What is the Nature of Mathematical–Logical Objects?

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• 作者：Stathis Livadas
• 关键词：A priori possibility ; Infinite totality ; Intentional consciousness ; Objectivity of understanding ; Predicative activity ; Set constitution ; Temporal unity
• 刊名：Axiomathes
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：27
• 期：1
• 页码：79-112
• 全文大小：
• 刊物类别：Humanities, Social Sciences and Law
• 刊物主题：Philosophy, general; Ontology; Linguistics, general; Cognitive Psychology; Logic;
• 出版者：Springer Netherlands
• ISSN：1572-8390
• 卷排序：27

文摘

This article deals with a question of a most general, comprehensive and profound content as it is the nature of mathematical–logical objects insofar as these are considered objects of knowledge and more specifically objects of formal mathematical theories. As objects of formal theories they are dealt with in the sense they have acquired primarily from the beginnings of the systematic study of mathematical foundations in connection with logic dating from the works of G. Cantor and G. Frege in the last decades of the nineteenth century. Largely motivated by a phenomenologically founded view of mathematical objects/states-of-affairs, I try to consistently argue for their character as objects shaped to a certain extent by intentional forms exhibited by consciousness and the modes of constitution of inner temporality and at the same time as constrained in the form of immanent ‘appearances’ by what stands as their non-eliminable reference, that is, the world as primitive soil of experience and the mathematical intuitions developed in relations of reciprocity within-the-world. In this perspective and relative to my intentions I enter, in the last section, into a brief review of certain positions of G. Sher’s foundational holism and R. Tieszen’s constituted platonism, among others, respectively presented in Sher (Bull Symb Log 19:145–198, 2013) and Tieszen (After Gödel: platonism and rationalism in mathematics and logic, Oxford University Press, Oxford, 2011).