Clark, List,
Just a couple of points to take up something that Clark says within the more general context of logic and formal mathematics, and, in this case, its relation to physics, but still very Peircean I think. See below. John Collier Professor Emeritus, UKZN http://web.ncf.ca/collier From: Clark Goble [mailto:[email protected]] Sent: Friday, 11 December 2015 19:41 To: Peirce-L Subject: Re: [PEIRCE-L] in case you were wondering I tend to see “in the long run” as more a regulatory concept rather than something actual. For a long time I did worry about how the “in the long run” worked and raised concerns similar to yours. The question of whether it really functions the way Peirce needs it to function if it’s not potentially actual in some sense is still a big issue I think gets neglected too much. So don’t think I’m brushing that aside. I do share some of your concerns there. I’ve just come to think that for Peirce the fundamental issue is the meaning of truth which then brings in the issues I raised as regulatory concepts. [JDC] Agreed. There are a number of counter-examples to convergence that are worrisome, such as counter-induction, sets that show arbitrarily long patterns for finite stages that aren’t reflected in the overall statistics of the whole set, and so on. All that said, I’m not sure infinity works quite the way you suggest simply because Peirce is not dealing with a normal potentially countable infinity. That is his continuity ends up dealing with higher order infinities - even if he does differ from the typical cardinal/ordinal sets we deal with in mathematics. Now I’ll confess it’s been more than 10 years since I last studied Peirce on these particular issues or where he differs from Cantor and company. So my memories are a tad fuzzy. Forgive me for errors. I think however that if there’s a potential countable infinity of the sort [\aleph_0] that Peirce’s in the long run in his semiotics allows this to be dealt with by semiotics running in higher orders like [\aleph_1] or so on. I’m curious as to what others thing here. My logic professor, George Boolos, dreamed up a being he called Zeus (so-called as to not pre-empt contemporary religious concerns) that got better at processes if it repeats them. The idea is that in calculating an infinite series, Zeus could do each step twice as fast as the previous one, and be able to complete a series in finite time. Obviously, neither we nor any other finite system could be a Zeus demon, but it does give a way to interpret infinite convergence. I see this as a case of going from [\aleph_0] to [\aleph_1] . The relevant set becomes the cross-product. I came up with a somewhat similar demon I called the Hermes demon, which can make every increasingly accurate measurements in the same way. It can achieve [\aleph_1] accuracy in measurement. Combine the two, and you have a Laplacean demon, making some sense of an otherwise somewhat mysterious idea. We could carry this to higher levels by calculating over all functions possible on an [\aleph_1] sized set, and so on, if necessary. Is this outside of the range of semiotics because it uses infinite methods and assumes creatures that could not exist (Peircean sense)? I think not, since it is an extension of ideas in the finite realm to the continuous in a fairly straight-forward way that is already pretty well understood. The second issue is whether we really need this. The concern ends up being more or less a common critique of convergence theories. That is you might test out to Tx but that the pattern completely shifts at Tx+1. I think Peirce’s conception works simply because in the long run is regulative as I mentioned but also because what is doing the testing is an infinite community rather than a finite one. That is the way Peirce attempts to get out of this is via his Hegelian/neoPlatonic like conception of the universe as an argument working itself out. So when we talk about truth it’s this universe that counts. That is we can maintain Peirce’s notion without having to deal with a practical knowing community. I think we need it. Testing the infinite community of functions seems to me to require at least two levels past [\aleph_0] , with the set of possible functions, as I mentioned above. The order, in this case, becomes irrelevant, because all orderings are included if the demon is carefully rendered. As someone who doesn’t find the Hegelian/neo-Platonic outlook very perspicuous (though for a time I thought it solved all outstanding metaphysical problems). In any case, from my current perspective to understand even the problem requires going to higher order infinities, let alone to understand how we might deal with actual cases. I am pretty sure that the Axiom of Choice, or one of many equivalent forms (well-ordering, basically) is required for bringing the abstract Laplacean demon I outlined down to earth. It doesn’t help a lot to know there are suitable functions to describe any pattern if one can’t find them. For any finite community then (i.e. any practical community we worry about) we’re always fallible from Peirce’s conception. What I sense you wanting isn’t a point of relative stability in our beliefs through continued inquiry. Rather I think you’re looking for something more akin to what Putnam takes up against Peirce. A kind of warranted assertability ala Dewey’s change from Peirce. If we’re looking for that sort of strong warrant then I’d probably agree we may not get it. I’m not sure we need that but I can completely understand why many might find Peirce ultimately unsatisfactory relative to these finite groups. He can offer inquiry but not certainty. (I’m not sure in practice Dewey/Putnam can do better mind you) I think it is obvious that we cannot meet Putnam’s requirement. It is also not necessary unless we assume some version of verificationism, however weak. Fallibilism undermines verificationism as a determiner of meaning. The result of Putnam’s (very weak – “meaning is something that we determine if anything does”) verificationism is the Quine-Duhem Thesis (see chapter 1 of my PhD thesis). I really should write this up. Best, John
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