Philosophy of Arithmetic

Centrone, Stefania

doi:10.1007/978-90-481-3246-1_1

Stefania Centrone¹⁰

Part of the book series: Synthese Library ((SYLI,volume 345))

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Abstract

The Philosophy of Arithmetic,¹ Husserl’s youthful work dedicated to a philosophical, or better, epistemological foundation of mathematics, shows the shift in his interests from more properly mathematical issues to those regarding the philosophy of mathematics. Husserl strives to understand and clarify what numbers and numerical relations are, a problem that he recasts in terms of the subjective origin² of the fundamental concepts of set theory and finite cardinal arithmetic. We will try to show that on the whole this work of Husserl’s does not deserve the criticism and ensuing neglect that it suffered from, ever since Frege published his well-known Review.³ Besides its hotly contested psychologism, we find ideas and conceptualizations that not only were original then, but are still interesting today, such as those concerning the autonomy of the formal-algorithmic aspect of abstract algebra and mathematics. Moreover, it is here that the Husserlian idea of a universal arithmetic receives its first formulation, the full elaboration of which will take at least ten more years, until his research on these topics reaches its stable form in 1901.⁴

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Notes

1.
Husserl, Philosophie der Arithmetik. Mit ergänzenden Texten (1890–1901), Huss XII, 1–283. Henceforth cited as PdA. English translation cited as PoA.
2.
Cp. Tieszen 1996: “Husserl thinks that arithmetical knowledge is originally built up in founding acts from basic, everyday intuitions in a way that reflects our a priori cognitive involvement” (304).
3.
Frege 1894. Cp., for example: Osborn 1949; Picker 1962, 289; Beth 1966, 353. Among the interpretations that give a positive re-evaluation of some aspects of the PdA: Farber 1943; Føllesdal 1958; Haddock 1973 (especially Ch. VI: however, his focus is mainly on Husserl’s logical theories in his later works, in particular in the Logical Investigations and Formal and Transcendental Logic); Miller 1982; Willard 1974, 97 f. & 1984; Tieszen 1990; Ortiz Hill 1994a & b.
4.
See Das Imaginäre in der Mathematik (December/January 1901/02), PdA, App. 430–451, PoA 409–432, and the new critical edition Schumann & Schumann 2001. Willard 1984 rightly stresses that Husserl shared “the general persuasion of mathematicians of the time that a rigorous development of higher analysis – arithmetica universalis in Newton’s sense – would have to emanate from elementary arithmetic alone.” However, few lines later he writes that “these further matters – intended is the foundation of the whole of mathematics on the elementary arithmetic – never received any detailed response from Husserl” (22). Though it is true that Husserl‘s “inquiry into the theory of number led him into general epistemological investigations that occupied him for the remainder of his life,” it should not be neglected that Husserl‘s Double Lecture on the Imaginary in Mathematics is a non-trivial attempt at dealing with the reduction of other number systems (the wholes, the rationals, the reals) and of their properties to the naturals and thus to give an answer to some of the problems left unsolved in PdA. From Miller 1982, too, one gets the impression that Husserl did not achieve “the philosophical project he had begun under the inspiration of Weierstrass” (9). Miller argues that “one can only conjecture about Husserl‘s reasoning here. Perhaps his view was simply this: Since even our most elementary number concept is largely ‘symbolic’, there is no intrinsic mystery regarding the introduction of other ‘symbolic’ concepts, such as those of negative, rational, irrational and imaginary numbers. The original or ‘authentic’ number concept has already been broadened to include numbers not actually given to us, so why should we not broaden it further? We are perfectly justified in taking this step …” What Miller does not seem to pay sufficient attention to is that in the Double Lecture Husserl’s philosophical problem is one of a conceptual kind: formally we can extend the natural number system by dropping certain restrictions to the executability of certain operations, but we cannot expand the concept at the basis of a specific numerical field (cp. our account of Husserl´s critique of Dedekind in ch. 3 below). So Husserl’s reasoning seems to be just the opposite of what Miller suggests.
5.
See PdA App. Abhandlung I, Zur Logik der Zeichen (Semiotik), 340–373; II, Begriff der allgemeinen Arithmetik, 374–379; III, Die Arithmetik als apriorische Wissenschaft, 380–384; V, Zum Begriff der Operation, 408–429; IX, Die Frage der Aufklärung des Begriffes der “natürlichen” Zahlen, als “gegebener”, “individuell bestimmter”, 489–492; X, Zur formalen Bestimmung einer Mannigfaltigkeit, 493–500. See Eley, Textkritischer Anhang, 521–562. A separate treatment has to be reserved for Abhandlung V, Zur Lehre vom Inbegriff, 385–407, see below Appendix 3.
6.
PoA Foreword 7; PdA Vorrede 7.
7.
“A part of the psychological investigations in the present volume was already included, almost word-for-word, in my Habilitationsschrift, from which a booklet four galley sheets in length, titled “On the Concept of Number: Psychological Analyses” was printed in the fall of 1887 but was never made available in bookstores” (PoA Foreword 8; PdA Vorrede 8). See Miller 1982, 11; Willard, 1984, 39; Ierna 2005: “Husserl’s Habilitationsschrift was never published and the work now known as Über den Begriff der Zahl is in fact just the first chapter of the Habilitationsschrift” (8).
8.
To be understood as ‘finite cardinal number’ or ‘natural number’. “E. Schröder introduced this term (natürliche Zahl) … it is apparently intended to mark the distinction of the cardinal numbers (Anzahlen) over against the other forms of number which come into play in arithmetic: the rational and irrational, the positive, negative and imaginary numbers. Moreover the term ‘Anzahl’ is not totally univocal, since it has sometimes been used to designate the concepts of numbers in series. … Nevertheless, we have thought it most suitable in this work to adhere to the older and almost universally customary use of language” (PdA 114 n., PoA 120 n.).
9.
“In the first of its two parts, the Volume I before us deals with the questions, chiefly psychological, involved in the analysis of the concepts multiplicity, unity, and number, insofar as they are given to us authentically (eigentlich) and not through indirect symbolizations. The second part then considers the symbolic representations of multiplicity and number, and attempts to show how the fact that we are almost totally limited to symbolic concepts of numbers determines the sense and objective of number arithmetic” (PdA 7; PoA 7).
10.
See Ortiz Hill 2002, 81.
11.
As is well known, one of the traits that distinguish the mathematics of the nineteenth century from the mathematics of the preceding century, is the birth of that movement, often called the ‘critical movement’, characterized by the need to provide rigorous concepts and proofs for vast branches of analysis and, later on, to reconsider the foundation of mathematics. The arithmetization of analysis initiated by Weierstrass concludes with the simultaneous publication in 1872 of the foundations of the system of real numbers by Georg Cantor (1845–1918) and Richard Dedekind (1831–1916). See Kline 1972, 947–978; Casari 1973, 1 ff.
12.
Cf. Schuhmann 1977, 7.
13.
Eley, Einleitung des Herausgebers, PdA xxi–xxii; K. Schuhmann 1977, 6–9; Miller 1982, 2–3; Ierna 2005, 5.
14.
“me totum abdidi in studia philosophica duce Francisco Brentano” (Schuhmann 1977, 13).
15.
Husserl in: Kraus 1919, 153–154.
16.
“It was my great teacher Weierstrass who, through his lectures on the theory of functions, aroused in me during my years as a student an interest in a radical grounding of mathematics. I acquired an understanding for his attempts to transform analysis – which was to such a very great extent a mixture of rational thinking and irrational instinct and knack – into a rational theory. His goal was to set out its original roots, its elementary concepts and axioms, on the basis of which the whole system of analysis could be constructed and deduced by a fully rigorous, thoroughly evident method” (Schuhmann, 7). Among the interpreters who see in Weierstrass the source of Husserl’s interest for “a radical grounding of mathematics” we find Willard 1984, 21–23; Miller 1982, 3 ff.; Ortiz Hill 1994a, 2 ff. & 1997b, 139 & 2004, 123–124; Ierna 2005, 3 ff.
17.
As Miller 1982 puts it “… Husserl’s philosophy of arithmetic took shape as an attempt to address the non-mathematical issues to which the program of arithmetizing analysis inevitably led”. (4) See also ibid., 19.
18.
Cp. Miller 1982, 4, 6, 8.
19.
Weierstrass, Einleitung in die Theorie der analytischen Funktionen (lecture of May 6, 1878), notes by Husserl, English transl. from Ierna 2006, 36 f. Cp. also Miller 1982, 3.
20.
Weierstrass 1966, 78.
21.
In his early writings Husserl “speaks of number, the concept of number, and the representation of number, in quite the same way” (Willard 1984, 26). (Hence we disagree with the following contention in Miller 1982, 22: “for the Husserl of PoA numbers are not presentations; they are rather concepts which are ‘contained’ in certain presentations”.) Willard also gives a justification for Husserl’s interchangeable use of the expressions “the concept of number” and “number”, and of “analysis of the concept of number” and “analysis of number”. This use has been largely followed in the secondary literature, and is adopted here as well. “Conceptual analysis is simultaneously an analysis of the essence of an object insofar as it is an object of the concept in question. … The literature of recent philosophy contains many passages where these terms are used interchangeably, and this also occurs in Husserl” (loc. cit.).
22.
PoA 15; PdA 14. It is noteworthy that in §82 of the Wissenschaftslehre Bolzano defines ‘collection (Inbegriff)’ as the comprehensive union (Zusammenfassung) of at least two arbitrary objects (concrete or abstract) – called parts of the collection – in a whole. Furthermore, he defines the concepts of set (Menge), sum (Summe), series (Reihe) as suitable specializations – through the specification of the kind of connection – of the concept of ‘comprehensive union of the parts in a whole’. Here is a rough summary: A collection is a sum if and only if (i) it contains the parts of its parts, and (ii) it is invariant with respect to the permutations of its parts. An aggregate is a set if and only if (i) it does not contain the parts of its parts, and (ii) it is invariant with respect to the permutations of its parts. Finally, a collection is a series if and only if (i) it does not contain the parts of its parts, (ii) it is not invariant with respect to the permutations of its parts, and (iii) it has an ordering relation. Cp. Bolzano, WL I, §§82–85; and the detailed critical reconstruction in Krickel 1995. Cp. also Simons 1997 & Simons (ms.); Behboud 1997. While Husserl does not say so explicitly, these conceptual distinctions are at work in PdA – in a very similar way as Bolzano intends them.
23.
Cp. Ortiz Hill 2002, 80.
24.
Cp. Casari 2000, 107.
25.
PoA 162; PdA 155.
26.
Cp. Casari 1991, 35–49. For a different account of Husserl’s characterization of the content of a concept see Willard 1984, 27.
27.
Willard 1974, 106.
28.
For this interpretation see Casari 1997a, 553–552. By contrast, Tieszen 1990 finds the paradigm for this procedure in Kant: “on the Kantian strategy human subjects are viewed as so constituted that their fundamental cognitive processes are isomorphic” (152). Referring to this process in Formale und Transzendental Logic Husserl says: “I acquired a determined view of the formal and a first comprehension of its sense already in my Philosophy of Arithmetic. Though immature … it was a first attempt to obtain clarity on the proper and original sense of the fundamental concepts of set- and number-theory, by falling back on the spontaneous activities of collecting and counting in which collections (“aggregates”, “sets”) and numbers are given… It can be recognized a priori that each time the form of this spontaneous activities remains the same, correlatively, the form of their constructions remains the same” (FTL 76, my emphasis).
29.
“Mathematicians have followed the principle of not regarding mathematical concepts as fully legitimized until they are well distinguished by means of rigorous definitions. But this principle, undoubtedly quite useful, has not infrequently and without justification been carried too far. In over-zealousness for a presumed rigor, attempts were also made to define concepts that, because of their elemental character, are neither capable of definition nor in need of it” (PoA 101; PdA 96 (my italics)). Simons (ms.) embraces Husserl’s position when he says: “It is impossible to define the general notion of a collection in terms of anything conceptually more simple, so let us simply give some examples.”
30.
“No one hesitates over whether or not we can speak of a multiplicity in the given case. This proves that the relevant concept, in spite of the difficulties in its analysis, is a completely rigorous one, and the range of its application precisely delimited. Therefore we can regard this extension as a given. …” (PoA 16; PdA 15). Cp. Ortiz Hill 2002, 81.
31.
For the distinction between proper and improper (symbolic) presentations, see §8 of this chapter.
32.
PoA 16; PdA 15. Cp. Ortiz Hill 2002, 81.
33.
PoA 19; PdA 18. Cp. Fine 1998, 600 (where, however, the characterization of the number concept through abstraction is discussed with reference to Cantor and Dedekind, and not to Husserl).
34.
Fine 1998, 602. Cp. Ortiz Hill 1997b & 2004; Simons (ms.).
35.
Cantor 1895, quoted after Fine 1998, 599. Cp. Ortiz Hill 1994b, 96.
36.
Dedekind 1888, 17. Cp. Fine 1998, 600.
37.
Fine 1998, 600. Fine also shows how, once the specific ontology underlying these characterizations of the concept of number by abstraction has been determined, it is possible to obtain an equally plausible conception – though this does not imply its correctness tout court – as the more familiar one of Frege-Russell on the one hand and of von Neumann-Zermelo on the other. Cp. also Ortiz Hill 1994b, 96 & 1997a, 67 & b, 141–143 & 2004, 109–114.
38.
I borrow here terminology from Lewis 1991, 3.
39.
PoA 18–22; PdA 17–21. Cp. Ortiz Hill 2002, 81–82.
40.
PoA 19; PdA 18.
41.
PoA 78; PdA 75.
42.
Dedekind 1888, III–IV.
43.
Willard 1984, 30 ff. provides a detailed account of this notion, in particular tracing it back to earlier sources like Lotze’s account of the psychological origins of representations of relations: “In a chapter to which Husserl makes explicit reference Lotze presents his general view of how relations come before consciousness in activities of ‘higher order’ … the activity of representing a relation is called ‘higher’ by him in a sense that precisely coincides with what Husserl later meant by the terms ‘founded’ and ‘higher order’ as applied to acts … of consciousness. It is … ‘higher in that determinate sense in which the higher has the lower for its necessary presupposition’ …” (30). The interest of this issue is mainly historical, so we shall not engage with it here.
44.
PoA 23–65; PdA 22–63.
45.
PoA 67–79; PdA 64–76.
46.
I borrow this terminology from Lewis 1981, 6.
47.
Terminology again from Lewis 1981, 6.
48.
PoA 77; PdA 74.
49.
“There is de facto so much in common between the primary relation and the psychical relation, as to their essential Moment (Hauptmoment), that I fail to see why a common term would not be justified here” (PoA 76, n. 11; PdA 73, n. 1).
50.
Cp. Ortiz Hill 2004, 126.
51.
In On the Concept of Number Husserl characterizes the abstractive process that yields the concepts of set and number the same way: “It is easy to characterize the abstraction which must be exercised upon a concretely given Vielheit in order to attain the number concepts under which it falls. One considers each of the particular objects merely insofar as it is a something or a one herewith fixing the collective combination; and in this manner there is obtained the corresponding general Vielheitsform, one and one … and one, with which a number is associated. In this process there is total abstraction form the specific characteristics of the particular objects … To abstract from something merely means to pay no special attention to it. Thus in our case at hand, no special interest is directed upon the particularities of the content in the separated individuals” (Husserl 1887, 116–117). Quoted after Ortiz Hill 2002, 82.
52.
PoA 83; PdA 79. Cp. Ortiz Hill 1994b, 96–98.
53.
Frege 1894, 181. Frege has developed this criticism already 1884 in his Grundlagen, §§29–44, and in his hilarious 1899 he directs it against a contribution to the Enzyclopädie der mathematischen Wissenschaften by a Gymnasialprofessor in Hamburg. On Frege against psychological abstraction see esp. Dummett 1991a, 49–50 & 1991b, Chapters 8, 12, and pp. 167–168 where psychological abstraction is carefully distinguished from logical abstraction as used, for example, in Frege’s contextual definition of the direction-operator. Cp. also Ortiz Hill 1994b, 96–98 & 2004, 114; Tieszen 1994, 318.
54.
Cp. Ortiz Hill 1992b, 98 & 1997, 66.
55.
Cf. Fine 1998, 604–605.
56.
In his Review Frege ironically concludes that, from what Husserl writes, to obtain the concept of number, one must exercise abstraction only up to a certain point, i.e. the point at which the members of the set no longer have any specific properties, but are nevertheless still distinct. “Number-abstraction simply has the wonderful and very fruitful property of making things absolutely the same as one another without altering them. Something like this is possible only in the psychological wash-tub” (Frege 1894, 188). Cp. Ortiz Hill, 1994b, 97.
57.
Cantor 1991, 365.
58.
I borrow here terminology from Simons 2007, 233.
59.
Loc. cit.
60.
Cp. Ortiz Hill 2002, 95–96 & Tieszen 1994, 320.
61.
PoA 125; PdA 119. Cp. Tieszen 1990, 152 & 1994, 320.
62.
Cf. PoA 125; PdA 119.
63.
Definitio prima, in Leibniz 1687.
64.
Grassmann 1844 represents an important moment in the history of the development of mathematics from a theory of magnitudes to a theory of forms. The substitution of the concept of magnitude with that of form in the definition of mathematics is not, however, to be considered an anticipation of the modern conception of mathematics as theory of structures, but, rather, in the sense of a conception of the objects of mathematics as forms of thought, objects of thought. On Grassmann also cp. Webb 1980, 44 ff. On Grassmann’s influence on Husserl see e.g. Hartimo 2007, 292 ff.
65.
Leibniz's and Frege's reasons for restricting the substitutivity claim are explained and discussed in Künne 2009, Ch. 1, §5. Here one also finds additional reasons that Leibniz and Frege did not yet take into account.
66.
PoA 102; PdA 97.
67.
Cp. Ortiz Hill 1994a, 5–11.
68.
PoA 103; PdA 98. Otto Stolz (1842–1905) was an influential Austrian mathematician (professor at the University of Innsbruck since 1872 until his death), with major interests in algebraic geometry and analysis. Husserl’s quotation is taken from Stolz 1885. Incidentally, Stolz is the first mathematician who wrote a paper on Bolzano: see Stolz 1881.
69.
“We can see that the presentation of ‘more’ and ‘less’ is already included in the definition of equality, while these themselves … cannot be conceived without presentations of equality. When we say that the bijection must not leave any element unconnected, then this is just a different way of saying that on neither side there can be an element more or less. Thus the circularity is obvious” (PoA 103–104; PdA 98–99).
70.
Cp. Tieszen 1990, 153.
71.
“If there is equality in the (internal or external) characteristics that at that moment constitute the focal point (Mittelpunkt) of our interest” (PoA 105; PdA 100).
72.
“What is simpler than comparing the two multiplicities with respect to their number by counting them both in the symbolic sense? Hence, we obtain not only the assurance of the equality (or inequality) of the numbers, but also these numbers themselves. That the mechanical process of counting, already for sets with a relatively low number, proceeds in an incomparably faster and more certain way than that apparently simple process of bijection, surely does not need a demonstration” (PoA 109–110; PdA 104–105).
73.
PoA 110; PdA 105.
74.
PoA 104; PdA 99.
75.
“Although it is not necessary to make the comparison through bijection, we are able to do it in all cases… In this consists, accordingly, the only meaningful and useful application of the ‘definition’” (PoA 110; PdA 105; my italics).
76.
According to the editor of PdA this study is from 1891. However, as we will show later, this dating is clearly mistaken.
77.
PoA 359–383; PdA, App. 385–407.
78.
PoA 374; PdA, App. 399.
79.
Among which Husserl explicitly mentions Stolz 1885.
80.
Cp. Tieszen 1994, 319.
81.
Here we still have the assumption that A and B would be finite sets. Moreover, Husserl’s tacit assumption is that all sets under consideration contain at least two distinct elements. On this latter point see Appendix 3.
82.
Obviously, there is here the tacit assumption that some (equivalently: every) set M in K has at least three elements.
83.
On the fact that Husserl starts counting from the number 2, see below. Furthermore, note the similarities with Schröder’s idea that the number represents (not means) the counted objects: “To obtain a sign, capable of expressing how many of those unities are present, we direct our attention step by step to each of the units under consideration, and we represent them with a small bar (Strich): 1 (i.e. with a numeral 1, a one). Then we put them in line, and to avoid conflating them into a number 111, they are connected with the + sign. Hence we obtain a number of the type: 1 + 1 + 1 + 1 + 1” (Schröder 1873). Cp. the following passage from the Philosophy of Arithmetic: “We take the sets 11, 111, 1111, … , obtained by repeating the bar (Strich) ‘1’ or the sound complex ‘one’, or (to avoid any confusion with certain composite signs of the decimal number system) the sets 1 + 1, 1 + 1 + 1, 1 + 1 + 1 + 1, … , as representatives of the classes and name them according to the series 2, 3, 4 …” (PoA 110; PdA 105).
84.
PoA 119; PdA 113.
85.
Loc. cit.
86.
PoA 122; PdA 116.
87.
PoA 122; PdA 116. Cp. Tieszen 1990, 156–157.
88.
PoA 123; PdA 117.
89.
“Nevertheless, this mathematical genius in no way belongs to the tendency the criticized above, as is apparent from all his later publications.” PoA 121, note 3; PdA 115, note 2.
90.
Cantor 1887–1888. As Ortiz Hill 1994a states, “enough kinship is apparent between Husserl’s and Cantor’s work to have prompted scholars to speak of the influence Husserl may have had on Cantor’s work”. Indeed such an influence has been suspected e.g. by Cavaillès 1962 and Casari 1991. We do not agree, for Cantor’s letter to Lasswitz dates from 1884, whereas Husserl’s first explicit definition of sets and cardinal numbers by Cantorian abstraction dates from 1887. One should rather assume an influence in the opposite direction.
91.
Cp. Ortiz Hill 1994b, 103 & 1997b, 143 & 2004, 112–114.
92.
On Kerry cp. Picardi 1994.
93.
Extensionally speaking. Actually, for Frege, numbers are extensions of second-level concepts, i.e. if F is a concept, the number of F is the extension of the concept ‘concept equinumerous to F.’
94.
“A foundation of arithmetic on a series of formal definitions out of which all the theorems of that science could be deduced purely syllogistically is Frege’s ideal” (PoA 124; PdA 118). Cp. Ortiz Hill 1994b, 101–104; Tieszen 1994, 318.
95.
Frege, Grundlagen der Arithmetik, §27. By way of an example, in the first volume of the Grundgesetze der Arithmetik (1893), criticizing the fact that Dedekind’s notion of ‘system’ seems to admit a psychological foundation, even though Dedekind was not interested to give such a foundation, Frege writes: “This holds especially of what mathematicians like to call a ‘set.’ Dedekind [Was sind und was sollen die Zahlen] uses the word ‘system’ with much the same intention. But despite the explanation that appeared in my Foundations four years earlier, he lacks any clear insight into the heart of the matter, though he sometimes comes close to it, as when he says (p. 2): ‘Such a system S … is completely determined if, for every thing, it is determined whether it is an element of S or not. The system S is therefore the same (dasselbe) as the system T, in symbols S = T, if every element of S is also an element of T, and every element of T is also an element of S.’ In other passages, however, he goes astray, e.g., in the following (1–2): ‘it very often happens that different things a, b, c … regarded for some reason from a common point of view, are put together (zusammengestellt) in the mind, and it is then said that they form a system S.’ A hint of the truth is indeed contained in talk of the ‘common point of view’; but ‘regarding’, ‘putting together in the mind’ is no objective characteristic. I ask: in whose mind? If they are put together in one mind, but not in another, do they then form a system? What may be put together in my mind must certainly be in my mind. Do things outside me, then, not form systems? Is the system a subjective construction in the individual mind? Is the constellation Orion therefore a system? And what are its elements? The stars, the molecules or the atoms?” (Frege 1893, 1–4, transl. Beaney 1997, 208–211).
96.
Op. cit., 218.
97.
“Surely no extensive discussion is necessary to show why I cannot share this view, especially since all the investigations which we have carried out up to this point present nothing but arguments in refutation of it” (PoA 124; PdA 119).
98.
Cp. Simons 2007, 229 ff.
99.
Frege 1884, §63; Beaney 1997, 110.
100.
Besides ‘triangular form.’
101.
Cp. Simons 2007: “As an inspection of Hume shows, it is not close to any principle actually formulated by Hume, and the name follows a somewhat misleading historical footnote to Hume’s Treatise in Grundlagen §63. In fact the principle … is clearly stated by Cantor in 1895” (246).
102.
Neo-Fregeans like Hale & Wright 2001 try to exorcize the spectre of Russell’s paradox by adjoining Hume’s principle to second-order logic and showing that the result is a consistent system in which all the fundamental laws of arithmetic are derivable as theorems.
103.
PoA 128; PdA 122. Cp. Tieszen 1990, 154; Ortiz Hill 1994a, 5 & b, 100.
104.
PoA 128, n. 14; PdA 122, n. 1. Cp. Ortiz Hill 2002, 96.
105.
Op. cit., 306. Perhaps it is useful to remember that in §2 of Concept and Object, while answering a series of objections by Benno Kerry, Frege writes – among other things – that Kerry, falsely, thinks that in the Grundlagen the concepts ‘concept’ and ‘extension of a concept’ had been identified, and that interpreting them thus would be to misunderstand the content of his book. He then explicitly states: “I merely expressed my view that, in the expression ‘the [natural] number that belongs to the concept F is the extension of the concept equinumerous to the concept F’, the words ‘extension of the concept’ could be replaced by ‘concept’.” (Frege 1892, 199; transl. Beaney 187). Cp. Tieszen 1990, 154.
106.
Frege 1884, §56.
107.
PoA 171; PdA 163.
108.
“The number is univocally determined when the collection upon which we exercise that abstraction process is determined” (PoA 172; PdA 163).
109.
PoA 172; PdA 164.
110.
PoA 177; PdA 168.
111.
“Unit (Einheit) in contrast to multiplicity is not the same as unit in the multiplicity. Along with the concept of the multiplicity (or number) the concept of the unit is inseparably given. But in no way this is true of the concept of the number one. The latter is only a later result of technical developments (ein späteres Kunstprodukt)” (PoA 141; PdA 134).
112.
“It will often be necessary for us to lose ourselves in what seem to be linguistic investigations concerning the meanings of the terms in order to put an end to obscurities and misinterpretations of the concepts that interest us. We find ourselves in such a position also with the question that is to employ us now: namely, that about the relationship between the concepts or terms one and unit” (PoA 143; PdA 136).
113.
Frege, Letter to Edmund Husserl, 24 May 1891.
114.
“No-many, or no multiplicity [keine Vielheit], is not a special case of many. One object is not a collectivity of objects. Therefore the assertion that there is one thing here is no assertion of number. And likewise, no object is not a collectivity, and therefore the assertion that there is no thing here is no assertion of number” ( PoA 138; PdA 131).
115.
PoA 141; PdA 133.
116.
“Certainly the decimal number system … would be unthinkable without this momentous expansion of the concept of number” (PoA 139; PdA 132).
117.
PoA 140; PdA 133.
118.
PoA 135; PdA 128.
119.
PoA 135; PdA 128.
120.
In the Introduction to the Philosophy of Arithmetic, Husserl indeed had postponed such a justification to the end of the discussion about the concepts of unity, multiplicity and number, being convinced that the subsequent considerations about the constitution of such concepts would be valid independently of this choice.
121.
Cp. Frege’s criticism of Helmoltz in 1903, 139–140.
122.
PoA 191; PdA 181.
123.
Webb 1980, 44.
124.
I borrow here terminology from Cantor 1991, 365.
125.
PoA 195–196; PdA 185.
126.
PoA 196; PdA 186.
127.
e.g., 4·3 = 3 + 3 + 3 + 3 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.
128.
Miller 1982, 9 observes that “… one of the eight theses [Husserl] chose to defend in a formal disputation in the summer of 1887 was the following ‘in the authentic sense one can barely count beyond three’ (PoA 357, PdA 339)”. Cp. Tieszen 2004, 32.
129.
Cp. Weyl’s remarkable summary of Husserl’s Philosophy of Arithmetic in his Habilitationsvortrag in Göttingen (Weyl 1910, 302).
130.
PoA 201–202; PdA 193. Cp. Tieszen 1996, 304: “It is a basic epistemological fact, for example, that we are finite beings. Elementary finitary processes such as counting objects in everyday experience, collecting them, or correlating them one-to-one, are clear and familiar to nearly everyone. It is not necessary to know any set theory in order to be able to do these things. And recall Poincaré’s characterization of mathematical induction: it is ‘only the affirmation of the power of the mind which know it can conceive of the indefinite repetition of the same act, once that act is possible.’ On Husserl’s view, number theory is founded on processes of this type.”
131.
“To [Brentano] I owe the deeper understanding of the vast significance of inauthentic presentations for our whole mental life; this is something which, so far as I can see, no one before him had grasped” (PoA 205; PdA 193). Cp. Miller 1982, 9.
132.
Leibniz 1684.
133.
LU II, §14a, 141; LI 367.
134.
LU VI, §45, 144 ; LI 785–786.
135.
PoA 205; PdA 194.
136.
A general representation cannot represent a particular object; e.g., the representation ‘man’ cannot be considered an unequivocal sign for a specific individual, e.g., Peter. In order to transform it into such a sign, we have to add to the general representation some distinctive features (Merkmale) that unequivocally identify that unique individual.
137.
Cp. Ortiz Hill 1997b 147 & 2002, 82–83 & 84; Tieszen 1994, 332.
138.
Cp. Tieszen 1990, 151.
139.
PoA 232; PdA 220.
140.
Tieszen 2001, 238.
141.
Ortiz Hill 1997b, 147 & 2002, 83.
142.
Cp. Ortiz Hill 2002, 83.
143.
PoA 242; PdA 229.
144.
Compare Tieszen 2000, 256.
145.
PoA 243; PdA 230.
146.
PoA 244; PdA 230–231.
147.
PoA 244; PdA 231.
148.
PoA 246; PdA 233.
149.
PoA 247–248; PdA 233–234.
150.
Webb 1980 observes that “Husserl’s theory of calculation has some of the flavour of Church’s calculi of λ-conversion: ‘systematic numbers’ (e.g. Arabic numerals) result from a series of rule governed ‘reductions’ of ‘unsystematic numbers’ (terms compounded out of numerals with function symbols), also called ‘symbolische Bildungen’” (25).
151.
Cp. Hartimo 2007, 288.
152.
PoA 251–252; PdA 237–238.
153.
PoA 251 ff.; PdA 237 ff.
154.
Casari 2000, 105.
155.
Casari 1973, 8–9.
156.
See Cantini 1979, 41 ff.
157.
Boole 1847. Cp. also Webb 1980, 79; Hartimo, 2007, 285 ff.
158.
PoA 273; PdA 258.
159.
See e.g. Tieszen 1996, 312–313 & 2000, 9.
160.
Cp. Centrone 2005.
161.
“Most researchers – guided by the general prejudice that every scientific methodology operates with the respective intended concepts – have also held the arithmetical operations to be abstract-conceptual, in spite of all clear indications” (PoA 272; PdA 257).
162.
Indeed, in both earlier definitions, calculating was defined as a deriving of numbers from numbers, and by numbers were meant the numerical concepts.
163.
PoA 273; PdA 258 (italics in the original). Cp. Hartimo, 2007, 289 f.
164.
As Webb 1980 puts it: “Husserl … attempted a complete development of the algorithmic conception of arithmetic, which required “die logische Untersuchung des arithmetischen Algorithmus”. The notion of algorithm, Husserl felt, had to be bound up with that of a ‘mechanical process’” (24–25).
165.
PoA 273; PdA 258 (italics in the original). Husserl does not fail to stress how important a good choice of the system of signs is, in terms of efficiency, for all three of these phases (encoding – calculation – decoding) of the solution of a problem.
166.
PoA 273; PdA 258 (italics in the original).
167.
“If we loose the number signs from their conceptual correlates, and work out, totally unconcerned with conceptual application, the technical methods which the sign system permits, then we have extracted the pure calculational mechanism that underlies arithmetic and constitutes the technical aspect of its methodology” (PoA 274; PdA 259).
168.
“To each non-systematic number there corresponds a univocally determinate systematic number that is equal to it, i.e. one which symbolizes the same authentic [i.e. proper] number concept” (PoA 276; PdA 261). Cp. Hartimo 2007, 290.
169.
PoA 277; PdA 262.
170.
Cp. Webb 1980 25.
171.
For a precise definition of the notion of partial numerical function and for a formal reconstruction of Husserl’s attempt see Appendix 1 below.
172.
PoA 282; PdA 267.
173.
The method of finding the systematic number corresponding to the construction a:b consists in reducing every division to a series of elementary divisions. Nevertheless, while for calculating addition and multiplication it is sufficient to use a table of all elementary additions and multiplications between an i and a j belonging to {0, 1, … , X}, for calculating division we need a table of all elementary divisions of the form a:c where c is a number between 1 and X and a is a “two-digit” number with respect to the basis X.
174.
PoA 285; PdA 269. As Webb 1980 rightly says “especially remarkable is [Husserl’s] suggestion that the question whether an arbitrary ‘Rechnungsaufgabe’ is always defined (Bedeutung haben) for any number will require a deep analysis” (25).
175.
PoA 283; PdA 268.
176.
Presburger 1929.
177.
Church 1936.
178.
PoA 385–408; PdA 408–429.
179.
PoA 407–408 (my emphasis); PdA 429.
180.
Cp. Webb 1980, 25.
181.
PoA 292; PdA 276–277.
182.
Webb 1980 points out that H. Grassman “was the first mathematician both to approach arithmetic axiomatically and to employ recursive definition for the basic arithmetic operations. … Recursive definitions for the basic arithmetic operations began to appear frequently in the literature after Grassman…” (44).
183.
For this interpretation see also Casari 1991.
184.
See Appendix 1 for a precise definition. Intuitively, a primitive recursive definition of a unary function f (the n-ary case, with n > 1, being analogous) consists of (i) an explicit definition of the value of f for the argument 0, and (ii) the definition of the value of f for an arbitrary argument distinct from 0, i.e. for an argument of the form x + 1, in terms of the value that f assumes for the argument x.
185.
It is reasonable to maintain that Husserl’s theorizations here do not go beyond ‘Ackermann’s succession’. There is no evidence at all that Husserl had realized the gap, both from the conceptual point of view and in terms of arithmetical complexity, between the succession f₀, f₁, f₂, … and the so-called ‘Ackermann’s function’, i.e. the ternary function A defined by:
A(n,m,r) = f_n(m,r)
A is clearly a computable function, due to the fact that each function in Ackermann’s succession is computable; one can prove however (as Ackermann did) that A is not primitive recursive. A recursive definition of A requires essentially a nested double induction, which does not fit into the primitive recursion schema.
186.
Webb 1980 recalls that also Dedekind used “recursion to provide a precise mathematical basis for the systematic introduction of new arithmetical operations”. Furthermore, concerning Ackermann’s sequence Webb rightly stresses: “Of course, none of these f_k, beyond f₂ [notice that in Webb’s definition f₀ is taken to be the successor function] have any use in the market place, as already by f₄, called ‘elevation’ by Husserl, we encounter a growth rate so steep as to make its calculation a practical impossibility for all but the smallest arguments. By f₅ even our standard notation conventions begin to buckle, while for f₆ we presumably will have to remain for ever content with a recursion formula [of the kind exemplified by the scheme] as our only feasible description of it” (51–52).
187.
Read: each operation.
188.
PoA 293 (my emphasis); PdA 277.
189.
PoA 293; PdA 277–278.
190.
PoA 293; PdA 277–278.
191.
PoA 294 (my emphasis); PdA 278.
192.
PoA 297; PdA 281.
193.
PoA 297; PdA 281.
194.
As regards systems of equations, we observe that it is always possible to associate with every system of equations an appropriate equation, equivalent to the first in the sense that a number x solves each equation of the system if and only if it solves the equation associated with the system. The case in which a number is determined by a system of equations, rather than by one, is therefore, as Husserl stresses, a case that “in spite of the greater degree of complication, offers nothing essentially new from the logical perspective” (PoA 297; PdA 281).
195.
But we can presume that Husserl also has in mind inversion, for example, of the direct operation of tetration (or super-exponentiation), i.e. the one that immediately follows the power function in Ackermann’s succession.
196.
PoA 297; PdA 281–282.
197.
That is, the ‘conditions’ of the ‘second sequence’ above.
198.
PoA., 298; PdA 282.
199.
PoA 299; PdA 283 (italics in the original).
200.
This question was originally raised in Casari 1991: “It would be really worth to further investigate Husserl’s attempt to dominate the totality of all conceivable arithmetical operations, as Husserl calls it. For, we also believe not to get wrong by saying that this is, most likely, the first characterization of the class of functions nowadays known as the class of partial recursive functions” (46). Indeed, our investigations concerning this issue, which finally led to the result presented here, originate from Casari’s insightful suggestions to attempt at a mathematical reconstruction of Husserl’s intuition.
201.
Equivalently, the closure of the class Pμ under unbounded minimization may be formulated as follows: given a k+1-ary relation R which is recursive (that is, such that its associated characteristic function χ_R is in Pμ), the k-ary function f such that f(x₁,…, x_k) ≈ μy(R(x₁,…, x_k, y)) belongs to Pμ.
202.
Robinson 1950. For a clear presentation, see also Yasuhara 1971, 110–117.
203.
A total function f is primitive recursive if it can be obtained from the initial functions Z, s and p_n,i by means of the operators S e R. In the expression defining J, note that the numerator is always an even number, so that dividing by 2 makes sense.
204.
Notice that – the assumption that f is injective in both places separately, like the exponential function, being too restrictive – we have replaced the operator ι, ‘the unique … such that …’, with the operator μ, ‘the minimum … such that …’. We will come back later on this rather delicate point.
205.
The reason being the following: H, modified as indicated, turns out to be still closed under minimization and to contain a β-function, and these features are sufficient to express primitive recursion.
206.
Note that, in case there is more than one z such that P(z) holds, ιzP(z) turns out to be undefined whereas μzP(z) is defined. For instance, let h(0, x) = 3 and h(1, x) = 3: then ιz(h(z, x) = 3) is undefined and μz(h(z, x) = 3) is defined and equal to 0.
207.
While it is clear that inversion I, once it is defined with ι in place of μ and not restricted to injective functions, doesn’t preserve computability.
208.
PdA App. 408–429; PoA 385–408. Cp. Centrone 2005 for the following discussion.
209.
PdA App. Textkritische Anmerkungen 538.
210.
Loc. cit.
211.
PdA App. 422; PoA 400.
212.
PdA App. 427; PoA 405.
213.
PR VII, PRe 42.
214.
PdA App. 408; PoA 385.
215.
Loc. cit.
216.
PdA App. 409; PoA 386.
217.
PdA App. 408; PoA 385.
218.
PdA App. 427; PoA 404.
219.
PdA App. 420; PoA 397–398.
220.
PdA App. 423; PoA 401.
221.
PdA App. 425; PoA 402.
222.
PdA App. 426; PoA 404.
223.
PdA App. 427; PoA 404.
224.
PdA App. 425; PoA 403.
225.
PdA App. 426; PoA 404.
226.
PdA App. 420; PoA 397.
227.
PdA App. 419; PoA 396.
228.
Faute de mieux I use ‘cognitions’ as my rendering of ‘Erkenntnisse (propositions that have become contents of knowledge)’.
229.
PdA App. 408; PoA 385.
230.
PdA App. 411; PoA 388.
231.
PdA App. 413; PoA 390.
232.
As to Husserl’s intellectual heritage in this respect, see the following section.
233.
Loc. cit.
234.
PdA App. 419; PoA 396.
235.
PdA App. 418; PoA 395.
236.
Loc. cit.
237.
PdA App. 414; PoA 391.
238.
PdA App. 415; PoA 392.
239.
Loc. cit.
240.
PdA App. 415–416; PoA 393.
241.
PdA App. 416–417; PoA 394.
242.
PdA App. 417; PoA 394.
243.
PdA App. 417; PoA 395.
244.
PdA App. 419; PoA 396.
245.
Loc. cit.
246.
PdA App. 419; PoA 397.
247.
PdA App. 416 PoA 393.
248.
Webb 1980, xii.
249.
Loc. cit.
250.
See above §1.2 (question 2).
251.
I borrow here terminology from Remnant & Bennett 1996, 10.
252.
Leibniz 1704, book IV, ch. viii, §2.
253.
This formulation is taken from Künne 2009, 253.
254.
Leibniz 1704, book IV, ch. viii, §4. Kant calls them tautologisch, in Bolzano they are called identisch oder tautologisch. Cp. Künne 2009, 253, fn. 52 & 53.
255.
Leibniz 1704, book IV, ch. viii, §5.
256.
Leibniz 1704, book IV, ch. ii, §1.
257.
Loc. cit.
258.
Loc. cit.
259.
Leibniz acknowledges that the Parisian philosopher Pierre de la Ramée (Petrus Ramus) was already aware of (b).
260.
PdA App. 412–413; PoA 389–390 (my emphasis).
261.
PoA 273; PdA 258.
262.
Loc. cit.
263.
Boole 1854.
264.
LV 96, App. 305–328.
265.
LV 96, App. 305.
266.
LV 96, App. 306.
267.
LV 96, App. 307.
268.
LV 96, App. 308.
269.
LV 96, App. 312.
270.
LV 96, App. 309.
271.
LV 96, App. 314.
272.
LV 96, App. 310.
273.
Frege 1990, 103–111.
274.
LV 96, App. 311.
275.
PdA App. 414; PoA 391.
276.
LV 96, App. 322–323.
277.
PdA App. 385–407; PoA 359–383; Textkritische Anmerkungen, 530–533.
278.
“Bernstein has demonstrated … a sufficient condition … On this point we still must have an exchange with Bernstein.” PdA App. 394; PoA 369.
279.
Felix Bernstein (Halle 1878 – Zürich 1956) studied with Cantor in Halle, then with Hilbert and Klein at Göttingen, where he obtained his doctorate with a dissertation entitled Untersuchungen aus der Mengenlehre. After his habilitation (Halle 1903), Bernstein taught at Göttingen from 1907 to 1932. After moving to the United States he taught for sixteen years at various universities (Columbia University, New York University, Syracuse University), and in 1949 returned to Göttingen. According to various biographic notes, Bernstein already started to follow Cantor’s seminars at the university of Halle while still attending the gymnasium; on the other hand, a precise indication of the year in which contact with Cantor began is unavailable (this information is not to be found in the most complete biography: Frewer 1981). However, it is reasonable to suppose that it did not begin before 1894–95 (Bernstein obtained his Abitur in 1896).
280.
“Read to Cantor when he told me of a treatise of Schröder’s for the Leopoldina” (PdA App. 399, n. 1; PoA 374, n. 9). The reference is to Schröder 1898.
281.
Loc. cit.
282.
Cp. Ortiz Hill 1994a, 3 & b, 96 & 1997b, 137 ff. & 2004, 110–114.
283.
In one of the many possible equivalent formulations: For every family F of non-empty and disjoint sets, there is at least one set X having one and only one element in common with each of the sets belonging to F.
284.
See Tarski 1925.
285.
PdA App. 385; PoA 359.
286.
PdA App. 385 f.; PoA 359 f.
287.
PdA App. 385; PoA 359.
288.
Bolzano 1975. Cp. WL I, §§84–86.
289.
“By a ‘set (Menge)’ we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and separate objects m of our intuition or our thought. These objects are called the ‘elements’ of M” (Cantor 1895, 481; transl. 1955, 85). “It very frequently happens that different things a, b, c, … for some reason can be considered from a common point of view, can be associated in the mind, and we say that they form a system S” (Dedekind 1888; 1–2).
290.
This condition on the concept of collection is taken from § 82 of Bolzano’s Wissenschaftslehre.
291.
PdA App. 387; PoA 361.
292.
PdA App. 386 ff.; PoA 360 ff.
293.
Husserl rewrites axiom [HU.3] so: (I + I′) – I′ ≡ I, (I – I′) + I′ ≡ I. But he observes that if I is a collection and I′ is either a single object or a collection, then “I + I′” is meaningful if and only if I′ does not belong to I, resp. if and only if I and I′ are disjoint. Analogously, “I – I′” is meaningful only when I′ is a proper part of I.
294.
Husserl alludes at this point (PdA. App. 389 n.) to Bolzano. Cp. now Bolzano, BGA 2A, 8, 15ff.
295.
Dedekind 1888.
296.
“With respect to this process through which we free the elements from every other content (abstraction), we can correctly affirm that numbers are a free creation of the human mind” (Dedekind, op. cit.).
297.
Cantor 1895, 481; transl. 1955, 86.
298.
PdA App. 394; PoA 369. Interpretation: supposing that the class of all the permutations would always be a set leads to a contradiction. In other words, Husserl shows here that he is aware of the need to distinguish between proper multiplicities (sets) and multiplicities that are “inconsistent” or “too extended” to be considered objects. Keep in mind that Cantor, already in 1895, identifies the paradox that Burali-Forti will make known in 1897, writing about it to Hilbert (1896) and Dedekind. In particular, in the famous letter to Dedekind of 1899, Cantor indicates the origin of certain difficulties that had been found in set-theory (specifically, the Burali-Forti paradox) due to the missing distinction between “absolutely infinite” or “inconsistent” multiplicities, for which “the assumption that all of its elements ‘are together’ leads to a contradiction, so that it is impossible to conceive of the multiplicity as a unity, as ‘one finished thing’,” and consistent multiplicities or sets for which the totality of the elements “can be thought without contradiction as ‘being together’, so that their collection into ‘one thing’ is possible” (Letter to Dedekind, 28 July 1899, in Cantor 1932, 443; transl. in van Heijenoort 1967, 114).
299.
Cantor asserts the property of trichotomy, without proving it, already in his 1878. In the first of his 1895 papers he explicitly acknowledges the need for (and the difficulties of) a proof of this property (see next footnote). In the letter to Dedekind of 28 July 1899, Cantor sketches a “proof” of the theorem of trichotomy (or rather, of the theorem that every cardinal is an aleph, from which follows as corollary the trichotomy), which, however – as Zermelo will observe – is not convincing as it appeals, tacitly and intuitively, to some sort of “principle of choice”. Indeed, the first correct proof of the theorem, in the context of the explication of the axiom of choice from which it depends, is given by Zermelo in 1904.
300.
More correctly, this theorem should be called the theorem of Dedekind–Schröder–Bernstein. Cantor, in fact, obtained it as a corollary of the (never proved) theorem of comparability or trichotomy, until the young Bernstein – in a seminar held around Easter 1897 – gave a demonstration of it, obtained in the previous year, completely independently from the comparability (see the letter to Dedekind of 30 August 1899). Bernstein’s proof was relayed by Cantor to E. Borel, who published it in his Leçons of 1898. In 1896, independently, also Schröder had tried to give a proof of the theorem of equivalence, but (as Korselt observed in 1911) this attempt contains an error. Finally, also Dedekind had found, already in 1887, a proof of the theorem (in the equivalent form: if A ⊆ B ⊆ C and A ≅ C then B ≅ C): Dedekind told Cantor in his letter of 29 August 1899, but the proof was published only in his 1931.
301.
This is further evidence for our contention that the study under consideration could not have been written in 1891.
302.
Cantor 1895, 484; transl. 90.
303.
Tieszen 1990, 153 only alludes to this point without elaborating it any further.
304.
“A system S is said to be infinite when it is similar to a proper part of itself; in the contrary case S is said to be a finite system” (Dedekind 1888, 18, def. n. 64; transl. 63). In Dedekind’s terminology, a representation φ of a system (or set) S in itself (i.e. a function from S to S) is called similar when it is injective, i.e. when to different elements a, b of system S correspond different φ-images.
305.
That is, equivalent.
306.
PdA App. 395; PoA 369 ff.
307.
PdA App. 404; PoA 379 ff.
308.
About the ordering in a series, Husserl makes a remark that might seem marginal, but that is of great importance from a philosophical point of view: the ordering in a series is not something extrinsical to numerical concepts, but it is a priori and intrinsical to the very nature of these concepts. This constitutes “the fundamental fact of arithmetic”. Fields of knowledge for which the order of the elements is analogous to that of the numerical field give rise to theories that are “equiform” or potentially identical to arithmetic. The relations and connections among the elements of these fields can be interpreted arithmetically. See PdA App. 398–399; PoA 373.
309.
PdA App. 397; PoA 371 ff.
310.
This is the approach already delineated informally in the Begriffschrift (1879) and pursued formally in the Grundgesetze (1893). As is well-known, the system of the Grundgesetze is inconsistent, and hence from this point of view the provability of a theorem of infinity, as that of any other proposition, is obvious. But what is interesting is that the proof – as we find it in the Grundgesetze – of the so-called ‘Theorem of Frege’ (i.e. of the fact that the system of natural numbers as defined by Frege verifies Peano’s axioms) can be reproduced in a consistent system obtained by substituting Hume’s Principle (that Frege proves using his Basic Law V, the one responsible for the contradiction) for Basic Law V; cp. Heck 1993; Hale & Wright 2001.
311.
Cp. Simons 2007, 231.
312.
Also see PdA 109; PoA 114.
313.
Here Husserl seems to answer one of the objections that Frege made in his Review of the Philosophy of Arithmetic, i.e. that if one maintains that the fastest way to compare the cardinal numbers of two sets is counting the elements of each set, and furthermore, if one maintains that the actual reason for which two sets turn out to be in bijection is that they have the same number and not the other way around, one commits the error of neglecting that “counting the elements” means precisely to put them in bijection with a segment of the series of naturals.
314.
Kline 1972, 947–978.
315.
Mulligan 2004 (unpublished).
316.
See §15.

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Centrone, S. (2009). Philosophy of Arithmetic. In: Logic and Philosophy of Mathematics in the Early Husserl. Synthese Library, vol 345. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3246-1_1

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