Hello all, > What you have given is the "handwaving" version of the proof. The > trouble is that human imagination can easily get us into trouble when > dealing with infinities, which is necessarily involved in dealing with
I disagree - why was that a handwaving proof? It was exactly the way someone _understands_ what the proofs are about. Mathematical notation is only meaningless symbolism unless it is interpreted. It is interpreted by our intuitions (visualization, relation to other, more basic concepts etc). Mathematical notation is good for a number of things: 1) define your concepts exactly (again, somewhere it has to bottom out intuitively like in the concept of set membership or the rules of inference) 2) use a convenient shorthand (=math notation) which let's us reason more easily about the concepts than in natural language. Good math notation captures some intuitive reasoning analogy in our brains about the subject - no platonic reality about the structural relation in itself. 3) Mathematics is then used to reason about ever more complex subjects. The notation has been developed in a way that inferential validity is preserved when mindless symbol shunting is correctly followed. This let's us "reason" about things where our intuition _fails_ to preserve inferential validity. So, actually, there is no _magic_ in math or in the notation: it is just a very clever way of performing reasoning. But in essence, a three page proof in english (if diligently written) differs not from a two paragraph proof in algebra (which is just more condensed). That is actually the reason (I think) why some people who are very intelligent fail at math: not because they are to dumb, but because somewhere in their education they had bad math teachers who failed to teach the intuition/understanding on a certain essential and basic formalism. As maths will build on this formalism in more complex situations, everybody who has failed to grasp the grounding "shorthand" will fail to grasp anything else (or it will appear like magic anyway). > Handwaving arguments are good for developing intuition. Great for > teaching during a lecture, and get the students to study the rigorous > proof later. Similarly, they're good for scientific seminars, but not > scientific papers. I'm not sure - I think the focus on formalism and the deprecatory attitude which one regards intuition nowadays is actually bad for mathematics. For a refreshingly different approach read for instance Needham: Visual complex analysis http://www.usfca.edu/vca/ which shows that you do not have to sacrifice rigor by being intuitive (on the contrary!). Cheers, Günther -- Günther Greindl Department of Philosophy of Science University of Vienna [EMAIL PROTECTED] http://www.univie.ac.at/Wissenschaftstheorie/ Blog: http://dao.complexitystudies.org/ Site: http://www.complexitystudies.org ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
