To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality, personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about ‘truth’, but the assignment of mathematics to informal reality is (...) not itself formally analysable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed. (shrink)

Abstract The first fractal constructions appeared in mathematics in the second half of the 19th century. Their history is divided into two periods. The first period lasted 100 years and is a good example of the method of proofs and refutations discovered by Lakatos. The modern history of these objects started 20 years ago, when Mandelbrot decided to create fractal geometry, a general theory concentrated on specific properties of fractals. His approach has been surprisingly effective. The aim of this paper (...) is to examine the reasons for Mandelbrot's success and for the present popularity of fractals. They are now known not only in mathematics but also in many fields of natural, social and applied sciences, and even in the arts. (shrink)

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.

A key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of ‘mathematical importance’, broadly (...) within the framework of progress Maddy offers, is developed with a special focus on mathematicians’ peer-review practice. (shrink)

I examine the classical idea of ‘algorithm’ as a sequential, step-by-step, deterministic procedure (i.e., the idea of ‘algorithm’ that was already in use by the 1930s), with respect to three themes, its relation to the notion of an ‘effective procedure’, its different roles and uses in logic, computer science, and mathematics (focused on numerical analysis), and its different formal definitions proposed by practitioners in these areas. I argue that ‘algorithm’ has been conceptualized and used in contrasting ways in the above (...) areas, and discuss challenges and prospects for adopting a final foundational theory of (classical) ‘algorithms’. (shrink)

In this paper, I highlight the significance of practices of _refutation_ in philosophical inquiry, that is, practices of showing that a claim, person or theory is wrong. I present and contrast two prominent approaches to philosophical refutation: refutation in ancient Greek dialectic (_elenchus_), in its Socratic variant as described in Plato’s dialogues, and as described in Aristotle’s logical texts; and the practice of providing counterexamples to putative definitions familiar from twentieth century analytic philosophy, focusing on the so-called Gettier problem. Moreover, (...) I discuss Lakatos’ method of proofs and refutations, as it offers insightful observations on the dynamics between arguments, refutations, and counterexamples. Overall, I argue that dialectic, in particular in its Socratic variant, is especially suitable for the philosophical purpose of questioning the obvious, as it invites reflection on one’s own doxastic commitments and on the tensions and inconsistencies within one’s set of beliefs. By contrast, the counterexample-based approach to philosophical refutation can give rise to philosophical theorizing that is overly focused on hairsplitting disputes, thus becoming alienated from the relevant human experiences. Insofar as philosophical inquiry treads the fine line between questioning the obvious while still seeking to say something significant about human experiences, perhaps a certain amount of what Lakatos describes as ‘monster-barring’—a rejection of overly fanciful, artificial putative counterexamples—has its place in philosophical argumentation. (shrink)

The alleged emergence of a ‘post-truth’ regime links the rise of new forms of social media and the reemergence of political populism. Post-truth has theoretical roots in the interdisciplinary field of Science and Technology Studies, with sociologists of science arguing that both true and false claims should be explained by the same kinds of social causes. Most STS theorists have sought to deflect blame for post-truth, while at the same time enacting a normative turn, looking to deconstruct truth claims and (...) subject expertise to criticism. Steve Fuller has developed a positive case for post-truth in science, arguing that post-truth democratizes science. I criticize this argument and suggest an alternative approach that draws on the prehistory of the field in the 1930s and 1940s, when philosophers and sociologists sought to define the social conditions necessary for reliable knowledge production that might stem mass media irrationalism. (shrink)

In recent years, there has been increasing discussion of whether philosophy actually makes progress. This discussion has been prompted, in no small part, by the depth and persistence of disagreement among philosophers on virtually every major theoretical issue in the field. In this paper, I examine the role that the Method of Cases – the widespread philosophical method of testing and revising theories by comparing their verdicts against our intuitions in particular cases – plays in creating and sustaining theoretical disagreements (...) in philosophy. Drawing on work from cognitive psychology, I argue that there is a fundamental incompatibility between the structure of the theories that philosophers seek to construct using the MoC and the structure of the concepts on which our case-specific intuitions are based. This incompatibility renders MoC-based philosophical theorizing unable ever to succeed by the very standards of adequacy that it sets for itself. And this, in turn, helps to explain the depth and persistence of theoretical disagreements – and, in certain ways, the lack of progress – in the many areas of philosophy where the MoC plays an important role. (shrink)

A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.

In recent years, substructural approaches to paradoxes have become quite popular. But whatever restrictions on structural rules we may want to enforce, it is highly desirable that such restrictions be accompanied by independent philosophical motivation, not directly related to paradoxes. Indeed, while these recent developments have shed new light on a number of issues pertaining to paradoxes, it seems that we now have even more open questions than before, in particular two very pressing ones: what (independent) motivations do we have (...) (if any) for restrictions on structural rules, and what to make of the plurality of new logics emerging from these restrictions, i.e. how to ‘choose’ among the different options. In this paper, we address these two questions from the perspective of a dialogical conception of logic that we've been advocating in recent years. We will argue that dialogical interpretations of structural rules, that is, as rules determining specific properties of the dialogues in question, provide a conveniently neutral framework to adjudicate between the different substructural proposals that have been made in the literature on paradoxes. (shrink)

This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and assessing (...) the relation between cognitive processes and certain properties of the stimuli at which they are directed. (shrink)

We report on an exploratory study of the way eight mid-level undergraduate mathematics majors read and reflected on four student-generated arguments purported to be proofs of a single theorem. The results suggest that mid-level undergraduates tend to focus on surface features of such arguments and that their ability to determine whether arguments are proofs is very limited -- perhaps more so than either they or their instructors recognize. We begin by discussing arguments (purported proofs) regarded as texts and validations of (...) those arguments, i.e., reflections of individuals checking whether such arguments really are proofs of theorems. We relate the way the mathematics research community views proofs and their validations to ideas from reading comprehension and literary theory. We then give a detailed analysis of the four student-generated arguments and finally analyze the eight students' validations of them. (shrink)

ABSTRACT This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and (...) assessing the relation between cognitive processes and certain properties of the stimuli at which they are directed. (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)

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)

One of the distinguishing features of mathematics is the exceptional level of consensus among mathematicians. However, an analysis of what mathematicians agree on, how they achieve this agreement, and the relevant historical conditions is lacking. This paper is a programmatic intervention providing a preliminary analysis and outlining a research program in this direction.First, I review the process of ‘negotiation’ that yields agreement about the validity of proofs. This process most often does generate consensus, however, it may give rise to another (...) kind of disagreement: whether the original and new proof are effectively the same, leading to potential disagreement about the validity of the original proof.Second, I historicize the phenomenon of consensus. I show that in earlier European mathematics and other mathematical cultures, consensus about the validity of arguments was substantially weaker or conceived differently than it is today. This means that contemporary consensus about the validity of mathematical proofs should be explained by historical changes in mathematical practice.Finally, I conjecture what brought about the contemporary form of mathematical consensus. Since a sharp rise in consensus occurs around the turn of the 20th century, it makes sense to explain this consensus by the concurrent formalization of mathematics. However, this explanation has a major flaw: it explains a really existing phenomenon (consensus) by something that hardly ever happens (writing proofs in formal languages). I will therefore explain the ways in which formalization does enter mathematical practice so as to account for contemporary forms of mathematical consensus. (shrink)

This work gives a new argument for ‘Empirical Philosophy of Mathematical Practice’. It analyses different modalities on how empirical information can influence philosophical endeavours. We evoke the classical dichotomy between “armchair” philosophy and empirical/experimental philosophy, and claim that the latter should in turn be subdivided in three distinct styles: Apostate speculator, Informed analyst, and Freeway explorer. This is a shift of focus from the source of the information towards its use by philosophers. We present several examples from philosophy of mind/science (...) and ethics on one side and a case study from philosophy of mathematics on the other. We argue that empirically informed philosophy of mathematics is different from the rest in a way that encourages a Freeway explorer approach, because intuitions about mathematical objects are often unavailable for non-mathematicians. This consideration is supported by a case study in set theory. (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 explore the prospects of a monist account of explanation for both non-causal explanations in science and pure mathematics. Our starting point is the counterfactual theory of explanation for explanations in science, as advocated in the recent literature on explanation. We argue that, despite the obvious differences between mathematical and scientific explanation, the CTE can be extended to cover both non-causal explanations in science and mathematical explanations. In particular, a successful application of the CTE to mathematical explanations requires us to (...) rely on counterpossibles. We conclude that the CTE is a promising candidate for a monist account of explanation in both science and mathematics. (shrink)

To adequately model mathematical arguments the analyst must be able to represent the mathematical objects under discussion and the relationships between them, as well as inferences drawn about these objects and relationships as the discourse unfolds. We introduce a framework with these properties, which has been used to analyse mathematical dialogues and expository texts. The framework can recover salient elements of discourse at, and within, the sentence level, as well as the way mathematical content connects to form larger argumentative structures. (...) We show how the framework might be used to support computational reasoning, and argue that it provides a more natural way to examine the process of proving theorems than do Lamport’s structured proofs. (shrink)

I investigate the construction of the mathematical concept of quaternion from a methodological and heuristic viewpoint to examine what we can learn from it for the study of the advancement of mathematical knowledge. I will look, in particular, at the inferential microstructures that shape this construction, that is, the study of both the very first, ampliative inferential steps, and their tentative outcomes—i.e. small ‘structures’ such as provisional entities and relations. I discuss how this paradigmatic case study supports the recent approaches (...) to problem-solving and philosophy of mathematics, and how it suggests refinements of them. In more detail, I argue that the inferential micro-structures enable us to shed more light on the informal, heuristic side of mathematical practice, and its inferential and rational procedures. I show how they enable the generation of a problem, the construction of its conditions of solvability, the search for a hypothesis to solve it, and how these processes are representation-sensitive. On this base, I argue that: 1.the recent development of the philosophy of mathematics was right in moving from Lakatos’ initial investigation of the formal side of a mathematical proof to the investigation of the semi-formal, heuristic side of the mathematical practice as a way of understanding mathematical knowledge and its advancement. 2.The investigation of mathematical practice and discovery can be improved by a finer-grained study of the inferential micro-structures that are built during mathematical problem-solving. (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)

An overview is provided of how the concept of the thought experiment has developed and changed for the natural sciences in the course of the 20th century. First, we discuss the existing definitions of the term 'thought experiment' and the origin of the thought experimentation method, identifying it in Greek Presocratics epoch. Second, only in the end of the 19th century showed up the first systematic enquiry on thought experiments by Ernst Mach's work. After the Mach's work, a negative attitude (...) towards thought experiments came in the beginning of the 20th century, which went on until the Thomas Kuhn's and Karl Popper's work on thought experiments. Only from the mid-1980s did thought experiments begin to be considered relevant to scientific enterprise. Finally, we show the existing empirical and 'functional' theories which have developed about the nature and purpose of thought experiments. (shrink)

Is there anything like an experiment in mathematics? And if this is the case, what would distinguish a mathematical experiment from a mathematical thought experiment? In the present paper, a framework for the practice of mathematics will be put forward, which will consider mathematics as an experimenting activity and as a proving activity. The relationship between these two activities will be explored and more importantly a distinction between thought-experiments, real experiments, quasi experiments and proofs in pure mathematics will be provided. (...) To do so, three examples of experimenting with triangles will be introduced and discussed. (shrink)

As the world economy has for better or worse become more and more dependent on the financial markets, a rethinking of the role of finance in both theory and practice is necessary. I argue that such a rethinking requires a new look at the theories of finance that is philosophical in kind. In effect, as Martha Nussbaum claims, if the absence of philosophy in economics is arguably one of the main reasons for the flaws in certain economic theories, the absence (...) of philosophy in finance is one the main reasons for the flaws in our theories on financial systems. In this paper I discuss the mutual relations and benefits between finance and philosophy. First, I examine the contribution that philosophy can offer to finance by analyzing a few critical issues in financial ontology, financial methodology and financial mathematics. I argue that philosophy is essential to enabling finance to achieve the goals for which it was designed, not only because it is a valuable external addition, but also internally, since philosophy is the proper tool for bringing about new theories and approaches. Then, I examine the contribution that finance can offer to philosophy by analyzing the relationship between theory and practice, data and hypothesis, and prediction, description and control in the context of financial systems. I argue that finance can help us to rethink some philosophical tenets on these issues. (shrink)

This title offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

I present a thought experiment in quantum mechanics and tease out some of its implications for the doctrine of “peaceful coexistence”, which, following Shimony, I take to be the proposition that quantum mechanics does not force us to revise or abandon the relativistic picture of causality. I criticize the standard arguments in favour of peaceful coexistence on the grounds that they are question-begging, and suggest that the breakdown of Lorentz-invariant relativity as a principle theory would be a natural development, given (...) the general trend of physics in this century. (shrink)

Two old problems in probability theory involving the concept of randomness are considered. Data obtained with one of them--Bertrand's chord problem--demonstrate the equivocality of this term in the absence of a definition or explication of assumptions underlying its use. They also support two propositions about probabilistic thinking: (1) upon obtaining an answer to a question of probability, people tend to see it as the answer, overlooking tacit assumptions on which it may be based, and tend not to consider the possibility (...) of other assumptions from which different answers would follow; (2) perceptual balance--moderate uniformity of density of distribution--is a determinant of the degree to which a two-dimensional pattern is perceived as random. (shrink)

Scientific anomalies are observations and facts that contradict current scientific theories and they are instrumental in scientific theory change. Philosophers of science have approached scientific theory change from different perspectives as Darden (Theory change in science: Strategies from Mendelian genetics, 1991) observes: Lakatos (In: Lakatos, Musgrave (eds) Criticism and the growth of knowledge, 1970) approaches it as a progressive “research programmes” consisting of incremental improvements (“monster barring” in Lakatos, Proofs and refutations: The logic of mathematical discovery, 1976), Kuhn (The structure (...) of scientific revolutions, 1996) observes that changes in “paradigms” are instigated by a crisis from some anomaly, and Hanson (In: Feigl, Maxwell (eds) Current issues in the philosophy of science, 1961) proposes that discovery does not begin with hypothesis but with some “problematic phenomena requiring explanation”. Even though anomalies are important in all of these approaches to scientific theory change, there have been only few investigations into the specific role anomalies play in scientific theory change. Furthermore, much of these approaches focus on the theories themselves and not on how the scientists and their experiments bring about scientific change (Gooding, Experiment and the making of meaning: Human agency in scientific observation and experiment, 1990). To address these issues, this paper approaches scientific anomaly resolution from a meaning construction point of view. Conceptual integration theory (Fauconnier and Turner, Cogn Sci 22:133–187, 1996; The way we think: Conceptual blending and mind’s hidden complexities, 2002) from cognitive linguistics describes how one constructs meaning from various stimuli, such as text and diagrams, through conceptual integration or blending. The conceptual integration networks that describe the conceptual integration process characterize cognition that occurs unconsciously during meaning construction. These same networks are used to describe some of the cognition while resolving an anomaly in molecular genetics called RNA interference (RNAi) in a case study. The RNAi case study is a cognitive-historical reconstruction (Nersessian, In: Giere (ed) Cognitive models of science, 1992) that reconstructs how the RNAi anomaly was resolved. This reconstruction traces four relevant molecular genetics publications in describing the cognition necessary in accounting for how RNAi was resolved through strategies (Darden 1991), abductive reasoning (Peirce, In: Hartshorne, Weiss (eds) Collected papers, 1958), and experimental reasoning (Gooding 1990). The results of the case study show that experiments play a crucial role in formulating an explanation of the RNAi anomaly and the integration networks describe the experiments’ role. Furthermore, these results suggest that RNAi anomaly resolution is embodied. It is embodied in a sense that cognition described in the cognitive-historical reconstruction is experientially based. (shrink)

This paper provides a detailed study of David Hilbert’s axiomatization of the theory of plane area, in the classical monograph Foundation of Geometry. On the one hand, we offer a precise contextualization of this theory by considering it against its nineteenth-century geometrical background. Specifically, we examine some crucial steps in the emergence of the modern theory of geometrical equivalence. On the other hand, we analyze from a more conceptual perspective the significance of Hilbert’s theory of area for the foundational program (...) pursued in Foundations. We argue that this theory played a fundamental role in the general attempt to provide a new independent basis for Euclidean geometry. Furthermore, we contend that our examination proves relevant for understanding the requirement of “purity of the method” in the tradition of modern synthetic geometry. (shrink)

In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

The paper offers a new understanding of induction in the empirical sciences, one which assimilates it to induction in geometry rather than to statistical inference. To make the point a system of notions, essential to logically sound induction, is defined. Notable among them are arbitrary object and particular property. A second aim of the paper is to bring to light a largely neglected set of assumptions shared by both induction and deduction in the empirical sciences. This is made possible by (...) appealing to the logic of common nouns and applying it to the logic of natural-kind terms.This strategy yields a new insight into the concept of natural kinds. While the strategy reveals deep affinity between empirical induction and deduction it also reveals two problems peculiar to induction. This helps to explain the intuition that induction is the more problematic of the two. The paper does not set out ‘to solve the problem of induction’. 1The paper has benefited from the support and critical comments of several friends: Steven Davis, Kevin Dunbar, Anil Gupta, Michael Hallett, Ray Jackendoff and Storrs McCall. Two friends, however, deserve special thanks, Michael Makkai and Gonzalo Reyes. Many of the points made in the paper, otherwise unacknowledged, are due to their generosity and patience. In fact, the paper can be seen as an application of the logic of kinds on which Gonzalo Reyes has been working for several years. (shrink)

Abstract Proofs and Refutations is Lakatos's masterpiece. This article investigates some of its central themes, in particular: the nature of proofs ('Proofs do not prove, they improve'); the nature of definitions (real, not nominal); and the consequences of all this for ontology (platonism vs Popper's World Three).

Scientific discourse leaves implicit a vast amount of knowledge, assumes that this background knowledge is taken into account – even taken for granted – and treated as undisputed. In particular, the terminology in the empirical sciences is treated as antecedently understood. The background knowledge surrounding a theory is usually assumed to be true or approximately true. This is in sharp contrast with logic, which explicitly ignores underlying presuppositions and assumes uninterpreted languages. We discuss the problems that background knowledge may cause (...) for the formalization of scientific theories. In particular, we will show how some of these problems can be addressed in the context of the computational representation of scientific theories. (shrink)