List, Ben: citing HP: >> HP: This "strange rule" illustrates Poincaré's criticism of logic as an >> impoverishment of natural language that can neither count nor tell time. >>
With respect to the contrast between mathematics and logic, a sharper argument is possible. A priori, mathematicians tend to profess that mathematics does not relate to the real world, the qualia of mathematics are held to relate only to mathematical qualia and nothing else. Hilbert professed to a mathematics as a symbolic game; no reality necessary, no meaning necessary, just symbolic manipulations with rules. Thus, the pure mathematics of Poincare does not recognize the quale of time per se, just a continuous variable. By way of contrast, the entelechy of applied mathematics demands a correspondence of meaning between the objects of the world and the symbols used as representations of the world. Thus, within this context of meaning of symbols, CSP's realism and pragmaticism leads to writings that generally view mathematics as applied. Numerous exceptions, of course. Cheers Jerry On Feb 15, 2015, at 11:36 AM, Benjamin Udell wrote: > Howard, > > It's not just my instinct, it's Peirce's that there's something strange about > an equivalence between where one _seems_ to formulate talk of one thing and > where one _seems_ to formulate talk of possibly two things. > The problem is that symbolic logic requires us to contract ideas into > concentrated forms where we tend to forget some of the structure of > alternatives in them. Deductive logic and related fields such as probability > theory can be peculiarly tricky, a bit _Twilight Zone_-ish at times, because > they are specially about abstract structures of alternatives, alternatives > among universes in a sense, rather than about structures of, per se, space, > measure, group, or order. Also, conveniences of logical verbal expression can > require a little getting used to (e.g., 'some' is usually taken in the sense > of 'at least some'; and, very notably, 'if p then q' just means 'not both p > and not q' because logicians don't like to invent funny expressions like "p > nandn't q" or "p nayor q" to say it instead). > First-order logic with identity and with axiomatic completeness can > distinguish 'at least one,' 'at most one,' 'exactly one,' 'at least two,' 'at > most two,' 'exactly two,' etc., and I've read that it allows time expressions > (e.g., where one treats "Jack at t₁" as an object, or something like that). > I'm no logician, but I think that one could allow numeric predicates as in > '5xyzwu'; I don't know whether they can be indefinite (even if still finite) > in valence without losing axiomatic completeness (i.e., moving us into > 2nd-order logic), and to treat them as variables would require 2nd-order > logic at least, I think. > If you want to make good on the instinctive sense that the 'strange rule' > should be avoidable, other than by thinking of veiled or unknown constants > (amounting to 2nd-order variables much like Quine's "dummy letters" in > schemata for 1st-order logic), then you'll need some sort of modal logic (of > the 2nd-order kind, I think), or branching quantifiers logic (2nd-order), or > some other area in 2nd-order logic. More generally, if that which Poincaré > desired was really mathematics, then why did he complain? He already has lots > of it. Axiomatically complete, first-order logic with identity is about all > things in a sense, but its completeness of axioms keeps it from being > powerful enough to be all things useful to theory. Think of it as a sandbox. > Best, Ben > > On 2/14/2015 11:19 PM, Howard Pattee wrote: > >> At 09:30 PM 2/14/2015, Benjamin Udell wrote: >> >>> The strange rule really isn't so strange. In CP 4.569 Peirce (without >>> calling it the 'strange rule') says: "The logical Principle is that to say >>> that there is some one individual of which one or other of two predicates >>> is true is no more than to say that there either is some individual of >>> which one is true or else there is some individual of which the other is >>> true." >>> >> HP: This "strange rule" illustrates Poincaré's criticism of logic as an >> impoverishment of natural language that can neither count nor tell time. >> >>> BU: In other words, 'there is something round or blue' is equivalent to >>> 'there is something round or there is something blue'. It seems >>> kind of strange because in the first case one seems to mention one >>> individual thing, while in the second case one seems to mention possibly >>> two things. >>> >> HP: I think Ben's instinct is correct. It is indeed strange that the >> difference between two different numbers, like one and two, are >> ignored. Only using logic rules is such indifference possible. The past and >> future succumb to the same logical indifference. >> >> Howard >> > > ----------------------------- > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu > . To UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu > 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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