Abstract
The analysis of the concept of number is indeed the ultimate goal of Frege’s scientific program, already stated in the Vorwort of the Begriffsschrift. We begin by considering Frege’s negative statements about numbers; numbers are neither external things nor subjective entities. This is an opportunity to see an interesting limitation in Frege’s system of concepts: he is not clear about subjective. This is why the dichotomy mentioned above (external thing-subjective) is not capable of covering all past philosophies of number. Something like transcendental subjectivity should be taken into account (10.1).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
GRL, §§ 21–25.
Concepts are for the author of GRL etwas Unsinnliches (GRL, p. 31 below; cf. also GRL, §§ 48, 87, 92; for later works: GRG II, p. 110).
This argument is given in § 24 and especially on page 31. Cf. a rather similar argument in Klemke [1], p. 511: “In other words, an object has properties. Hence, if the object has the right to be called an existent, then surely its properties are existent, too”.
That Frege is thinking of individual accidents, is clear from the following lines: “Wenn wir eine blaue Fläche sehen, so haben wir einen eigenthümlichen Eindruck, der dem Worte “blau” entspricht; und diesen erkennen wir wieder, wenn wir eine andere blaue Fläche erblicken” (GRL, p. 31). Also GRL, p. 34 below, implies that properties of external things (for instance Farbe) are sinnlich wahrnehmbar. Cf. GRL, § 87: “denn in der Außenwelt […] giebt es keine Begriffe, keine Eigenschaften von Begriffen”,as if “Eigenschaften von Dingen” were not excluded from the “external world”.
Actually Frege gives this “good” formulation in the title of § 24 (in the analysis of contents of GRL): “Wenn die Zahl etwas Sinnliches wäre, könnte sie nicht Unsinnlichem beigelegt werden.”
We shall consider them in Section 10.2.
GRL, p. 33.
GRL, § 26.
GRL, § 27.
GRL, p. X.
Cf. title of § 26 in the analysis of contents of GRL.
GRL, p. VI.
GRL, § 25.
“Während für Mill die Zahl etwas Physikalisches ist, besteht sie für Locke und Leibniz nur in der Idee.”
GRL, § 93.
“In diesem Gedankengange [Leibniz’s view, cf. note 14] kommt man leicht dazu, die Zahl für etwas Subjectives anzusehen.”
Leibniz [1], V (Nouv. Ess.),p. 76; “estre dans l’entendement” is obviously objective.
Cf. Section 2.51.
GRG I, p. 2: “In wessen Geiste?” This question is addressed to Dedekind, who spoke of sets as being “in the mind”.
Frege usually characterizes subjective as that which is Verschieden für verschiedene Menschen (BG, Vorwort, first page; SUB, pp. 29, 31; GRL, p. 37, etc.). This is precisely psychological subjectivity as opposed to transcendental subjectivity, cf. Eisler [1], “Subjektiv” (p. 516). Psychological subjectivity is also the strict sense of “subjective” (ibid.). I think it is appropriate to call this the “strong” sense as well. It seems that Frege has understood “subjective” only in this way, leaving aside the minor exceptions we refer to in the present section. “Subjective” means for Frege psychology,personal sensations, personal feelings, and so on (cf. for instance GRL, § 93). GED indicates that this was Frege’s understanding of that term until the end of his life. Of course in this sense numbers are “outside the subject” because they are in fact accessible to many thinkers.
“Wir beschäftigen uns in der Arithmetik mit Gegenständen, die uns nicht als etwas Fremdes von außen durch Vermittelung der Sinne bekannt werden, sondern die unmittelbar der Vernunft gegeben sind, welche sie als ihr Eigenstes völlig durschschauen kann. Und doch, oder vielmehr grade daher sind diese Gegenstände nicht subjective Hirngespinnste. Es giebt nichts Objectiveres als die arithmetischen Gesetze” (GRL, § 105). Frege acknowledges and introduces what philosophy has called transcendental subjectivity, but he does not wish to employ the term “subjective” to designate it. We could ask: “in whose Vernunft?” (cf. note 19). Cf. also: “So verstehe ich unter Objectivität eine Unabhängigkeit von unserm Empfinden, Anschauen und Vorstellen, von dem Entwerfen innerer Bilder aus den Erinnerungen früherer Empfindungen, aber nicht eine Unabhängigkeit von der Vernunft; denn die Frage beantworten, was die Dinge unabhängig von der Vernunft sind, hieße urtheilen, ohne zu urtheilen, den Pelz waschen, ohne ihn naß zu machen” (GRL, § 26). “Und wir kommen zu dem Schlusse, daß die Zahl weder räumlich und physikalisch ist E...] noch auch subjectiv wie die Vorstellungen, sondern unsinnlich und objectiv. Der Grund der Objectivität kann ja nicht in dem Sinneseindrucke liegen, der als Affection unserer Seele ganz subjectiv ist, sondern soweit ich sehe, nur in der Vernunft” (GRL, § 27).
“Dieses Subjektive der Erkenntnis [i.e., “subjectiv” in the kritisch-transzendentalen Sinne] ist zugleich objektiv [again, in the critical-transcendental sense] (Eisler [1], “Subjektiv”).
GRL, p. 38, § 45, § 93 (title). In GED, Frege will apply the same dichotomy to Gedanken.
Cf. Section 10.52.
Cf. Section 10.3.
Cf. Section 10.52.
Cf. Section 2.7. Frege associates both issues in GRL, § 106.
“. läßt sich die Zahl als eine Eigenschaft desjenigen Begriffes auffassen, unter welchem die gewählten Individuen vereinigt werden.” (Hilbert-Ackermann [1], pp. 115–116).
Aristotle [3] (Anal. Priora A, Ch. 27).
Anscombe-Geach [1], p. 130.
GRL, p. 5.
Ibid. Up to now Frege has not provided any hint as to what number is, except a vague reference to numbers “lying beyond intuition” (DISSE, p. 1) and the statement of his program of research in BG (Vorwort).
GRL, § 12, in fine, p. 57, § 89.
“Spricht man nicht in einem ganz andern Sinne von 1000 Blättern als von grünen Blättern des Baumes?” (GRL, p. 28; cf. also p. 29, in the middle).
“In dem Satze “dem Begriffe F kommt die Zahl 0 zu” ist 0 nur ein Theil des Prädicates, wenn wir als sachliches Subject den Begriff F betrachten. Deshalb habe ich es vermieden, eine Zahl wie 0, 1, 2 Eigenschaft eines Begriffes zu nennen” (GRL, § 57, italics Frege’s). Cf. § 46: “Aber dem Begriffe Venusmond wird dadurch eine Eigenschaft beigelegt, nämlich die, nichts unter sich zu befassen.”
Cf. Section 6.6.
Cf. Section 6.43.
Cf. Chapter 6.
Cf. Chapter 6.
“Trotz dieser Unräumlichkeit und Unwirklichkeit ist 1/2 kein Begriff in dem Sinne, daß Gegenstände unter ihn fallen könnten. Man kann nicht sagen: “dies ist eine 1/2”, wie man sagen kann: “dies ist ein rechter Winkel”, ebensowenig sind Ausdrücke wie “alle 1/2”, “einige 1/2” zulässig; sondern 1/2 wird als bestimmter einzelner Gegenstand behandelt…” (UFT, pp. 103–104). For an interesting criticism of Frege, cf. Kneale [1], p. 458f.
It is curious to observe that authors such as Wittgenstein [1] (p. 186) or Carnap [4] (p. 301) have erroneously assigned to Frege the thesis that number is a property of a concept, as if it were a second-level concept (cf. also Goodstein [1], p. 3). Numbers appear, so to speak, at the second level; but this means only that their names are proper names which occur in the name of a second-level property. Of course Frege’s language in GRL is in many instances loose enough to suggest that numbers are properties of concepts. For example, “beilegen” numbers to concepts is a frequent turn in his text (GRL, §§ 46, 53). Numbers also “zukommen” to concepts (GRL, §§ 51, 54). Frege also speaks of the Träger or Substrat of number (GRG II, p. 150), and this may erroneously suggest that concepts or classes are “subjects” of a property like two.
Cf. Section 10.4.
Cf. Section 10.51.
Benacerraf [1] and a paper mentioned therein by Ch. Parsons (unpublished) indicate an increasing interest in Frege’s intuitive philosophy of number.
I arrange the various possibilities indicated by Frege without considering his division given at the end of GRL § 28; this division will become a distinction of sub-cases in the interpretation of the Euclidean definition.
I use the following terminology (The Concise Oxford Dictionary): “unit” “individual entity regarded for purposes of calculation as single and complete”; “unity” “oneness, being formed of parts that constitute a whole” (indivisum). Frege uses “Einheit”, which covers both, as the term “unitas”.
Also in English “(the) unity” (The Concise Oxford Dictionary).
Cf. Section 8.1, especially note 5 of Chapter 8.
Cf. Section 10.11.
Aristotle [2], I, 6, 1057a, 2.
“unde etiam multitudo absolute dicta videtur comparari ad quantitatem discretam per modum generis, seu praedicati superioris essentialiter…” (Suarez [1], 41, sectio 1).
Cf. Section 3.62.
GRL, § 28 in fine or § 29.
Cf., for instance, GRL § 51.
Where numbers are defined as classes of concepts.
Cf.: “numerum definio unum et unum et unum, etc.”, Leibniz quoted in Achsel [1], p. 11.
GRL, § 51. Cf.: “Ferner soll dasselbe Element nicht mehrfach vorkommen dürfen” (Kamke [1], § 1).
P. 8.
“... if a = b, we have {a, b}={a, a}={a} (Sierpinski [1], p. 7, example 2). Frege has often attacked the notion of a set of units (or abstract units, i.e., “number ones”) conceived as an abstract “equivalent” of a concrete set of entities. Precisely, Cantor [1] (p. 283) speaks of lauter Einsen: (1, 1,…, 1). Frege in CANT, p. 270 calls them “jene unglücklichen Einsen”, and he asserts that Cantor asks impossible abstractions. Cf. Zermelo in Cantor [1], p. 351. For more recent implications, see Beth [2], p. 259.
GRL, p. 40. Cf. GRL, § 49 (in fine), now for the “transcendental” ens.
This is the core of Frege’s argument from § 34 to § 45. The dilemma is stated in § 39.
GRL, p. 58.
GRL, p. 50.
“Wenn wir die Zahl durch Zusammenfassung von verschiedenen Gegenständen entstehen lassen wollen, so erhalten wir eine Anhäufung, in der die Gegenstände mit eben den Eigenschaften enthalten sind, durch die sie sich unterscheiden, and das ist nicht die Zahl” (GRL, p. 50 (italics mine); cf. also p. 51).
Especially GRL, p. 57 bottom. Cf. pp. 49, 53.
The argument is reiterated for instance in SCHU, pp. 8–12.
GRL, § 45.
GRL, § 54; Frege’s speculation is familiar in classical ontology: “Sed Plato ex consideratione universalium deveniebat ad ponendum principia sensibilium rerum. Unde, cum diversitatis multorum singularium sub uno universali causa sit divisio materiae, posuit diversitatem ex parte materiae, et unitatem ex parte formae” (Aquinas [1], n. 168).
Cf. note 57.
GRL, § 51.
Cf. GRL, § 54.
Cf. Section 8.31.
Of course, this does not mean that numbers are second-level or n-level predicates. Frege requires, in my view, that they be individuals but individuals “appearing” at the second level. That numbers appear (whatever this may mean) at the second level is a traditional doctrine. Cf. Springer [1], p. 33, where, nevertheless, Maimonides’ having in mind a second-level predicate is not duly appreciated. Maimonides, like Geach [1], p. 159, speaks in ambiguous terms of being incidental to, or being supervenient upon. This is justifiable in Maimonides’ traditional approach to the higher predicates, but one wonders why Prof. Geach deliberately uses the Avicennian phraseology. Burkamp [1] is an important criticism of Frege’s view on numbers as they concern concepts. But, unfortunately, I do not understand what Burkamp (p. 203) means by “synthetic” and “analytic”, or “(un)wesentliche” predicates of predicates. He is aware of the distinction Merkmal-Eigenschaft (p. 304) and thus he should not use “unwesentlich” as traditional philosophers used “accidere” (cf. Section 9.2, note 18), but on p. 205 (lines 10–12 from the bottom) we are surprised to find that a “synthetic” higher predicate is that which is not… a mark! But of course, by this brief negative remark I do not intend to diminish the contribution of Burkamp [1], a book worthy of a special analysis, both for its interest and because it is dedicated to Frege. Krenz [1] gives a presentation of some of Burkamp’s ideas.
GRL, 55f.
Cf. Section 2.7.
Cf. Section 10.13.
Which may be illustrated by Frege’s first phrases in GRL § 62: “Wie soll uns denn eine Zahl gegeben sein, wenn wir keine Vorstellung oder Anschauung von ihr haben können?”
Marcel [1], p. 162. Cf. Geach [1], p. 158. Professor Geach probably has in mind GRL, § 107 (in fine), but perhaps Frege is referring there to his proposal of substituting “Begriff” for “Umfang des Begriffes” (GRL, p. 80 note). Still, I fully agree with Prof. Geach’s interpretation that the famous definition is “only a secondary and doubtful point”. I would add only the following qualification: as “essential” definition it is unimportant (i.e., it is not intended by Frege to express what numbers really are), but as “coffret” it is essential, as indicated by the last words of the Nachwort (GRG II).
GRL, § 62f.
Cf. Section 10.3, in particular note 63.
Cf. Section 10.2.
GRL, p. 80 note. GRL, p. 65 provides an example of what number one would be if identified with the concept having as many individuals as the concept moon of the Earth. Number one would be the second-level concept Einzigkeit, under which the concept moon of the Earth falls. But on p. 65 Frege does not say that Einzigkeit is number one. There he is trying to see; on such a phenomenological level he cannot say that Einzigkeit is 1. But on page 80 his method is different: he is trying the “logistic” definition. In my interpretation of Frege (10.2) such a thing as “instances” of numbers were excluded, but now the “definition” — if taken as a statement about the real nature of numbers — calls for a new discussion. I do not think that the Frege who suggests that numbers are ultimate subjects of predication (10.2) would readily admit that a trio of men is an instance of number 3 (to use Russell [3]’s example). Of course it is not thereby implied that a trio of men is an instance of number, which would be the rough traditional view of “concrete numbers” (cf. 10.51). Still, to view a trio of men as an “instance” of 3 is obviously a partial relapse into that traditional view, a relapse of course with respect to the Frege we have tried to understand in 10.2.
GRG II, p. 265, Frege says “fassen”.
For instance, Becker [3], p. 20, n. 3.
Quoted by Gilson [3], p. 284. Incidentally, the same famous authors are mentioned by Cantor [1] (p. 396), in a context of philosophy of number.
Cf. Chapter 3.
Cf. preceding section.
Sanchez Sedeíïo [1], Liber V, quaestio XXIII (“dissoluuntur argumenta”).
Cornford [2], p. 8, note 2.
This is suggested by an examination of books on the history of mathematics (M. Cantor’s, to begin with). It is most significant that the Euclidean definition is not apparent in Becker [2]. In mathematically oriented research such as Levi [1] (p. 87), the definition does not lead to any perplexities. Probably the only impact of that definition on mathematics has been the traditional exclusion of 0, and perhaps 1, from the class of numbers. Simplicius [2], (220 a 7) clearly explains that 0 and 1 cannot be numbers if number is a set of units. Cf. Ross in Aristotle [4] (p. 604). Chrysippus is said to have introduced 1 as a number (Becker [1], p. 45). All this, and recent history as well (cf. Gigli [1], p. 105), enables one to understand Frege’s concern in saying that 0 and 1 are numbers (for instance, title of GRL, § 44). I am grateful to Dr. F. Olivieri for valuable information on this matter.
Aristotle [2] Z, 13, 1039a, 11–14. Heath [1] (p. 113) is richly informative on this. The mass of commentaries on this passage would provide an interesting liaison between traditional metaphysics and GRL. Aristotle’s solution is to distinguish units in act and in potency. But I believe that act and potency are used here as a psychologistic expedient. Are they not the same as “paying attention to” (i.e., the units are in act) or “not paying attention to” (i.e., the units are in potency)? The culmination of traditional philosophy of number, Husserl [4] (p. 84, 156), proposes a solution (now, of Frege’s aporia in GRL) quite like that of Aristotle. Frege’s reply, in HUSS (p. 325), is an accusation of psychologism. Whitehead—Russell [1] (I, p. 72, note) prefer to eliminate classes.
This definition will also be successful (Euler, Lagrange…) and its effect will be to weaken the Aristotelian separation between discrete and continuous quantity.
Cf. Schütz [1], “numerus”.
Cf. Fernandez Garcia [1], “numerus”.
“Difficultas in qua discordant diversi” (ibid.).
Ens if and only if unum; but cf. Frege’s lack of interest in such transcendental properties, note 60.
Cf. Section 10.51.
As far as I know, the term “class” originally designated the Aristotelian ten categories. (Cf. for instance Eust. a Sto Paulo, quoted by Gilson [3], p. 35f.)
Keckermann [1], Lib. I, Sectio I, cap. 8.
“… unde quis non immerito concludat numerum non esse ens verum et per se…” (ibid.).
Fonseca [2], Tomus II, in Met. 5, cap. XIII, qu. IV, sectio IV: “Posteriorem sententiam [the subjectivist one] ex recentioribus quidem scholasticis plurimi amplectuntur…”. Our author remarks that even within a single lifetime (that of Duns Scotus) there was a similar process. And Fonseca himself adheres to this subjectivist interpretation, according to which in rerum natura there are only “heaps” (of substances or other real entities), and it is our mind which transforms a mere acervum lapidum aut frumenti into a number (always, of course, within the Euclidean definition).
If number “outside the mind” is only a “heap” (cf. preceding note), then: “dices, ergo, numerus ut numerus, et ut aliquid unum, est ens rationis, et non rei, et consequenter non poterit dici quantitas realis”. Answer: “Nam licet multitudo unitatum, ut multitudo, a parte rei sit, tarnen, ut habens aliquam unitatem per se, secundum aliquem ordinem vel constitutionem, non est nisi per intellectum, et dependenter ab intellectu” (Suarez [1], 41).
Cf. Sections 10.12 and 10.13.
Cf. Section 1.43.
The number three is supposed to be a property (an accident) inhering in more than one subject: “numerus trinarius non habet esse in uno subiecto sed in tribus.” In a reply to this argument, someone said: “Dico quod illa tria habent rationem illius subiecti respectu numeri, quae [ratio] est forma discreta; quia haec est ratio formae discretae quod habeat esse in subiecto diviso in actu in plures partes non copulatas in aliquo termino communi” (Nicolaus D’Orbellis [1] “quantitas discreta”).
Fonseca [2] (ibid., as in note 13, sectio II) mentions the accidens discretum, but he finds this meaningless: “… tarnen quia hoc ipsum mens capere non omnino videtur, ut unius accidentis non sit unum subiectum…” The notion of an “accidens partialiter existens in pluribus subiectis” is indeed hardly conceivable: “You will not, I believe, admit an accident which is in two subjects at once” (Russell [1], p. 206; the text is Russell’s translation of Leibniz). Incidentally, this is a good occasion to observe the divergency between the two ontological dimensions (cf. Chapter 1); per defznitionem, a property (predicate, universal) is something which may exist in several subjects.
UFT, GRG II, p. 156 (“Formal” “rein logisch”).
“In der That kann man so ziemlich Alles zählen, was Gegenstand des Denkens werden kann- Ideales so gut wie Reales, Begriffe wie Dinge, Zeitliches so gut wie Räumliches, Ereignisse wie Körper, Methoden so gut wie Lehrsätze; auch die Zahlen selbst kann man wieder zählen” (UFT, p. 94).
UFT, pp. 94–95: “Daraus ist doch wohl so viel zu entnehmen, daß die Grundsätze, auf denen sich die Arithmetik aufbaut, sich nicht auf ein engeres Gebiet beziehen dürfen […] sondern jene Grundsätze müssen sich auf alles Denkbare erstrecken; und einen solchen allgemeinsten Satz zählt man doch wohl mit Recht der Logik zu.” This has been clearly appreciated by Bochenski [1], 39.04. To be able to appreciate Frege’s logicism, it is necessary to know that it is not trivial. And it is not trivial with respect to the philosophical past, which Frege (along with other philosophers) combatted in order to vindicate the metaphysical significance of numbers (cf. next section). Outside of this perspective, I believe that “logicism” (vs. intuitionism, etc.) loses its original significance.
Suarez [1], 41, sect. II. Cf. Martin [1], § 8, and Maritain [2].
Ibid.
There will be sets of supra or non-sensible entities (“multitudo, quatenus ex unitatibus transcendentalibus consurgit”, ibid.); but the question remains whether such abstract sets may be called numbers. Our use of “proper” finds support, for instance, in Fonseca’s distinction between “unum transcendens” and “unum proprie dictum”. At least in the 16th century this was a rather artificial situation, because ordinary language, as Suarez points out, allows one to speak of the number of angels and to compare that number with the number of human beings (ibid.).
I mean those philosophers supporting the sententia negans (the universal applicability of number proper, Suarez [1], ibid.). There were also many who held the opposite position (cf. ibid.).
Cf. Section 3.62.
Cf. Suarez [1], ibid.
Leibniz’s view is quoted by Frege in GRL, § 24.
Husserl [3], § 24, p. 68 note.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1967 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Angelelli, I. (1967). Number. In: Studies on Gottlob Frege and Traditional Philosophy. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3175-1_11
Download citation
DOI: https://doi.org/10.1007/978-94-017-3175-1_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8326-5
Online ISBN: 978-94-017-3175-1
eBook Packages: Springer Book Archive