List, Jon Your post wrt Number Theory is very revealing concerning the origins of your beliefs with respect to matter / material world / reality in contrast to the world of perceptions, thoughts about the world "out-there." Can it be re-evaluated from an alternative perspective of the notion of (biological) relations?
You appear to base your suppositions on the logical operations of Division as opposed to the natural science concept of atoms as being indivisible objects. Indivisible matter means not separable into equal size parts (to be compared with division as merely logical separation of parts of a whole.) The concept of irreducible atoms (as non-divisible objects) is accepted today and manifests itself in the chemical table of elements as being unique objects, each defined in terms of its perceived and measured qualia, AND as separable into part-whole relations (ie, Quantum mechanical descriptions may correspond with some data and organization of part-whole relations.) By way of reference, in the short little book on "Number Theory," by George E Andrews (Dover, 1971), Number Theory is separated into two mathematical forms. One form is the theory of division and focuses on the qualia of integers as Prime Numbers and Factorization. These aspects of number theory are critical to group theory, linear algebra, topology, etc., and, for science, the factorization of polynomials so that collections of connected qualia can be separated. The second form of number theory is addition. The number theory of addition focuses on the study of part-whole relations among the integers. These two basic forms of number theory are causes for deep conundrums between the logic of the continuum and the propositional logic of discrete terms, as we have previously discussed here. The example to ponder is the nature of biological relatives. I would conjecture two questions concerning the general nature of your posts, which often befuddle me (and others as well.) My conjectures are both wide and wild (breadth and comprehension of information) so feel free to alter them by offering more realistic/pragmatic hypotheses. >From this perspective, is it possible that the focus of your life-long >writings seeks to illate the pathways from Peician writings to set theoretic >terminology? (Possibly your purpose could be to demonstrate a consistency >between Shannon's notions used in computer science logic and CSP?) And that the division operations are the principle mode of constructing such a path, in parallel to the methodology used by Cantor? (Possibly, your beliefs could be that Peircian logic is different from but the same as set theory logic. Just tweak the definitions a tad here, a tad there, another tad here, another tad there, and so forth, until... Pesto! the "big pictures" coincide and, extremely roughly and metaphorically speaking, CSP = Cantor/Russell. Cheers Jerry On Feb 16, 2015, at 11:00 PM, Jon Awbrey wrote: > Submitted for your approval, two ways of looking at the > “divisor of” relation, one of the most fundamental examples > of a binary (dyadic or two-place) relation in number theory. > > Table 1 shows the first few ordered pairs of the relation > on positive integers that corresponds to the relative term, > “divisor of”. Thus, the ordered pair i:j appears in the > relation if and only if i divides j, for which the usual > notation is i|j. > > Table 1. Elementary Relatives for the “Divisor Of” Relation > ☞http://inquiryintoinquiry.com/elementary-relatives-for-the-divisor-of-relation/ > > Table 2 shows the same information in the form of a logical matrix. > This has a coefficient of 1 in row i and column j when i|j, otherwise > it has a coefficient of 0. (The zero entries have been omitted here > for ease of reading.) > > Table 2. Logical Matrix for the “Divisor Of” Relation > ☞http://inquiryintoinquiry.com/logical-matrix-for-the-divisor-of-relation/ > > Just as matrices in linear algebra represent linear transformations, > these logical arrays and matrices represent logical transformations. > > Regards, > > Jon > > -- > > academia: http://independent.academia.edu/JonAwbrey > my word press blog: http://inquiryintoinquiry.com/ > inquiry list: http://stderr.org/pipermail/inquiry/ > isw: http://intersci.ss.uci.edu/wiki/index.php/JLA > oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey > facebook page: https://www.facebook.com/JonnyCache > <Elementary Relatives for the “Divisor Of” Relation.png><Logical Matrix > for%20%74%68%65%20%E2%80%9C%44%69%76%69%73%6F%72%20%4F%66%E2%80%9D%20%52%65%6C%61%74%69%6F%6E%2E%70%6E%67.png> > ----------------------------- > 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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