Numbers as Tools

'What is a number?' would be a perfect meaningful question for Socrates to ask. But Wittgenstein stress that 'number' is a name for many different, yet interrelated, things that do not have a single thing in common. Take for example 5 and pi. While 5 is an odd, prime number, it is nonsensical to ask whether pi is even or odd, prime or non-prime. Acknowledging the fact that there are different kinds of numbers, Wittgenstein drops the questions 'What is a number?', 'What is the meaning of numbers?' and even “What is the meaning of the word 'five'?” (PI §1). Wittgenstein reverses Socrates maxim 'Don't look, think!' into “[First,] look and see whether there is anything common to all” (PI §66). Therefore, in order to know the meaning of 'five', one has to look at the different language-games in which the word 'five' is employed, like teaching numbers, using number in groceries, mathematics, etc.

In the first paragraph of Philosophical Investigation Wittgenstein tries to show how the Augustinian picture of language may be right by providing an example of how names stand for objects. In doing so he acknowledges that 'five', 'apple' and 'red', while all names, are names of different types. The use of 'apple' is different from the use of 'five'. While 'apple' is the name of an object that the shopkeeper takes in his hand to put it in the basket, the word 'five' is used in a series of actions in which the shopkeeper takes an apple for every word he recites (aloud or just for himself) and he stops at the word 'five'. Similarly, he can just take at once five apples because he knows that such configuration of objects correspond to the sign 'five'. Is the order of natural numbers or the fact that 5 is odd and prime playing any role in his actions? It may or it may not. The shopkeeper does not need to have the answer of what 'five' is the name. He just knows his use and that is enough for Wittgenstein to say that the shopkeeper knows what 'five' is. Wittgenstein does not pretend to have explained what 'five' means or what are the grounds for the shopkeeper's actions. He only shows the direction in which the explanation of 'five' should be looked for, i.e. in the common practices that employ this number.

The knowledge of what 'five' means is not beyond the practice of the shopkeeper when he fulfils the request 'five red apples'. To add at the description given by Wittgenstein that 'five' is 'a number' would not be at all informative. The explanation will be only more complicated because it will make use of a name more in need of definition, i.e. 'number', than the initial name, i.e. 'five'. “Explanations come to an end somewhere” (PI §1). We do not have access to the shopkeeper's brain and even if we did our explanation would not be more complex or more illuminating. Even if we have had looked in his brain we would have found only a different story of the same events and not an entity like the meaning of 'five'.

For Augustine and his subsequent tradition the word 'five' stands for a mental object or for another word in the 'language of thought'. “Wittgenstein always rejected both the formalist and nominalist tendency to identify numbers with numerals and the Platonist contention that numerals stand for abstract numbers [...] Numbers are what numerals signify, but the meaning of numerals is given not by abstract entities, but by the rules of their use.” (Glock, 1996, 267) Even such mental objects existed they would add nothing relevant to the description in §1, because what is important in everyday usage is the role and the use that number-words play in counting or picking the relevant objects.

What Wittgenstein is doing is to give the simplest examples of using the words, in order to rule out any metaphysical adds-on that make the use of concepts unclear. Such metaphysical additions are the presuppositions of necessary order and the a priori infinity of the series of natural numbers. The expansion of the language (2) in paragraphs 8 and 9 gives away with such presuppositions. Andrew Lugg affirms that “Nothing of significance turns on his [Wittgenstein's] use of 'a' to signify one, 'b' to signify two, 'c' to signify three and 'd' to signify four” (2004, 27), while acknowledging that previously Wittgenstein used '1', '2', '3' and '4'. The reason for the use of “the series of letters of the alphabet” is twofold. First, contrary to the series of natural numbers, the succession of letters in the alphabet is a contingent one. It is not a necessary feature that 'c' is after 'b' and before 'd'. Second, the series of letters do not bring about the presupposition of an infinite series. Moreover, had Wittgenstein used the name of cardinal points for numerals, the series would have been strictly reduced to four. The moral of using the letters of alphabet for numerals is that one should not view in a language-game more than is specifically described. One should stop adding unnecessary presuppositions when one is after the basic structures of grammar (in Wittgenstein's sense).

Enriching the language (2) Wittgenstein shows that there are different kinds of names, for building blocks, for numerals, for demonstratives and for colour samples. These words are used in many respects in similar ways in obeying orders, but there are many differences also. It is important to mention that when introducing the numerals Wittgenstein stresses that they are used in the same manner as the shopkeeper used numerals and not the way we do. This is to show that the use of numerals should be restricted to an activity-based use without any theory that underlies the use of number-words in more complex language-games. The language evidently could lack a general concept of numbers and also it may lack any words for designating numbers other than 'a', 'b', 'c' and 'd'. What the practice of obeying the order 'd-slab-there' could not lack is the knowledge by heart of the succession of letters in the alphabet. “From the stock of slabs B takes one for each letter of the alphabet up to 'd'” (PI §8). The knowledge of the alphabet is a circumstance that should obtain for every individual that can carry on the activities in §8 correctly. But, as shown in §9, this characteristic is not necessary either. A previous knowledge of any series could be missing as well.

Using the letters for signifying numbers Wittgenstein distinguishes between two ways of learning the numerals/numbers: teaching by counting and ostensive teaching. One way to learn numbers is to memorise a series of names, 'a', 'b', 'c', 'd', and when a 'c' command is given to recite the series up to 'c' doing for each letter/name/numeral the required action (the action may be just to say the letter). This seems to be a very natural way for Augustinian picture. One already knows the series of natural numbers in his thought language and all he has to do is to learn the names of those numbers. Wittgenstein makes this picture more appealing by showing that one has to know a series, for example the letters of alphabet, and to associate this series with the activity of counting. In other words, at a certain moment of time the activity of reciting the series of letters should become another activity, that of counting. Anyway this picture shows how the Augustinian conception can fail. The number-words are not linked to any kind of image whatsoever. The number-words are just parts of a complex activity.

But this is not the only thing one can do. One can learn numbers by learning to associate some configuration of objects with some specific names. This is to show that natural numbers could also be taught ostensively. This seems to be possible only for the first five or six numbers, but it is enough to dismiss the idea that numbers cannot be taught ostensively. The ostensive teaching is more like labelling objects: “Something more like the ostensive teaching of the words "block", "pillar", etc. would be the ostensive teaching of numerals that serve not to count but to refer to groups of objects that can be taken in at a glance.” (PI §9) In this case the whole idea of numbers being part of a series disappears. In this case 'c' is not between 'b' and 'd' but is a name similar to 'b' and ‘d’ that single out a certain configuration of objects of the same kind. “As in the case of the word 'slab', the child learns the use of a single word rather than a series of words.” (Lugg, 2004, 28) Such ostensive definition of numbers corresponds to the definition of names as labels. But some features that seemed necessary in the previous example are dismissed in this new practice of teaching. In this case the prerequisite knowledge is not that of a series of any kind but the ability to single out the kind that is shown to one. To say “This is c” pointing to a group of three slabs require that there is a place of 'c' in the language-game, i.e. pointing to that group of objects single out their number and not other feature of the group. For Wittgenstein the action of pointing is part of language, of the language-game. But again, pointing, the ostensive definition, does not give the whole meaning of number-words. To learn 'c' ostensively does not imply that one will be able to follow the order 'c-slab-there'. One will be able to identify correctly what 'c' stands for, he will know only part of the language-game, only one use. “Ostensive definitions specify only one rule among others for the use of a word. Indeed they presuppose the grammatical category of the word defined.” (Baker & Hacker, 2005, 16) For the language-game to be effective it should make sense to point to a group of objects and to associate the word said with the gesture. The student should know or guess to what kind the teacher is pointing. To say 'c is a number' can be used to define 'c' as a name for a particular of a kind one already knows, i.e. number, or to build up a new kind given that one already knows what 'c' is and how it is used.

Wittgenstein has constructed in §9 two ways in which numbers can be learned and used. He has shown that number-words are different from other kinds of words in this language-game. And that the number-words have a similar use. But is this use identical in the two cases? Let numbers1 be the numbers taught as members of a series and numbers2 the numbers taught as configuration of groups of objects. Does 'c' as number1 plays exactly the same role as 'c' as number2? It may be. But if the order 'c-slab-there' is given the results can differ in the two cases. In the case of numbers as members of series a man who cannot hold three slabs in his hands will probably bring only one slab three time in a row, namely bringing 'a-slab-there', then bringing 'b-slab-there' and then bringing 'c-slab-there'. But for the man who knows numbers only as configurations of objects the order may be simply disobeyed. If I am to use a Kantian vocabulary, numbers1 have a temporal meaning, of going through a series of action till the last member of that series, while numbers2 have a spatial meaning of objects being simultaneously present. But this kind of explanation adds nothing to the practices in language-game (8) and so it does not enrich the meaning of 'c'.

In normal cases however both number1 and number2 function in the same manner. It is true that for classical physics it is important that any number corresponds to a physical quantity and so physicists may prefer the ostensive numbers, while for pure mathematics only the relations between numbers are important. That does not mean that their numbers differ. They use numbers for slightly different ends. One can stress the differences or the similarities. But these considerations will not modify the use or the meaning of numbers. Extraordinary events like the impossibility to carry on an order may give rise to the question that the interlocutor of Wittgenstein asks: “Now what do the words of this language signify?” (PI §10) These words signify “the kind of use they have”. To ask if numbers1 differ or not from number2 is a metaphysical question that tries to go with the explanation further than is required by a normal use of language. This kind of questions is struggling for one right definition, for strict boundaries of a concept, strict boundaries that Wittgenstein dismissed by talking about family resemblance. Such a metaphysical question could bring about a new definition of numbers, that anyway adds nothing to the use of language as described in §9. “Wittgenstein noted [that] an exciting new definition of number is of no philosophical concern (MS 110 (Vol. VI), 222). For if it is new, it is not a rule that we use and that guides us in our use of number-words” (Baker & Hacker, 2005, 32n)

The 'significance of a word', say 'c', is an appropriate tool in distinguishing things inside one category, for example to say that 'c' signifies a different number, a different configuration, a different member of a series than 'd'. Also it may be used to distinguish between categories: 'c' cannot be a name for a building block. To state these differences does not require a sharp definition of numbers. It is part of the language-game that 'c' and 'd' are different and also that 'c' and 'slab' are different, yet 'different' in different ways. And to state this difference one does not have to have a metalanguage, a superior level of explanation. The competent speakers could state these differences without the need of a metaphysical theory that state a difficult entity for which the name stands. What 'c' signifies is not something outside the language-game, something hidden from the view of competent speakers. A competent speaker recognises the correct use of 'c' and can give definitions of 'c' by pointing or by counting objects.

Bibliography

Baker, G.P., Hacker, P.M.S. 2005. Wittgenstein: Understanding and Meaning, Part I: Essays. Blackwell.

Glock, H.J. 1999. A Wittgenstein dictionary. Blackwell.

Schroeder, S. 2006. Wittgenstein. The Way Out of the Fly-Bottle. Polity Press.

Wittgenstein, L. 1999. Philosophical Investigations. Blackwell.