David Bloor

Wittgenstein and Mannheim on the sociology of mathematics

ONE of the central problems of the sociology of knowledge is the status of logic and mathematics. These branches of knowledge are so impersonal and objective that a sociological analysis scarcely seems applicable. Time and again, in his Ideology and Utopia Karl Mannheim’s determined advocacy of the sociology of knowledge stops short at this point.’ He could not see how to think sociologically about how twice two equals four. The argument in this paper is that Wittgenstein’s Remarks on the Foundations of Mathematics2 shows how sociology can penetrate to the very basis of these topics. Wittgenstein solves Mannheim’s problem.3

The Strengths of the Strong Programme

Professor Laudan’s strictures on the strong programme in the sociology of knowledge do not, I think, call for a defence of that programme so much as a thorough counter-attack. Nevertheless I am grateful to him for raising a range of objections and queries of a kind which are typical of philosophers. I shall therefore combine defence and attack. To ease the burden on the reader I shall discuss Laudan’s points in the order in which he raises them. The only departure from this procedure concerns some general points from the end of his paper that I shall take up immediately. These deal with what Laudan sees as the excessive emphasis on sociological approaches and the neglect of the pragmatic success of science.

Polyhedra and the Abominations of Leviticus

How are social and institutional circumstances linked to the knowledge that scientists produce? To answer this question it is necessary to take risks: speculative but testable theories must be proposed. It will be my aim to explain and then apply one such theory. This will enable me to propose an hypothesis about the connexion between social processes and the style and content of mathematical knowledge.

Sociology of Scientific Knowledge

The work of sociologists of knowledge and socially oriented historians of science should be of interest to epistemologists for one clear and overriding reason. It furnishes a theory of knowledge which exhibits knowing as a social process, and knowledge as a collective accomplishment. Such a claim should not be underestimated. The sociology of knowledge challenges much that has been put forward in the name of epistemology. There are a number of dimensions along which that challenge proceeds. First, the work, which has generated a social conception of knowledge is concrete rather than abstract. All too often philosophers have distanced themselves from the contingencies of real, historical cases in favour of logical formalism and displays of technical virtuosity. Second, the sociological approach is naturalistic rather than normative. The word ‘normative’ is not the opposite of ‘naturalistic’, but one way to evade the discipline of naturalistic enquiry is to retreat from the world of fact, the ‘is’, into a world of unsituated ‘oughts’, ideals and free-floating values. Concern with how a ‘true’ or ‘ational’ scientist ought to behave can be an excuse for avoiding the question of how actual passages of scientific work proceed. Third, and most important of all, the sociology of knowledge challenges the widely held individualism that permeates epistemology.

Essay Review: Popper's Mystification of Objective Knowledge

I will first make some preliminary points about the word ’objective’. This will give substance to issues which are in danger of becoming too rarefied. Second, I will outline Popper’s account of objectivity. Third, I will argue that despite the value of what he says, his approach is seriously misleading. I will propose a formula for systematically transforming Popper’s theses and exposing what is important in them. This ’transformative method’ points the way towards an entirely different conception of what makes knowledge objective.

The Sociology of Reasons: Or Why “Epistemic Factors” are Really “Social Factors”

The critics of the sociology of knowledge have produced a variety of interesting responses and counter arguments. Their objections are launched from very different standpoints, and the profile of their concessions is different in each case. On the one hand there is the sympathetic treatment of Gutting, who accepts the ‘symmetry’ requirement for sociological explanation, but denies its relativist implications. On the other hand, there is the more sweeping dismissal of Jarvie, who wants to sever the mere beliefs of scientists from what he calls ‘science as such’. Perhaps the line of critical commentary that contains most promise is that of Nicholas. His observation, that there is a considerable overlap between the ‘interest model’ of belief and recent work in decision and confirmation theory, is a valuable one. The programme of relating different inductive strategies to socially structured utilities is certainly one in which sociologists, historians and philosophers could cooperate.1 Gaston’s down-to-earth emphasis on the learning process indicates a similar point of contact between the disciplines. The most intriguing and colourful of the criticisms, however, comes from Butts.

Remember the Strong Program

Before arriving at the Science Studies Unit in 1967 I took a degree in experimental psychology at Cambridge. Richard Gregory and Donald Broadbent were just two of the fine group of experimentalists Cambridge housed at that time. My exposure to the Psychological Laboratory and the Applied Psychology Unit was intellectually exciting, and I acquired a great respect for the discipline. It provided me with an orientation I have never lost. I want to make use of that orientation now, as a way of introducing a discussion of the sociology of knowledge and, in particular, of the so-called strong program.1 This may seem an odd procedure because it straddles disciplinary boundaries, but I hope I can show that such worries are misplaced.

Essay Review: Two Paradigms for Scientific Knowledge?:

The growing interest in the sociology of science makes the publication of this collections of papers’ particularly timely because, in broad outline, it deals with the clash which occurs when the sociological approach makes incursions into the field normally occupied by philosophers of science. On the one side we have the Popperians-I. Lakatos, J. W. N. Watkins and Sir Karl himself. On the other, contributing the opening and closing papers, we have T. S. Kuhn, forcibly (though not uncritically) assisted by Miss Margaret Masterman. Sniping from between the trenches we have S. E. Toulmin, P. K. Feyerabend, and L. Pearce Williams (offering a very short paper). The fight that follows has all the vigour that one would expect from the impressive list of contributors, and some interesting changes of position emerge.

Boundaries, Contingencies and Rigor Thoughts on Mathematics Prompted by a Case Study in Transonic Aerodynamics

A case study in the history of transonic aerodynamics, circa 1950, is used as a basis for reflecting on the character of the distinction between pure and applied mathematics, including the mathematics used by engineers. The case study is set against an historical background of disciplinary confrontation led by such eminent representatives of mathematics and aerodynamics as Garrett Birkhoff and Theodore von Kármán. The successful attempt to construct an adequate account of the aerodynamics of the transonic realm highlighted some sharp differences in the procedures and preferences of mathematical practitioners operating in different fields. The existence and general character of these differences is already widely acknowledged but the task of exploring them from a sociological standpoint still requires much work. The nature of the disciplinary distinctions between different areas of mathematics is examined using Barnes’ theory of (idealized) natural-and social-kind terms (so-called N-and S-predicates). Although the ultimate status of the disciplinary boundaries turns out, uncontroversially, to be ‘conventional’, the attempt to make out and exhibit the conventionality in detail proves to be a non-trivial exercise. It transpires that a thorough study of the issues turns on deep questions about the nature of mathematical rigor and a process that might be called ‘the exploitation of contingency’. These points are illustrated in detail by reference to the technical work of the area (in which one of the authors was an active participant).

The dialectics of metaphor

Two points of contact are explored between contemporary philosophy of science and Dialectical Materialism. The first point deals with the interaction view of metaphor as an exemplification of the law of the unity of opposites. The contradiction is then noted between the strategy and tactics of much analytical philosophy and the lesson to be learnt from this account of metaphor. The concern to change category habits into category disciplines rules out the process of conceptual change of the interaction view. G. A. Pauls dismissal of Lenins theory of reflection is then criticized in the light of the interaction view.