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Mathematics as Syntax: Gödel's Critique and Carnap's Scientific Philosophy  

Lee, Jeongmin (Department of Philosophy, Seoul National University)
Publication Information
Korean Journal of Logic / v.21, no.1, 2018 , pp. 97-133 More about this Journal
Abstract
In his unpublished article, "Is Mathematics Syntax of Language?," $G{\ddot{o}}del$ criticizes what he calls the 'syntactical interpretation' of mathematics by Carnap. Park, Chun, Awodey and Carus, Ricketts, and Tennant have all reconstructed $G{\ddot{o}}del^{\prime}s$ arguments in various ways and explored Carnap's possible responses. This paper first recreates $G{\ddot{o}}del$ and Carnap's debate about the nature of mathematics. After criticizing most existing reconstructions, I claim to make the following contributions. First, the 'language relativity' several scholars have attributed to Carnap is exaggerated. Rather, the essence of $G{\ddot{o}}del^{\prime}s$ critique is the applicability of mathematics and the argument based on 'expectability'. Thus, Carnap's response to $G{\ddot{o}}del$ must be found in how he saw the application of mathematics, especially its application to science. I argue that the 'correspondence principle' of Carnap, which has been overlooked in the existing discussions, plays a key role in the application of mathematics. Finally, the real implications of $G{\ddot{o}}del^{\prime}s$ incompleteness theorems - the inexhaustibility of mathematics - turn out to be what both $G{\ddot{o}}del$ and Carnap agree about.
Keywords
$G{\ddot{o}}del$; Carnap; incompleteness theorems; consistency; correspondence principle;
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