Boudewijn de Bruin - Wittgenstein's Objections Against the Frege-Russell Definition of Number
Published as "Wittgenstein's Objections Against the Frege-Russell Definition of Number." Proceedings of the International Wittgenstein Symposium 1999. Kirchberg/Wechsel, 1999. 109-113.
What is a number actually? A question of great mathematical significance, yet only vaguely understood in philosophy, dealt with by people of different background and persuasion, and even by now without a final answer. A well-known approach is the Frege-Russell definition of number. It defines the "cardinality" of a concept C as the class of all concepts that have a one-one correlation (also called a "one-one correspondence," or a "bijection") with C, which is, intuitively speaking, the class of all concepts that have the same number of objects falling under them as C has. The collection of numbers is then defined to be the collection of such classes -- a collection that can be explicitly characterized by inductively defining (what I will call) "standard" concepts of arbitrary cardinality. (For Frege's original definition see (Frege 1950), or the more formal (Frege 1967); for Russell's improvements see (Russell&Whitehead 1925).)
Several philosophers have, with varying degrees of success, impugned this definition. Its most famous critic, the French philosopher Henri Poincare, blamed the principle of mathematical induction for a circulus vitiosus in the definition (see his (Poincare 1952)), while other critics simply observed that if for instance the number two were to be the class of all classes of two elements, it would change every time a pair of twins were born. Throughout his philosophical carrier, Wittgenstein, too, has been concerned with the notion of number. He first explained it by means of the "general form of a natural number" (TLP 6.03), a view he later gave up in favor of an account in terms of family resemblance (PI 67-68). I will not deal in this paper with these positive contributions, but rather discuss two explicit formulations of objections against the Frege-Russell definition. The first one appears in Friedrich Waismann's notes of a meeting of the Vienna Circle on January 4, 1931 (see (Waismann 1967)), while the second -- and considerably different -- objection is taken from Wittgenstein's 1939 lectures at Cambridge (in (Diamond 1975)). While Wittgenstein's positive contributions to the problem of number have received attention by a number of commentators (see, e.g., (Intisar-ul-Haque 1986), and the entry "number" in (Glock 1996)), to my knowledge there has been no discussion of his negative criticism among Wittgenstein scholars. Perhaps this is due to the questionable status of the textual sources: no written statements by Wittgenstein himself appear to be available. Another reason, however, may be that it is extremely difficult to make sense of the arguments. In particular the older critique is highly dubious since before attacking the Frege-Russell definition Wittgenstein rephrases and reinterprets it along modal lines. What I present here will be a kind of "reconstruction" of Wittgenstein's arguments. Furthermore, I will show by means of a counterexample that one of the objections is inconsistent. For more detailed treatment the reader is referred to my (De Bruin 1998).
The criticism on the Frege-Russell definition in Wittgenstein and the Vienna Circle (abbrev.: "WVC") is directed at the following version in terms of sets: A set S has cardinality n if and only if ("iff" henceforward) there exists a one-one correspondence between S and a particular standard set of cardinality n. Analyzing the conditions under which we are justified to assert the existence of such a one-one correspondence, Wittgenstein states that "A correlation only obtains if it has been produced" (WVC 165), that is, if the sets have actually been correlated by arranging their elements in a particular manner, or by providing a description of such an arrangement, etc. This observation is taken care of by the following modifition of the original definition: A set S has n elements iff a one-one correspondence between S and the standard set of n elements has been produced. This modification, however, has the undesirable consequence that the cardinality of a set will not be determined until someone has actually established a bijection, thus making it quite difficult to ascertain the truth value of statements like "By the beginning of the semester there were twenty students in the class," since we have to find out if someone has actually produced a bijection between the set of students and the standard set of 20 at the beginning of the semester. To solve this problem, Wittgenstein presents us with an example in which he asks how he can, given he has 12 cups, show that he has equally many spoons.
"If I had wanted to say that I allotted one spoon to each cup, I would not have expressed what I meant by saying that I have just as many spoons as cups. Thus it will be better for me to say, I can allot the spoons to the cups." (WVC 164, my emphasis)
This means that sentences stating an actual demonstration of a one-one correspondence between two sets are not needed to state their equinumerosity, and that consequently the Frege-Russell definition should rather refer to possible demonstrations. This is accounted for by the following second modification: A set S has n elements iff it is possible to produce a one-one correspondence between S and the standard set of n elements.
According to Wittgenstein, this twice modified definition "presupposes number" (WVC 165), which as it stands is not too informative. Although at one place it is claimed that to "explain" the possibility of correlation the concept of number must be presupposed (WVC 164), an interpretation of presupposition in terms of explanation would definitely miss the gist of Wittgenstein's argument. A systematically as well as textually more promising interpretation can be obtained from (Waismann 1959), in which the author discusses criticism of the Frege-Russell definition taken from "an unpublished manuscript of Ludwig Wittgenstein" (1959, 245). Indeed, Waismann's statement that "in order to recognize whether the correspondence is possible, I must already know that the sets are numerically equivalent" (1959, 109) shows that presupposition of knowledge is at stake. Wittgenstein's criticism of the Frege-Russell definition can now be phrased thus: If a person x knows whether it is possible to establish a one-one correspondence between two sets V and W, then x knows the cardinalities of V and W.
If this last sentence were true, the Frege-Russell definition would indeed be severely wounded. It is, however, false. First observe that the antecedent of the sentence is, in fact, a disjunction of "x knows that it is possible to establish a one-one correspondence between V and W," and "x knows that it is not possible to establish a one-one correspondence between V and W." To see that the first disjunct does not imply that x knows the cardinalities of V and W, let V be a large set of saucers, and W a large set of cups, and suppose that x has established a bijection between V and W by putting cups on saucers in the usual way. Now x need not know their respective cardinalities (which was suggestively ruled out by assuming them to be large), which shows that the first disjunct does not imply the consequent, and hence that the whole sentence is false. Although redundant, it is not devoid of any intrinsic interest to show that the second disjunct does not imply knowledge of the cardinalities either. Suppose x is given a large set of cherry-bobs, and that from each cherry-bob he throws one cherry in basket V and the other in basket W. Furthermore, suppose that after finishing the procedure he added one single cherry to V, leaving W untouched. Then x knows that V and W cannot be one-one correlated, but he does not know their cardinalities.
I will now turn to the objections against the Frege-Russell definition as they are formulated in Wittgenstein's Lectures on the Foundations of Mathematics ("LFM," for short) . In contrast with the preceding objections, it is the original Fregean version in terms of concepts that he criticizes, not sets: A concept C has n objects falling under it iff there exists a one-one correspondence between C and the standard concept of cardinality n. This definition requires three ingredients: a concept C, a standard concept, and a bijection between them. According to Wittgenstein, Russell claimed the third ingredient to be taken care of by the following sequence of relations Ri
R1(x,y) iff (x=a & y=b),
R2(x,y) iff (x=a & y=b) v (x=c & y=d),
R3(x,y) iff (x=a & y=b) v (x=c v y=d) v (x=e & y=f),
one-one correlating equinumerous concepts of any cardinality, provided the objects falling under them have been given the names a, b, c, etc. (See LFM 161.) Since the names for the objects falling under the standard concepts are known already, to show that n is the cardinality of C we only need to provide names for the objects falling under concept C. However, in the following argument Wittgenstein shows that naming objects falling under any concept V in order to one-one correlate it to any concept W, involves counting, and so in particular naming the objects falling under C involves number considerations, thus showing that the Frege-Russell definition presupposes number. What Wittgenstein in fact shows is that unless we allow for counting, we will not be able to rule out the unacceptable possibility of correlating any two concepts V and W (of different cardinalities) by means of any relation Ri. Suppose, he argues, that there are two boxes, one of them containing two apples, the other only one. How can we see that, say, R1 does not hold between them?
"Well, suppose we give them [the apples] all names and correlate the names. The first relation [i.e., R1] would be all right if we gave both the apples in one box the name "a''. -- We don't know how to apply a name to an apple, whether to apply two names "a" and "a" to an apple. You might say, "Oh, we mustn't do that.'' But how are we to find out whether we are doing that, except by counting?" (LFM 165)
This is the problem: there is no way to exclude one object from getting two names, or two objects from getting one name, except by counting. That is, the above "Oh, we mustn't do that" boils down to "Oh, we mustn't give one name to two objects." This concludes the argument. The Frege-Russell definition presupposes number.
Boudewijn de Bruin
Boudewijn de Bruin is assistant professor in the Faculty of Philosophy of the University of Groningen. De Bruin did his undergraduate work in musical composition at Enschede, and in mathematics and philosophy at Amsterdam, Berkeley, and Harvard. He obtained his Ph.D. in philosophy from The Institute for Logic, Language, and Computation in Amsterdam. His doctoral dissertation, under supervision of Johan van Benthem and Martin Stokhof, was on the logic of game theoretic explanations. read more
De Bruin, B. (1998), Wittgenstein's Philosophy of Mathematics. The Objections Against the Frege-Russell Definition of Number. M.A. Thesis, University of Amsterdam, ms.
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Waismann, F. (1959), Introduction to Mathematical Thinking: The Formulation of Concepts in Modern Mathematics. New York: Harper & Brothers Publishers, tr. T. Bolduc.
Waismann, F. (1967), Ludwig Wittgenstein and the Vienna Circle. Oxford: Basil Blackwell, tr. L. Schulte, B. McGuiness.
I would like to thank Professors Charles Chihara, Hans Sluga, and Martin Stokhof for their stimulating discussions of the subject.