In his general classification of the sciences[36], Peirce placed mathematics in a strategic position as the most theoretical of all the theoretical sciences. This positioning was consistent with his mathematician father, Benjamin Peirce's prior definition of mathematics as "the science which draws necessary conclusions", a definition which Peirce by and large took as his own. Later, he also went on to redefine mathematics as "the study of what is true of hypothetical states of things", being the province of "precise necessary reasoning", guided only by those hypothetical permissions and restrictions the reasoner chooses to introduce - which may of course differ from those of the real world.
Peirce's divisions of mathematics according to the degree and complexity of its hypotheses, or in other words, the different types of collections to be admitted "in the hypothetical universe of reasoning"[37], are as follows:
Figure 1: Peirce's divisions of mathematics[38]
This kind of systemic functional model allows the mathematician to define the hypothetical conditions which are to act as the defining entry conditions for any given set of problem-solving sequences of reasoning and attendant behaviour, and it is these hypothetical conditions which in turn frame, contextualizing in quite specific ways, the problems being investigated, as well as the type of results that emerge from the investigation. This is why Peirce insists that the division of mathematics should primarily be done according to the nature of its general hypotheses, rather than the types of methods used. Mathematics discovers what we are compelled to think, given that certain conditions, upon the selection of which conditions there exist no a priori restrictions, are admitted and taken as rules of the system under consideration.
Referring to Figure 1 above, we see that the most elementary systemic division of mathematics in Peirce's classification is the system of mathematics of finite collections, which in its turn is further divided up into the mathematics of logic and the general theory of finite groups. The sub-system of the mathematics of logic, in accordance with Kant's assertion that it is from within the most fundamental of the sciences that the table of categories is to be disclosed, was held by Peirce to be concerned with the simplest possible of all hypotheses, and thus the one from within which his own categories would be disclosed. While probably of little interest to the mathematician because of its simplicity from a mathematical point of view, for Peirce the logician and semiotician, the mathematics of logic is nonetheless fundamental for the development of logic per se, due precisely to this fundamental simplicity.
Within the mathematics of logic, Peirce made two main subdivisions, which he referred to as dichotomic and trichotomic mathematics. A monotomic mathematics, the mathematics of a one-object universe is of course imaginable, but it is at the same time a functionally empty notion, and is not really worth going into in detail. As Peirce points out: "Were nothing at all supposed, mathematics would have no ground at all to go upon. Were the hypothesis merely that there was nothing but one unit, there would not be the possibility of a question, since the only one answer would be possible. Consequently the simplest possible hypothesis is that there are two objects."[39]
Dichotomic mathematics although in Peirce's view, "a form of mathematics rather poverty-stricken as to ideas"[40], is nevertheless important since its application in the form of traditional two-valued logic allows us to deal with the truth or falsity of unknowns, and thus opens up for examination of dyadic relations. Trichotomic mathematics, on the other hand, introduces a third element, giving possibilities for further examination of complex series of relations which obtain between the strictly limited set of four dyadic relations made possible by dichotomic mathematics. This third element represents a further level of abstraction, opening up for the logic of relations, since triadic relations are not reducible to dyadic or monadic relations, while at the same time indefinitely expandable to further levels of complexity which nonetheless are all reducible to triads - a fundamental notion for Peirce's phenomenology and his general semeiotic, with its interpretant of the sign in a potentially unending process of semiosis seen as part of the growth of concrete reasonableness; concepts which Peirce in the later years of his life worked into his mature scientific metaphysics and cosmology.
We shall not go into all of this in any more detail for now. Those who may be interested in doing so will find a lot more detailed and thoroughly documented discussion of this particular issue in both Parker's and Murphey's work[41]. We shall only mention here in accordance with Parker's view of things, one I find both plausible and useful: in the same way that the two value system of dichotomic mathematics may be interpreted as switching between mutually exclusive states of true and false, the three value system of trichotomic mathematics may also be interpreted in terms of truth values, whereby the excluded middle is suspended, giving us, in Peirce's words: "the possibility of an intermediate state between the two, say the satte of being sometimes true and sometimes false."
In relation to our discussion of true continua previously, it has already been established that for Peirce, the principle of the excluded middle does not apply to assertions about the state of affairs at the limit point. Carolyn Eisele points out that in such a situation Peirce would conclude, as he did in a draft version of his Lowell Lectures from 1903 that "were a proposition to be true up to a certain instant and thereafter to be false, at that instant it would be both true and false."[42] The instant of change from one state to another is in other words a limit-case which is a general. Having recall to such a third truth value is necessary if there is, as Peirce believed there to be, real indeterminacy in the universe which our logical system is attempting to describe. "Triadic Logic is", he wrote, "that logic which, though not rejecting entirely the Principle of Excluded Middle, nevertheless recognizes that every proposition, S is P, is either true, or false, or else has a lower mode of being such that it can be neither determinately P nor determinately not P, but is at the limit between P and not P." In Peirce's system of categories, Thirdness is important precisely because it is that which mediates between conceptions of lower-order relations. Mediation of this kind requires a form of mind, which itself always involves a triadic element.
This formal category of mediation by mind allows us to deal more adequately with observed or thought states of affairs which cannot easily be handled by a system of two-valued logic, i.e. those situations where we are confronted with evidence that objectively indeterminate states of affairs really do exist. This is where I believe that its more general usefulness as a conceptual model, with regard to the further clarification and explication of the notoriously indeterminate nature of human communication in general, and as we shall also discuss below, of negotiation of meaning in distributed virtual environments, should be easily retrievable from what now follows.
[36] See Peirce's 1902 application to the Carnegie Institution, MS L75, and especially the online version at http://members.door.net/arisbe/menu/library/bycsp/l75/ver1/l75v1-02.htm, as well as Parker 1998, 28-58, Coppock 1997b, Liszka 1996, 1-17, Murphey 1993, 330-331
[37] CP 4.248
[38] Figure adapted from that of Parker 1998, 40
[39] CP 4.250
[40] NEM 4:165
[41] Parker 1998, especially 60-126, Murphey 1993, 321-407
[42] NEM 3:xvii