Kuhn, Lakatos, and the Image of Mathematics

EDUARD GLAS*

Because of both serious intrinsic difficulties and extrinsic difficulties in ac-counting adequately for the dynamic nature of scientific change, the re-ceived, logical-positivist view of science had to be abandoned. In its place came a widely varying range of 'post-positivist' alternatives, whose com-mon denominator consists, roughly, in that they emphasize the communal nature of sciences as cognitive practices drawing on socially shared and socially transmitted goals, methods, standards and criteria. Attempts to expand the new views also over mathematics lag far behind what has been achieved already in empirical sciences. In this article I want to explore possibilities of bringing post-positivist views of science to bear also on the field of mathematics, by way of a convergent accommodation of the Kuh-nian sociohistorical perspective on science and the Lakatosian programme Of radical assimilation of mathematics to science.

Logical positivism held that not only mathematics but science as well must be amenable to axiomatization in a certain canonical form. The logico-mathematical vocabulary of scientific theories was construed as ana-lytic, devoid of falsifiable content, and had to be neatly demarcated from the (synthetic) observational and theoretical vocabularies, construed as disjoint classes of terms. Correspondence rules had to supply the theoretical voca-bulary with at least a partial interpretation (Suppe [1977], pp. 16-56). To begin with the correspondence question and the associated observational-theoretical distinction, virtually all aspects of this view have come under serious attack. Its very starting point, viz., the idea that scientific theories must be rationally reconstructible as partially interpreted axiomatic theo-ries, now belongs to a past age. The received view of mathematics, however, seems not to have been affected in any direct way by the criticisms levelled against the said received view of science. The analytic-synthetic dichotomy has been criticized, but mainly with regard to the question of the strict sep-arability of the logico-mathematical from the theoretical and observational terms in scientific theories, leaving 'pure' mathematics largely untouched. • Department of Mathematics, Delft University, P. 0 . Box 5031, 2600 GA Delft, The Netherlands.

Most post-positivist alternative views of science implicitly or explicitly ex-clude (pure) mathematics from their domain of validity. For instance, Kuhn has initially limited the scope of his developmental model to natural sci-ence. He does now allow, however, that in mathematics, too, revolutions may occur—in which case there is no analogue to the crises which usually (but not necessarily) precede scientific revolutions.1

Clearly, a theory that has been expressly developed to account only for the nature and development of natural science cannot simply be trans-planted or 'applied' in a different field. Therefore, the question is not whether the full set of Kuhnian concepts could be applied to mathematics (it obviously cannot, cf. Gillies [1992]); the relevant question is whether the typically Kuhnian approach to science, as a cognitive practice guided by 'paradigmatic' forms of problem and methods of reasoning, may be brought to bear also on mathematics. I have argued before (Glas [1989b], p. 171) that the notion of Gestalt switches characterizes psychological pro-cesses that may or may not accompany shifts of practice, but does very little to explain the latter. Whereas shifts of Gestalt, at least in the origi-nal sense of the term, are not consciously reasoned, for radical changes of practice there can be given very good reasons indeed, even though they need not be conclusive for adherents to the older practice (Glas [1993b]). Kuhn himself has insisted again and again (e.g., Kuhn [1983]) that there are extra-paradigmatic criteria for judging the rationality of revolutionary change, although they are not in general sufficiently coercive to enforce con-vergence of judgement when the change is radical enough. I think, therefore, that Kuhn's account allows of giving primacy to scientific practice; on that premise, the GestaJt in which the world appears to us is the product of our ways of interacting cognitively and experimentally with it rather than the other way around. By focussing on the level of cognitive practices rather than that of the psychology of perception, talk of Gestalts and the accom-panying notion of crises can be skipped, and the prima facie inapplicability of Kuhn's views to mathematics vanishes. In particular the much-contested thesis of incommensurability, if read at the cognitive-practical rather than perceptual-psychological level, can be given a sense which applies no less to mathematics than to science. Practices are incommensurable when no conclusive reasons can be produced to enforce a decision between them on penalty of irrationalism (this idea will be clarified in what follows).2

1 'In mathematics, crises must be foundation crises and they are at best very, very rare. Revolutions can and mostly do occur without them. But in physics... any problem that defies the best efforts of the profession for a number of years is ipso facto a foundation crisis. Crises are not the only prelude to revolution, but they are quite usual.' (Personal communication of 31-10-87.)

2 Recently, Michael Malone [1993] has has also argued that the GestaJt interpreta-tion misrepresents Kuhn's own more fundamental incommensurability thesis, which can in his view be explained and defended more adequately without invoking

perceptual-The main remaining obstacle to accommodating the Kuhnian perspec-tive to the field of mathematics seems to be the evident difference in logical status between empirical and mathematical statements. However, confer-ring a different logical status on mathematical as opposed to empirical statements is by no means sufficient for drawing a strict line of demarca-tion between mathematics and natural science along this difference. Neither science nor mathematics just consists of sets of statements, let alone state-ments of a singular logical kind, but of transient complexes of concepts, models, theorems, rules, etc. Here we obviously touch on the old analytic-synthetic distinction on which the received view depended.

It is precisely this distinction that has been challenged, conclusively in my view, by Lakatos's programme of radical assimilation of mathematics to science, as well as by numerous ensuing post-positivist accounts of mathe-matics in non-deductivist and quasi-empiricist terms.3 In this paper, I will be concerned mainly with Lakatos's version because his emphasis on the methodological 'science-likeness' of mathematics is more adequate to my purpose than, for instance, Kitcher's primarily epistemological project of founding mathematical knowledge directly on an empirical basis. Indeed, any straightforwardly empiricist account of mathematical knowledge faces problems at least as serious as those to which the logical-empiricist philos-ophy of science succumbed. Therefore, the viability of (quasi-)empiricist philosophies of mathematics depends also on whether they are able to cir-cumvent or solve the problems that vitiated the received empiricist account of science. 'The gravest danger then in modern philosophy of mathemat-ics is that those who recognize the fallibility and therefore the science-likeness of mathematics, turn for analogies to a wrong image of science', says Lakatos ([1978b], p. 42). The Lakatosian programme at least tack-les these problems simultaneously and symmetrically, and moreover in an objectivist fashion rather than in Kitcher's 'subjectivist' way in terms of 'psychologically' warranted beliefs. Since I consider psychological talk re-dundant for explaining scientific change, it would not be in line with my project to reintroduce it as a way of accounting for mathematics.

Not only is mathematics in a great number of relevant respects 'like' an empirical science, empirical science is also much more 'like' mathematics than logical positivists presumed. Empirical sciences are far from 'em-pirical' in the sense intended by the aforesaid logical distinction between mathematical and empirical statements. On the contrary, the conceptual apparatus and organization especially of the more advanced physical sci-ences is largely non-observational and self-supporting. It is no coincidence psychological issues, which he considers too confused and questionable in themselves to be of much use.

3 See, for instance, Kitcher [1984], Tymoczko [1985], Van Bendegem [1988, 1989], As-pray and Kitcher [1988].

that the most successful scientific theories came into being only after much of the relevant mathematics had already been developed. Whereas on the received view new informative input into theories can only consist of new observational material, it in fact very often consists of fruitful mathemat-ical or other 'abstract' ideas (as for instance the 'principle of relativity'). The derivation of testable consequences mostly requires much theoretical effort, and their confrontation with observational material depends itself on various assumptions concerning the meaning of theoretical terms, the inter-pretation of observations, theories of experimental design and procedure, numerous ceteris paribus clauses, etc.; briefly: on more basic 'observational' theories. Testing a scientific theory ultimately amounts to checking its con-sistency with respect to lower-level theories ('like' mathematics) and it is always possible (though not always rational) to resolve 'empirical' difficul-ties without abandoning the questioned theory. Once they are couched in formal language, physical theories are no more directly 'refutable' than their mathematical counterparts, yet nobody would deny them their empirical significance for that reason. Nor should the impossibility of empirically falsifying mathematical theories be regarded as sufficient reason for deny-ing mathematics all claims to empirical significance. Formal mathematics surely can be immunized effectively against all informal counter evidence, but to a considerable extent this holds also for theoretical science and in both domains our actually choosing to do so implies resigning ourselves to stagnation of the growth of knowledge.

1. Science-likeness of Mathematics

The most central analogy between science and mathematics in Lakatos's philosophy is that between an experimental test and an informal mathe-matical proof. As scientific experiments are intended to hook theories onto observational statements that are provisionally accepted as unproblematic, so informal mathematical proofs are intended to embed conjectured theo-rems in trivially true or already proven statements belonging to the pro-visionally taken-for-granted background. Such proofs therefore function as Gedankenexperimente (thought experiments). In Proofs and refutations (Lakatos [1976]), Lakatos showed how mathematical development can be reconstructed as a dialectic process of 'improving by proving and disprov-ing', consisting in quasi-experimental attempts at once to embed conjec-tures in accepted background knowledge ('proofs') and to construct heuris-tic counter examples in this background ('refutations'). The role of proofs in the development of mathematical knowledge hence is similar to that of critical experimental tests in natural science, very unlike the role that standard justificationist accounts assign to them. On Lakatos's account, proofs—always in the informal sense—do not prove infallibly, but despite their flaws are the vehicles of mathematical progress. In contradistinction

to the classical logic of justification, the 'developmental logic' of Proofs and refutations turns on the fallibility (in the sense of improvability) of mathe-matical claims to knowledge. It is a methodological falsificationism, aiming at methods that enable us to get at better and better theories, no theory being incorrigibly true. Progress rather than truth is the Lakatosian stan-dard of mathematical achievement, for it is precisely the dogmatization of mathematical truth that incites us to shield our theories from falsification rather than to use counter examples constructively in order to get at richer and deeper theories. Non-logical falsifications can always be countered by appropriate redefinition of terms, specification of extra conditions and ex-ceptions; briefly: by radical withdrawal within a safe domain. But such 'immunizing' is ad hoc, places possibly undue restrictions and limitations on the meaning, scope and heuristic power of otherwise promising theo-rems, hence tends to turn fertile conjectures into sterile truisms. Progress demands that we must never settle definitively on allegedly ultimate math-ematical truths.

So the science-likeness of mathematics was situated by Lakatos at the methodological level. The continuity between the methodological falsifica-tionism of Proofs and refutations and the later, sophisticated methodology of research programmes, shows conspicuously in a piece that in Lakatos's collected works bears the appropriate title 'How failed attempts at refu-tations may be heuristic starting points of research programmes' (Lakatos [1978b], chapter 5). In it, Lakatos literally reconnected his 'final' views (1973-74) to his first endeavours in the methodology of mathematics (1956-61). It concerns the 'method of analysis and synthesis', used by Greek mathematicians as early as the fourth century BC, expounded explicitly by Pappus in the fourth century AD, and revived in the seventeenth cen-tury in Galileo's 'resolution and recomposition' and in Descartes's 'univer-sal method'. Since Lakatos's method obviously goes on in the same vein, his 'historical' comments shed the necessary light on how he saw the rela-tions between the axiomatic ('Euclidean') and the quasi-experimental side of mathematics.

Analysis consists of the systematic derivation of all consequences of a tentative conjecture: what must also be true if the conjecture were true? If in so doing one reaches already proven or trivial truths, one can try to demonstrate that the conjecture in fact follows from the said truths (syn-thesis). If on the other hand some consequence turns out to be evidently false, the conjecture must have been false, too. According to Lakatos, the Greeks had inherited from the barbarians an enormous stock of geometri-cal conjectures, which they put to the test of quasi-experimental analysis in the sense depicted. After a great many of such tests (thought experi-ments) had been performed, some lemmas appeared retrospectively to be corroborated—they stood up to attempts at falsification—whereas their

al-ternatives remained sterile. The many tested conjectures in this way yielded series of corroborated analytical components which converged on a small number of evident truths. These 'elements' became the 'hard core' of pro-grammes of axiomatization such as Euclid's. Once the axiomatization had been accomplished, the testing of conjectures could be exchanged for direct axiomatic proofs: 'Analysis as a means of getting to novel lemmas and axioms lost its function; when used at all it was only a heuristic device for mobilizing the—already proven or trivially valid—lemmas necessary for the synthesis. Analysis was not any more a venture into the unknown; it was an exercise in mobilizing and ingeniously connecting the relevant parts of the known. The lemmas which were once daring and often falsified conjectures hardened into auxiliary theorems' (Lakatos [1978b], p. 100). Lakatos made it perfectly clear that the way in which in Proofs and refutations proofs are analyzed in order to trace errors and hidden assumptions, and thus to get at improved theorems-cum-proofs, is wholly in line with the historically approved method of analysis: 'Thus in analysis we test—Popperianwise—a conjecture; but if we fail to refute it, we may succeed in turning it first into a proof and then into a mathematical research programme' (ibid. p. 96).

So Lakatos construed axiomatic theories as axiomatizations of contentful theories. 'Geometry existed before the Euclidean system, number theory existed before Peano's axioms', etc. (ibid. p. 100). The informal devel-opment must retrospectively have corroborated the underlying principles before axiomatization can start. The axioms derive their weight from the corroborated contents of the theory which they must logically entail. The acceptability of axioms thus depends (under the side condition of consis-tency) on the extent to which they can account for these contents. There-fore, the informal development has not only temporal priority, but logical priority as well. The truth even of axioms must not be dogmatized; they must be 'improvable' in the light of the growth of contentful knowledge. Lakatos thus construed mathematics as a 'quasi-empirical science' in the following sense. Whereas the Euclidean ideal demands that the primary truth-value injection into mathematics is at the 'top' (the axioms), and the crucial logical flow is top-down (from the axioms into the theorems), on the empirical view of mathematics this order is reversed. In a quasi-empirical science the primary truth-value injection is at the bottom (the informal models or interpretations of the theory), and the crucial logical flow is the retransmission of error elimination at the informal basis up to the axioms that are supposed to capture and explicate the contents of the theory. Axiomatization is aimed at a more exact and consistent explication of the contentful theories we had in mind in the first place, not at their replacement by mechanical rule-following in a system of meaningless signs. The informal interpretations or models of the formal theory are the sources of potential falsifiers: it is by making up for errors at the interpretative

level of axiomatic theories that the informative contents of these theories are made to increase.

As 'quasi-empirical sciences', mathematics and natural science are thus placed on a common, 'fallibilist' epistemological basis. However rudimen-tary, this basis is sufficient to support the falsificationist methodology that in a perfectly symmetrical fashion applies to science and mathematics alike. Historically, 'sophisticated methodological falsificationism' derived from the transfer of methodological ideas from mathematics to science, but its up-shot in the form of the methodology of research programmes offers many methodological ideas that can be transferred perfectly well back to math-ematics. As Lakatos himself noted, in Proofs and refutations he had not yet made a clear terminological distinction between theories and research programmes, and 'this impaired my exposition of a research programme in informal, quasi-empirical mathematics' (Lakatos [1978a], p. 52). Indeed, what for lack of a better name is called 'Euler's conjecture' can be construed as the hard core of a research programme. The idea that the relationship V — E + F ~ 2 between the numbers of vertices (V), edges (E) and faces (F) expresses some fundamental feature of polyhedra, is retained through-out the series of modifications of the conjecture in the face of informal counterexamples. As the 'irrefutable' raison d'etre of the whole series, it is constitutive of the continuity of the 'programme' in which the conjecture is unfolded.

The positive and negative heuristics of MSRP can likewise be retraced already in P&R.4 If the 'hard core' (Euler's conjecture in P&R) is promis-ing enough, it must be protected from straightforward falsification lest all development stops even before the programme has been able to deploy its full scope and heuristic potential. So the critical confrontation must take place a sufficient number of argumentative steps away from the hard core, in a 'protective belt' of auxiliary assumptions. This negative heuristic— redirecting the modus tollens from the hard core to the protective belt— may entail losses of scope and content, but not more than is strictly required for the protection of the hard core. It is a strategic withdrawal, not a wild retreat via ad hoc redefinitions and the ad hoc introduction of restrictions and exceptions. On the other hand we also find in P&R examples of the positive heuristic—progressive extension and improvement of the protective belt—in the form of deductive guessing and theoretical concept stretching. Counter examples furnish us with indispensable clues for guessing a better conjecture to which they are counter examples no longer. To this end we can deliberately generate counter examples by stretching the scope of con-cepts beyond their proper extension. In this way we will get the very best information for achieving the greatest expansion of the content and scope

4 I will henceforth use these abbreviations for the 'Methodology of scientific research programmes' and for 'Proofs and refutations'.

of our theories.

In MSRP Lakatos formulated as a general criterion of progress the anti-cipation of novel facts that the theory was not specially designed to account for. Each theoretical shift should in this sense be content-increasing, while every now and then this increase of content should appear to be retrospec-tively corroborated. Lakatos thus required of research programmes that they show consistently progressive theoretical shifts and at least intermit-tently progressive empirical shifts (Lakatos [1978a], pp. 31-37, 49). Hallett [1979] has shown how the criterion of theoretical progress might be retrans-ferred to mathematics and then yields what is known as 'Hilbert's rule' to judge a new mathematical theory by its potential 'to solve problems that it was not specially designed to solve'. As a criterion corresponding to 'empirical' progress I would propose the requirement that new theories have surplus content over their predecessors, i.e., capture their unfalsified components and at least part of the ground on which the predecessors were falsified (for instance, Eudoxus's theory of proportionality encompassed the older Pythagorean doctrine for the rational case, but covered also the irra-tional case which constituted a counter example to the Pythagorean theory, cf. Glas [1993a], pp. 57-60). Thus mathematical research programmes are to be judged by whether they consistently show theoretical progress in the sense of Hallett/Hilbert, and intermittently also growth of quasi-empirical content in the sense depicted by me.

I think the scientific-mathematical symmetry of Lakatos's methodolo-gical falsificationism has now been established—or at least the possibility of reconstituting the symmetry between the Methodology of Scientific Re-search Programmes and the Logic of Mathematical Discovery (= P&R). We are now in a position to take the historical development of mathema-tics rather than science as a testing ground for Lakatos's claim that his methodology

is sufficient to escape Kuhn's strictures... my concept of a 'research programme'

may be construed as an objective, 'third-worid' reconstruction of Kuhn's socio-psychological concept of 'paradigm': thus the Kuhnian 'Gestait-switch' can

be performed without removing one's Popperian spectacles (Lakatos [1978a], p. 91).

2. Mathematical Development

On Lakatos's account, new mathematical theories are not generated and established by deduction from rock-bottom 'first principles', but designed quasi-experimentally and appraised 'relative' to their predecessors or com-petitors. The units of appraisal are programmes, series of successive ver-sions of corrigible theories drawing on a shared hard core of basic assump-tions. Pursuing a programme implies not allowing this hard core to be refuted. Programmes are judged by whether the successive shifts are in the

longer run progressive, especially in comparison with rival programmes. As only relative increase of content is assessed, the contents themselves of theories from different programmes cannot be directly compared. The epistemological basis of Lakatos's methodology does not allow of compar-ing the contents of theories, i.e., the sets of their potential falsifiers, across the boundaries between programmes. Prom the fact that a programme is degenerating (losing content) it cannot be inferred that its content has in fact dropped 'below' that of a progressive competitor. Also any stagnant or degenerative programme might some day be revived and appear to pos-sess more heuristic potential than the rivals that once seemed to be more progressive. Therefore, Lakatos is absolutely right in stating that 'it is not irrational (my emphasis) to stick to a degenerating programme until it is overtaken by a rival, and even after'1 (Lakatos [1978b], p. 117). The

'rela-tivity' of progress makes methodological appraisal irremediably ambiguous. Indeed, if choices could be unambiguously rationally argued, rational sci-entists would change their commitment all at the same time in the same sense, and there would be no serious rational competitors. But the method-ology makes rational choice entirely dependent on the comparative assess-ment of the progress of competing programmes. Indeed, the 'sophistication' that distinguishes Lakatosian from Popperian falsification consists in that counter examples are counted as falsifications only if they are positive ex-amples for a rival programme (Lakatos [1978a], p. 35). So any pretension of forward-directedness would be self-annihilating for Lakatos's methodology. It cannot but be backward-directed, a rationalization ex post facto, and has itself to be judged by its potential for explaining (reconstructing rationally) the historical development of mathematics and science (cf. Glas [1993a]).5

5 The title of T. Koetsier's book, Lakatos' philosophy of mathematics: A historical

approach (Amsterdam: North Holland, 1991), suggests that it is concerned with a

histo-riographic appraisal of Lakatos's philosophy. However, this philosophy is very seriously misrepresented and its confrontation with history is done in a way entirely alien to Lakatos's methodological intentions as expounded in 'History of science and its rational reconstructions' (Lakatos [1978a], pp. 102-138)—a crucial primary source totally ignored by Koetsier.

Koetsier wrongly construes the Lakatosian methodology as an ars inveniendi, a set of rules for making mathematical discoveries, which are tested historically on the following criteria; (a) Have they led to mathematical discoveries in the past? (b) Do mathemati-cians recognize in them their own successful procedures? (c) Can the methods applied by great historical figures be reconstructed as instances of these rules? Needless to say, Koetsier's ideas of what methodology is, what it is for, and how it is to be appraised historically, have nothing whatever in common with Lakatos's ideas.

The bulk of Koetsier's book consists of lengthy 'reconstructions' of historical episodes in terms of his own 'refined' version of the methodology, the 'methodology of research projects and traditions'. Mathematics is developed in 'traditions', characterized by a 'fundamental domain' consisting of the most general mathematical entities under inves-tigation. Within traditions we find 'research projects', characterized by a certain unity of approach. Mindful of Lakatos's warning that historiography is already impregnated with methodological views, Koetsier proposes to regard the consensus among established

his-It is obviously not the task of methodology to account for all tortuous ways in which claims to knowledge have in fact been generated, modified, corrected, etc. in actual history. However, a methodology must be able to explain rationally the development of knowledge at 'third-world' level, the level of depersonified, objectively articulated products (theorems, proofs, theories), distilled from the real history with all its personal and situational contingencies. Lakatos's methodology is aimed, not at historiogTaphic ex-planations of scientific development, but at the rational reconstruction of what is considered to be its third-world equivalent, to which alone methodo-logical standards apply. Methodology has itself to be appraised by the ex-tent to which it is able to perform this task without having to appeal to factors that are 'external' to it, hence 'non-rational' by its intrinsic metho-dological standards. 'Internal' history in this—unusual—sense is said to be self-sufficient, and 'external' history irrelevant for understanding the growth of disembodied objective knowledge. The Lakatosian reconstruction of the history of science and mathematics thus presupposes that its rational as-pect is fully accountable by the logic of discovery, whereas external factors are invoked only for the non-rational explanation of the remaining discre-pancies between real history and its rationally reconstructed logical sub-stitute. The programme thus depends crucially on the assumption that at a certain level of description, the level of objectively articulated products, mathematical development can indeed be reconstructed as resulting from a method manifesting a rationality which transcends all social and cultural dependencies.

I contend that this assumption is not borne out by historical scrutiny, and is moreover redundant for rationally explaining the growth of mathe-matical knowledge (Glas [1993b]). Why should non-methodological factors a priori be deemed non-rational and irrelevant for understanding mathe-matical growth? Perhaps veritable progress depends crucially precisely on non-methodological factors! Some significant changes in mathematics can in fact be shown to defy the Lakatosian method, cases which it would be exceedingly odd to classify for that reason, as Lakatos would be obliged to

torians as a basis of methodologically neutral 'factual statements'. He accordingly moulds his 'methodological' account on 'views regularly expressed in historiography', which he finds in the secondary and tertiary literature of the early 1970s and before (the vast more recent literature being almost entirely ignored). In fact his 'method' just consists in extracting the 'greatest common divisor' from the standard literature written under the received view, and translating it into talk about research traditions and projects.

A telling illustration is to be found in his 'rational reconstruction' of the development of mathematical analysis. He claims that the eighteenth century was dominated by the Formal Tradition, whose 'fundamental domain', i.e., its associated set of 'most general entities', is identified as consisting of formulas. This example is representative of the level of philosophical understanding to which the Koetsierian approach leads. I think this suffices as an explanation why I am regrettably unable to find in Koetsier's work a constructive contribution to the problematics dealt with in the present article.

do, as 'non-rational', 'external' and 'irrelevant for our understanding'! It is now necessary to recapitulate the essentials of some of those cases.

The opposition between the analytic and the synthetic approach to mathematics in the first half of the nineteenth century is well known. In mathematical physics, Grattan-Guinness [1981] has traced a dichotomy between 'analysis' and 'calculus/descriptive geometry'-directed practices. Less well known is that both differences were rooted in a highly conse-quential rupture in the mathematical community in the wake of the French Revolution, a 'culture clash' between the established analytical tradition headed by Lagrange and Laplace, and a new, geometrically oriented ap-proach, led by the revolutionary 'upstart' Monge and followed by his re-putedly non-conformist students at the 'republican' Ecole polytechnique of Paris (Glas [1986]).

These leading mathematicians of the 1790s had to account for their dif-ferent views of mathematics in response to the government's demand for normalization and rectification of the language of mathematics. The 'ana-lysts' stuck rigidly to the terms of this assignment, setting out to purge the discipline from all informal, quasi-empirical elements, including all appeals to geometrical and kinematical 'models'. They construed mathematics pri-marily as a computational device for the exposition of the mechanistic World System, conceived as the legitimatory model of Natural Law for an enlightened egalitarian society. According to Laplace, 'analysis has the in-estimable advantage of transforming reasoning into mechanical procedures' (Laplace [1796], p. 464), thus forgoing the discriminatory use of imagina-tion. Monge, too, emphasized the importance of the 'language' aspect for the communication of objective knowledge, yet emphatically denied the reducibility of mathematical reasoning to 'mechanical' rule-following. He insisted on the indivisibility of form and content, and denied that progress could be achieved through formal procedures. Lagrange and Laplace on the other hand believed that the development of mathematics was virtually completed, so that for them progress could consist only in further formal perfection.

Monge's descriptive geometry was not an application of extant theories to practice but, conversely, the exact and systematic theoretization of an extant practice which up to then had developed largely independently of main-stream mathematics. Thorough systematization and exact theoretiza-tion of this practice were urgently needed in view of the shifting exigen-cies of a society undergoing revolutionary political and industrial changes. The heuristic power of Monge's new approach depended on the combina-tion of impressive synthetic ability and analytical skill (rather than either pure analysis or pure geometric synthesis) which revolutionized projective, analytical and differential geometry, and also brought considerable progress

even in abstract analysis. The issue must not be interpreted simply in terms of trivial differences between methods in neatly demarcated fields of re-search (analysis and geometry), but was concerned with incompatible views of mathematics as a whole. The analysts regarded Monge as an eminent geometer, but considered geometry irrelevant for the development of the formal doctrine. Monge, on the other hand, was far from denying the me-rits of Lagrange's and Laplace's achievements, but considered them formal exercises rather than contributions to substantive mathematical progress. Nor must the issue be characterized in terms of the distinction between pure and applied mathematics. The analysts were no less concerned with applications than Monge, and the latter was as much theoretically inclined as Lagrange and Laplace. But they differed in their appreciation of where exactly the theoretical interests of mathematics lay, and of which sorts of applications (in mechanistic science or technical-industrial practice) were legitimate landmarks of mathematical attainment.

Monge on the one hand, and Laplace and Lagrange on the other, en-tertained incommensurably different images of mathematics. For Lagrange and Laplace, analytical expressions were syntactically ordered strings of signs, to which all mathematics had ultimately to be reduced. Geometry was just a form of ''facile' pictography suitable to practitioners. For Monge, on the other hand, analysis was just a suitable 'script' (Venture) to denote conceptual operations and transformations which could always be given a geometrical interpretation and ultimately derived their meaning from the spatio-temporal structure of reality. The two parties entertained incompat-ible views of the very basics of the field, as for instance the nature of mathe-matics (formal language or conceptual science), its object (logical perfection or functional expansion of knowledge) and methods (analytical reduction or quasi-empirical exploration). The approaches were incommensurable in the sense that the rational ground that they shared was insufficient as a basis for arguments coercive enough to settle the case rationally (cf. Glas [1989a]). So this notion of 'incommensurability' does not imply that logical contact was hampered by radical meaning variance between the vocabula-ries of the two parties, nor that rational arguments could not transcend in principle the boundaries between their different outlooks on the field. But logic could not force them to shift their positions and thereby to impair the intrinsic value of their own practices: by no rational standard could they have been compelled to abandon the framework in which they had invested their entire professional competence and credibility. It was not irrational for either party to stick to the 'image' that conferred the highest value on the achievements and proficiencies embodied in their own practice. In this sense it may be said that the 'image' of mathematics is the product of the mathematical practice rather than the other way around.

Much the same can be said about another case: the confrontation be-tween Klein (joined by Hilbert and others) and the 'neo-humanist' tradition that had dominated German mathematics throughout the nineteenth cen-tury. Neo-humanism, with WeierstraB as its champion, was strongly anti-utilitarian and preoccupied with the formative value of 'pure' mathemati-cal thought, embodied for instance in the programme of 'arithmetization' of mathematics. Lack of 'purity' was regarded as 'uncleanliness', and the 'purists' therefore divided the field into neatly delineated, self-contained specialties, capable of standing up to the most rigorous logical standards. In their quest for infallibility they shunned the grey areas between the spe-cialties where the intended logical closure and rigour were simply not to be had. Thus the pursuit of neo-humanist ideals tended to split up mathe-matics into formally self-contained specialties, insulated and immunized against social, political and other unclean sources of contamination. The purist image embodied its own paradigmatic forms of problem, patterns of reasoning and standards of appraisal all in one, as one normative whole. Consequently, it could only be seriously challenged by a 'whole' new image, with its own internally coherent and consistent set of normative problem forms, methods and evaluative criteria. These requirements were met by the 'structural' image that I have undertaken to reconstitute from Klein's works in and about mathematics (Glas [1993c]).

Trained as a mathematical physicist, Klein had a strong feeling for con-nections between problem fields that in the purists' perception were entirely unconnected. A case in point was his recognition of a connection between Cayley's concept of projective distance and hyperbolic non-Euclidean geo-metry. When he brought up this point in a seminar with WeierstraB, the latter gave as his verdict that 'these spheres of thought are evidently en-tirely unconnected', a verdict which did not prevent Klein from constructing his now famous model for embedding non-Euclidean geometry in projective geometry. He went on to show how both Euclidean and non-Euclidean ge-ometries can be subsumed under the more 'elevated' point of view of pro-jective geometry, the metric varieties being special cases or sub-geometries of projective geometry. In the subsequent Erlanger Programm, all varieties of geometry (some newly invented by the programme itself) were placed in the unitary structural perspective of the theory of groups—a concept ap-plied so far only to algebraic problems. Here again Klein saw connections between fields that in the purists' perception were entirely unconnected. He succeeded in characterizing the great variety of geometries in all-embracing structural terms as 'the invariant-theories of groups of transformations act-ing on a manifold'.

Klein's conception implied a radical break with the still prevalent (Kan-tian) mode of grounding 'pure' mathematics in 'pure' mental forms of ap-perception and reasoning. In Klein's conception, geometry is concerned

with relations of togetherness which are not 'in' (pre-given) space, but which constitute their own space of reference. There is not one uniquely or necessarily true geometry: alternative (for instance Euclidean and non-Euclidean) geometries are epistemologically equivalent ways of representing and modelling the relations constituting their associated space.

The group-theoretic characterization generates a taxonomy in which the various geometries appear as generically related theories and sub-theories of groups and sub-groups of space transformations. For instance, Euclidean and non-Euclidean metric geometries appear as sub-geometries of protective geometry, as the metric group (of translations, rotations and reflections) can be conceived as a sub-group of the projective group. This implies that all theorems of metric geometry can be construed as special cases of theorems of projective geometry. To put it in still another way: geometries may be handled as models for other geometries, at different levels of the taxonomic system. Such models are on Klein's view indispensable for the consistent elaboration of theories on higher levels of abstraction and generalization— 'having a model' ascertaining at least relative consistency—and at the same time warrant that they will have relevant conceptual interpretations. Klein kept viewing mathematics as a conceptual science, and formalization as a means of improving and refining conceptual knowledge, not as an end in itself.

Klein saw mathematics not primarily as a collection of statements de-rived from a pre-given set of necessary truths, but as a collection of concep-tual models to which through a process of logical analysis and explication a suitable axiomatic basis is supplied. The primary stuff of mathematics consists of models (of structures), and any decision concerning the accep-tability of a proposed set of axioms depends on how well they serve as premises for the deduction of the conceptual 'facts' about these models. In Klein's account of mathematics, the logical form of presentation of theo-ries is subservient to the adequate representation of their contents. Models at once provide the Indispensable heuristic clues for consistent axiomatiza-tion and act as intermediaries for reconnecting the abstract structures to concrete 'applications' (Glas [1993c], sect. 3).

Klein's model view thus revealed numerous overarching bridges between pure and applied mathematics, between its various subdisciplines and spe-cialties, and between mathematics, science and technology, where the 'pu-rist-specialists' could see only unbridgeable gulfs. With its new mode of conceptualizing both the differences and the interrelations between various (sub-)fields of research, Klein introduced a 'language' which differentiated between them in such a way as to allow for their reintegration in a unified whole on a higher level. The neo-humanist 'language' of classifying in closed compartments (pure/applied, formal/conceptual, analytic/synthetic, etc.) was thereby exposed as belonging to an obsolete image of mathematics,

which had to be exchanged for the new 'structural' image and language, in which they were 'seen' to be complementary rather than contradictory aspects of one integrative approach, as just two sides of one coin.

Lakatos's methodology may well be adequate for assessing the progress achieved within the programmes pursued on either side of the lines that separated the Mongian from the Laplacian approach, and the Klein-Hilbert 'Gottingen' from the 'Berlin' (Weierstrafi) tradition, especially from a later, detached and incomparably more comfortable point of view. But it is defi-nitely incapable of making rational comparisons in purely (methodo)logical terms across the boundaries between them, especially between the different images that informed those programmes, and so to account for the overall transformation of mathematics that ensued from their confrontation.

In fact Lakatos did not even claim that his methodology solved the prob-lem of rational comparison and choice between old and new programmes. Indeed, the methodology required that choices can be made differently, it being 'not irrational' to stick to the old programme even after it had been superseded by a new one! More particularly, the methodology fails to specify unequivocally at which 'critical level' of degeneration continued commitment to a programme's hard core would become irrational—almost the exact equivalent of the reproof often levelled at Kuhn's theory, viz., that it fails to specify unambiguously at which 'critical level' of accumulation of anomalies a paradigm ceases to support a normal practice adequately! What Lakatos offers as a rational reconstruction of the Kuhnian concept of 'paradigm' ultimately reduces to a rhetorical appeal for honesty:

My methodology... allows people to do their own thing, but only as long as

they publicly admit what the score is between them and their rivals. There is freedom... in creation and over which programme to work on but the products have to be judged. Appraisal does not imply advice. (Lakatos [1978b], p. 110) There is no rational warrant, though, that it is always possible to decide what this 'score' is, especially not when the 'game' is itself in question, i.e., when different practices are calibrated by standards that belong to incommensurable metaviews of the nature, objects, methods and values of the discipline.

However, lack of rational warrants is no good reason for resigning our-selves to a radical relativism that no longer allows for any possibility of impartial, objective judgement. What must be challenged is not the legi-timate pursuit of rational criteria but their characterization in exclusively internal 'logical' terms. What is characteristically rational about the de-velopment of mathematics is not restricted to 'logical' relations between successive theories, but encompasses also the 'functional' ways in which conceptual constructs and patterns of reasoning have been progressively better adapted to the ever changing problematics of mathematics in

va-rious practical contexts and cultural milieux. It is the Lakatosian strict separation between the rational and the social that must be challenged.

And so must the Kuhnian separation between the social and the ex-ternal, the processes inside scientific communities and the pressures of the society outside. This is tantamount to abandoning psychological talk about Gestalts and focussing instead on the practice of mathematics in its sociocultural environment. External factors were at least crucially co-determinant of the changes of mathematical practice described in my case histories, and hence also of the accompanying conceptual innovations.

The break-through of Monge's innovation depended not only on its in-trinsic merits but on external changes in the social environment as well. , Without the military imperatives of the revolution and politically induced changes in the power structure of the intellectual community, it proba-bly could not have overcome the competition of the established analytical tradition. The extraordinary quality of the competence and proficiency embodied in Monge's practice, however, showed to good advantage in the military and industrial uses to which he put it, especially in his highly ef-fective way of organizing the armament industry, on which the fate of the revolution depended. In recognition of these merits, he got the opportu-nity to create a Central School of Public Works (which became the Ecole polytechnique) after his own 'image'. The thorough educational reforms effected in this prestigious institution (which later would still inspire Klein) produced a generation of students who stuck to his 'paradigmatic' teachings because they embodied the special competences and proficiencies on which their further careers depended. Monge's approach typically was a venture in as yet unexplored territory, enabling the best of his students to build up in only a few years a splendid reputation as pioneers in various new branches of pure and applied mathematics. We find them in the polytechnic-like institutions which in the wake of the Napoleonic conquests arose in various European capitals. Although by 1806 the Paris Ecole polytechnique had been reformed into an 'imperial' institution under 'Laplacian' rule, by that time it had produced sufficient 'Mongian' offspring to establish a viable tra-dition (also in mathematical physics, as it appears from Grattan-Guinness's findings referred to above).

The trend towards formalization and purification, on the other hand, was taken up by German mathematicians in their struggle for enhanced sta-tus and authority. The mathematics departments at Prussian universities (among them Berlin) owed their much wished-for autonomy and autarky to the early-nineteenth-century secularization of teacher education (Schubring [1989]). The intimate bonds between the academicians and their clients, the Gymnasien (grammar schools), were constitutive of the neo-humanist ideology that impregnated German academic life in general and mathe-matics in particular. Any association with the utilitarian aims of politics

and economy was seen as a threat to their intellectual authority. This ex-plains at least partially their bias in favour of rigorously closed specialties, and their overt hostility towards more 'open', generalist approaches such as Klein's, which violated the splendid isolation of the academic ivory tower. Illustrative, in this respect, is the verdict of the committee that in 1897 had to advise the Minister about the succession of WeierstraB in Berlin:

Candidates must be capable of setting an example to the students of serious and disinterested immersion in the problems of mathematics. For this reason, Prof. Klein from Gottingen (born in 1849), whose scientific merits are much disputed among scholars, and whose literary and educational activities are in contradiction with the aforesaid tradition of our University, had to be left out of consideration. (Biermann [1973], p. 207)

Klein and other former students of Clebsch were after the latter's death virtually ostracized from the German mathematical community. His jour-nal, the Mathematische Annalen, was 'forced to rely almost exclusively on contributions from the Clebsch School and a handful of allies in order to survive' (Rowe [1989], p. 192). One of these allies was Hilbert, with whom Klein shared the overarching structural vision that enabled them to communicate in a common language and join forces. They 'forged an intel-lectual partnership that... overturned the balance of power within German mathematics' (ibid., p. 186). In regard to power, it was very helpful that Klein about 1909 had become Vice-President of the Teaching Committee of the Prussian Parliament and President of the International Committee of Mathematics Education (Tobies [1989], pp. 50-51). These positions en-abled him to win over influential politicians and captains of industry, who could be expected to be susceptible to the acute new imperatives of mili-tary and industrial development. In the face of the new internal and exter-nal demands and challenges, the nineteenth-century neo-humanist ideology eventually had to give way (Glas [1993c], pp. 625-627).

3. Toward a Post-positivist Account of Mathematical Development

The analyses and case studies presented so far substantiate Lakatos's gene-ral thesis of the methodological science-likeness of mathematics. (There is also a more structural analogy in the sense that scientific and mathema-tical theories may both be construed as families of models, of which some sub-structures are isomorphic to the appearances, but this point need not be pursued in the present context.) On the basis of the symmetry that can be established/reconstituted between the logic of mathematical discovery (P&R) and the methodology of scientific research programmes, the main features of the 'methodology of mathematical research programmes'—the MSRP reapplied to mathematics—have been laid down. This adapted ver-sion of the methodology has been shown to be adequate to account

retro-spectively for the development of theories that depend on a shared hard core of basic assumptions, but nothing more. In particular, it does not account for the genesis of a new programme nor for the transition from one programme to another. Attempts to judge conceptual novelties and to make rational comparisons between new and old programmes drive us beyond the scope of this methodology, especially in cases of meta-level inno-vations (such as Monge's and Klein's) that draw on the pooled resources of different programmes in order to tackle problem situations that fall across the boundaries between these programmes. What makes such changes ra-tional is not the observance of canonical (methodo)logical rules, but the functional way in which the conceptual and argumentative structure of the field has been progressively better adapted to the new cognitive and social demands of new internal and external circumstances.

So the Lakatosian methodological perspective must at least be comple-mented by a sociohistorical perspective in the Kuhnian vein in order to account for the latter sort of changes. Kuhn's theory is in itself too much confined to issues specific to natural science to be directly 'applicable' to mathematics. Indeed, Kuhn wrote me that 'only after Lakatos's death did it occur to me that much about which he and I differed can be explained by the fact that he was raised a mathematician, I a physicist' (letter of 31-10-87). However, as pointed out in the introduction, a slight shift of emphasis is sufficient to get a reading of Kuhn that is no longer specifically 'scientific' and allows of accommodation to the problems of mathematical change along the lines of the Lakatosian 'assimilation of mathematics to science'.

This 'convergent accommodation' may be presented in 'neutral' terms, as follows. The generation of mathematical knowledge depends on sets of structuring assumptions which act also as guiding assumptions for the rational appraisal of the resulting knowledge products.6 The communal practices and their associated sets of assumptions are liable to transforma-tion or replacement when internal stresses and/or external exigencies call for new cultural accommodations of social and cognitive, practical and the-oretical aims and ideals. Sometimes conceptual innovations are so radical as to relegate the older doctrines to considerably lesser positions in the new view of mathematics as a whole. In the new image,7 mathematics 'is' no

longer the same as before, but is 'seen' in a new way. Progressive changes

8 The general characterization of 'post-positivist' views owes much to Laudan [1986]. For an analysis of Kuhn's views and concepts, and a survey of their bearing on other fields than science, see Gutting [1980]. Regard is paid also to Kuhn's later views (Kuhn [1983]).

7 The use of the term image in this context is taken over from Corry [1993]. It denotes what I earlier called also 'overall view' or 'perception' of mathematics (e.g., in Glas [1986]), comprising the assumptions, aims and criteria that characterize a communal research practice.

of this kind are 'continuous' insofar as previous knowledge is not discarded, but 'discontinuous' in the sense that it is accounted for on a radically new conceptual level. The calculus, for instance, could account for all the suc-cesses and failures of previous practices in determining areas, tangents, arc lengths, maxima and minima, etc., on a new conceptual level of greater unifying and explicatory fertility, and as such placed all earlier results in a 'discontinuously' new perspective.8

Replacement entails not only gains, but losses as well, and the two can-not in general be weighed against each other on the same balance. The overall success of a novel set of structuring and guiding assumptions is not in general measurable with the same measure as was applied to the old set, because the order of cognitive priorities may have been changed in the process (as in my case histories). Thus incommensurability, i.e., rational undecidability between different practices, may arise. Incommensurability in the sense intended here does not imply that no independent rational ar-guments at all could be given for variations in mathematical practice. But their rationale consisted in the manner in which they responded progres-sively to new internal and external challenges, not in conformity to a fixed (methodo)logical organon.

Mathematical knowledge grows not only expansively but also reflexively, in depth as well as in breadth, in a manner reminiscent of what Kuhn has depicted as normal and extraordinary phases.9 On the one hand we have 'normal', disciplinary research (e.g., 'Laplacian' analysis, 'Weierstrassian' rigorous calculus) into the ramifications and uses of a given set of formal techniques. It fosters the autonomy of specialist fields in refining their own

8 See Kitcher's interesting reconstruction of the development of analysis in his [1984], chapter 10. Kitcher placed his account of mathematical development in the context of a prior discussion of epistemological issues which diverges from Lakatos's (and Kuhn's) assumptions on many points. As pointed out in the introduction, my concern in this article is not primarily with epistemological issues but with the methodological science-likeness of mathematics. In spite of the different epistemological starting points, there are many parallels at the methodological level of accounting for mathematical growth of knowledge. However, systematic point-by-point comparison is out of the question within the confines of a single article.

9 In Gillies [1992], the question whether or not there are 'revolutions' in mathematics is treated thoroughly historically by a number of important scholars. Evidently, the answer depends largely on definitions, in particular on the degree of discontinuity that would justify the use of the term 'revolution'. The historical cases invoked in favour of mathematical revolutions are mainly concerned with 'discontinuous' changes at higher (meta)levels, inducing new perspectives for handling questions at the object-level, where the same 'signs' are mostly retained and thus 'continuity' is suggested. According as emphasis is placed more on continuity or on discontinuity, the changes are described in a globally Lakatosian or Kuhnian vocabulary. As pointed out in the introduction, I am not primarily concerned with the question which theory of scientific change fits most cases of mathematical change best, but with exploring possibilities to bring their underlying methodological and sociocultural considerations to bear on mathematics. This line of inquiry is not pursued in Gillies's book, but lies in a different plane.

proper procedures and in attempting to capture rigorously and exhaus-tively the entire problem range of neatly circumscribed subjects. On the other hand we have 'extraordinary', innovative research (such as Monge's or Klein's) drawing on new conceptual and methodological resources in or-der to tackle problem situations that fall across the boundaries between separate specialties. The two kinds of activities are of course not strictly separated from one another: a major break-through in technical research may be the prelude to a radical change of the overall perception of the field, and conversely. Whether or not the change is 'revolutionary' depends on the degree to which it affects this image of the discipline. 'Revolution-ary' change is radical conceptual innovation, affecting the basic structuring and guiding assumptions that support a normal mathematical practice. As the term 'revolution' is tainted by the psychological connotation of Gestalt switches in response to preceding crises, I will avoid the term altogether and denote the intended 'revolutionary' changes as radical conceptual in-novations.

Characteristic of historical changes that in hindsight appear as cases of radical conceptual innovation was the posing of new kinds of questions that suggested new ways of looking at old problems and problem-solving meth-ods. In the cases under discussion, the new research perspective revealed unsuspected connections between problem fields that previously were con-sidered to stand entirely apart. It put mathematicians in a position to combine and unify problem fields in a more comprehensive conceptual and argumentative framework, and to characterize their relations at levels of abstraction not captured by the extant doctrines. The conceptual innova-tions implied the reintegration on another conceptual plane of differentiated (sub)fields, so as to bring their varied methods and theories to bear on new questions touching on and affecting mathematicians' reflexive understand-ing of their own problem situations.

The rationale of the innovations derived from their relevance for hand-ling new problem situations in a changed historical context, the 'priority' of which had to be weighed against the established excellence of the ex-tant practice in handling formal procedures. Observance of formal techni-cal standards cannot by itself do more than protect mathematicians from incoherency and inconsistency. Innovative changes of the methods of argu-mentation and inquiry therefore were mostly functional responses to new kinds of questions of superior relevance under the 'given' circumstances.

In the cases referred to, the competing groups appealed to different kinds of technical credentials and ideological motives to substantiate their claims to public recognition and support of their specific ways of doing mathema-tics. The main selective factor wasfthe competitive control of stable and institutionalized opportunities for the recruitment of allies, especially the training of students. Because acquiring certain skills and proficiencies is

tantamount to acquiring a normative image that incorporates the structu-ring and guiding assumptions of the practice in which one is schooled, the training and early professional experiences of mathematicians will have a lasting effect on their judgement. Because of this relative 'closure' of communal frameworks, alternative images can only be compared by their heuristic and problem-solving fertility in novel situations. In particular, innovators have to make a convincing (though not logically coercive) case that the aims, ideals, priorities and values associated with their new vision are in themselves fully worthy of pursuit, and that the new framework for research is more adequate than the old one to attaining them. The social selection of conceptual variations can be characterized in terms of the intrinsic and extrinsic interest of the competences that function as the intellectual and professional resources on which mathematicians have to rely in their competition with others in a transient sociocultural environment (cf. Glas [1988]).

4. Afterword

Having argued that too narrowly (methodo)logical reconstructions of ma-thematical development wrongly exclude broader functional considerations with respect to the social and cultural context, the purport of my argu-ments is no less opposed to a narrowly sociological analysis that precludes any independent rational standpoint for objective judgement. The sense of my convergent accommodation of Lakatosian and Kuhnian notions is pre-cisely that it links the cognitive level of objective knowledge to the social level of communities of practitioners committed to increasing this com-munal knowledge by subscribing to a shared set of structuring and guid-ing assumptions, incorporated in a shared 'image' of mathematics. The significance of Lakatos's quasi-empiricist methodology is its aiming at ob-jective growth of knowledge through quasi-experimental interaction with the (third) world. The decisive issues on which the shifts of mathematical practice in my case histories hinged, were inextricably social and rational. What, in these cases of radical conceptual innovation, effectively turned the scale was the weight of the reasons for transforming the mathematical practice so as to adapt it better, rationally and progressively, to both inter-nal and exterinter-nal stresses, challenges, imperatives and priorities. 'Reasons' are to a certain extent socially constructed, and their 'weighing' implies, among other things, processes of social negotiation. There is indeed a very strong social dimension to processes of mathematical development, and it is hardly possible to put too much research effort into attempts at elucidating this dimension of the process. But this does not commit us to a relativist or subjectivist position; Lakatos's accusing Kuhn of having replaced ratio-nality by 'mob psychology' is quite unjust. To be viable, cognitive goals, methods of reasoning, criteria of judgement and standards of achievement

cannot be 'chosen' at will, in an arbitrary or capricious fashion. They have to be integrated in a coherent and consistent image of mathematics as a whole, capable of accounting for all that we have already learned in a long historical process. This criterion is not 'softer', but 'harder' than the methodological criteria applied by Lakatos.

Mathematics is constructed on the basis of what we have already learned, and this process of construction consists not only in adding new knowledge, but also in increasing our capacity for learning, by extending our abilitity to 'see' things in relevant new ways. As Klein once noted, 'the inertia of conventional scientific ideas often prevented otherwise capable minds from transcending certain tacit assumptions in order to see an old problem in a new light' (Glas [1993c], p. 629). But unconventionally bears fruit only when we have at our disposal an appropriate, coherent and consistent new framework to supersede the old framework that restricted our predecessors' vision.

References

ASPRAY, W. and P. KITCHBR (eds) [1988]: History and philosophy of modern

mathematics, Minneapolis: University of Minnesota Press.

BENDEGEM, J.-P. VAN (ed.) [1988, 1989]: 'Recent issues in the philosophy of mathematics', Philosophica 42 no. 2 &; 43 no. 1.

BIERMANN, K. [1973]: Die Mathematik und ihre Dozenten an der Berliner

Uni-versitat, 1810-1920, Berlin: Akademie Verlag.

CORRY, L. [1993]: 'Kuhnian issues, scientific revolutions and the history of math-ematics', Studies in History and Philosophy of Science 24, 95-117.

GILLIES, D. (ed.) [1992]: Revolutions in mathematics, Oxford: Oxford University Press.

GLAS, E. [1986]: 'On the dynamics of mathematical change in the case of Monge and the French Revolution', Stud. Hist. Phil. Sci. 17, 249-268.

[1988]: 'Between form and function: Social issues in mathematical change', in Van Bendegem op. cit., pp. 21-41.

[1989a, b]: 'Testing the philosophy of mathematics in the history of mathematics, (a) The socio-cognitive process of conceptual change', Stud. Hist.

Phil. Sci. 20, 115-131; (b) 'The similarity between mathematical and scientific

growth of knowledge', ibid., 157-174.

[1993a, b]: 'Mathematical progress: Between reason and society, (a) The methodological model and its alternatives', Zeitschrift fur allgemeine

Wis-senschaftstheorie 24, 43-62; (b) 'The interplay of cognitive and social factors',

ibid., 235-256.

[1993c]: 'From form to function: A reassessment of Felix Klein's unified programme of mathematical research, education and development', Stud. Jfist.

Phil. Sci. 24, 611-631.

GRATTAN-GUINNESS, I. [1981]: 'Mathematical physics in France, 1800-1835', in Jahnke, H. N. and M. Otte (eds), Epistemological and social problems of the

GUTTING, G. (ed.) [1980]: Paradigms and revolutions, London: University of

Notre Dame Press.

HALLETT, M. [1979]: 'Toward a theory of mathematical research programmes',

British Journal of Philosophy of Science 30, 1-25, 135-159.

KITCHER, P. [1984]: The nature of mathematical knowledge, Oxford, New York: Oxford University Press.

KUHN, T. S. [1983]: 'Rationality and theory choice', Journal of Philosophy 80, 563-570.

LAKATOS, I. [1976]: Proofs and refutations: The logic of mathematical discovery, Cambridge: Cambridge University Press.

[1978a, b]: (Worrall, J. and G. Currie, eds) (a) The methodology of

scientific research programmes. Philosophical Papers, Vol.1, (b) Mathematics, science and epistemology. Philosophical Papers, Vol.2, Cambridge: Cambridge

University Press.

LAPLACE, P. S. [1796): Exposition du systeme du monde, Tome VI of the Oeuvres

completes, Paris: Gauthier-Villars, 1878-1892.

LAUDAN, L. et ai. [1986]: 'Scientific change: Philosophical models and historical research', Synthese 69, 141-223.

MALONE, M. E. [1993]: 'Kuhn reconstructed: Incommensurability without rela-tivism', Stud. Hist. Phil. Sci. 24, 69-93.

ROWE, D. E. [1989]: 'Klein, Hilbert, and the Gottingen mathematical tradition',

Osiris 5, 186-213.

SCHUBRING, G. [1989]: 'Pure and applied mathematics in divergent institutional settings: The role and impact of Felix Klein', in Rowe, D. E. and J. McCleary (eds), The history of modern mathematics, Boston: Academic Press, pp. 171-207.

SUPPE, F. (ed.) [1977]: The structure of scientific theories (second ed.), Urbana, Chicago, London: University of Illinois Press.

TOBIES, R. [1989], 'The activities of Felix Klein in the Teaching Commission of the Second Chamber of the Prussian Parliament', in Keitel, C. (ed.), Mathematics,

education, and society, Paris: UNESCO, pp. 50-51.

TYMOCZKO, T. (ed.) [1985]: New directions in the philosophy of mathematics, Boston: Birkhauser.

ABSTRACT. In this paper I explore possibilities of bringing post-positivist philoso-phies of empirical science to bear on the dynamics of mathematical development. This is done by way of a convergent accommodation of a mathematical version of Lakatos's methodology of research programmes, and a version of Kuhn's account of scientific change that is made applicable to mathematics by cleansing it of all references to the psychology of perception. The resulting view is argued in the light of two case histories of radical conceptual innovations.