minimal set of Hilbert axioms for plane geometry; Moore, Greenberg& Jahren

Is there a good treatment of a minimal version of Hilbert's axioms for
plane geometry, with proofs that this minimal version implies the
stronger set of axioms in Hilbert's book Foundations of Geometry and
in Greenberg's book Euclidean and Non-Euclidean Geometries?

I wrote such a paper myself
http://www.math.northwestern.edu/~richter/hilbert.pdf based on notes
by Bjorn Jahren http://folk.uio.no/bjoernj/kurs/4510/gs.pdf and
helpful conversations with him. I imagine Jahren would be a coauthor
if my paper was worth submitting.

I found my minimal version in Venema's book Foundations of geometry.
The Wiki link http://en.wikipedia.org/wiki/Hilbert's_axioms points
out that R. L. Moore showed that Hilbert's axiom II.4 was redundant,
but I know of no proof of this other than mine. Greenberg proves that
Hilbert's axiom II.2 is too strong in an exercise. Greenberg does not
list Hilbert's redundant axiom II.4, but he strengthens Hilbert's
axiom II.5, which says that if a line intersects an edge triangle, it
must intersect another edge as well. Greenberg however strengthens
this axiom to say that a line has exactly two side, and shows this
easily implies that a line cannot intersect all three edges of a
triangle. Jahren explained how Hilbert's unstrengthened axiom II.5
implies that a line cannot intersect all three edges of a triangle,
but this doesn't quite prove that a line only has two sides: we need
to handle the case of 3 collinear points. I did this, and this
implies proves Hilbert's redundant axiom II.4.

Let me explain my thinking about high school Geometry, as I wrote my
paper in order to teach to my son, who read it, and is working through
Greenberg's book. I learned that
1) Euclid wasn't too rigorous, as he superposed and missed
betweenness.
2) Birkhoff came up with a much shorter rigorous list of axioms than
Hilbert's by starting with the real line to measure lengths & angles.
3) High school Geometry textbooks more or less follow Birkhoff.
4) Kodaira wrote a very nice textbook on Hilbert's axioms that top
high school students could read, but the book was not translated from
Japanese and is now out of print.

The textbook my son is using seems particularly bad to me. They don't
even formally state Birkhoff's two real line axioms, and only mention
the axioms in remarks in the text. Their first theorem is that any
two right angles are congruent. Their proof is very simple:
90 degrees = 90 degrees!
The point is that Euclid took this result as an axiom, but Hilbert
gave a serious proof of it using his axioms.

 

I've substantially improved my expository paper
http://www.math.northwestern.edu/~richter/hilbert.pdf
giving I think a better proof of Greenberg's angle addition theorem.
Greenberg's version of the SAS axiom is stronger than Hilbert's, and I
included Hilbert's proof that his weak SAS axiom implies Greenberg's
strong SAS theorem. I have two important updates.


1) Miguel Lerma found a 1902 Notices article explaining R. L. Moore's
proof that Hilbert's axiom II.4 is redundant. Moore's proof is much
the same as mine, although I like the organization of mine.

2) I am amazed by Hartshorne's book Geometry, Euclid and Beyond.
Thanks to Bjoern Jahren, who suggested I read it. Hartshorne first
goes through 25 of the 1st 27 propositions of the Euclid's Elements,
explaining how Hilbert's betweenness axioms are necessary at times, I
now think that Euclid's work is great, and I never would have read it
without Hartshorne's expert guidance. Euclid has e.g. a simple proof
of the triangle inequality, not using the law of cosines and he gets
great use out of exterior angles, which I'd never considered.

Unfortunately Hartshorne also uses Greenberg's overly strong version
of Hilbert's axioms.

 

Bjoern Jahren pointed out that

1) two of the results in my (recently revised) paper
http://www.math.northwestern.edu/~richter/hilbert.pdf
were proved in a 1944 paper:
C. Wylie, Hilbert's axioms of plane order, Amer. Math. Monthly 51.
Wylie reproves Moore's result that Hilbert's axiom II.4 is redundant,
by a much different argument, and gives the same argument as Jahren's
that Hilbert's axiom II.5 implies Greenberg's stronger version.

2) Hilbert in 1930 published in German a new version of his
"Grundlagen der Geometrie" with I believe the minimal set of axioms.

Also Greenberg's SAS axiom C6 is stronger than Hilbert's, and I give
Hilbert's proof in my paper that the weaker SAS axiom suffices.

So the point I'm making can't be that I've proved new things about
Hilbert's axioms that aren't published anywhere. My point instead is

4) There should be a good treatment of axiomatic geometry at the level
of rigor of a junior level math course. I think my paper is a step in
that direction. The excellent books by Greenberg & Hartshorne seem
more suitable for graduate math classes, and their aims seem instead
to be non- Euclidean geometry and reviving interest in Euclid.

5) It's not enough for results to be published without meaningful Math
Reviews. People need to be able to find the papers and books! I'm
particularly grateful to Jahren as his nice 6 page paper comes up on
the first page when I google for "Hilbert Geometry axioms", and it's
by far the most useful link I've found.

 

Hi Bill,

Jack Lee (at Univ. of Washington) is writing a book like this, intended for math majors who plan to be high school math teachers. As such, it's based on the axioms which tend to be used in the high school geometry books -- Hilbert's axioms are pretty far removed from this. You could contact him and ask to look at a preprint.) Someone named Matthew Harvey has an unpublished manuscript on-line:

<http://www.mcs.uvawise.edu/~msh3e/resources/geometryBook/geometryBook.html>

I think he uses Hilbert's axioms. His book looks very pretty, but I haven't looked at it in enough detail to comment on its mathematical quality.

 

The Hartshorne's one is used for Berkeley's Math130 which is a "junior
level math course". It is mostly about Euclid/Hilbert geometry. For
most students, this is the first "non-layman" math class they ever
took.

Hope this helps,
Ilya

 

Thanks, John! Say hi to Jack Lee for me, and please ask him how his
book will compare to Venema's book (based on Birkhoff not Hilbert)
Foundations of Geometry, which I think is rigorous and quite suitable
for good high school students or junior math majors.

Thanks for the link to Matthew Harvey, who teaches at a branch of UVA.
His book is the link Geometry Illuminated on his web page
http://www.mcs.uvawise.edu/~msh3e/. He's using Greenberg's
non-minimal version of Hilbert's axioms, as we see from
http://www.mcs.uvawise.edu/~msh3e/resources/HilbertsAxioms.pdf

Ilya, I see my term "junior level math course" is fuzzy. When I was
in high school I read a number of "junior level math texts" texts,
such as Gillman & Jerison's Rings of Continuous Functions, which is in
the Springer GTM series. In college I was quite satisfied with my
"junior level math texts" in abstract algebra, algebraic topology and
real analysis. I don't include in this group of texts Hartshorne's
wonderful book, I don't think I could have read it when I was in high
school. I think his exposition is better suited to graduate students.
For instance I found his proof of Euclid's Proposition I.7 (p 35 &
Ex. 9.4) to be quite incomplete, and wrote up a proof myself in my
notes http://www.math.northwestern.edu/~richter/hilbert.pdf

Let me expand on my praise for Hartshorne. A week ago I would have
said that Greenberg's quote of Bertrand Russell

The value of Euclid's work as a masterpiece of logic has been very
grossly exaggerated.

completely summed up the Elements. But after reading the first part
of
Hartshorne, I see I was completely wrong. Sure, Euclid wasn't
rigorous,
as he used betweenness axioms he didn't state and superposition that
maybe
can't be stated. But the Elements looks like a work of genius, now
that Hartshorne explained it to me, and gave the Hilbert rigorization.