List, John: >From your responses below, it is clear that we hold radically different >notions about the nature of linguistic symbolizations of meaning and >mathematical symbolizations of meaning. I must re-iterate that my views of >mathematics are based on pragmatic realism that is meaningful for human weal.
Because other obligations are pressing, it is not possible to fully respond to your beliefs about the relationships between Peircian realism, modern mathematics and science. Our disagreements are sharp and well defined. I add a few comments below that address some key points. > On Sep 21, 2018, at 9:38 AM, John F Sowa <[email protected]> wrote: > > On 9/20/2018 5:33 PM, Jerry LR Chandler wrote: >> I ask that you clearly state the meaning of your notion of “pure >> mathematics”. I have taken it to mean the usual undergraduate >> level of mathematical philosophy... > > The scope of pure mathematics, as Peirce defined it, is infinitely > larger than whatever was or ever will be discovered, taught, or > applied by anyone anywhere. That includes all intelligent aliens > in any galaxy anywhere in the universe. > Wow! Wow! Wow! Even without the aid of "intelligent aliens”, I can wildly assert, even logically, that: “The scope of mathematics is constrained to physically meaningless symbolic relations." >> in [John's] posts here when viewed from the traditional views of >> the philosophy of Vienna Circle. > > Absolutely not! the logical atomism of Russell and the logical > positivism of the Vienna Circlers were a low point in the logic > of the early 20th century. They made the "grave errors" (schwere > Irrtümer) that Wittgenstein spent the second half of his life > trying to correct. This response is remote from my interpretations of your writings and PPTs. For example, decades before you joined this list, I was studying the chemical symbol system as a mode of representation. I studied the book “Knowledge Representation.” I was deeply impressed with Figure 6.14, Peirce’s three trichotomies of signs. page 397 and Fig. 6.15, p. 401, and text of section 6.6, p. 394-402. Do these pages continue to represent your philosophy of mathematics, either pure of applied? > >> A chemical atomic number... can never be a variable in the sense >> of the Cartesian axis system, which is > the sense of Quine's >> categorical error. > > The word 'variable', as used in mathematics, is a metalevel term > about the notation. It just means that letters like x, y, z may > be used to refer to different things on different occasions. If > you use x to refer to something, that does not imply that the > thing you designate by x would vary. > Many mathematical texts disagree with this view. > To emphasize the point, you can replace the word 'variable' with > a synonym, such as 'index', 'label', or 'line of identity’. Many math texts disagree with this view. > > Definition: For Peirce and modern mathematicians, pure mathematics > may be defined as the totality of all provably true statements of > the form "If [list of axioms], then [conclusion].” Some mathematician adopt the Hilbert perspective. Many others do not. Axioms, are, of course, merely assumptions. > > For more, Peirce's CP has 49 instances of "pure mathematics". > In CP 1.636, for example, he says that the goal of pure mathematics > is to discover pure possibilities: "that real potential world" of > which actual existence is "nothing but an arbitrary locus”: Really? The nature of counting of objects, such as exhibited by the abstract nature of biological reproduction of the same numerical objects, argues otherwise. Of course, if one ignores the methods mother nature uses to count objects, you can ignore my objectification and objection. Let’s just say that you and I differ about the nature of mathematics as well as applied mathematics. I will address your other claims about applied mathematics separately. Cheers Jerry > >> if you enjoy the good fortune of talking with a number of >> mathematicians of a high order, you will find that the typical >> pure mathematician is a sort of Platonist... The eternal is for >> him a world, a cosmos, in which the universe of actual existence >> is nothing but an arbitrary locus. The end that pure mathematics >> is pursuing is to discover that real potential world. (CP 1.646) > > He also said that there is no danger of pure mathematics "evaporating > into an airy nothingness... spun from the stuff dreams are made of" > because that is "precisely what mathematics ought to be": > >> mathematics is distinguished from all other sciences except only >> ethics, in standing in no need of ethics. Every other science... >> is in its early stages in danger of evaporating into airy >> nothingness, degenerating, as the Germans say, into an anachrioid >> film, spun from the stuff that dreams are made of. There is no >> such danger for pure mathematics; for that is precisely what >> mathematics ought to be. (CP 4.242). > > For 67 definitions of pure mathematics (which include CP 4.242), > https://todayinsci.com/QuotationsCategories/P_Cat/PureMathematics-Quotations.htm > > Peirce would agree with most them, quibble with some, and reject > a few. I think he would like the following by Albert Einstein: > >> Pure mathematics is, in its way, the poetry of logical ideas. >> One seeks the most general ideas of operation which will bring >> together in simple, logical and unified form the largest possible >> circle of formal relationships. In this effort toward logical >> beauty, spiritual formulas are discovered necessary for the >> deeper penetration into the laws of nature. > > JLRC >> This usage [in physics and chemistry] is much wider than predicate >> logic but excludes the various forms of para-consistent logics. > > Pure mathematics excludes nothing. Peirce rejected the idea that > mathematics depends on logic. Instead, every version of logic of > any kind is a special case of some theory of mathematics. > > In fact, every humanly conceived version of pure mathematics, > which includes every version of logic as a subset, can be > specified with predicate calculus or existential graphs as the > metalanguage. > > The nature of the metalanguage does not, in any way, constrain > or restrict the nature of the object language. For example, > the two-valued FOL can be used to specify 3-valued logic, > many-valued logic, fuzzy logic, modal logic(s), nonmonotonic > logic, intuitionistic logic, paraconsistent logic, etc. > > John > > ----------------------------- > 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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