کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4662868 1345201 2014 30 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Empiricism, probability, and knowledge of arithmetic: A preliminary defense
ترجمه فارسی عنوان
تجربه گرایی، احتمال و دانش ریاضی: دفاع اولیه
کلمات کلیدی
فلسفه و ریاضیات؛ منطق استقرایی؛ حساب پیانو
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات منطق ریاضی
چکیده انگلیسی

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgments of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity (Section 2.1), while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic (Section 2.2). Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the ω-rule (Section 3.1) or to the non-computability and non-continuity of probability assignments (Section 3.2). Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments (Appendix A–Appendix C), such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated (Theorem 4Appendix B). In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Applied Logic - Volume 12, Issue 3, September 2014, Pages 319–348
نویسندگان
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