“Wittgenstein is emphasising that mathematical language, like the rest of our language, is part of a practice, and a practice rests on contingencies.” (Gerrard, 1996, pg 192)
“If mathematics is to be regarded as a practice, and if a practice requires, among other things, the existence of regular use (PI, 198), then there must be enough regularity for their to be mathematics at all.” (Gerrard, 1996, pg 192)
- RFM, I, 4: "Counting (and that means : counting like this) is a technique that is employed daily in the most various aspects of our lives. And that is why we learn to count as we do: with endless practice with merciless exactitude.”
“Thus, in order for there to be correctness or incorrectness at all, there must be a practice; outside of the practice there is not falsehood, but nonsense.” (Gerrard, 1996, pg 192)
“Without a natural environment of a certain constancy, without a shared humanity of similar needs and reactions, unless we spoke of a shared language, unless there was enough agreement, then it would be meaningless [to speak credibly].” (Gerrard, 1996, pg 192)
“Mathematics is grounded, as it were, both in the biological and in the social. The rules of calculating and so on, established by human beings like ourselves with certain biological capabilities and limitations, are appealed to in judging the correctness of particular calculations and inferences.” (Phillips, 1979, pg 134 - 135)
“The above discussion leads to an interesting conclusion: measuring, calculating, inferring and so forth, are bounded by the facts of nature but particular systems of measurement, calculations and so on, are fully a matter of social convention.” (Phillips, 1979, pg 135)
“To say a reality corresponds to ‘2+2=4’ is like saying a ‘A reality corresponds to “two”.’ It is like saying a reality corresponds to a rule, which would come to saying: ‘It is a useful rule, most useful - we couldn’t do without it for a thousand reasons, not just one.’ (LFM, p. 249)” (Fogelin, 2009, pg 85)
“Natural numbers, like our language, are neither true nor false; they are used. And the fact that numbers, like language, can be used in a great variety of different situations and can function without special preparation for every eventuality is due to custom.” (Phillips, 1979, pg 125)
“Mathematics ... [shapes], and [is] shaped by, our contingently given ways of looking at and experiencing events, actions and experiences. They involve forms or arrangements of facts, ways of intuiting the particular.” (Floyd, 2010, pg 321)
“Every instance of the use of a formula is the culmination of a process of socialisation ... Training differs from explanation in that - at least among children - it is largely non-verbal and it is aimed at producing certain actions.” (Phillips, 1979, pg 126)
“We see one system of experience in another when we employ the calculus [of probability]. ‘Seeing-in’ implies that there is nothing intrinsically necessary that requires us to apply a concept to a particular situation, and that we therefore bear some responsibility for the application of a structure (here, a mathematical one) to the interpretation of experience. We may have good psychological, emotional, ethical or philosophical reasons for feeling that probability’s way of seeing a situation does not speak to our lives or interests at all.” (Floyd, 2010, pg 322)
“To apply probability we are obliged to regard our powers of representation as successful but conditional upon application, as open to further articulation and arrangement.” (Floyd, 2010, pg 320)
“[It] is itself constructive, sometimes pleasurable, and characteristic of certain kinds of significance we find and create in our lives.” (Floyd, 2010, pg 323)
“Here the variety of possible contexts into which the figure may be imagined fitting is even wider (a blueprint, a paradigm for a child ... ) so that what seems to engage us is less the representation itself than the worlds and activities with which we surround it.” (Floyd, 2010, 324)
“[A] method, which is more generally applicable, is to examine the training a child receives in developing proficiency in the use of various expressions ... we can first notice how difference in use can be reflected in differences in modes of training. Learning to count is the normal entryway into arithmetic. This usually starts with the rote memorisation of an ordered sequence of sounds and marks. Here learning to count is similar to learning one’s ABCs. But there are important differences between the two. With counting, at a certain point the student is expected to continue on her own.” (Fogelin, 2009, pg 87)
“The difference between numerals and letters becomes more striking when we examine the way in which students are taught to employ these two kinds of symbols ... The child is taught the use of numerals primarily from learning to count objects. There are all sorts of ways that the child can go wrong beyond simply forgetting how to recite the numerals in the proper order.” (Fogelin, 2009, pg 88)
“Yet she has not done what we wanted her to do; she has yet to master the technique that underlies our use of the expression ‘5x5.’ ... We might try and get around these difficulties by making out instructions more specific, and it’s a fact that sometimes making instructions more specific increases a student’s chances of getting things right ... however specific we make our instructions, there will be some interpretation of what we have said that will support the claim that she has not done what we told her to do.” (Fogelin, 2009, pg 99)
“We will suppose that the child goes on in the usual way to master the rules for the first basic arithmetic operations (addition, subtraction, multiplication, and division), and then learns to apply them in the standard ways in performing practical calculations. Once mastered, these rules will largely fall into the background and operate, we might say, invisibly.” (Fogelin, 2009, pg 89)
“On Wittgenstein’s approach, arithmetic starts out applied and then is purified by being decoupled from particular application.” (Fogelin, 2009, pg 90)
“The process of knowing does not follow only from the ‘nature of things’ or from ‘pure logical possibilities’ but is often influenced by ‘extra-theoretical’ factors of diverse sorts.” (Phillips, 1979, pg 122)
“The fact that we generally agree in our calculations is the result of how we learn mathematics, the way in which we are trained. As with any other language-game, the language-game of mathematics requires socialisation into the rules, standards, conventions, and grammar that one must master in order to play the game correctly. Part of this requires learning that there are ‘right’ and ‘wrong’ ways of playing the game. In mathematics, we learn that one can be ‘correct’ and that one can also make ‘mistakes.’ As Wittgenstein notes: ‘Our children are not only given practice in calculation but are also trained to adopt a particular attitude toward ... calculating.’ (RFM, V, 40)” (Phillips, 1979, pg 123)
“We only play the language-game of mathematics through an appeal to and reliance upon earlier acquired linguistic abilities. Our original, everyday language constitutes a meta-language in use which underlies our learning and practices in mathematics.” (Phillips, 1979, pg 128)
- TLP 3.11: We use the perceptible sign of a proposition (spoken or written, etc) as a projection of a possible situation. The method of projection is to think of the sense of the proposition.
- TLP 3.13: A proposition includes all that the projection includes, but not what is projected. Therefore, though what is projected is not itself included, its possibility is. A proposition, therefore, does not actually contain its sense, but does contain the possibility of expressing it. (‘The content of a proposition’ means the content of a proposition that has sense.) A proposition contains the form, but not the content, of its sense.
“What Wittgenstein has in mind ... is some notion of a meta-language in use or, more specifically, the human activities which allow us to conduct and frame our investigations. This ‘meta-language’, however, is not only linguistic, for it includes a host of activities directed toward coping with the world around us.” (Phillips, 1979, pg 129)
- RFM III 35. If I were to see the standard meter in Paris, but were not acquainted with the institution of measuring and its connection with the standard metre -- could I say, that I was acquainted with the concept of the standard metre?”
“Wittgenstein therefore points toward the importance of active arrangement of concepts and symbols, open-ended self-discovery, pleasure, and absorbed intuitive preoccupation with diagrams and symbols as ineradicable features of our philosophical, logical and mathematical activities.” (Floyd, 2010, pg 315)
“The idea [is] that the role of mathematics ... is to allow us to expand, rearrange, and interpret our expressive and representational powers.” (Floyd, 2010, pg 315)
“The brute fact that human beings, after undergoing common training, come to agree on many things; the tendency to assign a protected status to certain expressions and thereby give them the character of rules; the tendency to apply the rules (standards, paradigms) back upon themselves, this converting normativity to ideality; and the tendency to attribute to particular expressions the force that they derive from their role as part of a system of interlocking expressions.” (Fogelin, 2009, pg 109)
“For a platonist, mathematical proofs establish truths concerning ideal mathematical objects. For Wittgenstein, they are, or could also be viewed as, constructions within a system of mathematical rules.” (Fogelin, 2009, pg 115)
“What were the cultural conditions, for example, that help account for the changes from medieval to modern logic? One might expect more studies ... [that] consider the social-historical background of the number systems and mathematics.” (Phillips, 1979, pg 121)
References (back to top)
- Floyd, J. (2010). On being surprised: Wittgenstein on aspect-perception, logic and mathematics. In W. Day and V. Krebs (Eds), Seeing Wittgenstein anew. (pp. 314 - 337). Cambridge: Cambridge University Press.
- Fogelin, R. (2009). Wittgenstein on the philosophy of mathematics. In R. Fogelin, Taking Wittgenstein at his word: a textual study. (pp. 79 - 166). Princeton: Princeton University Press.
- Gerrard, S. (1996). A philosophy of mathematics between two camps. In H. Sluga, H. and D. Stern (Eds.), The Cambridge companion to Wittgenstein. (pp. 171 - 197) Cambridge: Cambridge University Press.
- Phillips, D. (1977). The social nature of mathematics. In D. Phillips, Wittgenstein and scientific knowledge. (pp. 119 - 141). London: MacMillan Press
- Wittgenstein, L. (2001). Tractatus Logico-Philosophicus. Translated by D.F. Pears and B.F. McGuinness. London: Routledge.
- _____________ (2001). Philosophical Investigations. 3rd Edition. Translated by G.E.M. Anscombe. Oxford: Blackwell Publishing.
- _____________ (1978). Remarks on the Foundation of Mathematics. (3rd Edition). Edited by G.H. von Wright, R. Rhees, and G.E.M. Anscombe. Translated by G.E.M. Anscombe. Chicago: University of Chicago Press.
- _____________ (1976). Wittgenstein’s Lectures on the Foundation of Mathematics, Cambridge, 1939. Edited by Cora Diamond. Ithaca: Cornell University Press.