to be certain that we have captured the 
concepts correctly, we need to "bracket" (in the phenomenological sense) 
the meanings so as to be certain that everything actually *does* fall 
out from the axioms.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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