In the Cambridge Conferences Lectures of 1898 Peirce defines a continuum as a "collection of so vast a multitude" that its elements "become welded into one another." He links the transinfinity (the "vast multitude") of a continuum to the confusion of its elements by a line of mathematical reasoning closely related to Cantor's Theorem. I trace the mathematical and philosophical roots of this conception of continuity, and examine its unresolved tensions, which arise mainly from difficulties in Peirce's theory of collections.
C.S. Peirce defines mathematics in two ways: first as "the science which draws necessary conclusions," and second as "the study of what is true of hypothetical states of things" (CP 4.227–244). Given the dual definition, Peirce notes, a question arises: Should we exclude the work of poietic hypothesis-making from the domain of pure mathematical reasoning? (CP 4.238). This paper examines Peirce's answer to the question. Some commentators hold that for Peirce the framing of mathematical hypotheses requires poietic genius but is not scientific work. I propose, to the contrary, that although Peirce occasionally seems to exclude the poietic creation of hypotheses altogether from pure mathematical reasoning, Peirce's position is rather that the creation of mathematical hypotheses is poietic, but it is not merely poietic, and accordingly, that hypothesis-framing is part of mathematical reasoning that involves an element of poiesis but is not merely poietic either. Scientific considerations also inhere in the process of hypothesis-making, without excluding the poietic element. In the end, I propose that hypothesis-making in mathematics stands between artistic and scientific poietic creativity with respect to imaginative freedom from logical and actual constraints upon reasoning.
It is part of the conventional wisdom about the James family that the elder Henry James (1811–82) had a large influence on his son, William James (1842–1910), in the direction of religious interests. But William neither adopted his father's spirituality nor did he regard it as a foil to his own secularity. Instead, after first rejecting the elder James's idiosyncratic faith, he became increasingly intrigued with his insights into the natural world, which were in turn shaped by the Swedenborgian philosophy of correspondences and use, which depict worldly facts as vessels of the spirit. The young science student drew upon this approach to nature as a resource for finding the operation of immaterial aspects within the world. The influence of the father emerges in William's emphasis on the will in human psychology, his eagerness to punctuate the striving of "the will to believe" with sessions of comforting conviction, his readiness to find "'piecemeal' supernaturalism" in subliminal psychology, his incorporation of idealism into his radical empiricism, and his openness to psychical experience. Without accepting the particulars of Henry James's faith, William James shared with his father a conviction that providential action in the universe, usually understood as the work of transcendental forces, was embedded within the natural world and within humankind.
The paper presents some mathematical aspects of the question of reducibility of relations. After giving a formal definition of reducibility we present the basic result (due to Herzberger) to the effect that relations of valency at least 3 are always reducible if the cardinality of the relation is at most equal to the cardinality of the underlying set (which is automatically the case if this set is infinite). In contrast to this, if the term "reduction" is given a practicable form, relations on finite sets are "generically" irreducible, as is shown by a simple counting argument. Next we discuss the question of an "intrinsic" criterion for reducibility. Finally we propose a scheme for the graphic representation of reductions.
Peirce was greatly influenced by Aristotle, particularly on the topic of final cause. Commentators are therefore right to draw on Aristotle in the interpretation of Peirce's teleology. But these commentators sometimes fail to distinguish clearly between formal cause and final cause in Aristotle's philosophy. Unless form and end are clearly distinguished, no sense can be made of Peirce's important claim that 'desires create classes.' Understood in the context of his teleology, this claim may be considered Peirce's answer to nominalists and sceptics on the possibility and status of scientific knowledge. On the basis of an improved view of Peirce's teleology, the objection that inorganic physical events do not admit of teleological explanation can be answered. I argue that the non-teleological alternative leaves the laws of nature and the actions of inorganic matter unexplained.
Thoreau's journal contains a number of passages which explore the nature of perception, developing a response to skeptical doubt. The world outside the human mind is real, and there is nothing illusory about its perceived beauty and meaning. In this essay, I draw upon the work of Stanley Cavell (among others) in order to frame Thoreau's reflections within the context of the skeptical questions he seeks to address. Value is not a subjective projection, but it also cannot be perceived without the appropriate kind of emotional orientation or attunement toward the world: that is, an attitude of trust or acceptance. Without this affective receptivity, or "perceptual faith," our knowledge of reality is limited. The beliefs we hold onto in the face of objective uncertainty establish the framework within which we make particular evaluations, and in this sense they are a necessary condition of practical reason. Every understanding has its mood.
C. S. Peirce had no theory of metaphor and provided only few remarks concerning the trope. Yet, some of these remarks seem to suggest that Peirce saw metaphor as fundamental to consciousness and thought. In this article we sketch a possible connection between metaphor and cognition; we understand Peircean metaphor as rooted in abduction; it is part of an intricate relation between experience, body, sign and guessing instinct as a semeiotic mechanism which can convey new insights.