blogをいくつか見させていただいて、そこからの情報で以下を入手。
- Paul Cohen “Skolem and pessimism”, in: Philosophical Transactions of the Royal Society A. Mathematical, Physical and Engineering Sciences, vol. 363, no. 1835, 2005
… We sketch the development from Frege’s first formalization, to the debates over intuitionism and other schools, through Hilbert’s program and the decisive blow of the Godel Incompleteness Theorem. A critical role is played by the Skolem–Lowenheim Theorem, … Skolem himself regarded this as a body blow to the belief that mathematics can be reliably founded only on formal axiomatic systems. In a remarkably prescient paper, he even sketches the possibility of interesting new models for set theory itself, something later realized by the method of forcing. This is in contrast to Hilbert’s belief that mathematics could resolve all its questions. We discuss the role of new axioms for set theory, questions in set theory itself, and their relevance for number theory. We then look in detail at what the methods of the predicate calculus, i.e. mathematical reasoning, really entail. The conclusion is that there is no reasonable basis for Hilbert’s assumption. …
- A. Macintyre “The mathematical significance of proof theory”, in: Philosophical Transactions of the Royal Society A. Mathematical, Physical and Engineering Sciences, vol. 363, no. 1835, 2005
Returning to old ideas of Kreisel, I discuss how the mathematics of proof theory, often combined with tricks of the trade, can occasionally be useful in extracting hidden information from informal proofs in various areas of mathematics.
- Frank P. Ramsey “Truth and Probability”, in: R.B. Braithwaite ed., The Foundations of Mathematics and other Logical Essays, Kegan Paul, 1931. This document is 1999 electronic edition.
- ditto “Further Considerations”, in: R.B. Braithwaite ed., The Foundations of Mathematics and other Logical Essays, Kegan Paul, 1931. This document is 1999 electronic edition.
- ditto “Probability and Partial Belief”, in: R.B. Braithwaite ed., The Foundations of Mathematics and other Logical Essays, Kegan Paul, 1931. This document is 1999 electronic edition.
F. Ramseyの3本は、Ramsey SocietyのHPからDL。しかし確かにこのRamsey SocietyのHPのデザインはすごい。‘distressing’とか‘horrible’とか言われてしまって気の毒である。Ramseyを賞賛しようという意欲は強く伝わってくるのだが、どうしたものか？ もう少し落ち着いたデザインにすればRamseyさんのイメージに合うのではないでしょうか？ 個人的にはRamseyさんって落ち着いた方というイメージを持っているもので…。
PS. 今日もQuineの“Three Grades ... ”を少しだけ翻訳した。もう寝ます。おやすみなさい。
