Discussion Question 2
Marty
axioms.
Look up Euclid's axioms. Doing research online, answer one or more of
the following questions, and give sources where appropriate:
(1) Do Euclid's axioms really suffice to prove his claims?
(1 1/2) At one point, Euclid classifies the platonic solids. Is he
using an axiomatic approach at this point?
(2) How does Hilbert's treatment differ from Euclid's? Do Hilbert's
axioms suffice to prove Euclid's claims?
(3) Look at Tarski's axioms for plane geometry (these will be placed
in the binder in JBE 356 on Thursday). These are first-order axioms.
Can you express Euclid's five axioms easily using Tarski's first-order
language of geometry?
(4) Look at a high-school geometry book. To what extent is an
axiomatic development carried out? What are the "postulates" or
"axioms" or "assumptions" used in high-school geometry?
Jacob West
> (1) Do Euclid's axioms really suffice to prove his claims?
I will first state Euclid's axioms, then consider his claims. Early
in Book I of his text "Elements", Euclid gives the following five
"postulates" [1,2]:
1. Any two points can be joined by a straight line.
2. Any straight line segment can be extended indefinitely in a
straight line.
3. Given any straight line segment, a circle can be drawn having the
segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines intersect a third in such a way that the sum of the
inner angles on one side is less than two right angles, then the two
lines inevitably intersect each other on that side if extended far
enough. (This is known as the Parallel postulate.)
In first proposition in Book I of Euclid's "Elements" states [2]:
"Let AB be the given finite straight line. It is required to
construct an equilateral triangle on the straight line AB. Describe
the circle BCD with center A and radius AB. Again describe the ACE
with center B and radius BA. Join the straight lines CA and CB from
the point C at which the circles cut one another to the points A and
B. Now, since the point A is the center of the circle CDB, therefore
AC equals AB. Again, since the point B is the center of the circle
CAE, therefore BC equals BA. But AC was proved equal to AB, therefore
each of the straight lines AC and BC equals AB. And things which
equal the same thing also equal one another, and therefore AC also
equals BC. Therefore, the three straight lines AC, AB, and BC equal
one another. Therefore, the triangle ABC is equilateral, and it has
been constructed on the given finite straight line AB."
Already we see several implicit geometric assumptions not explicitly
included in the axioms. For example, why are we guaranteed the
existence of the point C? The axioms do not provide its existence.
If a triangle is a plane figure, what justifies the implicit
assumption that the line segments AC, AB, and BC contain a plane
figure? It is also implicitly assumed that any line lies entirely in
a plane. Furthermore, how do we know that the lines AC and BC do not
meet before reaching the point C? That is, what justifies the
assumption that straight lines cannot have a common segment? Or
rather, that two straight lines can meet in at most one point? Also,
the argument involving equality of 'things' relies on Euclid's 'common
notions', but already implicit is the assumption of reflexivity of the
metric, that is, that the distance AB equals the distance BA. Several
of above examples were taken from [2]. So, already in Euclid's first
proposition, a wealth of implicit geometric assumptions were used
beyond those stated in his axioms showing their insufficiency in
proving his claims.
> (2) How does Hilbert's treatment differ from Euclid's? Do Hilbert's
> axioms suffice to prove Euclid's claims?
Whereas Euclid primarily sought to constructively formalize geometric
intuitions, Hilbert's aim was more so to provide an axiomatic system
for Euclidean geometry comprised of provably independent axioms that
were provably complete and consistent [3]. In part, he wanted the
significance of the various axioms and the scope of what could be
proved using them to be made very explicit. However, unlike Euclid's
approach, Hilbert's treatment of Euclidean geometry does not
constitute a first-order theory because some axioms cannot be
expressed in first order logic (implicitly relying on set theory) [4].
Despite this, Hilbert's axioms do suffice to axiomatize Euclidean
solid geometry and a minor reduction of the axioms yields an
axiomatization of Euclidean plane geometry [4].
> (3) Look at Tarski's axioms for plane geometry (these will be placed
> in the binder in JBE 356 on Thursday). These are first-order axioms.
> Can you express Euclid's five axioms easily using Tarski's first-order
> language of geometry?
I was going to attempt this here, but I think I will save it for
another post.
References:
[1] Euclidean geometry. (2007, October 4). In Wikipedia, The Free
Encyclopedia. Retrieved October 7, 2007, from
http://en.wikipedia.org/w/index.php?title=Euclidean_geometry&oldid=162184921
[2] Euclid's Elements, Introduction. Retrieved from October 7, 2007,
from http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
[3] The Foundations of Geometry, David Hilbert. Retrieved October 7,
2007, from http://www.gutenberg.org/etext/17384
[4] Hilbert's axioms. (2007, August 20). In Wikipedia, The Free
Encyclopedia. Retrieved October 7, 2007, from
http://en.wikipedia.org/w/index.php?title=Hilbert%27s_axioms&oldid=152486714
[5] Tarski's axioms. (2007, May 30). In Wikipedia, The Free
Encyclopedia. Retrieved October 7, 2007, from
http://en.wikipedia.org/w/index.php?title=Tarski%27s_axioms&oldid=134612314