I reply to recent criticisms by Uygar Abaci and Peter Yong, among others, of my reading of Kant's pre-Critical of God's existence, and of its fate in the Critical period. Along the way I discuss some implications of this debate for our understanding of Kant's modal metaphysics and modal epistemology generally.

Numbers are concepts whose content, structure, and organization are influenced by the material forms used to represent and manipulate them. Indeed, as argued here, it is the inclusion of multiple forms (distributed objects, fingers, single- and two-dimensional forms like pebbles and abaci, and written notations) that is the mechanism of numerical elaboration. Further, variety in employed forms explains at least part of the synchronic and diachronic variability that exists between and within cultural number systems. Material forms also impart characteristics (...) like linearity that may persist in the form of knowledge and behaviors, ultimately yielding numerical concepts that are irreducible to and functionally independent of any particular form. Material devices used to represent and manipulate numbers also interact with language in ways that reinforce or contrast different aspects of numerical cognition. Not only does this interaction potentially explain some of the unique aspects of numerical language, it suggests that the two are complementary but ultimately distinct means of accessing numerical intuitions and insights. The potential inclusion of materiality in contemporary research in numerical cognition is advocated, both for its explanatory power, as well as its influence on psychological, behavioral, and linguistic aspects of numerical cognition. (shrink)

In the Critique of the Power of Judgment, Immanuel Kant characterizes the sublime as a negative pleasure derived from the encounter with something absolutely great. Some scholars have claimed that Kant’s notion of the sublime only applies to objects in nature (Abaci 2008, 239). Others contend that the object that provokes the sublime experience is not confined to objects of nature since the sublime is an experience in the mind (Clewis 2010, 169). This paper argues that Kant’s Critique and (...) the Grateful Dead’s song “The Other One” provide a framework for understanding the artistic mathematical sublime in music, wherein the pleasure and displeasure experienced when engaging with music can be identified as instances of the sublime. Beginning with an elucidation of Kant’s concepts of judgments of taste and the sublime, the paper explores their similarities and differences. Notably, his theory of the sublime is less developed than his theory of the beautiful. This provides an opportunity to interrogate arguments surrounding which objects can elicit the sublime and how artworks represent or present it to their viewers. After demonstrating a textual disparity of the role of music in Kant’s Critique, I show how music without words, also known as jams, can fit the definition of free beauty and can be classified as art. The discussion further delineates the process of experiencing the mathematical sublime and describes indicators in music theory for discerning the sublime in musical compositions. I use the Grateful Dead’s performance of “The Other One” on April 7, 1972, to illustrate how particular works of art manifest the mathematical sublime. The paper concludes by reflecting on how the qualitative description of the piece highlights why “The Other One” should be seen as both artistic (because, while the piece is coherent, there is no discernible concept that seems to unify it) and sublime (because it is mind-blowingly complex). How we engage with music like the Grateful Dead can allow us to discuss why particular works of art are meaningful to us. (shrink)