FregeのlogicismとRussellのlogicismが、随分異なるものであることが、最近時々言われているものと思われます。
ではFregeのlogicismとDedekindのlogicismとでは、どうなのだろう？
今日入手した次の文献の‘Section 3. Logicism and Structuralism’の冒頭に、FregeとDedekindのlogicismに関する共通点と相違点が簡略に記されているので、そこを引用し、小見出しのようなものをつけて表わしてみよう*1。
- Erich Reck “Dedekind's Contributions to the Foundations of Mathematics”, in: The Stanford Encyclopedia of Philosophy, April 22, 2008
3. Logicism and Structuralism
So far Dedekind's contributions in his overtly foundational writings were in focus. We reviewed his set-theoretic accounts of the natural and real numbers; we also sketched his role in the rise of modern set theory. Along the way, philosophical issues have come up. A more extended reflection on them seems called for, however, especially a reflection on Dedekind's “logicism” and “structuralism”.
[DedekindとFregeの共通点。DedekindはFregeと同様に、logicを今で言うset theoryを含んでいるものと見ていた。そしてそのようなlogicは、直観に依拠した幾何学からは独立していると考えていた。]
Like for Frege, the other important logicist in the nineteenth century, “logic” is more encompassing for Dedekind than often assumed today (as comprising only first-order logic). Both thinkers take the notions of object, set, and function to be fundamental for human thought and, as such, to fall within the range of logic. Each of them develops a version of set theory (a theory of “systems”, “extensions”, or “classes”) as part of logic. Also for both, logic, even in this encompassing sense, is independent of intuitive considerations, and specifically of geometry (understood itself as grounded in intuition). What reducing analysis and arithmetic to logic is meant to establish, then, is that those fields too are independent of intuition.
[Analysisもarithmeticも直観に依拠した幾何学に基付くのではなく、logicに基付くというような考えは、既に当時GaussとDirichletが抱いていた。DedekindとFregeが新しかったのは、この考えを表明するのみならず、実際にそのことを細部に渡って実行してみせたことである。]
The view that analysis and arithmetic are not dependent of geometry, since they fall within the realm of purely logical thought, was not entirely new at the time—Gauss and Dirichlet already held such a view, as mentioned above. Dedekind's and Frege's original contributions consisted in particular, detailed reductions of, on the one hand, analysis to arithmetic and, on the other hand, arithmetic to logic; moreover, they complemented these reductions by, or grounded them in, systematic elaborations of logic. And as Dedekind's work was better known than Frege's at the time, perhaps because of his greater reputation as a mathematician, he was seen as the main representative of “logicism” by interested contemporaries, such as Charles Sanders Peirce and Ernst Schröder.
[DedekindもFregeも、証明できることは、いかなることも、証明なくして受け入れるべきではない、という方法論を堅持していた。彼らはこの方法論をともに当時の数学の歴史から学んだのである。]
Besides these commonalities between Dedekind's and Frege's versions of logicism, the two thinkers also agreed on a more general methodological principle, encapsulated in the following remark by Dedekind: In science, and especially in mathematics, “nothing capable of proof ought to be accepted without proof”. This principle ought to be adhered to not so much because it increases certainty; rather, it is often only by providing an explicit, detailed proof for a result that the assumptions on which it depends become evident and, thus, its range of applicability established. Both of them had learned this lesson from the history of mathematics, especially from nineteenth-century developments in geometry, algebra, and the calculus.
[DedekindとFregeの相違点。Fregeがlogicに対してなした貢献は、syntacticでproof-theoreticな面であったのに比べ、Dedekindがlogicに対してなした貢献は、semanticでmodel-theoreticな面であった。]
Beyond where they are in agreement, it is instructive to consider some of the differences between Dedekind and Frege as well. First and put in modern terminology, a major difference is that, while Frege's main contributions to logic concern syntactic, proof-theoretic aspects, Dedekind tends to focus on semantic, model-theoretic aspects. Thus, nothing like Frege's revolutionary analysis of deductive inference, by means of his “Begriffsschrift”, can be found in Dedekind. In turn, Dedekind is much more explicit and clear than Frege about issues such as categoricity, completeness, independence, etc., which puts him in proximity with a “formal axiomatic” approach as championed later by Hilbert and Bernays.
[DedekindはFregeと違い、infiniteについて詳しい考察を行っている。またKroneckerの提起した問題にも自覚的である。それにDedekindはordinal numbersを基礎にnatural numbersを理解しているが、Fregeはcardinal numbersを基礎にnatural numbersを理解している。さらにDedekindはnatural numbersをstructurallyに理解している一方で、Fregeはnatural numbersをintrinsicallyに理解している。]
Compared to Frege, Dedekind also has much more to say about the infinite (not just by formulating his explicit definition of that notion, but also by exploring the possibility of different infinite cardinalities with Cantor). And he shows more awareness of the challenge posed by Kroneckerian computational and constructivist strictures to logicism. In addition, the differences between Frege's and Dedekind's respective treatments of the natural and real numbers are noteworthy. As we saw, Dedekind conceives of the natural numbers primarily as ordinal numbers; he also identifies them purely “structurally”. Frege makes their application as cardinal numbers central; and he insists on building this application into the very nature of the natural numbers, thus endowing them with non-relational, “intrinsic” properties. The case of the real numbers is similar.
まとめてみよう。
DedekindとFregeの共通点。
- Logicはset theoryを含む。そのようなlogicは、直観に依拠した幾何学からは独立している。
- Analysisもarithmeticも直観に依拠した幾何学に基付くのではなく、logicに基付く。このことの詳細な論証を展開。
- 証明できることは、いかなることも、証明なくして受け入れるべきではない、という方法論の堅持。およびこの方法論の同時代における数学史からの学習。
DedekindとFregeの相違点。
- Dedekindのlogicに対する貢献: Semanticでmodel-theoreticな側面
- Fregeのlogicに対する貢献: Syntacticでproof-theoreticな側面
- Dedekindはordinal numbersを基礎にnatural numbersを理解している。
- Fregeはcardinal numbersを基礎にnatural numbersを理解している。
- Dedekindはnatural numbersをいわばstructurallyに理解している。
- Fregeはnatural numbersをいわばintrinsicallyに理解している。
また
1. Dedekindはinfiniteについて詳しい考察を行っているが、Fregeはそうではない。
1. DedekindはKroneckerの提起した問題に自覚的であるが、Fregeはそうではない。
*1:Reckさんによる、文献の参照を求める註はすべて省いて引用してある。
