In this paper, I explore an intriguing view of definable numbers proposed by a Cambridge mathematician Ernest Hobson, and his solution to the paradoxes of definability. Reflecting on König’s paradox and Richard’s paradox, Hobson argues that an unacceptable consequence of the paradoxes of definability is that there are numbers that are inherently incapable of finite definition. Contrast to other interpreters, Hobson analyses the problem of the paradoxes of definability lies in a dichotomy between finitely definable numbers and not finitely definable (...) numbers. To bypass this predicament, Hobson proposes a language dependent analysis of definable numbers, where the diagonal argument is employed as a means to generate more and more definable numbers. This paper examines Hobson’s work in its historical context, and articulates his argument in detail. It concludes with a remark on Hobson’s analysis of definability and Alan Turing’s analysis of computability. (shrink)

This article provides a detailed analysis of the transfer of a key cluster of ideas from mathematical logic to computing. We demonstrate the impact of certain of Turing’s logico-philosophical concepts from the mid-1930s on the emergence of the modern electronic computer—and so, in consequence, Turing’s impact on the direction of modern philosophy, via the computational turn. We explain why both Turing and von Neumann saw the problem of developing the electronic computer as a problem in logic, and we describe their (...) joint journey from logic to electronic computation. While much has been written about Turing’s and von Neumann’s individual contributions to the development of the computer, this article investigates less well-known terrain: their interactions and mutual influences. Along the way we argue against ‘logic skeptics’ and ‘Turing skeptics’, who claim that neither logic nor Turing played any significant role in the creation of the modern computer. (shrink)

In a recent paper, Francis Y. Lin proposes a “memory-challenge” to two main interpretations of Wittgenstein’s private language argument: the “no-criterion-of-correctness” interpretation and the “no-stage-setting” interpretation. According to Lin, both camps of interpretation fail to explain why a private language is impossible within a short time period. To answer the “memory-challenge”, Lin motivates a grammatical interpretation of the private language argument. In this paper, I provide a critical discussion of Lin’s objection to these interpretations and argue that Lin’s objection fails. (...) In the case of the “no-stage-setting” interpretation, Lin suggests that the private language user can establish a stage within a short time period. However, I show that this stage is insufficient for a private language to be used correctly. In the case of the “no-criterion-of-correctness” interpretation, Lin believes that since memory is reliable within a short period, no criterion is needed for the correct use of a private language. However, I argue that his objection attacks a strawman, since the interpretation concerns the structure of justification, rather than the weakness of memory itself. I conclude with a critical discussion on memory and primitive expressions, the latter of which are crucial to Wittgenstein’s approach to public language. This discussion will help to draw a sharp line between private language and public language, and cast some doubt on Lin’s grammatical interpretation. (shrink)

The present study provides evidence that the activation strength produced by emotional stimuli must pass a threshold level in order to be consciously perceived, contrary to the assumption of continuous quality of representation. An analysis of receiver operating characteristics for attentional blink performance was used to distinguish between two models of emotion perception by inspecting two different ROC’s shapes. Across all conditions, the results showed that performance in the attentional blink task was better described by the two-limbs ROC predicted by (...) the Krantz threshold model than by the curvilinear ROC implied by the signal-detection theory. (shrink)