Although understanding is the object of a growing literature in epistemology and the philosophy of science, only few studies have concerned understanding in mathematics. This essay offers an account of a fundamental form of mathematical understanding: proof understanding. The account builds on a simple idea, namely that understanding a proof amounts to rationally reconstructing its underlying plan. This characterization is fleshed out by specifying the relevant notion of plan and the associated process of rational (...) reconstruction, building in part on Bratman's theory of planning agency. It is argued that the proposed account can explain a significant range of distinctive phenomena commonly associated with proof understanding by mathematicians and philosophers. It is further argued, on the basis of a case study, that the account can yield precise diagnostics of understanding failures and can suggest ways to overcome them. Reflecting on the approach developed here, the essay concludes with some remarks on how to shape a general methodology common to the study of mathematical and scientific understanding and focused on human agency. (shrink)

Mathematical investigation, when done well, can confer understanding. This bare observation shouldn’t be controversial; where obstacles appear is rather in the effort to engage this observation with epistemology. The complexity of the issue of course precludes addressing it tout court in one paper, and I’ll just be laying some early foundations here. To this end I’ll narrow the field in two ways. First, I’ll address a specific account of explanation and understanding that applies naturally to mathematical reasoning: the (...) view proposed by Philip Kitcher and Michael Friedman of explanation or understanding as involving the unification of theories that had antecedently appeared heterogeneous. For the second narrowing, I’ll take up one specific feature (among many) of theories and their basic concepts that is sometimes taken to make the theories and concepts preferred: in some fields, for some problems, what is counted as understanding a problem may involve finding a way to represent the problem so that it (or some aspect of it) can be visualized. The final section develops a case study which exemplifies the way that this consideration – the potential for visualizability – can rationally inform decisions as to what the proper framework and axioms should be. The discussion of unification (in sections 3 and 4) leads to a mathematical analogue of Goodman’s problem of identifying a principled basis for distinguishing grue and green. Just as there is a philosophical issue about how we arrive at the predicates we should use when making empirical predictions, so too there is an issue about what properties best support many kinds of mathematical reasoning that are especially valuable to us. The issue becomes pressing via an examination of some physical and mathematical cases that make it seem unlikely that treatments of unification can be as straightforward as the philosophical literature has hoped. Though unification accounts have a grain of truth (since a phenomenon (or cluster of phenomena) called “unification” is in fact important in many cases) we are far from an analysis of what “unification” is.. (shrink)

The frame concept from linguistics, cognitive science and artificial intelligence is a theoretical tool to model how explicitly given information is combined with expectations deriving from background knowledge. In this paper, we show how the frame concept can be fruitfully applied to analyze the notion of mathematical understanding. Our analysis additionally integrates insights from the hermeneutic tradition of philosophy as well as Schmid’s ideal genetic model of narrative constitution. We illustrate the practical applicability of our theoretical analysis through a (...) case study on extremal proofs. Based on this case study, we compare our analysis of proof understanding to Avigad’s ability-based analysis of proof understanding. (shrink)

Prouver des théorèmes est une pratique mathématique qui semble clairement améliorer notre compréhension mathématique. Ainsi, prouver et reprouver des théorèmes en mathématiques, vise à apporter une meilleure compréhension. Cependant, comme il est bien connu, les preuves mathématiques totalement formalisées sont habituellement inintelligibles et, à ce titre, ne contribuent pas à notre compréhension mathématique. Comment, alors, comprendre la relation entre prouver des théorèmes et améliorer notre compréhension mathématique. J'avance ici que nous avons d'abord besoin d'une notion différente de preuve (formelle), qui (...) ne tienne pas la forme pour opposée au contenu. La pratique de la preuve algébrique au xviiie siècle fournit un exemple de preuve à la fois pleinement rigoureuse et porteuse de contenu, et dans ce cas, il est possible de voir comment une preuve mathématique apporte une compréhension mathématique. Il s'agira alors de mobiliser les enseignements de cet exemple pour étudier le type de raisonnement déductif à partir de concepts, qui a constitué la norme dans la pratique mathématique depuis le xixe siècle. (shrink)

Prouver des théorèmes est une pratique mathématique qui semble clairement améliorer notre compréhension mathématique. Ainsi, prouver et reprouver des théorèmes en mathématiques, vise à apporter une meilleure compréhension. Cependant, comme il est bien connu, les preuves mathématiques totalement formalisées sont habituellement inintelligibles et, à ce titre, ne contribuent pas à notre compréhension mathématique. Comment, alors, comprendre la relation entre prouver des théorèmes et améliorer notre compréhension mathématique. J'avance ici que nous avons d'abord besoin d'une notion différente de preuve (formelle), qui (...) ne tienne pas la forme pour opposée au contenu. La pratique de la preuve algébrique au xviiie siècle fournit un exemple de preuve à la fois pleinement rigoureuse et porteuse de contenu, et dans ce cas, il est possible de voir comment une preuve mathématique apporte une compréhension mathématique. Il s'agira alors de mobiliser les enseignements de cet exemple pour étudier le type de raisonnement déductif à partir de concepts, qui a constitué la norme dans la pratique mathématique depuis le xixe siècle. (shrink)

Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting (...) meanings of the word “argument”; to some of the specific argumentation-theoretic tools that have been applied to mathematics, notably Toulmin layouts and argumentation schemes; to some of the different ways that argumentation is implicated in mathematical practices; and to the social aspects of mathematical argumentation. (shrink)

Diagrams are ubiquitous in mathematics. From the most elementary class to the most advanced seminar, in both introductory textbooks and professional journals, diagrams are present, to introduce concepts, increase understanding, and prove results. They thus fulfill a variety of important roles in mathematical practice. Long overlooked by philosophers focused on foundational and ontological issues, these roles have come to receive attention in the past two decades, a trend in line with the growing philosophical interest in actual mathematical practice.

In a recent paper in this journal, Mark Balaguer develops and defends a new version of mathematical fictionalism, what he calls ‘Hermeneutic non-assertivism’, and responds to some recent objections to mathematical fictionalism that were launched by John Burgess and others. In this paper I provide some fairly compelling reasons for rejecting Hermeneutic non-assertivism — ones that highlight an important feature of what understanding mathematics involves (or, as we shall see, does not involve).

A lot of philosophical energy has been devoted recently in trying to determine if mathematics can contribute to our understanding of physical phenomena. Not many philosophers are interested, though, if the converse makes sense, i.e., if our cognitive interaction (scientific or otherwise) with the physical world can be helpful (in an explanatory or non-explanatory way) in our efforts to make sense of mathematical facts. My aim in this paper is to try to fill this important lacuna in the (...) recent literature. My answer to the question of this paper is negative. As I will argue, there are serious problems with the main reasons for believing in the first place that it is possible to have physical understanding of mathematical facts. (shrink)

In this paper, we wish to explore the role that textual representations play in the creation of new mathematical objects. We do so by analyzing texts by Joseph-Louis Lagrange (1736–1813) and Évariste Galois (1811–1832), which are seen as central to the historical development of the mathematical concept of groups. In our analysis, we consider how the material features of representations relate to the changes in conceptualization that we see in the texts.Against this backdrop, we discuss the idea that new mathematical (...) concepts, in general, are increasingly abstract in the sense of being detached from material configurations. Our analysis supports the opposite view. We suggest that changes in the material aspects of textual representations (i.e., the actual graphic inscriptions) play an active and crucial role in conceptual change.We employ an analytical framework adapted from Bruno Latour’s 1999 account of intertwined material and representational practices in the empirical sciences. This approach facilitates a foregrounding of the interconnection between the conceptual development of mathematics, and the construction, (re-)configuration, and manipulation of the materiality of representations. Our analysis suggests that, in mathematical practice, distinctions between the material and structural features of representations are not permanent and absolute. This problematizes the appropriateness of the distinction between concrete inscriptions and abstract relations in understanding the development of mathematical concepts. (shrink)

The philosophical analysis of mathematical explanations concerns itself with two different, although connected, areas of investigation. The first area addresses the problem of whether mathematics can play an explanatory role in the natural and social sciences. The second deals with the problem of whether mathematical explanations occur within mathematics itself. Accordingly, this entry surveys the contributions to both areas, it shows their relevance to the history of philosophy and science, it articulates their connection, and points to the philosophical (...) pay-offs to be expected by deepening our understanding of the topic. (shrink)

Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, how do (...) they relate to the sorts of explanation encountered in philosophy of science and metaphysics? (shrink)

Visual thinking in mathematics is widespread; it also has diverse kinds and uses. Which of these uses is legitimate? What epistemic roles, if any, can visualization play in mathematics? These are the central philosophical questions in this area. In this introduction I aim to show that visual thinking does have epistemically significant uses. The discussion focuses mainly on visual thinking in proof and discovery and touches lightly on its role in understanding.

In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting (...) for the role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role. (shrink)

In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, although they do not fall squarely under a traditional philosophical purview. The goal of this article is to try to articulate some of these questions more clearly, and assess the philosophical methods that may be brought to bear. In Section 3, I note that most of the issues can be classified under two headings: (...) some deal with the ability of computers to deliver appropriate “evidence” for mathematical assertions, a notion that is explored in Section 4, while others deal with the ability of computers to deliver appropriate mathematical “understanding,” a notion that is considered in Section 5. Final thoughts are provided in Section 6. (shrink)

Definitions are traditionally seen as abbreviations, as tools for notational convenience that do not increase inferential power. From a Philosophy of Mathematical Practice point of view, however, there is much more to definitions. For example, definitions can play a role in problem solving, definitions can contribute to understanding, sometimes equivalent definitions are appreciated differently, and so on. This chapter reviews the literature on definitions and (to a certain extent) concepts in mathematical practice. It is structured according to four themes (...) through which definitions (and concepts) in mathematical practice have been studied. These themes concern (1) the nature of definitions, (2) whether and how concepts evolve, (3) definitions and concepts from a communal perspective, and (4) different values relating to definitions and concepts. (shrink)

Concerns the ‘problem of existence’ in mathematics: the problem of how to understand existence assertions in mathematics. The problem can best be understood by considering how Mathematical Platonists have understood such existence assertions. These philosophers have taken the existential theorems of mathematics as literally asserting the existence of mathematical objects. They have then attempted to account for the epistemological and metaphysical implications of such a position by putting forward arguments that supposedly show how humans can come to (...) know of the existence of mathematical objects. The reasoning of the most prominent philosophers in this Literalist tradition, Quine and Gödel, are critically discussed. By way of contrast, an alternative position, Heyting's Intuitionism, is examined. (shrink)

Philosophers have paid little attention to mathematical explanations . I present a variety of examples of mathematical explanation and examine two cases in detail. I argue that mathematical explanations have important implications for the philosophy of mathematics and of science. ;The first case study compares many proofs of Pick's theorem, a simple geometrical result. Though a simple proof surfaces to establish the result, some of the proofs explain the result better than others. The second case study comes from George (...) Polya's Mathematics and Plausible Reasoning. He gives a proof that, while entirely satisfactory in establishing its conclusion, is insufficiently explanatory. To provide a better explanation, he supplements the proof with additional exposition. ;These case studies illustrate at least two distinct explanatory virtues, and suggest there may be more. First, an explanatory improvement occurs when a sense of "arbitrariness" is reduced in the proofs. Proofs more explanatory in this way place greater restrictions on the steps that can be used to reach the conclusion. Second, explanatoriness is judged by directness of representation. More explanatory proofs allow one to ascribe geometric meaning to the terms of Pick's formula as they arise. ;I trace the lack of attention to mathematical explanations to an implicit assumption, justificationism, that only justificational aspects of mathematical reasoning are epistemically important. I propose an anti-justificationist epistemic position, the epistemic virtues view, which holds that justificational virtues, while important, are not the only ones of philosophical interest in mathematics. Indeed, explanatory benefits are rarely justificational. I show how the epistemic virtues view and the recognition of mathematical explanation can shed new light on philosophical debates. ;Mathematical explanations have consequences for philosophy of science as well. I show that mathematical explanations provide serious challenges to any theory, such as Bas van Fraassen's, that considers explanations to be fundamentally answers to why-questions. I urge a closer interaction between philosophy of mathematics and philosophy of science; both will be needed for a fuller understanding of mathematical explanation. (shrink)

Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that (...) visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking. (shrink)

As in science and philosophy, thought experiments in mathematics link a problem to new epistemic resources that are unavailable in a given practice, e.g., Euclidean geometry. Thought experiments invite us to perform an imaginary scenario involving counterfactual, deductive and sensory elements. This chapter aims to pinpoint the beneficial peculiarities of thought experiments in mathematics in comparison with inferences, diagrams and calculative procedures. Reflection about thought experiments assists us to realize both the limits and opportunities in mathematical thinking. Henceforth, (...) the analysis of examples suggests a broader understanding of mathematical, thought and experiment. (shrink)

Over the past half-century, formal, machine-executable proofs have been developed for an impressive range of mathematical theorems. Formalists argue that such proofs should be seen as providing the fully worked out proofs of which mathematicians’ proofs are sketches. Nonformalists argue that this conception of the relationship of formal to informal proofs cannot explain the fact that formal proofs lack essential virtues enjoyed by mathematicians’ proofs, the fact, for example, that formal proofs are not convincing and lack the explanatory power of (...) their informal counterparts. Formal proofs do not yield mathematical insight or understanding, and they are not fruitful in the way that informal proofs can be, for instance, in suggesting novel approaches to old problems. And yet, there is a clear sense in which the formalist is right: Formal proofs do seem to make explicit what is otherwise taken for granted in the mathematician’s reasoning. Very recent work uncovers the source of the dilemma by showing that rigor does not entail formalism insofar as rigor, unlike formalism, is compatible with contentfulness. Mathematical proofs, were they to be recast in a Leibnizian language, at once a characteristica and a calculus, would be, or could be made to be, completely gap-free and hence machine checkable. Because as so recast they would be grounded in mathematical ideas, they would not be machine executable. It is on just this basis that a compelling account can be given of the fact that formal proofs are mostly irrelevant to mathematicians in their practice. (shrink)

The role of audiences in mathematical proof has largely been neglected, in part due to misconceptions like those in Perelman and Olbrechts-Tyteca which bar mathematical proofs from bearing reflections of audience consideration. In this paper, I argue that mathematical proof is typically argumentation and that a mathematician develops a proof with his universal audience in mind. In so doing, he creates a proof which reflects the standards of reasonableness embodied in his universal audience. Given this framework, we can better understand (...) the introduction of proof methods based on the mathematician’s likely universal audience. I examine a case study from Alexander and Briggs’s work on knot invariants to show that we can fruitfully reconstruct mathematical methods in terms of audiences. (shrink)

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The (...) starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. (shrink)

Marcus Giaquinto presents an investigation into the different kinds of visual thinking involved in mathematical thought, drawing on work in cognitive psychology, philosophy, and mathematics. He argues that mental images and physical diagrams are rarely just superfluous aids: they are often a means of discovery, understanding, and even proof.

Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two (...) cases: Cramér’s random model of the prime numbers and the function field model of the integers. These cases show that mathematicians, like empirical scientists, rely on unrealistic models to gain understanding of complex phenomena. They also have important implications for some much-discussed theses about scientific understanding. First, modeling practices in mathematics confirm that one can gain understanding without obtaining an explanation. Second, these cases undermine the popular thesis that unrealistic models confer understanding by imparting counterfactual knowledge. (shrink)

This paper argues for two related theses. The first is that mathematical abstraction can play an important role in shaping the way we think about and hence understand certain phenomena, an enterprise that extends well beyond simply representing those phenomena for the purpose of calculating/predicting their behaviour. The second is that much of our contemporary understanding and interpretation of natural selection has resulted from the way it has been described in the context of statistics and mathematics. I argue (...) for these claims by tracing attempts to understand the basis of natural selection from its early formulation as a statistical theory to its later development by R.A. Fisher, one of the founders of modern population genetics. Not only did these developments put natural selection of a firm theoretical foundation but its mathematization changed the way it was understood as a biological process. Instead of simply clarifying its status, mathematical techniques were responsible for redefining or reconceptualising selection. As a corollary I show how a highly idealised mathematical law that seemingly fails to describe any concrete system can nevertheless contain a great deal of accurate information that can enhance our understanding far beyond simply predictive capabilities. (shrink)

The core idea of social constructivism in mathematics is that mathematical entities are social constructs that exist in virtue of social practices, similar to more familiar social entities like institutions and money. Julian C. Cole has presented an institutional version of social constructivism about mathematics based on John Searle’s theory of the construction of the social reality. In this paper, I consider what merits social constructivism has and examine how well Cole’s institutional account meets the challenge of accounting (...) for the characteristic features of mathematics, especially objectivity and applicability. I propose that in general social constructivism shows promise as an ontology of mathematics, because the view can agree with mathematical practice and it offers a way of understanding how mathematical entities can be real without conflicting with a scientific picture of reality. However, I argue that Cole’s specific theory does not provide an adequate social constructivist account of mathematics. His institutional account fails to sufficiently explain the objectivity and applicability of mathematics, because the explanations are weakened and limited by the three-level theoretical model underlying Cole’s account of the construction of mathematical reality and by the use of the Searlean institutional framework. The shortcomings of Cole’s theory give reason to suspect that the Searlean framework is not an optimal way to defend the view that mathematical reality is socially constructed. (shrink)

In his 1939 Cambridge Lectures on the Foundations of Mathematics, Wittgenstein proclaims that he is not out to persuade anyone to change their opinions. I seek to further our understanding of this point by investigating an exchange between Wittgenstein and Turing on contradictions. In defending the claim that contradictory calculi are mathematically defective, Turing suggests that applying such a calculus would lead to disasters such as bridges falling down. In the ensuing discussion, it can seem as if Wittgenstein (...) challenges Turing's claim that such disasters would occur. I argue that this is not what Wittgenstein is doing. Rather, he is scrutinizing the meaning and philosophical import of Turing's claim—showing how Turing is wavering between making an empirical prediction and a logical observation, and that it is only through this wavering that Turing can believe that he has provided a proper explanation of why contradictory calculi are mathematically defective. (shrink)

Mathematics is one of the core STEM (science, technology, engineering, and mathematics) subject disciplines. Engaging students in learning mathematics helps retain students in STEM fields and thus contributes to the sustainable development of society. To increase student engagement, some mathematics instructors have redesigned their courses using the flipped classroom approach. In this review, we examined the results of comparative studies published between 2011 and 2020 to summarize the effects of this instructional approach (vs. traditional lecturing) on (...) students’ behavioral, emotional, and cognitive engagement with mathematics courses. Thirty-three articles in K–12 and higher education contexts were included for analysis. The results suggest that the use of the flipped classroom approach may increase some aspects of behavioral engagement (e.g., interaction and attention/participation), emotional engagement (e.g., course satisfaction), and cognitive engagement (e.g., understanding of mathematics). However, we discovered that several aspects (e.g., students’ attendance, mathematics anxiety, and self-regulation) of student engagement have not been thoroughly explored and are worthy of further study. The results of this review have important implications for future flipped classroom practice (e.g., engaging students in solving real-world problems), and for research on student engagement (e.g., using more objective measures, such as classroom observation) in mathematics education. (shrink)

The paper is concerned with the way in which “ontology” and “realism” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possible coherent positions, their strengths and weaknesses. I shall also discuss related but different aspects of these problems. The terms in the title are the common thread that connects the various sections.

The article is devoted to the applicability of Wittgenstein’s following the rule in the context of his philosophy of mathematics to real mathematical practice. It is noted that in «Philosophical Investigations» and «Remarks on the Foundations of Mathematics» Wittgenstein resorted to the analysis of rather elementary mathematical concepts, accompanied also by the inherent ambiguity and ambiguity of his presentation. In particular, against this background, his radical conventionalism, the substitution of logical necessity with the «form of life» of the (...) community, as well as the inadequacy of the representation of arithmetic rules by a language game are criticized. It is shown that the reconstruction of the Wittgenstein concept of understanding based on the Fregian division of meaning and referent goes beyond the conceptual framework of Wittgenstein language games. (shrink)

Mathematical explanations are poorly understood. Although mathematicians seem to regularly suggest that some proofs are explanatory whereas others are not, none of the philosophical accounts of what such claims mean has become widely accepted. In this paper we explore Wilkenfeld’s suggestion that explanations are those sorts of things that generate understanding. By considering a basic model of human cognitive architecture, we suggest that existing accounts of mathematical explanation are all derivable consequences of Wilkenfeld’s ‘functional explanation’ proposal. We therefore argue (...) that the explanatory criteria offered by earlier accounts can all be thought of as features that make it more likely that a mathematical proof will generate understanding. On the functional account, features such as characterising properties, unification, and salience correlate with explanatoriness, but they do not define explanatoriness. (shrink)

Mathematicians’ use of external representations, such as symbols and diagrams, constitutes an important focal point in current philosophical attempts to understand mathematical practice. In this paper, we add to this understanding by presenting and analyzing how research mathematicians use and interact with external representations. The empirical basis of the article consists of a qualitative interview study we conducted with active research mathematicians. In our analysis of the empirical material, we primarily used the empirically based frameworks provided by distributed cognition (...) and cognitive semantics as well as the broader theory of cognitive integration as an analytical lens. We conclude that research mathematicians engage in generative feedback loops with material representations, that they use representations to facilitate the use of experiences of handling the physical world as a resource in mathematical work, and that their use of representations is socially sanctioned and enabled. These results verify the validity of the cognitive frameworks used as the basis for our analysis, but also show the need for augmentation and revision. Especially, we conclude that the social and cultural context cannot be excluded from cognitive analysis of mathematicians’ use of external representations. Rather, representations are socially sanctioned and enabled in an enculturation process. (shrink)

This volume of essays addresses the main problem confronting an epistemology for mathematics; namely, the nature and sources of mathematical justification. Attending to both particular and general issues, the essays, by leading philosophers of mathematics, raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And of what epistemological importance is the formalizability of proof? The editor, Michael (...) Detlefsen, has brought together an outstanding collection of essays, only one of which has previously appeared. It will be essential for philosophers and historians of mathematics, as well as philosophically inclined logicians and philosophers interested in the nature of reasoning and justification. A companion volume entitled Proof, Logic and Formalization, edited by Michael Detlefsen, is also available from Routledge. (shrink)

After a brief outline of the topic of non-language thinking in mathematics the central phenomenological tool in this concern is established, i.e. the eidetic method. The special form of eidetic method in mathematical proving is implicit variation and this procedure entails three rules that are established in a simple geometrical example. Then the difficulties and the merits of analogical thinking in mathematics are discussed in different aspects. On the background of a new phenomenological understanding of the performance (...) of non-language thinking in mathematics the well-known theses of B. L. van der Waerden that mathematical thinking to a great extent proceeds without the use of language is discussed in a new light. (shrink)

In the article the problem of independence in mathematics is discussed. The status of the continuum hypothesis, large cardinal axioms and the axiom of constructablility is presented in some detail. The problem whether incompleteness is really relevant for ordinary mathematics and for empirical science is investigated. Another aim of the article is to give some arguments for the thesis that the problem of reliability and justification of new axioms is well-posed and worthy of attention. In my opinion, investigations (...) concerning the status of independent sentences give insight into our understanding of mathematical concepts, of mathematical knowledge and of the role of mathematics in empirical science. (shrink)

The paper is devoted to the reconstruction of some stage of the proces leading to the emergence in modern science the concept of Infinite „Euclidean” space to geometry of the Elements in late antiquity and the Middle Ages. Some historical medieval sources and views concerning Archytas, Cleomedes, Proclus, Simplicius, Aganis, al-Nayrizi and the Arabs, Boetius, Gerard of Cremona, Albertus Magnus et al., are described analyzed and compared. The small changes in the understanding of geometry in the Elements during the (...) ages are reconstructed up to the first explicit use of the concept of infinity in geometry by Nicole Oresme. (shrink)

The present research seeks to utilize Implicit Theories of Intelligence and Achievement Goal Theory to understand students’ intrinsic motivation and academic performance in mathematics in Singapore. 1,201 lower-progress stream students, ages ranged from 13 to 17 years, from 17 secondary schools in Singapore took part in the study. Using structural equation modeling, results confirmed hypotheses that incremental mindset predicted mastery-approach goals and, in turn, predicted intrinsic motivation and mathematics performance. Entity mindset predicted performance-approach and performance-avoidance goals. Performance-approach goal (...) was positively linked to intrinsic motivation and mathematics performance; performance-avoidance goal, however, negatively predicted intrinsic motivation and mathematics performance. The model accounted for 35.9% of variance in intrinsic motivation and 13.8% in mathematics performance. These findings suggest that intrinsic motivation toward mathematics and achievement scores might be enhanced through interventions that focus on incremental mindset and mastery-approach goal. In addition, performance-approach goal may enhance intrinsic motivation and achievement as well, but to a lesser extent. Finally, the study adds to the literature done in the Asian context and lends support to the contention that culture may affect students’ mindsets and adoption of achievement goals, and their associated impact on motivation and achievement outcomes. (shrink)

This paper pursues two goals. Its first goal is to clear up the “identity problem” faced by the structuralist interpretation of mathematics. Its second goal, through the consideration of examples coming in particular from the theory of permutations, is to examine cases of misunderstandings in mathematics fit to cast some light on mathematical understanding in general. The common thread shared by these two goals is the notion of setting. The study of a mathematical object almost always goes (...) together with the choice of a particular setting, and the understanding of the workings of mathematical settings is an essential component of mathematical knowledge. It is claimed that the recognition of mathematical settings, as features distinct from both mathematical structures and the systems which instantiate those structures, allows one to classify most of understandable misunderstandings in mathematics, and also to solve the identity problem. (shrink)

ACMES is a multidisciplinary conference series that focuses on epistemological and mathematical issues relating to computation in modern science. This volume includes a selection of papers presented at the 2015 and 2016 conferences held at Western University that provide an interdisciplinary outlook on modern applied mathematics that draws from theory and practice, and situates it in proper context. These papers come from leading mathematicians, computational scientists, and philosophers of science, and cover a broad collection of mathematical and philosophical topics, (...) including numerical analysis and its underlying philosophy, computer algebra, reliability and uncertainty quantification, computation and complexity theory, combinatorics, error analysis, perturbation theory, experimental mathematics, scientific epistemology, and foundations of mathematics. By bringing together contributions from researchers who approach the mathematical sciences from different perspectives, the volume will further readers' understanding of the multifaceted role of mathematics in modern science, informed by the state of the art in mathematics, scientific computing, and current modeling techniques. (shrink)

This paper is an updated translation of an article published in French in the Journal Lato Sensu (I, 2014, p. 70-80). We study here the so-called 'Mathematical part' of Plato's Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of integers. As the most ancient text on the subject, and on Greek mathematics and mathematicians as well, its historical importance is enormous. The difficulty to understand (...) it lies in the close intertwining of different fields we found in it: philosophy, history and mathematics. But conversely, correctly understood, it gives some evidences both about the question of the origins of the irrationals in Greek mathematics and some points concerning Plato's thought. Taking into account the historical context and the philosophical background generally forgotten in mathematical analyses, we get a new interpretation of this text, which far from being a tribute to some mathematicians, is a radical criticism of their ways of thinking. And the mathematical lesson, far from being a tribute to some future mathematical achievements, is ending on an aporia, in accordance with the whole dialogue. Résumé. Cet article est une traduction anglaise révisée d'un article paru en français dans la revue Lato Sensu (I, 2014, p. 70-80). (shrink)

Research into mathematics often focuses on basic numerical and spatial intuitions, and one key property of numbers: their magnitude. The fact that mathematics is a system of complex relationships that invokes reasoning usually receives less attention. The purpose of this special issue is to highlight the intricate connections between reasoning and mathematics, and to use insights from the reasoning literature to obtain a more complete understanding of the processes that underlie mathematical cognition. The topics that are (...) discussed range from the basic heuristics and biases to the various ways in which complex, effortful reasoning contributes to mathematical cognition, while also considering the role of individual differences in mathematics performance. These investigations are not only important at a theoretical level, but they also have broad and important practical implications, including the possibility to improve classroom practices and educational outcomes, to facilitate people's decision-making, as well as the clear and accessible communication of numerical information. (shrink)

This paper surveys central justifications and approaches adopted by educators interested in incorporating history of mathematics into mathematics teaching and learning. This interest itself has historical roots and different historical manifestations; these roots are examined as well in the paper. The paper also asks what it means for history of mathematics to be treated as genuine historical knowledge rather than a tool for teaching other kinds of mathematical knowledge. If, however, history of mathematics is not subordinated (...) to the ideas and methods at the heart of the usual mathematical curriculum – algebraic equations, functions, derivatives, analytic geometry – if it becomes an essential subject for mathematics, then the attempt to find a place for history of mathematics requires refining our understanding of the nature of mathematics education itself. Thus, the paper asks not only how can history of mathematics be incorporated into mathematics education but also how the idea of mathematics education may need to be adjusted to accommodate history. (shrink)

This is not a mathematics book, but a book about mathematics, which addresses both student and teacher, with a goal as practical as possible, namely to initiate and smooth the way toward the student’s full understanding of the mathematics taught in school. The customary procedural-formal approach to teaching mathematics has resulted in students’ distorted vision of mathematics as a merely formal, instrumental, and computational discipline. Without the conceptual base of mathematics, students develop over (...) time a “mathematical anxiety” and abandon any effort to understand mathematics, which becomes their “traditional enemy” in school. This work materializes the results of the inter- and trans-disciplinary research aimed toward the understanding of mathematics, which concluded that the fields with the potential to contribute to mathematics education in this respect, by unifying the procedural and conceptual approaches, are epistemology and philosophy of mathematics and science, as well as fundamentals and history of mathematics. These results argue that students’ fear of mathematics can be annulled through a conceptual approach, and a student with a good conceptual understanding will be a better problem solver. The author has identified those zones and concepts from the above disciplines that can be adapted and processed for familiarizing the student with this type of knowledge, which should accompany the traditional content of school mathematics. The work was organized so as to create for the reader a unificatory image of the complex nature of mathematics, as well as a conceptual perspective ultimately necessary to the holistic understanding of school mathematics. The author talks about mathematics to convince readers that to understand mathematics means first to understand it as a whole, but also as part of a whole. The nature of mathematics, its primary concepts (like numbers and sets), its structures, language, methods, roles, and applicability, are all presented in their essential content, and the explanation of non-mathematical concepts is done in an accessible language and with many relevant examples. (shrink)