Logical Thinking in the Pyramidal Schema of Concepts: The Logical and Mathematical Elements by Lutz Geldsetzer & Richard L. Schwartz

Author:Lutz Geldsetzer & Richard L. Schwartz
Language: eng
Format: epub
Publisher: Springer Netherlands, Dordrecht

Corollaries and Annotations to the Paragraphs

0.That logic is the “organon” or “instrument” of all sciences and erudition was always felt – although not always taken seriously – in the history of occidental science. Aristotle, who assigned it that role, did not clearly say whether logic itself should be a science (episteme) or not, and, if not, whether it could be carried on as a practical art as distinct from a (scientific) technique, since in Aristotle’s time and for long afterwards the same term was used for both. (Greek: “techne”  =  Latin: “ars”). Only the Stoics expressly denominated logic a science. Consequently, logic developed in different times and contexts into all of these. But Aristotle did clearly assign mathematics as a “second episteme” to the theoretical sciences and situated it between metaphysics (or ontology) and physics. This ­suggested that mathematics also required logical instruments for its constitution. That suggestion was neglected for ages, but then adopted by modern “mathematical logicism”. – Perhaps because Aristotle’s logic has always been well-known and intensively studied, his teachings concerning mathematics, in contrast to those of Plato and Euclid, have been underestimated by the historians of philosophy. Some references: Joseph Biancani, Aristotelis loca mathematica ex omnibus eius operibus collecta, Bologna 1615; A. Görland, Aristoteles und die Mathematik, (Diss.) Marburg 1899; J. L. Heiberg, “Mathematisches in Aristoteles”, in: Abhandlungen zur Geschichte der mathematischen Wissenschaften 18, Leipzig 1904, p. 1–49; Th. Heath, Mathematics in Aristotle, Oxford 1949; H. G. Apostle, Aristotle’s philosophy of mathematics, Chicago 1952; I. Mueller, Aristotle on geometrical objects, in: Archiv für Geschichte der Philosophie, 52, 1970, p. 150–71; J. Lear, Aristotle’s philosophy of mathematics, in: Philosophical Revue 91, 1982, p. 161–192; J. Barnes, Aristotle’s arithmetic, in: Revue de philosophie ancienne 3, 1985, p. 97–133; E. Hussey, Aristotle on mathematical objects, in: I. Mueller (ed.), Peri ton mathematon, Edmonton 1991. – See also M. Cantor, Vorlesungen über Geschichte der Mathematik, vol. I, 3. ed. Leipzig 1907, p. 251–256.

0.1.On the problem of formalism in logic and mathematics see: L. Brouwer, Intuitionistische Betrachtungen über den Formalismus, in: Sitzungsberichte der Preußischen Akademie der Wissenschaften 1928, p. 48–52; R. Carnap, Formalization of Logic. Studies in Semantics II, Cambridge, Mass. 1943, 2. ed. 1959; S. Krämer, Symbolische Maschinen. Die Idee der Formalisierung in geschichtlichem Abriß, Darmstadt 1988; T. Stoneham, Logical Form and Thought Content, in: Analysis 59, 1993, p. 183–185; L. Horsten, Platonistic Formalism, in: Erkenntnis 55, 2001, p. 173–194; G. Brun, Die richtige Formel. Philosophische Probleme der logischen Formalisierung, Frankfurt a. M.-London 2003.

0.1.1.Inaugurators of “ideal languages” were G. Dalgarno (1626–1687), Ars signorum vulgo character universalis et lingua philosophica, London 1661, and J. Wilkins (1614–1672), An Essay towards a Real Character and a Philosophical Language, 1668, with their versions of “Characteristica universalis”. See L. Couturat and L. Léau, Histoire de la langue universelle, Paris 1903. – G. Frege, Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle 1879, B. Russell and A. N. Whitehead, Principia Mathematica, Cambridge 1910–1913, and L. Wittgenstein, Tractatus logico-philosophicus (3.325), 1921, continued to promote this view of logic, and it is due to their influence that it has become almost universal in modern logic.

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