- B. Rang and W. Thomas “Zermelo's Discovery of the “Russell Paradox””, in: Historia Mathematica, vol. 8, issue 1, 1981
In his 1908 paper on the Well-Ordering Theorem, Zermelo claimed to have found “Russell's Paradox” independently of Russell. Here we present a short note, written by E. Husserl in 1902, which contains a detailed exposition of Zermelo's original version of the paradox. We add some comments concerning the date of Zermelo's discovery, the circumstances which caused Husserl to write down Zermelo's argument, and the argument itself.
- Gregory H. Moore and Alejandro Garciadiego “Burali-Forti's Paradox: A Reappraisal of Its Origins”, in: Historia Mathematica, vol. 8, issue 3, 1981
Using both published and unpublished letters and manuscripts, this article shows that Burali-Forti's paradox, which has long been regarded as the first of the set-theoretical paradoxes to be discovered, was not created by either Burali-Forti or Cantor. It arose gradually and began to take recognizable form only in Russell's The Principles of Mathematics of 1903. Russell's long-standing predisposition to seek paradoxes was a vestige of the Kantian and Hegelian philosophical traditions in which he was schooled. Between 1904 and 1906, Burali-Forti's paradox was nurtured by Jourdain and Poincaré, both of whom considered it to be more fundamental than Russell did. To the end, both Burali-Forti and Cantor maintained that there was no such paradox.
- Michael Dummett “What is Mathematics About?”, in Dale Jacquette ed., Philosophy of Mathematics, Blackwell Publishing, Blackwell Philosophy Anthologies Series, 2001. Originally published in Alexander George ed., Mathematics and Mind, Oxford University Press, 1994.
- Philip Wadler “Proofs are Programs: 19th Century Logic and 21st Century Computing”, 2000
As the 19th century drew to a close, logicians formalized an ideal notion of proof. They were driven by nothing other than an abiding interest in truth, and their proofs were as ethereal as the mind of God. Yet within decades these mathematical abstractions were realized by the hand of man, in the digital stored-program computer. How it came to be recognized that proofs and programs are the same thing is a story that spans a century, a chase with as many twists and turns as a thriller. At the end of the story is a new principle for designing programming languages that will guide computers into the 21st century.
For my money, Gentzen’s natural deduction and Church’s lambda calculus are on a par with Einstein’s relativity and Dirac’s quantum physics for elegance and insight. And the maths are a lot simpler. I want to show you the essence of these ideas. I’ll need a few symbols, but not too many, and I’ll explain as I go along.
To simplify, I’ll present the story as we understand it now, with some asides to fill in the history. First, I’ll introduce Gentzen’s natural deduction, a formalism for proofs. Next, I’ll introduce Church’s lambda calculus, a formalism for programs. Then I’ll explain why proofs and programs are really the same thing, and how simplifying a proof corresponds to executing a program. Finally, I’ll conclude with a look at how these principles are being applied to design a new generation of programming languages, particularly mobile code for the Internet.
- “An International Bibliography of Works by and Selected Works about Nelson Goodman”
- 志賀浩二 『無限への飛翔 集合論の誕生』、シリーズ《大人のための数学 3巻》、紀伊國屋書店、2008年