- Vincent F. Hendricks and Hannes Leitgeb ed. Philosophy of Mathematics 5 Questions, Automatic Press / VIP, 2007
何とはなしに上記の本の Albert Visser さんの文章を読むと、中々面白い。
私は知らなかったのですが、Visser さんは Frege に興味をお持ちで、彼の自然数理解を高く評価されており、Neo-Fregeanism の行く末に関心をお持ちのようである。
Visser さんの経歴においては、最初 Russell, Popper, Quine さん達の本を読み、特に Quine さんに最も影響を受けたらしい。但し今では Quine さんの哲学観には反対の立場を堅守されているようである。その後、Kripke さんの theory of truth に痛く感銘を受け、このことは一つの‘revelation’だったらしい。そしてそのあと dynamic predicate logic へと関心を広げられたみたいである。
そこで興に任せて、氏が Frege について語っているところを引いてみよう*1。原文中に付されている註は、一部を除き、省いて引用します。
Let me finally mention the philosophical question what is number? Frege came up with an answer that I always found strongly compelling. His idea was that if you have an equivalence relation, there are ipso facto ‘abstract objects’ filling the bill for the phrase: that which equivalent objects have in common. Thus, for the equivalence relation parallelism in Euclidean Geometory, we have abstract objects: directions, for the equivalence relation equidistance between pairs of points in Euclidean Geometory, we have objects that are the distances between those points. Etcetera. One attractive feature of Frege's principle is that we can imagine the objects to be sui generis [Note 6: This is not Frege's idea but my peferred way of looking at it.]. In contrast, e.g. equivalence classes in a set theoretical reconstruction are a modeling of the idea, but they carry the excess of a specific implementation.
The idea of abstracts being sui generis suggests that, always when we create abstracts, they form a type distinct from the already existing objects. However, in the case that we consider numbers and extensions, we run into a problem,[sic] It seems undeniable that we can, in principle, count or collect anything, whatever it is. Thus, e.g., the number 3 can be also applied the concept 1, 2, 3, and, thus, the idea of the type of the abstracts being brand new seems to founder here. Surprisingly, Frege's abstraction principle as applied to equinumerosity of concepts −this is called Hume's principle− is paradox-free, when taken on its own. (This can be shown by an easy argument that goes back to Peter Geach.) However, when we bring in extensions-as-objects, as is does in Frege's Axiom V, the Russell Paradox strikes.
I guess one should consider the state of affairs outlined above as a skandalon of Philosophy[sic]. Thus, a careful reflection on all steps of the argumet is in order. The Neo-Fregean school and Kit Fine did just that. The technical side of the work of the Neo-Fregeans has been presented in lucid detail by John Burgess in a wonderful book [Fixing Frege]. I think that, even if the conclusion of Burgess' book is that a Fregean Program that does not import substantial further non-Fregean principles does not go very far, the path we take towards this conclusion is illuminating.
As so often with these things, the Neo-Fregean program has features that are very interesting from a more internal logical point of view. One of these is the precise determination of the strength of various predicative systems. Mihail Ganea showed in his paper [his unpublished manuscript] that the fisrt level of the predicative Frege hierarchy is mutually interpretable with Robinson's arithmetic Q. I showed that the n＋1-th level of the hierarchy is equivalent to Q plus the n-fold iterated consistency of Q. See [Visser, “The Predicative Frege Hierarchy”].
PS
一年の終わりに当たって。
この一年も色々あった。
特に後半は忙しくて大変だった。
よかったことは、大変遅まきながら、いわゆる comprehension principle を残したままでも無矛盾な集合論があり得、その上に(いくらかの)古典数学を建設することができ、しかもそのような試みの中には Frege の Logicism の再生とも考えられ、希望が持てるかもしれないような、そのような理論があり得るということを知ったことです。随分気が付くのが遅くなって我ながらあきれるが、しかしいずれにせよ、いつからでも勉強は始められるのだから、かなりの周回遅れだけれど、ぼちぼちこの辺を勉強して行きたいです。
しかし残念だったこともある。それは同年代の知人が亡くなってしまったことである。あんなに元気だったのに、しばらく前から病気だったとはいえ、1〜2ヶ月で急に逝ってしまうなんて、とても shocking だった。そして怖かった。この世は仮の住まいだと実感した。残された私達は後を受けてできる限りのことをするべきなんだと思った。あの世ではゆっくり休んで下さい。
*1:Hendricks and Leitgeb, pp. 269-70.
