In "The Fixation of Belief" [CP 5.377f], first published in 1877, Peirce argues for the superiority of the "method of science", contrasting this with three other historically established methods of fixing beliefs. These are: "The method of tenacity" (believing that which happens to suit us at the time and not being prepared to renounce it even in the face of conflicting evidence), "the method of authority" (institutionalisation of doctrines and subsequent elimination of dissenters) , "the a priori method" (adopting those opinions which seem "agreeable to reason"). Building on his conception of the Real, developed earlier as part of his theory of reality, and now extending it to be taken as an ultimate premiss, Peirce argues for the need of a method of inquiry by which "our beliefs may be caused by nothing human, but by some external permanency - by something upon which our thinking has no effect" [CP 5.384] This optimal method is "the method of science," the hypothesis of which is formulated by Peirce as follows:
"There are Real things, whose characters are entirely independent of our opinions about them; whose realities affect our senses according to regular laws, and, though our sensations are as different as our relations to the objects, yet, by taking advantage of the laws of perception, we can ascertain how things really are; and any man, if he have sufficient experience and reason enough about it, will be led to the one True conclusion" [CP 5.384]
Murphey (1993 p. 167-171) points out with reference to what Manley Thompson called "the paradox of Peirce's realism"[37] that the problem with this formulation is that it actually presupposes that inquiry must go on for ever in order for reality to be guaranteed at all. Peirce is essentially defining reality in purely phenomenological terms, as the possibility of permanent sensation, while at the same time maintaining that the real object is something more than merely a regularity of sensation. If inquiry does not go on infinitely, something that cannot actually be proved to be the case since it is a projection of a hypothetical postulate to an infinite future, then no finite final opinion can ever be reached, and The Real becomes wholly an object of faith. Peirce however was content to leave this particular problem hanging in the air for a time, while he went on to revise his categories in accord with the new theory of meaning which he had developed as a result of the incorporation of the theoretical consequences evoked by discovery of the logic of relatives into his general philosophical scheme.
To this end, he abandoned the notions of Being and Substance, and instead developed further the concept of Thirdness - as the mediating representation or interpretant that brings things into relation - into the general relation that connects any two terms. He went on to try and show how Thirdness is the most important of the three categories since it corresponds to relation, and also that it was related to the notion of continuity - "the direct instrument of the finest generalizations" [CP 2.646]. The idea of continuity is in turn related to the notion of the infinitesimal, the reciprocal of the infinite, which at that time was considered an essential notion in calculus, and which constituted part of the origins of modern set theory[38], discussions of which Peirce had already introduced around 1873 as an element in the development of his theory of cognition (and he in fact continued to defend the reality of the infinitesimals for the remainder of his life[39]). It was this notion of continuity and the associated extended notion of Thirdness which would form the basis of his later notion of Synechism, the development of which into his cosmology was in many respects one of the last important revisions of his philosophical system that he made.
Before we go on to discuss these matters further, mention should be made of Peirce's notion of community, which is central in his theory of inquiry, and also in some way related to the concept of continuity mentioned above. In "How to Make our Ideas Clear?", published in 1878. Peirce asks rhetorically: "in what does the reality of mind exist?", replying that
"We have seen that the content of consciousness, the entire phenomenal manifestation of mind, is a sign resulting from inference. Upon our principle, therefore, that the absolutely incognizable does not exist, so that the phenomenal manifestation of a substance is the substance, we must conclude that the mind is a sign developing according to the laws of inference" [CP 5.313]
This represents in one sense an expression of Peirce's belief in the idea of the divine mind, since in this context man's individuality only exists insofar as "his separate existence is manifested only by ignorance and error, so far as he is anything apart from his fellows, and from what he and they are to be, is only a negation" [CP 5.327].
The reference to "what he and they are to be", in this case implying some kind of ideal community, or merging of opinion, with the divine mind. Though individuality is solely expressed in terms of man's individual ignorance and error, identity remains, since it "consists in the consistency of what he does and thinks, and consistency is the intellectual character of a thing, that is, expressing something" [CP 5.315], where by "expressing something" is meant being the embodiment of an abstraction: "When we think, then, we ourselves are a sign" [CP 5.283]. In the long run, provided inquiry is carried out in the correct way, both the individual and the community will come more and more into conformity with the divine mind through the gradual elimination of ignorance and error, this being the sole aim of inquiry.
"The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of you and me. Thus, the very origin of the conception of reality shows that the conception essentially involves the notion of COMMUNITY, without definite limits, and capable of a definite increase in knowledge." [C.P 5.311]
Since Peirce's theory of number is one of the main bases for his later philosophy, it seems appropriate at this point to make a brief examination of his more general conceptions of what mathematics is, and its role in the scheme of the sciences. Again, time and space constraints do not allow for any deep-going discussion of this particular issue here, and those who might be interested in looking into this topic more closely are referred to Murphey's (1993, pp. 183-288) comprehensive and interesting background discussion of the main developments in nineteenth century mathematics, and Peirce's own mathematical theories and the role they played in his time in this particular arena of science[40].
Peirce's father, Benjamin Peirce, defined mathematics as "the science which draws necessary conclusions"[41] Charles, who had worked with his father on this particular paper enthusiastically adopted this definition, and commented in his notes to his father's paper that "In truth, no two minds could be more directly opposite than the cast of mind of the logician and the mathematician... The mathematician's interest in reasoning is as a means of solving problems... The logician, on the other hand, is interested in picking a method to pieces and finding out what its essential ingredients are."[42] Murphey argues convincingly that Peirce largely regarded mathematics as a purely formal deductive system. In such a system, premisses of any propositions made within the framework of the system are to be regarded as "pure hypotheses". The inherent meanings of the symbols and terms that are used is irrelevant - they are mere uninterpreted variables. The truth or falsehood of the premisses forming the basis of mathematical propositions is not an issue here, and thus these propositions "may be taken as definitions of the objects under the consideration of the mathematician without involving any assumption in reference to experience or intuition" [CP 3.20]
Peirce harboured a strictly Platonic view of number, in the sense that he maintained that numbers are ideas belonging to a different universe of experience (a "Platonic world of pure forms" [CP 4.118]) from facts and laws. They were nonetheless real, in that "their Being consists in mere capability of getting thought, not in anybody's Actually thinking them" [CP 6.455]. Murphey (1993, p. 239) observes that mathematical entities thus form a quite unique class in Peirce's ontology, as abstract entities they are never to be known in their entirety, being realised only in instances whence their reality consists precisely in their possibility through "getting", rather than "Actually " being thought.
Peirce derived his notion of continuity from Cantor, who had, with others, conceived that set theory might provide a way to unify arithmetic and geometry. Continuum belongs to geometry, whereas the collection or set belongs to arithmetic. Cantor's theory was based on the idea that a collection of entities having the multitude of the set of real numbers, over which certain order relations are defined, possesses every property of the continuum, and may be treated as the continuum. Where entities are points on a real line, the continuum is a geometrical continuum. Using this principle geometrical entities may be defined in terms of point sets, and space, too, becomes a kind of set.[43] Since he held infinitesimals to be reals, Peirce did not agree that the multiplicity of the continuum could in any case be equal to any discrete multitude, and he maintained therefore that every interval on a line must be capable of further division. His concept of Kanticity was developed to support this principle. Something is kantistic if and only if "every part has itself parts of the same kind". [CP 3.538-539] Peirce therefore maintained on his own theory of the continuum, that:
"a continuum, where it is continuous and unbroken, contains no definite parts; that its parts are created in the act of defining them and the precise definition of them breaks the continuity" [CP 6.168]
Peirce used this principle to point out that the fallacy of Zeno's paradox of Achilles and the tortoise is precisely the assumption of the necessity of having to pass through an infinite series of discrete segments - the point of the paradox being that if there are an infinite number of such segments, then they will never be able to be passed through in a finite time, and Achilles will never overtake the tortoise. In Peirce's view, the introduction of a discrete value measurement schema to define Achilles' position at any given time introduces the fiction that there are discrete points on the line corresponding to the measurement scale values, which there are not since the continuum is wholly non-discrete, pure possibility, and thus independent of both measurement and any notions of magnitude. The paradox arises from the confusion of the thing measured with measurement.
Murphey concludes in this connection that:
"Peirce's theory thus stands between those of the intuitionists and the logistic school. Whereas the latter regard elements as a pre-existing totality, and the former deny the existence of any elements not constructible, Peirce is willing to admit a pre-existing totality of possibilities, but recognizes only a small part of these as actually existent." (Murphey 1993, p. 288)
This basic question of how actuality emerges from mere possibility, or in other words how actually existent things are possible, is more than anything else the fundamental problem that provoked Peirce to go on to develop his cosmology.