Nicholas Thompson wrote circa 04/25/2010 01:50 PM: > I was talking about plain old vanilla philosophical induction: The > fallacy is that without deduction, induction can't get you anywhere, > and that people who think they are getting somewhere through induction > alone are so caught up in an ideology that they cannot see their > dependency on deduction. > > As I have watched the thread develop, I have been less and less sure > that my comment was relevant.
I think it is. (But as the thread develops, I'm less and less confident that it'll come to anything... Aaaaaaa! I can't believe I might agree with Doug on something. ;-) Going back to: 1) Grant's branch on relations vs. components, 2) Lee's further branch on unity (universe, closure), 3) Russ' inclusion of time and Eric's inclusion of consequence, and 4) Nick's inclusion of fallacious generalization. Sarbajit was right to consult the dictionary. Every valid statement is a "theorem". But when it's just a small stepping stone, we call it a lemma or some other diminutive term. Why? Because math constructs (proofs) are rhetoric. That's all they are, presentations meant to persuade and communicate. Math is a language, first and foremost. And it is used to communicate. So, the main question for "why theorems" has its answer in the larger question, "why communicate?". Any structures that seem to obtain and persist through our acts of communication is a psych- and socio-logical effect of the underlying _carrier_ of that communication. As Lee points out, these "persistent" structures will obviously reflect (if not be total slaves of and epiphenomenal to) the humans that participate. It's a direct effect of the _intent(s)_ of the participants. The concept of psychological induction is necessary in order to follow the rhetoric of whatever proof you're examining. What is the argument (or communication) the author is pursuing? Therein lies your structure, its persistence, its interestingness, relevance, etc. So, Russ' question is way too vague to be answered. You can't generalize across all of math/logic to talk about "why theorems?" any more than you can generalize over all of natural language and ask "why sentences?", unless you're willing to accept the equally vague and useless answer "because we use sentences/theorems to communicate." You have to talk about specific rhetoric. E.g. "why War & Peace?" or at least something like "why number theory?" That way, you can go further and ask what the author's(s') intentions are when building upon that rhetoric. Looking back, I see I didn't explicitly tie in relations vs. components or closure. [sigh] To make it as short as my limited skills allow: o Consequence is composition (even temporally); hence, an author can start with relations or components but, in the end, both are always necessary. o Closure (or circumspection) is necessary for any rhetoric to convince (convict, capture, imprison) the audience. But, more ontologically, any rhetoric is bounded by its context. As Goedel and Tarski point out, a (nontrivial) pure syntax cannot be closed; a higher order language is necessary to complete it. And the highest order language we have is the context in which we're embedded: reality. -- glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com ============================================================ 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
