[FOM] On Euclidean and Hilbertian geometry and predicativeness

Aatu Koskensilta aatu.koskensilta at xortec.fi
Tue May 20 03:04:21 EDT 2003

In Euclid's Element's the axioms and postulates are stated in terms of 
constructions; given this (two points) you can construct that (a 
circle). In the Hilbertian axiomatisation of Euclidean geometry we have 
instead that the existence of certain things implies the existence of 
other things.

This seems like a very clear cut constructivist/platonistic constrast; 
in Euclid's geometry we may only consider things we could construct (out 
of thin air, so to speak, but construct nevertheless) whilst in 
Hilbertian geometry we may quantify over a collection of 
points/lines/circles/etc that is only "implicitly defined" by the axioms.

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 contrast it seems that prima facie Hilbert's 
geometry allows for "essentially impredicative" things to exist; there 
may be points or lines which can only be defined by referring to (a 
subset, containing the point or line in question, of) the totality of 
things geometric. The completeness axiom in particular seems to rule out 
the picture of geometric space as being the closure of some set of 
points and lines under the operations of Euclidean construction.

Of course, theories which at first look impredicative might very well be 
equivalent to predicative theories after all. It seems to me that the 
completeness axiom rules out any "purely predicative" theory equivalent 
to the Hilbertian axiomatisation, but I could be wrong.

Surely there has been research into these questions. I'd be grateful if 
someone could point me to the relevant literature. Comments in the FOM 
list are appreciated, too.

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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