In 2013 the Health Research Council of New Zealand began a stream of funding titled 'Explorer Grants', and in 2017 changes were introduced to the funding mechanisms of the Volkswagen Foundation 'Experiment!' and the New Zealand Science for Technological Innovation challenge 'Seed Projects'. All three funding streams aim at encouraging novel scientific ideas, and all now employ random selection by lottery as part of the grant selection process. The idea of funding science by lottery has emerged independently in several corners (...) of academia, including in philosophy of science. This paper reviews the conceptual and institutional landscape in which this policy proposal emerged, how different academic fields presented and supported arguments for the proposal, and how these have been reflected in actual policy. The paper presents an analytical synthesis of the arguments presented to date, notes how they support each other and shape policy recommendations in various ways, and where competing arguments highlight need for further analysis or more data. In addition, it provides lessons for how philosophers of science can engage in shaping science policy, and in particular highlights the importance of mixing complementary expertise: it takes a village to raise policy. (shrink)

The ArgumentMost lay users of statistics think in terms of means (averages), variances or the square of the standard deviation, and Gaussians or bell-shaped curves. Such conventions are entrenched by statistical practice, by deep mathematical theorems from probability, and by theorizing in the various natural and social sciences. I am not claiming that the particular conventions (here, the statistics) we adopt are arbitrary. Entrenchment can be rational without its being as well categorical (excluding all other alternatives), even if that entrenchment (...) claims also to provide for categoricity. 1 provide a detailed description of how a science is “normal” and conventionalized. A characteristic feature of this entrenchment of conventions by practice, theorems, and theorizing, is its highly technical form, the canonizing work enabled by apparently formal and esoteric mathematical means. (shrink)

I begin by asking whether there was a Fregean revolution in logic, and, if so, in what did it consist. I then ask whether, and if so, to what extent, Russell played a decisive role in carrying through the Fregean revolution, and, if so, how. A subsidiary question is whether it was primarily the influence of _The Principles of Mathematics_ or _Principia Mathematica_, or perhaps both, that stimulated and helped consummate the Fregean revolution. Finally, I examine cases in which logicians (...) sought, in the years immediately following publication of the _Principles_ and _Principia_, to integrate traditional logic into the Fregean paradigm, focusing on the case of Henry Bradford Smith. My proposed conclusion is that there were different means adopted for rewriting the syllogism, in terms of the logic of relations, in terms of the propositional calculus, or as formulas of the monadic predicate calculus. This suggests that the changes implemented as a result of the adoption of the Russello-Fregean conception of logic could more accurately be called by Grattan-Guinness’s term _convolution_, rather than _revolution_. (shrink)

Van Heijenoort’s account of the historical development of modern logic was composed in 1974 and first published in 1992 with an introduction by his former student. What follows is a new edition with a revised and expanded introduction and additional notes.

Stephen Toulmin once observed that ”it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate’ [Toulmin et al., 1979, An Introduction to Reasoning, Macmillan, London, p. 89]. Might the application of Toulmin’s layout of arguments to mathematics remedy this oversight? Toulmin’s critics fault the layout as requiring so much abstraction as to permit incompatible reconstructions. Mathematical proofs may indeed be represented by fundamentally distinct layouts. However, cases of genuine conflict characteristically reflect an underlying (...) disagreement about the nature of the proof in question. (shrink)

We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our (...) main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the “great triumvirate” of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Rather, following Cohen’s lead, the Marburg philosophers sought to clarify Leibniz’s principle of continuity, and to exploit it in making sense of infinitesimals and related concepts. (shrink)

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.

The Research Assessment Exercise was introduced in 1986 by Thatcher, and was continued by Blair. So it has now been running for 21 years. During this time, the rules governing the RAE have changed considerably, and the interval between successive RAEs has also varied. These changes are not of great importance as far as the argument of this paper is concerned. We will concentrate on the main features of the RAE which can be summarised as follows.

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 discusses the connection between the actual history of mathematics and Lakatos's philosophy of mathematics, in three parts. The first points to studies by Lakatos and others which support his conception of mathematics and its history. In the second I suggest that the apparent poverty of Lakatosian examples may be due to the way in which the history of mathematics is usually written. The third part argues that Lakatos is right to hold philosophy accountable to history, even if Lakatos's (...) own view of mathematics fails that test. (shrink)

A mathematical matrix is usually defined as a two-dimensional array of scalars. And yet, as I explain, matrices are not in fact two-dimensional arrays. So are we to conclude that matrices do not exist? I show how to resolve the puzzle, for both contemporary and older mathematics. The solution generalises to the interpretation of all mathematical discourse. The paper as a whole attempts to reinforce mathematical structuralism by reflecting on how best to interpret mathematics.

In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathematics has important consequences on our conception (...) of mathematical knowledge, in particular mathematical growth. (shrink)

The aim of the paper is to clarify Kuhn’s theory of scientific revolutions. We propose to discriminate between a scientific revolution, which is a sociological event of a change of attitude of the scientific community with respect to a particular theory, and an epistemic rupture, which is a linguistic fact consisting of a discontinuity in the linguistic framework in which this theory is formulated. We propose a classification of epistemic ruptures into four types. In the paper, each of these types (...) of epistemic ruptures is illustrated by examples from physics. The classification of epistemic ruptures can be used as a basis for a classification of scientific revolutions and thus for a refinement of our view of the progress of science. (shrink)

Cieľom predkladanej state je pokus o upresnenie Kuhnovej teórie vedeckých revolúcií. Navrhujem rozlíšiť pojem vedeckej revolúcie, ktorý označuje sociologický fakt zmeny postoja vedeckého spoločenstva vo vzťahu k určitej teórii a pojem epistemickej ruptúry, ktorý označuje lingvistický fakt diskontinuity jazykového rámca, v ktorom je táto teória formulovaná. Analýzou zmien jazykového rámca možno získať klasifikáciu epistemických ruptúr na štyri typy, nazvané ideácia, re-prezentácia, objektácia a re-formulácia. V stati je každý z týchto typov epistemických ruptúr ilustrovaný na sérii príkladov z dejín fyziky. Uvedené (...) typy epistemických ruptúr úzko súvisia s vedeckými revolúciami. Klasifikácia epistemických ruptúr sa tak dá použiť ako východisko pri klasifikácii vedeckých revolúcií. Jednotlivé revolúcie možno klasifikovať podľa toho, akého druhu je epistemická ruptúra, ktorá príslušnú revolúciu sprevádza. (shrink)

The question whether Kuhn's theory of scientific revolutions could be applied to mathematics caused many interesting problems to arise. The aim of this paper is to discuss whether there are different kinds of scientific revolution, and if so, how many. The basic idea of the paper is to discriminate between the formal and the social aspects of the development of science and to compare them. The paper has four parts. In the first introductory part we discuss some of the questions (...) which arose during the debate of the historians of mathematics. In the second part, we introduce the concept of the epistemic framework of a theory. We propose to discriminate three parts of this framework, from which the one called formal frame will be of considerable importance for our approach, as its development is conservative and gradual. In the third part of the paper we define the concept of epistemic rupture as a discontinuity in the formal frame. The conservative and gradual nature of the changes of the formal frame open the possibility to compare different epistemic ruptures. We try to show that there are four different kinds of epistemic rupture, which we call idealisation, re-presentation, objectivisation and re-formulation. In the last part of the paper we derive from the classification of the epistemic ruptures a classification of scientific revolutions. As only the first three kinds of rupture are revolutionary (the re-formulations are rather cumulative), we obtain three kinds of scientific revolution: idealisation, re-presentation, and objectivisation. We discuss the relation of our classification of scientific revolutions to the views of Kuhn, Lakatos, Crowe, and Dauben. (shrink)

One way to determine the quality and pace of change in a science as it undergoes a major transition is to follow some feature of it which remains relatively stable throughout the process. Following the chosen item as it goes through reinterpretation permits conclusions to be drawn about the nature and scope of the broader change in question. In what follows, this device is applied to the change which took place in logic in the mid-nineteenth century. The feature chosen as (...) the focal point is the categorical syllogism. (shrink)

I argue that the construction of representation theorems is a powerful tool for creating novel objects and theories in mathematics, as the construction of a new representation introduces new pieces of information in a very specific way that enables a solution for a problem and a proof of a new theorem. In more detail I show how the work behind the proof of a representation theorem transforms a mathematical problem in a way that makes it tractable and introduces information into (...) it that it did not contain at the beginning of the process. (shrink)

In this paper I explore possibilities of bringing post-positivist philosophies 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. (shrink)

This paper considers whether philosophy of mathematics could benefit by the introduction of some sociology. It begins by considering Lakatos's arguments that philosophy of science should be kept free of any sociology. An attempt is made to criticize these arguments, and then a positive argument is given for introducing a sociological dimension into the philosophy of mathematics. This argument is illustrated by considering Brouwer's account of numbers as mental constructions. The paper concludes with a critical discussion of Azzouni's view that (...) mathematics differs fundamentally from other social practices. (shrink)

1 Mathematical practice: a short overview This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of 'practices' in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic...

ABSTRACT This paper examines consequences of the computer revolution in mathematics. By comparing its repercussions with those of conceptual developments that unfolded in the nineteenth century, I argue that the key epistemological lesson to draw from the two transformative periods is that effective and successful mathematical practices in science result from integrating the computational and conceptual styles of mathematics, and not that one of the two styles of mathematical reasoning is superior. Finally, I show that the methodology deployed by applied (...) mathematicians in modern scientific computing is a paradigmatic instance of this key lesson. (shrink)

In his writings about hypergeometric functions Gauss succeeded in moving beyond the restricted domain of eighteenth-century functions by changing several basic notions of analysis. He rejected formal methodology and the traditional notions of functions, complex numbers, infinite numbers, integration, and the sum of a series. Indeed, he thought that analysis derived from a few, intuitively given notions by means of other well-defined concepts which were reducible to intuitive ones. Gauss considered functions to be relations between continuous variable quantities while he (...) regarded integration and summation as appropriate operations with limits. He also regarded infinite and infinitesimal numbers as a façon de parler and used inequalities in order to prove the existence of certain limits. He took complex numbers to have the same legitimacy as real quantities. However, Gauss’s continuum was linked to a revised form of the eighteenth-century notion of continuous quantity: it was not reducible to a set of numbers but was immediately given. (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)

Lakatos’ (Lakatos, 1976) model of mathematical conceptual change has been criticized for neglecting the diversity of dynamics exhibited by mathematical concepts. In this work, I will propose a pluralist approach to mathematical change that re-conceptualizes Lakatos’ model of proofs and refutations as an ideal dynamic that mathematical concepts can exhibit to different degrees with respect to multiple dimensions. Drawing inspiration from Godfrey-Smith’s (Godfrey-Smith, 2009) population-based Darwinism, my proposal will be structured around the notion of a conceptual population, the opposition between (...) Lakatosian and Euclidean populations, and the spatial tools of the Lakatosian space. I will show how my approach is able to account for the variety of dynamics exhibited by mathematical concepts with the help of three case studies. (shrink)

The aim of this paper is to provide a unified account of scientific and mathematical change in a thoroughly empiricist setting. After providing a formal modelling in terms of embedding, and criticising it for being too restrictive, a second modelling is advanced. It generalises the first, providing a more open-ended pattern of theory development, and is articulated in terms of da Costa and French's partial structures approach. The crucial component of scientific and mathematical change is spelled out in terms of (...) partial embeddings. Finally, an application of this second pattern is made, with the examination of the early formulation of set theory. (shrink)