Phil, There are many solutions* to the problem as written. By focusing on A solution, we lose sight of alternative, perhaps equally valid solutions - which was the point of my post. For example, d may be a complex number (say, representing day.ergs), not the assumed integer. I have just finished writing a book on modeling complex systems. In it I cite Dennett and Kinsbourne's** studies of perception between what they call the Cartesian Theater model and the Multiple Drafts model. When one views a model, or when one views a mathematical relationship, there are resonances caused by consonance and dissonance in the understanding of the symbology. These resonces are the source of emergent concepts from the Multiple Drafts. You speak of divergence. I see this more as a process of entanglement - leading to what Prigogine refers to as a dissipative system, or Large Poincare System. Until the model is realized, the temporal symmetry is not broken and you can have "do-overs" or reversibility. Once the model has been physically realized, having been realized in time, it cannot be totally reversed in configuration space-time - so-called Arrow of Time. *Possibly infinitely many. **Time and Observer http://ase.tufts.edu/cogstud/papers/time <http://ase.tufts.edu/cogstud/papers/time&obs.htm> &obs.htm Ken
_____ From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of Phil Henshaw Sent: Thursday, October 02, 2008 8:34 AM To: 'The Friday Morning Applied Complexity Coffee Group'; [EMAIL PROTECTED] Subject: Re: [FRIAM] Wittgenstein Ken, To make that divergent math work, your 2 + 2 = n + d is just the kind of dilemma with modeling the emerging divergent systems of nature that not studying divergent sequences distracts us from. There's a solution. Can you guess? Phil From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of Kenneth Lloyd Sent: Thursday, October 02, 2008 10:12 AM To: [EMAIL PROTECTED]; 'The Friday Morning Applied Complexity Coffee Group' Subject: Re: [FRIAM] Wittgenstein Nick, First, 2 + 2 does not equal 4 in base 3. Second, equality only works in equilibrium. What if our mathematics rule stated for every day d that passed, 2 + 2 = n + d? The mathematics would be linearly dynamic. There are subtle cultural assumptions being imbued upon mathematics that may not hold. Mathematics is designed to communicated the least false concept with the least information content at some maximum entropy. Thus 2 apples + 2 oranges does not equal 4 orapples, but 1 fruit basket, yet generally 2 + 2 = 4 (in equilibrium). Ken _____ From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of Nicholas Thompson Sent: Wednesday, October 01, 2008 11:18 PM To: Friam@redfish.com Subject: [FRIAM] Wittgenstein I have put the following material in an email message because is suspect it would fascinate some of you., and given that you are mostly people with real jobs and given that the information comes from the guts of a 700 page book, I suspect that many of you would be unlikely to stumble on it on your own. I have, as I have said, been reading Monk's biography of W. In it we learn many weird things, for instance, that W. turned up at Russell's door in Cambridge in 1911 or so, an callow Austrian lad, who had graduated from a technical school and got a job making kites in Manchester. Within a year, Russell was ruminating about whether he should turn his entire project in the foundations of mathematics over to W. and do something else himself. By 1937, W. had developed enormous contempt for the whole foundationalist project. As luck would have it, both he and Turing were giving relevant lectures at Cambridge and Turing came to hear W. talk. W. (never a particularly nice man) took the occasion to beat on Turing about the absurdity of the foundationalist project Here is a quote from Monk, p. 418. "Wittgensteins technique was not to reinterpret certain particular proofs, but, rather, to redescribe the whole of mathematics in such a way that mathematical logic would appear as the philosophical aberration he believed it to be, and in a way that dissolved entirely the picture of mathematics as a science which discovers facts about mathematical objects . I shall try again and again, he said, to show that what is called a mathematical discovery had much better be called a mathematical invention. There was, on his view, nothing for the mathematician to discover. A proof in mathematics does not establish the truth of a conclusion; if fixes, rather, the meaning of certain signs. The inexorability of mathematics, therefore, does not consist in certain knowledge of mathematical truths, but in the fact that mathematical propositions are grammatical. To deny, for example, that two plus two equals four is not to disagree with a widely held view about a matter of fact; it is to show ignorance of the meanings of the terms involved. Wittgenstein presumably thought that if he could persuade Turing to see mathematics in this light, he could persuade anybody." Turing apparently gave up on W. a few lectures later. I have to admit the distinction that W. is making here does not move me particularly. It seems to me as much of a discovery to find out what is implied by the premises of a logical system as to find out how many electrons there are in an iron atom, and since logic is always at work behind empirical work, I cannot get very excited about the difference. Perhaps because I am dim witted. No response necessary. Nick Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University ([EMAIL PROTECTED])
============================================================ 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
