22 research outputs found
Singular Thought
A singular thought can be characterized as a thought which is directed at just one object. The term âthoughtâ can apply to episodes of thinking, or to the content of the episode (what is thought). This paper argues that episodes of thinking can be just as singular, in the above sense, when they are directed at things that do not exist as when they are directed at things that do exist. In this sense, then, singular thoughts are not object-dependent
The Rule-Following Paradox and the Impossibility of Private Rule-Following
Kripkeâs version of Wittgensteinâs rule-following paradox has been influential. My concern is with how itâand Wittgensteinâs views more generallyâhave been perceived as undercutting the individualistic picture of mathematical practice: the view that individualsâRobinson Crusoesâcan, entirely independently of a community, engage in cogent mathematics, and indeed (more generally) have âprivate languages.â What has been denied is that phrases like âcorrectly countingâ can be applied to such individuals because these normative notions (so the Wittgensteinian analysis is taken to show) can only be applied cogently in a context involving community standards. I attempt to show that this shocking corollary doesnât follow even if Kripkeâs Wittgensteinian objections to dispositional approaches to rule-following are largely right. My reason for claiming this is that there is another (âscepticalâ) solution to the rule-following paradox, one that doesnât favor community standards over individual ones. Furthermore, it doesnât replace truth conditions with assertability conditions; and this latter maneuver is essential to Kripkeâs sceptical solution favoring the community over the individual
Tracking reason: proof, consequence, and truth
When ordinary people - mathematicians among them - take something to follow (deductively) from something else, they are exposing the backbone of our self-ascribed ability to reason. This book investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians
The Rule-Following Paradox and the Impossibility of Private Rule-Following
Kripkeâs version of Wittgensteinâs rule-following paradox has been influential. My concern is with how itâand Wittgensteinâs views more generallyâhave been perceived as undercutting the individualistic picture of mathematical practice: the view that individualsâ<em>Robinson Crusoes</em>âcan, entirely independently of a community, engage in cogent mathematics, and indeed (more generally) have âprivate languages.â What has been denied is that phrases like âcorrectly countingâ can be applied to such individuals because these normative notions (so the Wittgensteinian analysis is taken to show) can only be applied cogently in a context involving community standards. I attempt to show that this shocking corollary doesnât follow even if Kripkeâs Wittgensteinian objections to dispositional approaches to rule-following are largely right. My reason for claiming this is that there is another (âscepticalâ) solution to the rule-following paradox, one that doesnât favor community standards over individual ones. Furthermore, it doesnât replace truth conditions with assertability conditions; and this latter maneuver is essential to Kripkeâs sceptical solution favoring the community over the individual
Recommended from our members
Donald Mac Kenzie. Mechanizing Proof: Computing, Risk, and Trust. Cambridge, Mass.: MIT Press, 2001. Pp. xi + 427
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The Algorithmic-Device View of Informal Rigorous Mathematical Proof
A new approach to informal rigorous mathematical proof is offered. To this end, algorithmic devices are characterized and their central role in mathematical proof delineated. It is then shown how all the puzzling aspects of mathematical proof, including its peculiar capacity to convince its practitioners, are explained by algorithmic devices. Diagrammatic reasoning is also characterized in terms of algorithmic devices, and the algorithmic device view of mathematical proof is compared to alternative construals of informal proof to show its superiority