Jerry, I specifically referred to the Newtonian interpretation as an example, which you excised. My point was that the ordering here does not imply any intuitive order in terms of greater or less than.
The order of the numbers in the domains is a separate issue from the ordering of the parametres in the Newtonian function. Everything you say I can accept as accurate inasmuch as I understood it, but my reaction was why is this relevant? Any finite set can be put into a transitive relation (well-ordering theorem), but there are so many it is trivial. Most of these orderings would not be considered intuitive orderings. This would differ if you make restrictions on the domain that are strong enough, but I think that this would ultimately lead to presupposing the ordering (presumably on the basis of something else, either mathematical or empirical), which is where the real issue lies. On some interpretations the result will be an intuitive greater than or less than relation; other times it will not be. The overall point is that transitivity is not best interpreted in terms of greater than or less than, although you could arbitrarily (and thus vacuously) define it that way. Sometimes this is done in set theory to guide intuitions, such as in fixed point theorems (there is an x such that f(x) = x) and the like when we use iterations to find the fixed point, getting closer and closer. This works best (as a practical method) when the domains and ranges are already well-ordered, or at least partial ordered. I taught set theory from Suppes' Axiomatic Set Theory, but it was some time ago, and I am a bit rusty, but I don't think my set theoretic intuitions have gone off. It just takes a lot more effort now to do a proof and make myself analytically clear. I think I see your point about eh two ways of representing chemical processes. It seems to me that one has an intuitive order, but the other does not. Digging into my long unpracticed mineralogy, it seems to me that the atomic weights approach would constrain the chemical formula approach, but there are a lot of other constraints like valence and so on. Best, John -----Original Message----- From: Jerry LR Chandler [mailto:[email protected]] Sent: January 29, 2015 11:07 PM To: Peirce-L Cc: Benjamin Udell; John Collier Subject: Re: [PEIRCE-L] Triadic Relations John, List: > For example in a function, like f=ma, <m,a> is an ordered pair, one from one > domain and another from another domain such that their product is in another > domain which is the range of the function. Huh? Yes, as stated, I agree with your sentence. And that a function can be defined as an order pair between two mathematical objects. But, there is nothing in either mathematics or physics that requires the pair of symbols, "m" and "a" to be ordered as "m multiplied by a" as to be distinguished from the pair "a multiplied by m". Both calculations give the same number for force, do they not? The usual requirement for ordering, at least as I understand it, is the sense of a transitive relation, a<b<c or a>b>c. A practical example of ordering is the counting all possible combinations of pairs of parentheses, aligned along the a number line such as the Catalan numbers are generated. Further, the concept of ordered pairs or ordered triplets, or... does NOT REQUIRE an extension to higher counts, such as the set of all integers. I point these distinctions out because the concept of order is used in a DRAMATICALLY different manner in the chemical sciences, even in the 19th Century views of CSP, as evidenced by the routine use of "molecular formula" as contrasted with "molecular weights". Think about it... do these two chemical terms imply the same concept of order or ordered pairs? By way of contrast, physical symbols, such as those you use, as well as many, many other physical symbols, pre-suppose that the concept of order and number are virtually synonymous, as amply documented by the International System of Units. At least until catastrophe/chaos/fractal theories arrived in the 1960s - 1970s. In pure philosophical discourse (metaphysics?) it would appear to me that the concept of "order" is a matter of personal judgment. Consequently, the discipline has an unending source of stimulations about the relative importance of different ideas and the "difference that makes a difference". Would you disagree with this generality? Cheers Jerry On Jan 29, 2015, at 2:52 AM, John Collier wrote: > Ben, List, > > I believe that a weaker is required for an ordered triple. Any finite set can > be ordered. The Axiom of Choice, which is controversial, implies that any set > including infinite ones can be ordered. The order need not be anything like > 'more' or 'less' in any intuitive sense. For example in a function, like > f=ma, <m,a> is an ordered pair, one from one domain and another from another > domain such that their product is in another domain which is the range of the > function. Obviously, under the Newtonian interpretation m and a are not > either more or less than the other in any intuitive (or even nondegenerate) > sense. I think that this is worth remembering when thinking of Peircean > triads in particular. I would go further than saying that we should not think > of object, sign and interpretant as "falling dominos", since I am not at all > clear that there is a unique "order of semiotic determination". This follows > from the way I understand irreducible triads as not fully computable, and > hence inherently open-ended. > > Best, > John > > -----Original Message----- > From: Benjamin Udell [mailto:[email protected]] > Sent: January 28, 2015 7:07 PM > To: [email protected]; 'Peirce-L' > Subject: Re: [PEIRCE-L] Re: Triadic Relations > > Jeff, Jon, lists, > > I think that all that is required for an ordered triple, or an ordering of > any length, is a rough notion of 'more' or 'less', for example an ordering of > personal preferences, and this is enough for theorems, for example > http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem. > Exact quantities are not required. In the case of object, sign, interpretant, > insofar as the object determines the sign to determine the interpretant to be > determined by the object as the sign is determined by the object, the order > of semiotic determination is 'object, sign, interpretant', although object, > sign, interpretant are not to be understood as acting like successive falling > dominoes. > > Best, Ben > > On 1/27/2015 2:08 PM, Jeffrey Brian Downard wrote: > > [....] > Here is the starting question: Doesn't the notion of an ordered triple > require that we already have things sorted out in such a way that we are able > to ascribe quantitative values to each subject that is a correlate of the > triadic relation? > [....] > > > > > ----------------------------- > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] > . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] > with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at > http://www.cspeirce.com/peirce-l/peirce-l.htm . > > > >
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