From: cet1-nospam@cam.ac.uk.invalid (Chris Thompson) Subject: Re: Not in books: Solid proof of the Duality Theorem for projective geometry Date: 23 Oct 2000 23:22:31 GMT Newsgroups: sci.math Summary: [missing] In article <8t121q$n9g$1@newsg3.svr.pol.co.uk>, Max137 wrote: > > wrote in message news:8strn8$b26$1@nnrp1.deja.com... >> A while ago I started a thread about the duality principle of projective >> geometry and got refered to at least one excellent book that I now have >> (Coxeter's Projective Geometry). In fact I have _two_ books from Coxeter >> on projective geometry, for I also have The Real Projective Plane by >> him. However, of the vast majority of books on projective geometry I >> found at my university's library, none of them gave proof of the duality >> principle. They all just stated it and said simply to the effect that it >> was "a consequence of the form of the axioms." I guess proof is absent >> from all these books because the duality principle is a theorem about >> theorems and so a proof would likely require a formal definition of >> "theorem" and "proof." Now, I am taking a logic class right now, and so >> I do have some understanding of elementary model theory, the formal >> definition of satisfaction and proof, and so forth. So how is a formal >> proof of the duality theorem given? Where oh where can I find it? >> Surely, some mathematician somewhere wrote it down in some book so that >> knowledge of how to do it would be preserved for future generations.... >> :-) >> >I thought the argument was simply that if you replaced "line" with "point" >and "point" with "line" consistently in all the axioms, you ended up with >the same set of axioms in a different order. Well, with the usual axioms, not quite. For a projective plane, these are 1. Any two distinct points are incident with exactly one line. 2. Any two distinct lines are incident with exactly one point. 3. There are four distinct points, no three of which are incident with any one line. (3) serves to outlaw "degenerate projective planes". (1) and (2) interchange under duality, but (3) becomes 3'. There are four distinct lines, no three of which are incident with any one point. 3' is a *theorem* when 1,2 & 3 are axioms; it's not an axiom itself. In fact it's often the first theorem proved in presentations of the subject, just so that duality can be got working as soon as possible. >Any proof of a theorem is a set of logically consistent statements following >from the axioms. If you make the same substitution throughout the theorem, >you will still have a set of logically consistent statements following from >axioms which are in your original set. So, the "substituted" conclusion is >still true. Any of course, this still works: just prefix the proof of 3' from 1,2,3 to the dualised proof. To prove (3'), let the points of (3) be A,B,C,D, and consider the lines (existing by (1)) AB,BC,CD,DA. Chris Thompson Email: cet1 [at] cam.ac.uk ============================================================================== From: default Subject: Re: Not in books: Solid proof of the Duality Theorem for projective Date: Mon, 23 Oct 2000 00:02:12 -0400 Newsgroups: sci.math lemma_one@my-deja.com wrote: > However, of the vast majority of books on projective geometry I > found at my university's library, none of them gave proof of the duality > principle. They all just stated it and said simply to the effect that it > was "a consequence of the form of the axioms." You can also think of duality as a semantic principle, not dependent on the specific axiomatization. Given a projective plane P (collection of lines and points and a notion of incidence between point and line) you can construct the dual plane Q, whose points are, by definition, the lines of P and whose lines are the points of P, and with the same relation of incidence. Any configuration of points and lines in P can be viewed as a dual configuration of lines and points in Q. Any statement (involving only the concepts point, line, and incidence) about a configuration in P is true if and only if the dual statement is true of the dual configuration in Q. Introducing quantifiers does not violate this. So whatever the axiomatization of projective geometry, duality will hold. Also, Given a conic in P, polarization with respect to this conic gives a specific identification between P and Q (aka the identification of underlying dual vector spaces via a quadratic form). This allows to "visualize" duality to some extent.