> At 10:12 PM 12/22/2011, Dan Christensen wrote: > > >The axioms for geometry and real numbers are not > >built into the DC Proof system. The numbers of > >axioms alone would overwhelm the beginner. For > >the axioms of geometry, see the work of Tarski > >or Hilbert. You can find a version of the axioms > >for the real numbers in the DC Proof/Samples > directory. > > > >Beware, though. In my experience, even > >elementary results in the Euclidean plane can > >quickly explode into proofs of several hundred, > >even thousands of lines. So, I really can't > >recommend geometry for the beginner just > >learning how to write mathematical proofs. > >Better to stick to logic, set theory, and > >elementary number theory, the axioms of which > >ARE built into DC Proof. (See "To the Educator" at > my homepage.) > > All true but I come to exactly the opposite > conclusion about beginners just learning how to > write mathematical proofs. I continue to believe > (as with far better mathematicians than either of > us over the last couple millennia) that Euclidean > plane geometry remains a wonderful place to > introduce mathematical proof to > beginners.
Studies have shown that proof-writing skills learned in one branch of mathematics such as geometry may not be easily transferred to other branches such as abstract algebra and analysis. F. A. Ersoz (2009) suggests that the many informal "axioms" of Euclidean geometry, as usually taught, are based largely on personal intuition and imagination (p. 163). While this may serve as a productive basis for some discussion, it can blur the boundary between the formal and informal, and lead to confusion as to what constitutes a legitimate proof in other domains (branches) of mathematics. Ersoz also suggests that introductory geometry courses seldom present many of the methods of proof used in more abstract courses ? methods like proofs by induction, contrapositive or contradiction (p. 164). http://22.214.171.124/~icmi19/files/Volume_1.pdf
> Beginning as you propose strikes me as being > perceived by most students as an exercise in > abstract nonsense, not in providing some level of > understanding of the solid foundation for > mathematics that proof provides.
I use the simplest possible examples to illustrate various methods of proof in my tutorial. There is no "abstract nonsense."
> Beyond that, > except for mathematical logic itself, no > professional mathematician appeals to axiomatic > set theory or formalistic logic in their thinking > nor in their publications.
True, but the methods of proof I illustrate with formal logic and set theory, and elementary number theory can be applied in every branch of mathematics, whether in formal or informal proofs. Every mathematician, for example, must understand how to apply the rule of the contrapositive, how to manipulate quantifiers, etc. Like they say in the arts, you have to know the rules before you can break them with impunity.