Hi Charles, List, Nice questions. Let me start by saying that I am no expert in number theory, and that my email response to Jon was motivated by some research I've been doing for the sake of writing an essay that takes up the questions Peirce asks on the opening page of “The Logic of Mathematics; an attempt to develop my categories from within." Here they are:
1. what are the different systems of hypotheses from which mathematical deduction can set out, 2. what are their general characters, 3. why are not other hypotheses possible, and the like. Despite the length of the essay and the rather involved analyses contained therein, it is not easy to determine whether or not he has supplied the reader with adequate answers to any of the three questions. Instead, it looks to me like he is providing just part of an answer to each, and that the fuller answers depend on quite a lot of his other work--especially what he has written on the phenomenological and logical categories, the nature of monadic, dyadic and triadic relations, and the character of the different definitions, postulates and axioms that serve as starting points for the different systems of mathematics. Peirce takes himself to have given in this essay at least a part of answer to the second question of what the general characters are for the starting points of number theory, and he is focusing his analyses on the numbers 1, 2 and 3. He also takes himself to have given part of an answer to the third question as to why all number systems require the kinds of relations he is studying. There is a reason, I think, that he suggests that the number 1 is closely associated with the character of the monadic relation, and the number 2 is associated with the character of the dyad, and the number 3 is associated with the character of the triad. Appreciating the argument he is developing requires, I think, some understanding of what he thinks the relationships are between the different systems of numbers. As such, we'd need to take a closer look at what he says about the nature of monadic, dyadic and triadic relations. The dyadic case is, I think easier. Having said that, I'm finding his remarks about materially and formally ordered dyads hard to make out, and it is even hard to square these remarks with what he says in "Nomenclature and Divisions of Dyadic Relations" about the kinds of order that can be found in certain kinds of dyadic relations. This doesn't answer the questions you've raised, but this where I'd start in developing an answer. --Jeff Jeff Downard Associate Professor Department of Philosophy NAU (o) 523-8354 ________________________________________ From: charles murray [[email protected]] Sent: Saturday, July 04, 2015 8:55 AM To: Jeffrey Brian Downard Cc: Peirce List Subject: Re: [PEIRCE-L] Re: Survey of Relation Theory • 1 Jeff, I would like to communicate a few questions about your June 20 post copied below, in spite of severe misgivings about my competence to do so. It seems that, for a system with only positive integers starting with 1, equations of the form x-y=z can after all include 1 as a minuend, provided that the variables can be substituted with mathematical expressions, or otherwise put, that the variables can range over results of the subtraction operation. Thus if 'y' can be replaced by '(n - m)', where n and m are positive integers n=m, we could have for example: 1 - (1 - 1) = 1. Similarly, for a system with 0 and positive integers, where n, m, o, and p are positive integers such that n=m and o=p we could have 0 - (n - m) = (o - p) . In this case, unlike the one above, we can express the fact that where n=m, n - m = 0. Are these cases degenerate in a Peircean sense? Do they help explain the roles played by 0 and 1 in systems containing them? In any case, could you say a little more about how understanding the special roles played by 0 and 1 in systems containing them might illuminate the roles played in semiosis by the object, the sign and the interpretant? - Charles On Jun 20, 2015, at 11:16 AM, Jeffrey Brian Downard wrote: > Jon, List, > > The arithmetic example you offer in "Relations & Their Relatives: > 9" is quite clear. Having said that, what should we say about a > number system that only allows positive integers starting with 1? > In this number system, the number 1 can serve in the role of a > subtrahend or a difference. It cannot, however, serve the role of a > minuend. What does this show us about the number 1 in the system of > positive integers starting with 1? > > How do things change when we work with a number system allowing only > positive integers and 0? In this case, the number 1 can now serve > in any of the three places. The same follows for the case of 0. > But 0 can be the minuend only where it is also the subtrahend and > the difference. What does this show us about the numbers 0 and 1 in > the system of positive integers that include 0? > > Off the top of my head, I would think that the unit has a special > place in all three number systems we are considering (all integers > including 0 and negative, only positive integers starting with 1, > and all positive integers including 0) because the basic operations > of addition and subtraction are both understood in terms of the > operation of adding one more or taking one away. What is more, the > number 0 has a very special place in the number systems that allow > such an expression. How might we explain the special role that 0 > and 1 play in such systems? > > --Jeff > > > Jeff Downard > Associate Professor > Department of Philosophy > NAU > (o) 523-8354
----------------------------- 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 .
