At that same session, I was going on about the Kochen-Specker theorem, asking for references, on the basis of Baez's comment about it in "This Week's Finds" at http://www.math.ucr.edu/home/baez/week257.html. He was discussion some ideas around the concept of a "topos":
"It basically means this: using topos logic, we can talk about a classical space of states for a quantum system! However, this space typically has "no global points" - that's called the "Kochen-Specker theorem". In other words, there's no overall classical reality that matches all the classical snapshots." Since I'd been thinking about locality and structure lately (thus the interest in topoi), that statement seemed kind of remarkable to me, so after asking around I finally got around to looking it up. Wikipedia had little to say, but the Stanford Encyclopedia of Philosophy had a nice discussion: http://plato.stanford.edu/entries/kochen-specker/#intro Anyhow, based on Nick's intro, parts of that discussion now look germane (situated equivalence!) to this discussion. Carl Nicholas Thompson wrote: > > */“The truth arises from arguments amongst friends”/* -- *David Hume* > > One of my goals at Friam, believe it or not, is actually to get some > fundamental issues settled amongst us. We had, last week, a brisk > discussion about causality. I don’t think I was particularly > articulate, and so, to push that argument forward, I would like to try > to state my position clearly and succinctly. > > The argument was between some who felt that causality was “real” and > those that felt that it was basically a figment of our imaginations. > The argument may seem frivolous, but actually becomes of consequence > anytime anyone starts to think about how one proves that X is the > cause of Y. Intuitively, X is the cause of Y if Y is X’s “fault”. To > say that X is the cause of Y is to accuse X of Y. Given my current > belief that story-telling is at the base of EVERYTHING, I think you > convince somebody that X is the cause of Y just by telling the most > reasonable story in which it seems obvious that Y would not have > occurred had not X occurred. But there is no particular reason that > the world should always be a reasonable place, and therefore, it is > also ALWAYS possible to tell an UNREASONABLE story that shows that Y’s > occurrence was not the responsibility of X, no matter how reasonable > the original causal attribution is. One of us asked for a hammer and > nail, claiming that if he could but drive a nail into the surface of > one of St. John’s caf? tables, none of us would be silly enough to > doubt that his hammering had been the cause of the nails penetration > of the table. Not withstanding his certainty on this matter, several > of us instantly offered to be JUST THAT SILLY! We would claim, we > said, that contrary to his account, his hammering had had nothing to > do with the nail’s penetration, but that the accommodating molecules > of wood directly under the nail had randomly parted and sucked the > nail into their midst. > > How validate a reasonable causal story against the infinite number of > unreasonable causal stories that can always be proposed as > alternatives. By experience, obviously. We have seen hundreds of cases > where nails were driven into wood when struck by hammers (and a few > cases where the hammer missed the nail, the nail remained where it > was, and the thumb was driven into the wood.) Also, despite its > theoretical possibility, none of us has EVER seen a real world object > sucked into a surface by random motion of the surface’s molecules. So > it is the comparative analysis of our experience with hammers and > nails that would have convinced us that the hammering had driven in > the nail. > > So what is the problem? Why did we not just agree to that proposition > and go on? The reason to me is simple: the conventions of our language > prevent us from arriving at that conclusion. We not only say that > Hammers Cause Nails to embed in tables, which is what we know to be > true, we also say that THIS Hammer caused THIS nail to be embedded in > the wood. Thus our use of causality is a case of misplaced > concreteness. Causality is easily attributed to the pattern of > relations amongst hammers and nails, but we err when we allow > ourselves to assert that that higher order pattern is exhibited by any > of its contributory instances. In fact, that in our experience the > missed nails have not been driven into the wood is as much a real part > of our notions of causality and hammering as the fact that a hit nail > is. Causality just cannot be attributed to an individual instance. > > The fallacy of misplaced concreteness is so widespread in our > conversation that we could barely speak without it, but it is a > fallacy all the same. Other instances of it are intentions, > dispositions, personality traits, communication, information etc., > etc., and such mathematical fictions as the slope of a line at a > point. Whenever we use any of these terms, we attribute to single > instances properties of aggregates of which they are part. > > Now, how do we stop arguing about this? First of all, we stop and give > honor to the enormous amount of information that actually goes into > making a rational causal attribution that hammering causes embedding, > information which is not available in any of its instances. Second, we > then stop and give honor to the incredible power of the human mind to > sift through this data and identify patterns in it. Third, and > finally, we stop and wonder at whatever flaw it is in our evolution, > our neurology, our cognition, our culture, or our language that causes > us to lodge this knowledge in the one place it can never be … single > instances. > > Are we done? > > Nick > > Nicholas S. Thompson > Research Associate, Redfish Group, Santa Fe, NM ([EMAIL PROTECTED] > <mailto:[EMAIL PROTECTED]>) > Professor of Psychology and Ethology, Clark University > ([EMAIL PROTECTED] <mailto:[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 ============================================================ 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
