Abstract
Wittgenstein harshly criticized what he considered “mythologies” infecting our understanding of how mathematics work. He was especially reluctant to the idea that they describe a realm of abstract entities influencing our cognitive capacities. Several Wittgenstein scholars have drawn the conclusion that his analysis results in a sociological interpretation of mathematics. The present paper aims at demonstrating that although mathematical practices deserve to be called “social” this doesn’t necessarily leads us to the conclusion that they are offered to sociological explanation.
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Notes
A similar point is made by Anthony Giddens: “Human history is created by intentional activities, but is not an intended project” (Giddens, 1984, p. 27).
It can also be said that rules structure practices. Chess game can be described both as an institution and as a practice. But many institutions do not lend themselves to this double description. A court of justice is an institution but certainly not a practice, although its existence allows the implementation of many different practices. The idea that practices are rule-guided makes all the difference between practices and activities. To use Wittgenstein’s words in his Philosophical Investigations (Wittgenstein, 2009, § 199), we can imagine one single person conducting an activity only once in a lifetime, but such a single occurrence is unconceivable in the case of practices, because they are rule-guided in nature: as Wittgenstein reminds us, following a rule is a custom, an institution.
The previous distinctions explain my reservations about the definition provided by Anthony Giddens: “Those practices which have the greatest time-space extension within such [societal] totalities can be referred to as institutions”. (Giddens, 1984, p. 17) If Giddens’ purpose is to characterize practices and if he suggests, accordingly, that practices having a large extension in space and time deserve to be called institutions, his assertion is acceptable. Conversely, defining institutions as practices is highly questionable. In another paper (Giddens, 1979, p. 96), he borrows a definition from Radcliffe-Brown (Radcliffe-Brown, 1940, p. 9) and maintains that institutions are “standardised modes of behaviour”. This is a curious claim. The truth is rather that such modes of behaviour can sometimes be found within institutions.
The view according to which institutions are abstract models devised to explain various experiences has famously been defended by Karl Popper (Popper, 1957, § 29).
G. P. Baker and P.M.S. Hacker explain that in his Philosophical Investigations Wittgenstein combats successively the Bedeutungskörper mythology and the Regelskörper mythology. (Baker and Hacker, 1984, p. 17)
Bloor picks up this example from O. Neugebauer’s book, The Exact Sciences in Antiquity (1952).
Can an arithmetical practice significantly differ from our own? The question whether an individual (a kid, for instance, in the course of his training) can deviate from common practice is quite a different issue. We most probably would be reluctant to describe him as entering an alternate practice. We would probably simply consider him mistaken. In short, discovering a community entertaining a practice we would feel inclined to describe as “arithmetic” but nevertheless deviating from our “own” would raise an issue that the observation of a misdirected individual wouldn’t raise. Figuring out such a community is, for sure, an interesting Gedankenexperiment, but its interest doesn’t prove that it corresponds to something we could come across.
A similar observation about color terms is made by Nelson Goodman in his book Languages of Art (Goodman, 1976, p. 78). In what way, he asks, must all (literally) green things, for instance, be similar?
David Bloor, Ibid. Bloor alludes here to what has been called the public-check argument. The claim behind this argument is that the way an agent follows a rule can, in principle, be subjected to an objective control. In other words, the agent cannot decide individually if the step he takes in following a rule is correct or not. If such a public check didn’t exist, no distinction between following a rule and one thinking that one follows a rule could be established. Robert Fogelin thinks that this argument fails (See Fogelin, 1987, p. 167-183). If he is right, even a whole community can mistake thinking that one follows a rule for following a rule. The point is that the public-check argument seems to result in what has been called a community view (= the correct way of following a rule is the majority way to follow it). But this is not plausible, as no logical connection seems to exist between being the majority way and being the correct way. The majority use of technical or specialized concepts can be inaccurate or incorrect and, in such cases, the “community” legitimated to assess the layman’s use of these concepts is the body of experts or specialists who make use of them on a daily basis.
Another terminological choice would be to call this normative ingredient societal (Mandelbaum, 1955). In this classical paper, Mandelbaum writes: “My aim is to show that one cannot understand the actions of human beings as members of society unless one assumes that there is a group of facts which I shall term ‘societal facts’ which are as ultimate as those facts which are ‘psychological’ in character. In speaking of ‘societal facts’, I refer to any fact concerning the forms of organization present in society”. (Mandelbaum, 1955, p. 307) The kind of normativity involved in rule following is certainly a societal fact, but when it comes to the basic rules of logic or arithmetic this “facts concerning the forms of organization present in society” cannot be facts differentiating this society from features exhibited by other societies.
Even rules considered patent examples of arbitrariness, such as chess game rules, prove difficult to change without altering the game’s Witz (Wittgenstein, 2009, § 564) and, perhaps, eventually rendering it unpracticable. And it is difficult to see whether our sense that arithmetical rules cannot be changed reveals that they differ in nature or only in degree from chess rules.
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Le Du, M. No Place for Private Practice. Topoi 42, 297–305 (2023). https://doi.org/10.1007/s11245-022-09863-5
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DOI: https://doi.org/10.1007/s11245-022-09863-5