Abstract
In the Foundations of Arithmetic 1 — announced in 1879 but first published in 1884 — Frege investigates the concept of number without the help of the tools of the Begriffsschrift, since the philosophical public then as now was not attracted by symbolism. This work was not intended solely for mathematicians. Frege makes explicit in the introduction that the investigation of the concept of number is “common to mathematics and philosophy” and the collaboration of the two sciences was not as close as it should be because the mathematicians rightly reject the “primacy of the psychological point of view in philosophy, even spilling over into logic” (GI., v).
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References
Frege, G., Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl [The Foundations of Arithmetic. A Logical-Mathematical Investigation of the Concept of Number], Breslau 1884 (2nd ed.: Breslau 1934); Photomechanical reproduction: Hildesheim 1961).
Frege, G., Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet [Basic Laws of Arithmetic. Ideographically deduced], Jena, I: 1893, II: 1903. (Reproduction: Hildesheim and Darmstadt 1962).
Gg., I, xv. The distinction between ‘natural law of thought’ and ‘normal law of thought’ is already to be found in M. W. Drobisch, Neue Darstellung der Logik nach ihren einfachsten Verhältnissen, mit Rücksicht auf Mathematik und Naturwissenschaft [A New Presentation of Logic in its Simplest Form, with Reference to Mathematics and Natural Science], in the second paragraph of the second edition: Leipzig 1851. (First edition: 1836).
Erdmann, B., Logik, I: Logische Elementarlehre [Logic, I · Elements of Logic], Halle a.S. 1892.
Papst, W., Gottlob Frege als Philosoph [Frege as Philosopher], Berlin 1932 (dissertation), 12. Like Osborn (p. 56f.), Mortan opposes Papst’s view on this point, in his Gottlob Freges philosophische Bedeutung [Frege’s Philosophic Importance], Jena 1954 (dissertation), 27f.
Frege’s view that there are not mutually opposed sentences of theoretical logic and sentences of normative logic but that the theoretical and normative are only different aspects of one and the same logical law, seems to us to be only superficially different from the view of Husserl, who, although he distinguishes normative and theoretical sciences, reduces the difference between normative and theoretical sentences to one of form (LU, I, 48, 49). If one puts for the judgement ‘p’ the ’p is true’ which both Frege and Husserl recognized as equivalent thereto, the second form becomes relevant to what Husserl called the ‘basic norm’ of normative logic, even if it is only the forms, ‘only he who judges “p” (and not “non p”) judges rightly’ and ‘you should judge “p”’, which are sufficiently rich.
Papst will have nothing to do with classifying Erdmann as psychologistic. But it is to be noted that her judgement is made on the basis of the second edition of Erdmann’s Logic, the year of publication of which (1907) is later than that of the first volume of Husserl’s Logische Untersuchungen. The latter also treats Erdmann as a representative of psychologistic logic (under the heading ‘Psychologism as Sceptical Relativism’, we find paragraph 40: ‘The Anthropologism in B. Erdmann’s Logic’).
Husserl, E., Philosophie der Arithmetik. Psychologische und logische Untersuchungen [Philosophy of Arithmetic. Psychological and Logical Investigations], First (and only) volume: Halle a.S. 1891.
Zeitschrift für Philosophie und philosophische Kritik (henceforward: ZPPK) 103 (1894), 313–332.
Cf. Osborn, A.D., Edmund Husserl and his Logical Investigations, 2nd ed., Cam- bridge, Mass., 1949 (1st ed.: 1934); Chapter 4: ‘Frege’s Attack on Husserl’. Also: Follesdal, D., Hissed and Frege. Ein Beitrag zur Beleuchtung der Entstehung der Phänomenologischen Philosophie [Husserl and Frege. A Contribution to the Clarification of the Origins of Phenomenological Philosophy], Oslo (Akad.) 1958, and: Farber, M., The Foundation of Phenomenology. Edmund Husserl and the Quest for a Rigorous Science of Philosophy, Cambridge, Mass., 1943, and the review thereof by Church in Journal of Symbolic Logic 9 (1944), 63–65. — It is noteworthy that none of the authors dealing with Frege’s influence on Husserl seemed to be aware of the existence of a (partially available) correspondence between them. It is also strange to see Follesdal’s efforts to establish that Husserl knew ‘all of Frege’s works up to 1893’. It is easy to check this in the Husserl Archives at Louvain, where many of Frege’s offprints are to be found with a dedication to Husserl. See Angelelli’s note in the second appendix of the new printing of the Begriffsschrift (Hildesheim and Darmstadt 1964, 117) on the same subject.
Gl., 3. Frege’s use here of the turn “to form the content of the sentence in the consciousness” openly contradicts Bierich’s view that Frege’s use of ‘content’ and ‘idea’ in the Foundations is a systematic separation of the objective from the subjective and of the logical from the psychological.
Mill, J. S., A System of Logic Ratiocinative and Inductive, Being a Connected View of the Principles of Evidence and the Methods of Scientific Investigation, London 1843.
On the terminology, see Mill, System, Book II, c. vi, beginning of the third paragraph. Rege, too, distinguishes numerical formulae which relate to specific numbers (e.g., “3 = 2 + 1”) from the general laws which are valid for all numbers (e.g., “a(b -f-c) = ab + ac”).
GI.,16f. Papst’s suggestion that Frege here is victim of “a confusion of empirical induc- tion with mathematical induction” (p. 14) is absurd in the face of the clarity of Frege’s text.
Locke, Essay, II, 16, paragraph 1.
Gl., 6. Cf. KRV, B 205–206.
Pg., A, 29; KRV, B 15.
The logistic point of view does not necessarily lead to the conclusion that mathematical sentences are ‘content-less’ (cf., e.g., Wittgenstein, Tractatus, 6.21: “The mathematical sentence expresses no thought”). The apparent difference disappears if one follows Carnap in distinguishing ‘logical value’ from ‘cognitive value’ (Log. Aufb., § 50, § 106f.).
GI., 101f. The existence of such clear statements makes it incomprehensible that people continue to claim that Frege wanted to prove “all of mathematics”, including geometry, to be purely logical or analytic. Even Natorp seems to share this view when he accuses Frege of wanting to keep all of mathematics free of pure intuition. And it is too much to say that Frege did not deal at all with the essence of geometry, even in his articles ‘On the Foundations of Geometry’ (Papst). A correct picture is provided by F. Waismann, Introduction to Mathematical Thinking. The Formation of Concepts in Modern Mathematics, New York 1951.
Nouveaux Essais, IV, § 10 (Erdm., 363).
“The doctrine that we can discover facts, detect the hidden processes of nature, by an artful manipulation of language, is so contrary to common sense, that a person must have made some advances in philosophy to believe it” (System, II, 6, § 2).
The so-called ‘principle of permanence’, introduced by Hankel under the name ‘principle of the permanence of the formal laws’. The validity of this principle — occasionally designated by Hankel himself as ‘metaphysical’ — is not supposed to be limited to mathematics. See: Hankel, H., Vorlesungen fiber die complexen Zahlen and ihre Functionen [Lectures on Complex Numbers and Their Functions], I, Leipzig 1867. On the principle of permanence, see: Stammler, G., Der Zahlbegriff seit Gauss. Eine erkenntnistheoretische Untersuchung [The Concept of Number since Gauss. An Epistemological Investigation], Halle a.S. 1926, 58ff.
Heine, E., Die Elemente der Functionenlehre’ [The Elements of the Theory of Functions], Journal für die reine and angewandte Mathematik (Crelle’s Journal) 74 (1872), 172–188. Quote from p. 173.
JDMV 15 (1906), 434, in almost literal agreement with Thomae, Elementare Theorie der analytischen Functionen einer complexen Veränderlichen [Elementary Theory of Analytic Functions of a Complex Variable], 2nd ed., Halle a.S. 1898. The formulation expresses the turning against Dedekind by means of a play on words in reference to his Was sind and was sollen die Zahlen? [’The Nature and Meaning of Numbers.’, in R. Dedekind, Essays on the Theory of Numbers, La Salle, Ill., 1901, 4th printing 1948], Braunschweig 1887 (9th ed.: 1961; reproduction).
Hilbert, D. ‘Neubegründung der Mathematik. Erste Mitteilung’ [A new Founding of Mathematics. A First Report], Abhandlungen aus dem Mathematischen Seminar der Hamburger Universität, 1 (1922), 157–177. Quoted from Hilbert, Ges. Abh., III, 165.
Ibid. However, it seems that Hilbert no longer keeps the two viewpoints apart in this work, in as far as the philosophical basis is concerned. He imports the philosophical standpoint of the first conception of formalism also into the second: “In taking this standpoint I am - in opposition to Frege and Dedekind - taking the sign itself as the object of the theory of numbers” (163). But it is clear that the signs are the objects of meta-mathematics and not of the theory of numbers. The difference between Frege and Hilbert here consists in the fact that Frege recognizes a content-ful arithmetic and a meta-mathematics (more exactly, meta-arithmetic), while Hilbert evidently only recognizes a meta-mathematics which is distinguished from arithmetic only in that it includes, in addition to the signs for numbers, sentences and proofs, etc., concerning the latter. One can certainly ask what has happened to the ‘higher’ level. It seems that the mystical saying “In the beginning is the sign” (loc. cit.) is not enough for the “firm philosophical position” which Hilbert thinks is his.
Loc. cit., 165.
The object of which is chess and not ‘practical playing of chess’ as Thomae once maintained in a long (and fruitless) discussion with Frege (Thomae, ‘Gedankenlose Denker. Eine Ferienplauderei’ [Thoughtless Thinkers: A Vacation-Talk], JDMV 15 (1906), 434–438; Frege, ‘Antwort auf die Ferienplauderei des Herrn Thomae’ [Answer to the Vacation-Talk of Mr. Thomae], Ibid., 586–590; Thomae, ’Erklärung’ [Declaration], Ibid., 590–592; Frege, ‘Die Unmöglichkeit der Thomaeschen formalen Arithmetik aufs Neue nachgewiesen’ [The Impossibility of the Formal Arithmetic of Thomae Newly Proved], JDMV 17 (1908), 52–55; Thomae, ’Bemerkung zum Aufsatze des Herrn Frege’ [Remark on the Article of Mr. Frege], Ibid., 56; Frege, ‘SchluBbemerkung’ [Concluding Remark], Ibid., 56.
Frege’s view was that meta-mathematical investigations neither replace nor facilitate the construction of content-ful arithmetic. Frege thought he had the means of constructing arithmetic in a non-contradictory way. This conviction was destroyed by the discovery of Russell’s antinomy, which also made demonstrations of non-contradictoriness with mathematical means a great necessity. The development of meta-mathematical procedures, e.g., Frege’s concern about the semantic proof of independence in his last article on the foundations of geometry, began at this time. - In their report, ‘The Scientific Heritage of Gottlob Frege’, at the International Congress for Scientific Philosophy (Paris 1935), Scholz and Bachmann revealed that “Löwenheim succeeded in 1909 - in a rich correspondence which was supposed to be published - in convincing Frege, on the basis of the Basic Laws of Arithmetic Vol. II paragraph 90, of the possibility of an unobjectionable construction of formal arithmetic” (loc. cit., 29. The report does not say if the correspondence existed in Münster at the time. We have thus far not been able to find a trace of this important material.
Cf. also Scholz, H., ‘Die Sonderstellung der Logik-Kalküle im Bereich der elementaren logistischen Kalkülforschung’ [The Special Place of Logical Calculi in the Domain of Elementary Logistic Investigation of Calculi], in Travaux du IXe Congrès International de Philosophie (Congrès Descartes), 6: Logique et Mathématiques, Paris 1937, 40–42.
JDMV 17 (1908), 55.
Here we must once again contradict Papst, who thinks that the modern form of formalism has rendered Frege’s objections irrelevant.
Gg., II 100. In fact, according to Carnap (Log. Synt., 254), “From the sentence, ‘in this room now there is Carl and Peter and nobody else’, we cannot derive the sentence, ‘in this room now there are two people’, with the help solely of the logical-mathematical calculus - as the formalists normally pretend: this can, however, be done with the help of the logistic system, i.e., on the basis of Frege’s definition for ‘2’.”
Natorp, P. Die logischen Grundlagen der exakten Wissenschaften [The Logical Foundations of the Exact Sciences]. Noted by P.E.B. Jourdain in Mind 20 (1911), 552–560 (review of Natorp’s book). Cf. J.J. Maxwell in Mind 21 (1912), 302–303, and Jourdain, ibid., 470–471.
Frege, G., Rechnungsmethoden, die sich auf eine Erweiterung des Grbßenbegrifes gründen [Methods of Calculation, Based on an Expansion of the Concept of Magnitude], Jena 1874 (Habilitationsschrift), p. 2. Cf. Gg., II, 70n (-71).
Scholz, H. and Schweitzer, H., Die sogenannten Definitionen durch Abstraktion. Eine Theorie der Definitionen durch Bildung von Gleichheitsverwandtschaften [The So-Called Definitions Through Abstraction. A Theory of Definitions Through Formation of Relationships of Equality], Leipzig 1935 (p. 102) (= volume 3 of the ‘Investigations on Logistics and the Foundations of the Exact Sciences’, edited by Scholz). A new printing of this useful work was an nounced in 1964 by the Wissenschaftliche Buchgesellschaft (Darmstadt), but dropped in 1966 because of the small number of subscriptions.
The two are to be distinguished. Contrary to Frege’s view in the Habilitationsschrift of 1874, a transposition provides no clear concept of magnitude, since one can, for example, pass from the congruence of two straight lines as well to the equality of their lengths as to the equality of surface of the square which can be constructed over them. But, even though one denies the definitional character of the transformations, the transitions themselves do not have to be dropped. A later point of view is to be found in Gg., II, 70n. (-71) as a remark on Peano’s use of the equal sign. When one says of bodies of equal volume that the volume of one is equal to that of the other, “the signs on both sides of the equal sign… are signs not for the bodies but for the volumes… or also [!] for the quantitative indices which result from measuring these volumes” (loc. cit.). Since volume and quantitative index are two different things, no clear concept of magnitude has been defined. As in the case of the directional length of vectors, there is only a ‘determination’: in the questionable transpositions ‘only new determinations (modi) are imposed on the objects in question“ (loc. cit.). This example antedates Padoa’s of 1904: cf. Scholz and Schweitzer, op. cit., 42f.
Beth, E.W., The Foundations of Mathematics, Amsterdam 1955, § 108 (pp. 356ff.).
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Thiel, C. (1968). Number and Concept. In: Sense and Reference in Frege’s Logic. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2981-9_3
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