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Collaborative articulation of how abstraction and language is employed in the computational manifestation of numbers -- including analysis of the role of syntax, semantics, and meaning in the specification and use of software interfaces.
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2008-06-14Turing Arrives: Petzold on ArithmeticTechnorati Tags: Charles Petzold, The Annotated Turing, Alan Turing, Andrew Hodges, Keith Devlin, Mathematical Realism
I pre-ordered my copy of Charles Petzold's The Annotated Turing on November 22, 2007. On May 24, I had to authorize a delay in the estimated ship date. I waited patiently. When I saw that Petzold had his copies, I wondered if my order would fall through the cracks. The amazon.com site listed the book as in stock, but I had no word. That was resolved this Monday, June 9, when I received notice of shipment. The book arrived two days later by postal mail. The problem with actually having The Annotated Turing in my possession is deciding when to start and clearing the time to do it. I did start reading at the end of the book, and I have nosed into a few other sections. Naturally, the book arrived at a moment when all of my projects are behind and I am already starting an important new one. A systematic reading is yet to come. I know I will love it if only for the historical threads and connections that Petzold traces in the book. As part of the tracing of connections, Petzold has been reading One to Nine by Turing's biographer, Andrew Hodges. There are numerous connections traced there, and I like it that Petzold finds himself arguing with Hodges as he works through the book. Yesterday, Petzold comments on Hodges' objection to memorization of arithmetic with recognition of his own experience in learning the multiplication tables. The interesting idiosyncrasy is how Petzold failed to have automatic memory of certain multiplication combinations and he would solve those cases by algebraic deduction when needed. That resonated for me. There are many cases where I did not remember a rule, but I could and did recreate it on demand. I also share Petzold's having done that long after simply memorizing the result would have been more productive. (This shows up in other activities of mine too, including re-inspection of already-written code to remind myself that it is sound and what the context is before adding more to it.) I wonder how much this ability to have abstracted an applicable principle (in my case, remembering the times-11 and times-12 cases in terms of times-10 plus times-1 or times-2) leads to algebraic facility and the handy use of identities and mathematical induction well before I developed anything like a fundamental understanding of number theory over the course of my adult years. I recall re-derivation as being valuable in test-taking and yet it is not as direct as having embodied the result for immediate availability. I can't tell you how many times I have verified for myself what the correct formula for the sum of the first n integers is by redoing the constructive derivation. My doubt is always between n(n+1)/2 and n(n-1)/2 and it is, of course [easy for me to say], the former. I say that not because I have memorized it but because I know how to tell quickly another way, the first way I ever saw it "proved." I suspect that I have just sped that up for myself by looking at it anew this time. Then there's the one about the sum of the first n powers of 2 and what it looks like in binary, etc. I suspect that our diminished respect for the teaching of arithmetic and how to verify arithmetic results is causing trouble for students and their teachers when it is time to approach algebra where one can't avoid dealing with ratios and fractions by using a calculator. Seeing this latest post from Petzold has me thinking of the connection with a recent paper by Keith Devlin (via Richard Zack), "The Useful and Reliable Illusion of Reality in Mathematics." Two connections that come to mind: how we might come to exercise our capacity for abstract, conceptual thinking as we develop our facility with language, and the tendency to see mathematical conceptions as real. In the second case, Petzold has observed that Turing's machine was his idea for a "real" computer, and I am surprised by that. There are deeper connections in the Devlin paper with how we end up regarding mathematical objects, and that is worthy of separate discussion with regard to what makes computers so successful and so devilishly difficult to deal with.
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