This paper clarifies and discusses Imre Lakatos’ claim that mathematics is quasi-empirical in one of his less-discussed papers A Renaissance of Empiricism in the Recent Philosophy of Mathematics. I argue that Lakatos’ motivation for classifying mathematics as a quasi-empirical theory is epistemological; what can be called the quasi-empirical epistemology of mathematics is not correct; analysing where the quasi-empirical epistemology of mathematics goes wrong will bring to light reasons to endorse a pluralist view of mathematics.

Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent (in an informal sense) and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

Complete inferential rigour is achieved by breaking down arguments into steps that are as small as possible: inferential ‘atoms’. For example, a mathematical or philosophical argument may be made completely inferentially rigorous by decomposing its inferential steps into the type of step found in a natural deduction system. It is commonly thought that atomization, paradigmatically in mathematics but also more generally, is pro tanto epistemically valuable. The paper considers some plausible candidates for the epistemic value arising from atomization and finds (...) that none of them fits the bill. In particular, atomized arguments do not ground the correctness of their unatomized counterparts; they are typically not credence-preserving; and they need not reveal the source of inferential disagreement. The moral this suggests is that complete rigour is not even a defeasible epistemic ideal. (shrink)

Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that (...) some of what mathematicians take to be deductive knowledge is in fact non-deductive. 1 Introduction2 Why It Might Matter3 Two Further Examples and Preliminaries4 An Exclusive Epistemic Virtue of Proof?5 Analyses of Knowledge6 The Inductive Basis of Deduction7 Physical to Mathematical Linkages8 Conclusion. (shrink)

This paper attempts to show that mathematical knowledge does not grow by a simple process of accumulation and that it is possible to provide a quasi-empirical (in Lakatos's sense) account of mathematical theories. Arguments supporting the first thesis are based on the study of the changes occurred within Eudidean geometry from the time of Euclid to that of Hilbert; whereas those in favour of the second arise from reflections on the criteria for refutation of mathematical theories.

This paper has two objectives, neither previously attempted in the published literature—first, to outline J. M. Keynes's theory of knowledge in some detail, and, secondly, to justify the contention that his epistemology is a variety of rationalism, and not, as many have asserted, a form of empiricism. Keynes's attitude to empirical data is also analysed as well as his views on prediction and theory choice. 1This paper is partly based on ideas initially advanced in O'Donnell [1982], a revised and expanded (...) version of which is to be published as O'Donnell [1989]. I should like to thank an anonymous referee for helpful comment, and King's College, Cambridge for permission to quote from the Keynes Papers. (shrink)

Abstract:According to Quine, Charles Parsons, Mark Steiner, and others, Russell’s logicist project is important because, if successful, it would show that mathematical theorems possess desirable epistemic properties often attributed to logical theorems, such as aprioricity, necessity, and certainty. Unfortunately, Russell never attributed such importance to logicism, and such a thesis contradicts Russell’s explicitly stated views on the relationship between logic and mathematics. This raises the question: what did Russell understand to be the philosophical importance of logicism? Building on recent work (...) by Andrew Irvine and Martin Godwyn, I argue that Russell thought a systematic reduction of mathematics increases the certainty of known mathematical theorems (even basic arithmetical facts) by showing mathematical knowledge to be coherently organized. The paper outlines Russell’s theory of coherence, and discusses its relevance to logicism and the certainty attributed to mathematics. (shrink)

. Heuristic is a central concept of Lakatos' philosophy both in his early works and in his later work, the methodology of scientific research programs. The term itself, however, went through significant change of meaning. In this paper I study this change and the ‘metaphysical’ commitments behind it. In order to do so, I turn to his mathematical heuristic elaborated in Proofs and Refutations. I aim to show the dialogical character of mathematical knowledge in his account, which can open a (...) door to hermeneutic studies of mathematical practice. (shrink)

Ethical intuitionists regard moral knowledge as deriving from moral intuition, moral observation, moral emotion and inference. However, moral intuitions, observations and emotions are cultural artefacts which often differ starkly between cultures. Intuitionists attribute uncongenial moral intuitions, observations or emotions to bias or to intellectual or moral failings; but that leads to sectarian ad hominen attacks. Intuitionists try to avoid that by restricting epistemically genuine intuitions, observations or emotions to those which are widely agreed. That does not avoid the problem. It (...) also limits epistemically genuine intuitions, observations or emotions to those with meagre content, and the intuitionists offer no plausible explanation for how inference from such insubstantial propositions can engender substantial moral knowledge. Instead of moral knowledge, intuitionism offers the prospect of mutual name-calling between intellectually stagnant groups. I criticise and reject the principle of phenomenal conservatism, to which intuitionists sometimes appeal. (shrink)

Kitcher and Aspray distinguish a mainstream tradition in the philosophy of mathematics concerned with foundationalist epistemology, and a ‘maverick’ or naturalistic tradition, originating with Lakatos. My claim is that if the consequences of Lakatos's contribution are fully worked out, no less than a radical reconceptualization of the philosophy of mathematics is necessitated, including history, methodology and a fallibilist epistemology as central to the field. In the paper an interpretation of Lakatos's philosophy of mathematics is offered, followed by some critical discussion, (...) and an extension to a social constructivist position (which might well have been unacceptable to Lakatos). (shrink)

After a brief discussion of Kreisel’s notion of informal rigour and Myhill’s notion of absolute proof, Gödel’s analysis of the subject is presented. It is shown how Gödel avoids the notion of informal proof because such a use would contradict one of the senses of “formal” that Gödel wants to preserve. This Gödelian notion of “formal” is directly tied to his notion of absolute proof and to the question of the general applicability of concepts, in a way that overcomes both (...) Kreisel and Myhill’s conceptions. This paper aims to contribute to the present-day debate on informal and epistemic mathematics, focusing on what appears necessary for a better understanding of the issues at stake. (shrink)

In mathematics, the deductive method reigns. Without proof, a claim remains unsolved, a mere conjecture, not something that can be simply assumed; when a proof is found, the problem is solved, it turns into a “result,” something that can be relied on. So mathematicians think. But is there more to mathematical justification than proof? -/- The answer is an emphatic yes, as I explain in this article. I argue that non-deductive justification is in fact pervasive in mathematics, and that it (...) is in good epistemic standing. (shrink)

Euclid’s Elements inspired a number of foundationalist accounts of mathematics, which dominated the epistemology of the discipline for many centuries in the West. Yet surprisingly little has been written by recent philosophers about this conception of mathematical knowledge. The great exception is Imre Lakatos, whose characterisation of the Euclidean Programme in the philosophy of mathematics counts as one of his central contributions. In this essay, we examine Lakatos’s account of the Euclidean Programme with a critical eye, and suggest an alternative (...) picture which builds on his work, but differs in a number of important respects. (shrink)

Laboratory experimentation was once considered impossible or irrelevant in economics. Recently, however, economic science has gone through a real ‘laboratory revolution’, and experimental economics is now a most lively subfield of the discipline. The methodological advantages and disadvantages of controlled experimentation constitute the main subject of this thesis. After a survey of the literature on experiments in philosophy and economics, the problem of testing normative theories of rationality is tackled. This philosophical issue was at the centre of a famous controversy (...) in decision theory, during which a methodology of normative falsification was first articulated and used to assess experimental results. In the third chapter, the methodological advantages of controlled experimentation are illustrated and discussed with examples taken from the experiments on the so-called ‘preference reversal’ phenomenon. Laboratory testing allows to establish with a high degree of certainty that certain phenomena lie behind the experimental data, by means of independent testing, elimination of alternative hypotheses, and the use of different instruments of observation. The fourth chapter is devoted to a conceptual analysis of the problem of ‘parallelism’. This is the problem of inferring from the occurrence of a phenomenon in the laboratory, to its instantiation also in non-laboratory environments. Experimental economists have discussed parallelism at length, and their views are presented and criticised. Eventually, it is argued that parallelism is a factual matter and as such can only be established on empirical grounds. The fifth chapter provides an example of how one can argue for parallelism, focusing on the case of experimentation on the ‘winner’s curse’ phenomenon. The role of experiments as ‘mediators’ between theoretical models and their target domain of application is illustrated, and the structure of parallelism arguments analysed in detail. Finally, in the last chapter, economic experiments are compared to simulations, in order to highlight their specific characteristics. (shrink)