Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

Written by experts in the field, this volume presents a comprehensive investigation into the relationship between argumentation theory and the philosophy of mathematical practice. Argumentation theory studies reasoning and argument, and especially those aspects not addressed, or not addressed well, by formal deduction. The philosophy of mathematical practice diverges from mainstream philosophy of mathematics in the emphasis it places on what the majority of working mathematicians actually do, rather than on mathematical foundations. -/- The book begins by first challenging the (...) assumption that there is no role for informal logic in mathematics. Next, it details the usefulness of argumentation theory in the understanding of mathematical practice, offering an impressively diverse set of examples, covering the history of mathematics, mathematics education and, perhaps surprisingly, formal proof verification. From there, the book demonstrates that mathematics also offers a valuable testbed for argumentation theory. Coverage concludes by defending attention to mathematical argumentation as the basis for new perspectives on the philosophy of mathematics. ​. (shrink)

In this article, against the background of a notion of ‘assembled’ truth, the evolutionary progressiveness of a theory is suggested as novel and promising explanation for the success of science. A new version of realism in science, referred to as ‘naturalised realism’ is outlined. Naturalised realism is ‘fallibilist’ in the unique sense that it captures and mimics the self-corrective core of scientific knowledge and its progress. It is argued that naturalised realism disarms Kyle Stanford’s anti-realist ‘new induction’ threats by showing (...) that ‘explanationism’ and his ‘epistemic instrumentalism’ are just two positions among many on a constantly evolving continuum of options between instrumentalism and full-blown realism. In particular it is demonstrated that not only can naturalised realism redefine the terms of realist debate in such a way that no talk of miracles need enter the debate, but it also promises interesting defenses against inductive- and under-determination-based anti-realist arguments. (shrink)

Karl Popper is famous for having proposed that science advances by a process of conjecture and refutation. He is also famous for defending the open society against what he saw as its arch enemies – Plato and Marx. Popper’s contributions to thought are of profound importance, but they are not the last word on the subject. They need to be improved. My concern in this book is to spell out what is of greatest importance in Popper’s work, what its failings (...) are, how it needs to be improved to overcome these failings, and what implications emerge as a result. The book consists of a collection of essays which dramatically develop Karl Popper’s views about natural and social science, and how we should go about trying to solve social problems. Criticism of Popper’s falsificationist philosophy of natural science leads to a conception of science that I call aim-oriented empiricism. This makes explicit metaphysical theses concerning the comprehensibility and knowability of the universe that are an implicit part of scientific knowledge – implicit in the way science excludes all theories that are not explanatory, even those that are more successful empirically than accepted theories. Aim-oriented empiricism has major implications, not just for the academic discipline of philosophy of science, but for science itself. Popper generalized his philosophy of science of falsificationism to arrive at a new conception of rationality – critical rationalism – the key methodological idea of Popper’s profound critical exploration of political and social issues in his The Open Society and Its Enemies, and The Poverty of Historicism. This path of Popper, from scientific method to rationality and social and political issues is followed here, but the starting point is aim-oriented empiricism rather than falsificationism. Aim-oriented empiricism is generalized to form a conception of rationality I call aim-oriented rationalism. This has far-reaching implications for political and social issues, for the nature of social inquiry and the humanities, and indeed for academic inquiry as a whole. The strategies for tackling social problems that arise from aim-oriented rationalism improve on Popper’s recommended strategies of piecemeal social engineering and critical rationalism, associated with Popper’s conception of the open society. This book thus sets out to develop Popper’s philosophy in new and fruitful directions. The theme of the book, in short, is to discover what can be learned from scientific progress about how to achieve social progress towards a better world. (shrink)

We argue that there are mutually beneficial connections to be made between ideas in argumentation theory and the philosophy of mathematics, and that these connections can be suggested via the process of producing computational models of theories in these domains. We discuss Lakatos’s work (Proofs and Refutations, 1976) in which he championed the informal nature of mathematics, and our computational representation of his theory. In particular, we outline our representation of Cauchy’s proof of Euler’s conjecture, in which we use work (...) by Haggith on argumentation structures, and identify connections between these structures and Lakatos’s methods. (shrink)

According to conventional wisdom, local gauge symmetry is not a symmetry of nature, but an artifact of how our theories represent nature. But a study of the so-called theta-vacuum appears to refute this view. The ground state of a quantized non-Abelian Yang-Mills gauge theory is characterized by a real-valued, dimensionless parameter theta—a fundamental new constant of nature. The structure of this vacuum state is often said to arise from a degeneracy of the vacuum of the corresponding classical theory, which degeneracy (...) allegedly arises from the fact that “large” (but not “small”) local gauge transformations connect physically distinct states of zero field energy. If that is right, then some local gauge transformations do generate empirical symmetries. In defending conventional wisdom against this challenge I hope to clarify the meaning of empirical symmetry while deepening our understanding of gauge transformations. I distinguish empirical from theoretical symmetries. Using Galileo’s ship and Faraday’s cube as illustrations, I say when an empirical symmetry is implied by a theoretical symmetry. I explain how the theta-vacuum arises, and how “large” gauge transformations differ from “small” ones. I then present two analogies from elementary quantum mechanics. By applying my analysis of the relation between empirical and theoretical symmetries, I show which analogy faithfully portrays the character of the vacuum state of a classical non-Abelian Yang-Mills gauge theory. The upshot is that “large” as well as “small” gauge transformations are purely formal symmetries of non-Abelian Yang-Mills gauge theories, whether classical or quantized. It is still worth distinguishing between these kinds of symmetries. An analysis of gauge within the constrained-Hamiltonian formalism yields the result that “large” gauge transformations should not be classified as gauge transformations; indeed, nor should “global” gauge transformations. In a theory in which boundary conditions are modeled dynamically, “global” gauge transformations may be associated with physical symmetries, corresponding to translations of these extra dynamical variables. Such translations are symmetries if and only if charge is conserved. But it is hard to argue that these symmetries are empirical, and in any case they do not correspond to any constant phase change in a quantum state. (shrink)

The late Imre Lakatos once hoped to found a school of dialectical philosophy of mathematics. The aim of this paper is to ask what that might possibly mean. But Lakatos's philosophy has serious shortcomings. The paper elaborates a conception of dialectical philosophy of mathematics that repairs these defects and considers the work of three philosophers who in some measure fit the description: Yehuda Rav, Mary Leng and David Corfield.

This thesis focuses on models of conceptual change in science and philosophy. In particular, I developed a new bootstrapping methodology for studying conceptual change, centered around the formalization of several popular models of conceptual change and the collective assessment of their improved formal versions via nine evaluative dimensions. Among the models of conceptual change treated in the thesis are Carnap’s explication, Lakatos’ concept-stretching, Toulmin’s conceptual populations, Waismann’s open texture, Mark Wilson’s patches and facades, Sneed’s structuralism, and Paul Thagard’s conceptual revolutions. (...) -/- In order to analyze and compare the conception of conceptual change provided by these different models, I rely on several historical reconstructions of episodes of scientific conceptual change. The historical episodes of scientific change that figure in this work include the emergence of the morphological concept of fish in biological taxonomies, the development of scientific conceptions of temperature, the Church-Turing thesis and related axiomatizations of effective calculability, the history of the concept of polyhedron in 17th and 18th century mathematics, Hamilton’s invention of the quaternions, the history of the pre-abstract group concepts in 18th and 19th century mathematics, the expansion of Newtonian mechanics to viscous fluids forces phenomena, and the chemical revolution. I will also present five different formal and informal improvements of four specific models of conceptual change. I will first present two different improvements of Carnapian explication, a formal and an informal one. My informal improvement of Carnapian explication will consist of a more fine-grained version of the procedure that adds an intermediate, third step to the two steps of Carnapian explication. I will show how this novel three-step version of explication is more suitable than its traditional two-step relative to handle complex cases of explications. My second, formal improvement of Carnapian explication will be a full explication of the concept of explication itself within the theory of conceptual spaces. By virtue of this formal improvement, the whole procedure of explication together with its application procedures and its pragmatic desiderata will be reconceptualized as a precise procedure involving topological and geometrical constraints inside the theory of conceptual spaces. My third improved model of conceptual change will consist of a formal explication of Darwinian models of conceptual change that will make vast use of Godfrey-Smith’s population-based Darwinism for targeting explicitly mathematical conceptual change. My fourth improvement will be dedicated instead to Wilson’s indeterminate model of conceptual change. I will show how Wilson’s very informal framework can be explicated within a modified version of the structuralist model-theoretic reconstructions of scientific theories. Finally, the fifth improved model of conceptual change will be a belief-revision-like logical framework that reconstructs Thagard’s model of conceptual revolution as specific revision and contraction operations that work on conceptual structures. -/- At the end of this work, a general conception of conceptual change in science and philosophy emerges, thanks to the combined action of the three layers of my methodology. This conception takes conceptual change to be a multi-faceted phenomenon centered around the dynamics of groups of concepts. According to this conception, concepts are best reconstructed as plastic and inter-subjective entities equipped with a non-trivial internal structure and subject to a certain degree of localized holism. Furthermore, conceptual dynamics can be judged from a weakly normative perspective, bound to be dependent on shared values and goals. Conceptual change is then best understood, according to this conception, as a ubiquitous phenomenon underlying all of our intellectual activities, from science to ordinary linguistic practices. As such, conceptual change does not pose any particular problem to value-laden notions of scientific progress, objectivity, and realism. At the same time, this conception prompts all our concept-driven intellectual activities, including philosophical and metaphilosophical reflections, to take into serious consideration the phenomenon of conceptual change. An important consequence of this conception, and of the analysis that generated it, is in fact that an adequate understanding of the dynamics of philosophical concepts is a prerequisite for analytic philosophy to develop a realistic and non-idealized depiction of itself and its activities. (shrink)

We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, which gave rise to this special issue. Lastly, (...) we provide a summary of the contributions to this issue. (shrink)

A parallel reading of ten powerful works of conceptual and analytical originality yields a novel epistemological method based on the possibility of automated experimentation in engineering and architecture. An initial protocol for Parametric Epistemic Things, a particular kind of assemblage (as postulated by DeLanda following Deleuze & Guattari) that builds on Rheinberger’s ideas of Epistemic Things, is outlined and conceptualised to allow for the possibility of such automation. Following the interpretations of fragments from the texts, a discussion examines future potentials (...) and identifies some of the conceptual and pragmatic thresholds to overcome in order to leverage the proposal and situate it within a general theory of science and research, while attempting to answer the question at the heart of this essay: what is the role of parametric epistemic things in contemporary evolutionary architecture? (shrink)

Imre Lakatos' views on the philosophy of mathematics are important and they have often been underappreciated. The most obvious lacuna in this respect is the lack of detailed discussion and analysis of his 1976a paper and its implications for the methodology of mathematics, particularly its implications with respect to argumentation and the matter of how truths are established in mathematics. The most important themes that run through his work on the philosophy of mathematics and which culminate in the 1976a paper (...) are (1) the (quasi-)empirical character of mathematics and (2) the rejection of axiomatic deductivism as the basis of mathematical knowledge. In this paper Lakatos' later views on the quasi-empirical nature of mathematical theories and methodology are examined and specific attention is paid to what this view implies about the nature of mathematical argumentation and its relation to the empirical sciences. (shrink)

. In Minding God Gregory Peterson takes a careful look at the kind of freedom that human persons have. He concludes that humans are constrained to be free and unpacks this into a version of compatibilism. That is, humans are not metaphysically free under current existence because of the causal determination inherent in their physical nature, but they can take credit for the origination of selfforming decisions because the causes occur inside of us. Peterson does advocate an eschatological hope looking (...) forward to the breaking of causal determination by God's own action. Thus, Minding God presents an eschatologically limited compatibilism. Compatibilism of any kind, however, presents serious challenges to most Christian theologies and to many religious traditions broadly considered. After I interpret Peterson's position I make the argument that compatibilism is neither desirable nor required for a theological anthropology intent on serious engagement of cognitive science. (shrink)

This article examines the connections among heuristics, the epistemological and ontological presuppositions that underlie theorizing, and substantive explanations in sociology. It develops and contrasts three heuristics: “doing as knowing” (DK), “categorizing as knowing” (CK), and “praxis as knowing” (PK). These are each composed of four dimensions: the theory of knowledge, the theory of reality, the theory of the growth of knowledge, and the theory of knowledge producers. The article then shows the importance of heuristics for empirical work by demonstrating how (...) they shape explanations in the sociological subfield of the historical sociology of knowledge. The essay draws two main conclusions: it argues that PK offers a more useful basis for developing explanations in sociology than either of the two alternatives (DK and CK) that currently shape substantive work; furthermore, it claims that the exposition of heuristic assumptions is an important task for sociology. (shrink)

Attacks the irrationalism of Lakatos's Proofs and Refutations and defends mathematics as a "last bastion" of reason against postmodernist and deconstructionist currents.

Abstract: Particularism is usually understood as a position in moral philosophy. In fact, it is a view about all reasons, not only moral reasons. Here, I show that particularism is a familiar and controversial position in the philosophy of science and mathematics. I then argue for particularism with respect to scientific and mathematical reasoning. This has a bearing on moral particularism, because if particularism about moral reasons is true, then particularism must be true with respect to reasons of any sort, (...) including mathematical and scientific reasons. (shrink)

In the philosophy of mathematical practice, the aim is to understand the various aspects of this practice. Even though definitions are a central element of mathematical practice, the study of this aspect of mathematical practice is still in its infancy. In particular, there is little empirical evidence to substantiate claims about definitions in practice. In this article, we address this gap by reporting on an empirical investigation on how mathematicians create definitions and which roles and properties they attribute to them. (...) On the basis of interviews with thirteen research mathematicians, we provide a broad range of relevant aspects of definitions. In particular, we address various roles of definitions and show that definitions are not just a product of mathematical factors, but also of social and contingent factors. Furthermore, we provide concrete examples of how mathematicians interact and think about definition. This broad empirical basis with a variety of examples provides an optimal starting point for future investigations into definitions in mathematical practice. (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)

I argue in this article that an aspect of Imre Lakatos’s philosophy has been largely ignored in previous literature. The key feature of Lakatos’s philosophy of the historiography of science is its non-representationalism, which enables comparisons of alternative ‘historiographic research programmes’ without implying that the interpretations of history re-present or mirror the past. I discuss some problems of this interpretation and show specifically that Lakatos’s philosophy does not distort the history of science despite its normative ambitions. The last section is (...) devoted to updating Lakatos’s programme to answer the needs of contemporary history and philosophy of science. The standard of rationality used in comparative assessments should be understood as a tool for measuring the coherence of an account of history with regard to the ‘actual history’. This standard takes two forms: framework-dependent and framework-independent rationality. The latter is decisive in comparative assessments. (shrink)

This article canvasses five senses in which one might introduce an historical element into the philosophy of mathematics: 1. The temporal dimension of logic; 2. Explanatory Appeal to Context rather than to General Principles; 3. Heraclitean Flux; 4. All history is the History of Thought; and 5. History is Non-Judgmental. It concludes by adapting Bernard Williams’ distinction between ‘history of philosophy’ and ‘history of ideas’ to argue that the philosophy of mathematics is unavoidably historical, but need not and must not (...) merge with historiography. (shrink)

The general theme of this article is the actual practice of how definitions are justified and formulated in mathematics. The theoretical insights of this article are based on a case study of topological definitions of chaos. After introducing this case study, I identify the three kinds of justification which are important for topological definitions of chaos: natural-world-justification, condition-justification and redundancy-justification. To my knowledge, the latter two have not been identified before. I argue that these three kinds of justification are widespread (...) in mathematics. After that, I first discuss the state of the art in the literature about the justification of definitions in the light of actual mathematical practice. I then go on to criticize Lakatos’s account of proof-generated definitions—the main account in the literature on this issue—as being limited and also misguided: as for topological definitions of chaos, in nearly all mathematical fields various kinds of justification are found and are also reasonable. (shrink)

A phenomenological approach to mathematical practice is sketched out, and some problems with this sort of approach are considered. The approach outlined takes mathematical practices as its data, and seeks to provide an empirically adequate philosophy of mathematics based on observation of these practices. Some observations are presented, based on two case studies of some research into the classification of C*-algebras. It is suggested that an anti-realist account of mathematics could be developed on the basis of these and other studies, (...) locating the substance of mathematics in the various informal argument methods used by mathematicians. (shrink)

Despite continued attention, finding adequate criteria for distinguishing “good” from “bad” scholarly journals remains an elusive goal. In this essay, I propose a solution informed by the work of Imre Lakatos and his methodology of scientific research programmes (MSRP). I begin by reviewing several notable attempts at appraising journal quality – focusing primarily on the impact factor and development of journal blacklists and whitelists. In doing so, I note their limitations and link their overarching goals to those found within the (...) philosophy of science. I argue that Lakatos’s MSRP and specifically his classifications of “progressive” and “degenerative” research programmes can be analogized and repurposed for the evaluation of scholarly journals. I argue that this alternative framework resolves some of the limitations discussed above and offers a more considered evaluation of journal quality – one that helps account for the historical evolution of journal-level publication practices and attendant contributions to the growth (or stunting) of scholarly knowledge. By doing so, the seeming problem of journal demarcation is diminished. In the process I utilize two novel tools (the mistake index and scite index) to further illustrate and operationalize aspects of the MSRP. (shrink)

Imre Lakatos gained fame in the English-speaking world as a follower and critic of philosopher of science Karl Popper. However, Lakatos’ background involved other philosophical and scientific sources from his native Hungary. Lakatos surreptitiously used Hegelian Marxism in his works on philosophy of science and mathematics, disguising it with the rhetoric of the Popper school. He also less surreptitiously incorporated, particularly in his treatment of mathematics, work of the strong tradition of heuristics in twentieth century Hungary. Both his Marxism and (...) his emphasis on heuristics contained a view of science and mathematics that contrasted with the mainstream of Anglo-American philosophy of science. Both involved a dynamic view of science, whether historical or psychological, and an emphasis on practice as opposed to static, formal representations of scientific theories. (shrink)

In this paper, I argue that Imre Lakatos’s account on the relation between the history and the philosophy of science, if properly understood and also if properly modified, can be valuable for the philosophical comprehension of the relation between the history and the philosophy of science. The paper is divided into three main parts. In the first part, I provide a charitable exegesis of the Lakatosian conception of the history of science in order to show that Lakatos’s history cannot be (...) a caricature. In the second part, I describe what I believe are the real problems of Lakatos’s account on a metaphysical and on a methodological level. In order to address these problems, I take advantage of the recent critique of the so-called “confrontation model”. Finally, in the third part, I suggest that a proper modification of Lakatos’s perspective can resolve the aforementioned problems without distorting Lakatos’s central aspiration, which is that the philosophical theories of scientific rationality can be assessed with the help of the history of science without losing their normative content. (shrink)

We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science​, which gave rise to this special issue. Lastly, (...) we provide a summary of the contributions to this issue. (shrink)

This paper addresses the actual practice of justifying definitions in mathematics. First, I introduce the main account of this issue, namely Lakatos's proof-generated definitions. Based on a case study of definitions of randomness in ergodic theory, I identify three other common ways of justifying definitions: natural-world justification, condition justification, and redundancy justification. Also, I clarify the interrelationships between the different kinds of justification. Finally, I point out how Lakatos's ideas are limited: they fail to show how various kinds of justification (...) can be found and can be reasonable, and they fail to acknowledge the interplay among the different kinds of justification. (shrink)

Resumen En la filosofía de Lakatos existe una tensión entre falibilismo y optimismo epistemológico. En el presente artículo propongo la reconstrucción de dos elementos clave para resolver positivamente esa tensión: una noción de verdad empírica, que neutralice el convencionalismo, y una noción de verdad parcial, que dé cuenta de la creciente verosimilitud de los programas de investigación. Para esto último, reviso un aspecto de la obra de Lakatos frecuentemente ignorado por los críticos: su posición en el debate acerca de las (...) leyes de la naturaleza.There is a tension between fallibilism and epistemological optimism in Lakatos’s philosophy. This article proposes the reconstruction of two key elements to positively resolve such tension: a notion of empirical truth that could neutralize conventionalism, and a notion of partial truth, capable to account for the increasing verisimilitude of research programs. For the latter, I review one aspect of Lakatos’s work often ignored by the critics: his position on the debate regarding laws of nature. (shrink)