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John,
Thank you kindly for your reply. I should say that I'm familiar with Tarski's work as well, though on a digression, I thought he committed to points, and nothing else, not vanishingly small spheres, i'll have to look over his axioms again. Moreover, i'm under the impression that Tarski and Hilbert's geometries are mutually interpretable, though I haven't come across a proof of this in logic, though many in geometric algebras exist.
The issue i'm grappling with is the degree of ontological commitment one makes when picking a relation vs an "entity" (something that we quantify over) to represent a concept.
I imagine each choice of representation has its own strengths and weaknesses, though i'm not entirely sure what these would be.
My question to the community is -- have you come across scenarios where it is more advantageous to commit to the existence of an entity as opposed to capturing the "behaviour" of an entity, implicitly via a relation? Why / why not?
Cheers,
Ali On Sat, Feb 14, 2009 at 10:37 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:
Ali,
Although I used a theory with a single axiom and predicate to
illustrate the idea, it is better to consider an entire theory
(defined by the total collection of axioms) to determine the
ontological commitment.
> For example, take the notion of linesegment in (or extending)
> Hilbert's geometry formalization. One might be tempted to
> implement it is as strictly a relation between 2 (or 3) points
> say in ontology O1 - i.e. (linsegment x y z) where (x,y,z) are
> all points. Another, might in ontology O2, be tempted to define
> a new entity "linesegment" which consists of points -i.e.
> (linesegment XY x y). Is one making a stronger ontological
> commitment than the other?
Since Hilbert's axioms already specify lines, points, and a lot
more, I suspect that your axioms (assuming that they are consistent
with Hilbert's) would be a rather straightforward application that
wouldn't add much, if anything to the ontological commitment.
For a more radical example, I suggest Tarski's version of solid
geometry, in which the only primitives are spheres of arbitrary
finite size:
Tarski, Alfred (1929) "Foundations of the geometry of solids,"
in Tarski (1982) _Logic, Semantics, Metamathematics_, Second
edition, Hackett Publishing Co., Indianapolis, pp. 24-29.
In that short paper, Tarski used a version of mereology instead
of set theory. That made a much smaller commitment right at
the beginning, since mereology does not have the generative
capacity of set theory -- i.e., it doesn't support the option
of building complex mathematical structures from iterations of
the empty set -- {}, {{}}, {{},{{}}}, {{},{{}},{{{}}}}...
For a brief summary, see the paragraph below.
For pretty pictures inspired by Tarski's geometry, see
http://frot.org/t/tarski/
Note that the paper is only 6 pages long. That is not long enough
to build up all Euclidean geometry. What Tarski did was to build
the foundation and demonstrate that it had a great deal of power.
Finally, he defined 'point' as the limit of a nest of spheres.
Then he showed that the axioms for Euclidean geometry could be
defined in terms of those points:
1. The only ontological commitment is to finite spheres.
2. Points, straight lines, and planes don't "truly" exist on
the same level as spheres. They are abstractions defined as
limiting cases of infinite series of spheres.
Since all physical structures are made of tiny atoms (or particles
even smaller than atoms), truly straight physical lines, planes, and
solids never occur in nature, and they're impossible to construct.
Therefore, all the constructs of point-based Euclidean geometry
are "imaginary" or "fictitious" structures that cannot exist
physically.
That was Whitehead's motivation for his system of "extensive
abstraction". He independently developed an approach that was
even more general than Tarski's because he started with arbitrary
four-dimensional blobs. But I recommend Tarski's approach for
an initial study, since his paper is only 6 pages long.
John
________________________________________________________________________
Source: http://en.wikipedia.org/wiki/Alfred_Tarski
In the 1920s and 30s, Tarski often taught high school geometry. In 1929,
he showed that much of Euclidean solid geometry could be recast as a
first order theory whose individuals are spheres, a primitive notion, a
single primitive binary relation "is contained in," and two axioms that,
among other things, imply that containment partially orders the spheres.
Relaxing the requirement that all individuals be spheres yields a
formalization of mereology far easier to exposit that Lesniewski's
variant. Starting in 1926, Tarski devised an original axiomatization for
plane Euclidean geometry, one considerably more concise than Hilbert's.
Tarski's axiomatization is a first-order theory devoid of set theory,
whose individuals are points, and having only two primitive relations.
In 1930, he proved this theory decidable because it can be mapped into
another theory he had already proved decidable, namely his first-order
theory of the real numbers. Near the end of his life, Tarski wrote a
very long letter, published as Tarski and Givant (1999), summarizing his
work on geometry.
-- (•`'·.¸(`'·.¸(•)¸.·'´)¸.·'´•) .,.,
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