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Later Wittgenstein on ‘Truth’ and Realism in Mathematics
Philosophy Pub Date : 2023-10-19 , DOI: 10.1017/s0031819123000281
Philip Bold

I show that Wittgenstein's critique of G.H. Hardy's mathematical realism naturally extends to Paul Benacerraf's influential paper, ‘Mathematical Truth’. Wittgenstein accuses Hardy of hastily analogizing mathematical and empirical propositions, thus leading to a picture of mathematical reality that is somehow akin to empirical reality despite the many puzzles this creates. Since Benacerraf relies on that very same analogy to raise problems about mathematical ‘truth’ and the alleged ‘reality’ to which it corresponds, his major argument falls prey to the same critique. The problematic pictures of mathematical reality suggested by Hardy and Benacerraf can be avoided, according to Wittgenstein, by disrupting the analogy that gives rise to them. I show why Tarskian updates to our conception of ‘truth’ discussed by Benacerraf do not answer Wittgenstein's concerns. That is, because they merely presuppose what Wittgenstein puts into question, namely, the essential uniformity of ‘truth’ and ‘proposition’ in ordinary discourse.



中文翻译:


后来维特根斯坦论数学中的“真理”和实在论



我表明维特根斯坦对 G.H.哈代的数学实在论自然延伸到了保罗·贝纳塞拉夫(Paul Benacerraf)颇具影响力的论文《数学真理》。维特根斯坦指责哈代仓促地类比数学和经验命题,从而导致了一幅在某种程度上类似于经验现实的数学现实图景,尽管这造成了许多谜题。由于贝纳塞拉夫依靠同样的类比来提出有关数学“真理”及其对应的所谓“现实”的问题,因此他的主要论点也受到了同样的批评。维特根斯坦认为,哈代和贝纳塞拉夫提出的数学现实的问题图景可以通过破坏产生它们的类比来避免。我将展示为什么塔尔斯基对贝纳塞拉夫讨论的“真理”概念的更新并没有回答维特根斯坦的担忧。也就是说,因为它们只是预设了维特根斯坦所质疑的内容,即普通话语中“真理”和“命题”的本质一致性。

更新日期:2023-10-19
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