The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...) of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...) from considering rigour of proof as a fixture, I discuss fixed models, invariant notions and fixed information about objects across theories. There are other fixtures, but it is enough to start with these. (shrink)

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)

ArgumentThis paper argues for the viability of a different philosophical point of view concerning classical Greek geometry. It reviews Reviel Netz's interpretation of classical Greek geometry and offers a Deleuzian, post-structural alternative. Deleuze's notion of haptic vision is imported from its art history context to propose an analysis of Greek geometric practices that serves as counterpoint to their linear modular cognitive narration by Netz. Our interpretation highlights the relation between embodied practices, noisy material constraints, and operational codes. Furthermore, it sheds (...) some new light on the distinctness and clarity of Greek mathematical conceptual divisions. (shrink)

Philosophers of information, according to Luciano Floridi (The philosophy of information. Oxford University Press, Oxford, 2010, p 32), study how information should be “adequately created, processed, managed, and used.” A small number of epistemologists have employed the concept of information as a cornerstone of their theoretical framework. How this concept can be used to make sense of seemingly intractable epistemological problems, however, has not been widely explored. This paper examines Fred Dretske’s information-based epistemology, in particular his response to radical epistemological (...) skepticism. We discuss the relationship between information, evidence and knowledge in relation to the problem of skepticism and the options available to an information-based epistemology for dealing with it. (shrink)

The canonical history of mathematics suggests that the late 19th-century “arithmetization” of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, (...) which reveals a reliance on dynamic conceptual resources. The second is a cognitive-historical case study of an incident in 19th-century mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice. (shrink)

It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it (...) accommodates the many mathematical proofs that include actions on objects other than propositions; (v) this conception of logic permits the articulation of project-sized tasks for the philosophy of mathematical practice, thereby supplying a partial characterisation of normal research in the field. (shrink)

This article discusses the role of diagrams in mathematical reasoning in the light of a case study in analysis. In the example presented certain combinatorial expressions were first found by using diagrams. In the published proofs the pictures were replaced by reasoning about permutation groups. This article argues that, even though the diagrams are not present in the published papers, they still play a role in the formulation of the proofs. It is shown that they play a role in concept (...) formation as well as representations of proofs. In addition we note that 'visualization' is used in two different ways. In the first sense 'visualization' denotes our inner mental pictures, which enable us to see that a certain fact holds, whereas in the other sense 'visualization' denotes a diagram or representation of something. (shrink)

Leibniz frequently argued that reasons are to be weighed against each other as in a pair of scales, as Professor Marcelo Dascal has shown in his article "The Balance of Reason." In this kind of weighing it is not necessary to reach demonstrative certainty – one need only judge whether the reasons weigh more on behalf of one or the other option However, a different kind of account about rational decision-making can be found in some of Leibniz's writings. In his (...) article "Was Leibniz's Deity an Akrates?" Professor Jaakko Hintikka has argued that Leibniz developed a new vectorial model for rational decisions which is better suited to complicated decisions, where values are complementary to each other. This model, related closely to his work in metaphysics and the philosophy of mind, is a heuristic device which helps in finding rational combinations - and in an ideal case an optimum - between plural inclinations to the good. I shall argue that Leibniz applies more or less implicitly both of these models in his practical rationality. In simple situations he applied the pair of scales model and in more complicated situations he applied the vectorial model. (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.

Just as mathematics helps us to represent and reason about the natural world, in its internal applications one branch of mathematics helps us to represent and reason about the subject matter of another. Recognition of the close analogy between internal and external applications of mathematics can help resolve two persistent philosophical puzzles concerning its applicability: a platonist puzzle arising from the abstractness of mathematical objects; and an empiricist puzzle arising from mathematical propositions’ lack of empirical factual content. In order to (...) see how this is the case, we will examine what it is to apply mathematics internally and describe examples. (shrink)

Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the potential to (...) broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding. (shrink)

This paper will not present a case study of the historical development of a virtual entity. Rather, I will develop an outlook on virtual entities in the sciences and propose a corresponding method for studying them (historically). In essence, my presentation can be considered a synthesis of different observations from the history and philosophy of science and has its roots in my dissertational research on the development of the virtual particle. Starting with a reflection on the role of presentism for (...) the study of concept formation and development processes, I will show, through the example of the virtual particle, how current debates and interpretations can inform our access to a historical reconstruction. Following these reflections, I will argue for a pragmatist account of concepts as tools for the scientific practitioners. According to the approach presented in my article, concepts perform their functions through representations, and I will lay special focus on verbal representations and their different functions within scientific reasoning. In conclusion, I will frame the outcome of my discussion in terms of a proposal that might, through further research, enrich our understanding of virtual entities in the sciences. (shrink)

I argue that mathematical representations can have heuristic power since their construction can be ampliative. To this end, I examine how a representation introduces elements and properties into the represented object that it does not contain at the beginning of its construction, and how it guides the manipulations of the represented object in ways that restructure its components by gradually adding new pieces of information to produce a hypothesis in order to solve a problem.In addition, I defend an ‘inferential’ approach (...) to the heuristic power of representations by arguing that these representations draw on ampliative inferences such as analogies and inductions. In effect, in order to construct a representation, we have to ‘assimilate’ diverse things, and this requires identifying similarities between them. These similarities form the basis for ampliative inferences that gradually build hypotheses to solve a problem.To support my thesis, I analyse two examples. The first one is intra-field, that is, the construction of an algebraic representation of 3-manifolds; the second is inter-fields, that is, the construction of a topological representation of DNA supercoiling. (shrink)

The idea that ambiguity can be productive in data science remains controversial. Efforts to make scientific publications and data intelligible to computers generally assume that accommodating multiple meanings for words, known as polysemy, undermines reasoning and communication. This assumption has nonetheless been contested by historians, philosophers, and social scientists, who have applied qualitative research methods to demonstrate the generative and strategic value of polysemy. Recent quantitative results from linguistics have also shown how polysemy can actually improve the efficiency of human (...) communication. I present a new conceptual typology based on a synthesis of prior research about the aims, norms, and circumstances under which polysemy arises and is evaluated. The typology supports a contextual pluralist view of polysemy’s value for scientific research practices: polysemy does both substantial positive and negative work in science, but its utility is context-sensitive in ways that are often overlooked by the norms people have formulated to regulate its use, including prior scholars researching polysemy. I also propose that historical patterns in the use of partial synonyms, i.e. terms with overlapping meanings, provide an especially promising phenomenon for integrative research addressing these issues. (shrink)

Both early analytic philosophy and the branch of mathematics now known as topology were gestated and born in the early part of the 20th century. It is not well recognized that there was early interaction between the communities practicing and developing these fields. We trace the history of how topological ideas entered into analytic philosophy through two migrations, an earlier one conceiving of topology geometrically and a later one conceiving of topology algebraically. This allows us to reassess the influence and (...) significance of topological methods for philosophy, including the possible fruitfulness of a third conception of topology as a structure determining similarity. (shrink)

In this excellent book Sebastien Gandon focuses mainly on Russell's two major texts, Principa Mathematica and Principle of Mathematics, meticulously unpicking the details of these texts and bringing a new interpretation of both the mathematical and the philosophical content. Winner of The Bertrand Russell Society Book Award 2013.

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

This paper is concerned with the status of mathematical fictions in Leibniz’s work and especially with infinitary quantities as fictions. Thus, it is maintained that mathematical fictions constitute a kind of symbolic notion that implies various degrees of impossibility. With this framework, different kinds of notions of possibility and impossibility are proposed, reviewing the usual interpretation of both modal concepts, which appeals to the consistency property. Thus, three concepts of the possibility/impossibility pair are distinguished; they give rise, in turn, to (...) three concepts of mathematical fictions. Moreover, such a distinction is the base for the claim that infinitesimal quantities, as mathematical fictions, do not imply an absolute impossibility, resulting from self-contradiction, but a relative impossibility, founded on irrepresentability and on the fact that it does not conform to architectural principles. In conclusion, this “soft” impossibility of infinitesimals yields them, in Leibniz view, a presumptive or “conjectural” status. (shrink)

We treat the problem of reasoning with ambiguous propositions. Even though ambiguity is obviously problematic for reasoning, it is no less obvious that ambiguous propositions entail other propositions, and are entailed by other propositions. This article gives a formal analysis of the underlying mechanisms, both from an algebraic and a logical point of view. The main result can be summarized as follows: sound reasoning with ambiguity requires a distinction between equivalence on the one and congruence on the other side: the (...) fact that \ entails \ does not imply \ can be substituted for \ in all contexts preserving truth. Without this distinction, we will always run into paradoxical results. We present the sequent calculus \, which we conjecture implements sound and complete propositional reasoning with ambiguity, and provide it with a language-theoretic semantics, where letters represent unambiguous meanings and concatenation represents ambiguity. (shrink)

In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. Against this, we show that (...) by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions—a conception he later characterized as “syncategorematic”. Thus, one cannot infer the existence of infinitesimals from their successful use. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero; and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom. (shrink)

Additional theorizing about mathematical practice is needed in order to ground appeals to truly useful notions of the virtues in mathematics. This paper aims to contribute to this theorizing, first, by characterizing mathematical practice as being epistemic and “objectual” in the sense of Knorr Cetina The practice turn in contemporary theory, Routledge, London, 2001). Then, it elaborates a MacIntyrean framework for extracting conceptions of the virtues related to mathematical practice so understood. Finally, it makes the case that Wittgenstein’s methodology for (...) examining mathematics and its practice is the most appropriate one to use for the actual investigation of mathematical practice within this MacIntyrean framework. At each stage of thinking through mathematical practice by these means, places where new virtue-theoretic questions are opened up for investigation are noted and briefly explored. (shrink)

In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the (...) distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education. (shrink)

What sorts of epistemic virtues are required for effective mathematical practice? Should these be virtues of individual or collective agents? What sorts of corresponding epistemic vices might interfere with mathematical practice? How do these virtues and vices of mathematics relate to the virtue-theoretic terminology used by philosophers? We engage in these foundational questions, and explore how the richness of mathematical practices is enhanced by thinking in terms of virtues and vices, and how the philosophical picture is challenged by the complexity (...) of the case of mathematics. For example, within different social and interpersonal conditions, a trait often classified as a vice might be epistemically productive and vice versa. We illustrate that this occurs in mathematics by discussing Gerovitch’s historical study of the aggressive adversarialism of the Gelfand seminar in post-war Moscow. From this we conclude that virtue epistemologies of mathematics should avoid pre-emptive judgments about the sorts of epistemic character traits that ought to be promoted and criticised. (shrink)

The purpose of this article is to analyse the mathematical practices leading to Rafael Bombelli’s L’algebra (1572). The context for the analysis is the Italian algebra practiced by abbacus masters and Renaissance mathematicians of the fourteenth to sixteenth centuries. We will focus here on the semiotic aspects of algebraic practices and on the organisation of knowledge. Our purpose is to show how symbols that stand for underdetermined meanings combine with shifting principles of organisation to change the character of algebra.

In a famous passage from his al-Bāhir, al-Samaw’al proves the identity which we would now write as (ab)n=anbn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(ab)^n=a^n b^n$$\end{document} for the cases n=3,4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=3,4$$\end{document}. He also calculates the equivalent of the expansion of the binomial (a+b)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a+b)^n$$\end{document} for the same values of n and describes the construction of what we now call the Pascal Triangle, showing (...) the table up to its 12th row. We give a literal translation of the whole passage, along with paraphrases in more modern or symbolic form. We discuss the influence of the Euclidean tradition on al-Samaw’al’s presentation, and the role that diagrams might have played in helping al-Samaw’al’s readers follow his arguments, including his supposed use of an early form of mathematical induction. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

This article closely compares the opposing foundations of theoretical philosophy in René Descartes and Ayn Rand. The developmental course of Rand's foundations, with their continual opposition to Descartes, is tracked. Arguments particularly against Descartes are assembled in this article, and the bountiful contemporary scholarship on Descartes is engaged.

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)

This is the first part of a two-part paper in which I conclude the process, initiated elsewhere, of tracking objective conditions of vagueness of representation from language to pictures, from philosophy to imaging science, from vagueness to approximation, from representation to reasoning, with a focus on the application of fuzzy set theory and its challenges.

The paper asks whether diagrams in mathematics are particularly fruitful compared to other types of representations. In order to respond to this question a number of examples of propositions and their proofs are considered. In addition I use part of Peirce’s semiotics to characterise different types of signs used in mathematical reasoning, distinguishing between symbolic expressions and 2-dimensional diagrams. As a starting point I examine a proposal by Macbeth. Macbeth explains how it can be that objects “pop up”, e.g., as (...) a consequence of the constructions made in the diagrams of Euclid, that is, why they are fruitful. It turns out, however, that diagrams are not exclusively fruitful in this sense. By analysing the proofs given in the paper I introduce the notion of a ‘faithful representation’. A faithful representation represents as either an image or as a metaphor. Secondly it represents certain relevant relations. Thirdly manipulations on the representations respect manipulations on the objects they represent, so that new relations may be found. The examples given in the paper illustrate how such representations can be fruitful. These examples include proofs based on both symbolic expressions as well as diagrams and so it seems diagrams are not special when it comes to fruitfulness. Having said this, I do present two features of diagrams that seem to be unique. One consists of the possibility of exhibiting the type of relation in a diagram—or simply showing that a relation exists—as a contrast to stating in words that it exists. The second is the spatial configurations possible when using diagrams, e.g., allowing to show multiple relations in a single diagram. (shrink)

Among the universal attributes of homo sapiens, several have become established as special fields of study—language, art and music, religion, and political economy. But mathematics, another universal attribute of our species, is still modeled separately by logicians, historians, neuroscientists, and others. Could it be integrated into “mathematics studies,” a coherent, many-faceted branch of empirical science? Could philosophers facilitate such a unification? Some philosophers of mathematics identify themselves with “positions” on the nature of mathematics. Those “positions” could more productively serve as (...) models of mathematics. (shrink)

An adequate account of how inferences and discoveries are made in modern biology is a difficult prospect for a philosopher. Do we really deduce conclusions from Darwin’s principles? Once Darwinian biology is integrated with molecular biology, can we deduce the organism from its DNA? What does induction look like in an era where data sets are often too large to be processed by a human being? What is the role of abductive explanatory claims that try to define the biological individual (...) in relation to the microbiome with which it may be associated, or to revise the notion of evolution when the interaction of organism and environment comes to seem much more complex than earlier generations imagined. How should we evaluate “origins of life” experiments conducted in the laboratory, where chemistry shifts to biology and we try to recreate early conditions on earth to which we have no empirical access? How are the carefully controlled conditions in the lab to be brought into productive relationship with the messy, contingent outdoor work of biologists in the field, studying crabs or eelgrass at the edge of the Pacific Ocean, or prairie plants at the end of woods, on the plains of the Midwest. To answer these questions, I sent my graduate student Tano Posteraro to work with Ted Grosholz, a marine biologist at the University of California / Davis, and my undergraduate student Alex Grigas to work with Ruth Geyer Shaw, a population geneticist at the University of Minnesota. They came back with complex and interesting answers to these questions. (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)

I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley. In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, and also how (...) what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures, that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forward. (shrink)

The way scientific discovery has been conceptualized has changed drastically in the last few decades: its relation to logic, inference, methods, and evolution has been deeply reloaded. The ‘philosophical matrix’ moulded by logical empiricism and analytical tradition has been challenged by the ‘friends of discovery’, who opened up the way to a rational investigation of discovery. This has produced not only new theories of discovery, but also new ways of practicing it in a rational and more systematic way. Ampliative rules, (...) methods, heuristic procedures and even a logic of discovery have been investigated, extracted, reconstructed and refined. The outcome is a ‘scientific discovery revolution’: not only a new way of looking at discovery, but also a construction of tools that can guide us to discover something new. This is a very important contribution of philosophy of science to science, as it puts the former in a position not only to interpret what scientists do, but also to provide and improve tools that they can employ in their activity. (shrink)

Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that (...) this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

© The Authors [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] Grosholz is interested in the growth of knowledge: what happens when reasoning not only orders what we already know, but adds to what we know? In her previous works, especially in her [2007], Grosholz insisted on the fact that working scientists and mathematicians, when they add to what we know, often combine different ‘modes of representation’, taking advantage of the ambiguity that arises when (...) features of one sort of discourse are grafted onto another. In her new book she goes on along the same path but brings an important addition to the pattern: ‘ampliative’ thinking takes place ‘when discourse whose main intent is to establish and clarify reference is yoked with discourse whose main intent is analysis’. According to Grosholz, the basis of productive thinking is the difference between... (shrink)

No século XVII, vemos a emergência de uma nova abordagem geométrica às seções cônicas. Desenvolvida inicialmente por Girard Desargues e por Blaise Pascal, tal geometria é herdeira do método de representação pela perspectiva linear a aponta na direção da geometria projetiva do século XIX. Estudos recentes de J. Echeverría e de V. Debuiche iniciaram a discussão da recepção de tais trabalhos por Leibniz, assim como a relação deles com os trabalhos do próprio Leibniz em perspectiva e com a Geometria situs. (...) Este artigo percorre algumas questões associadas a estes domínios, assim como a importância do paradigma da perspectiva para a filosofia de Leibniz. (shrink)

In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the “axiomatic conception” of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic text books.

There are three elements in this paper. One is what we shall call ‘the Hungarian project’. This is the collected work of Andréka, Madarász, Németi, Székely and others. The second is Molinini’s philosophical work on the nature of mathematical explanations in science. The third is my pluralist approach to mathematics. The theses of this paper are that the Hungarian project gives genuine mathematical explanations for physical phenomena. A pluralist account of mathematical explanation can help us with appreciating the significance of (...) the Hungarian project. The significance consists in the fruitfulness and spread of the project. The spread is wide because the explanations are written in the very familiar language of first-order logic with identity. For this reason, the explanation is understandable to many mathematicians. Because of the methodology adopted in the Hungarian project, the explanations are fruitful in another sense. In the Hungarian project certain questions are asked that would not be asked with a more usual methodology. The Hungarians are distinguished from other scientists in asking logical and mathematical questions, and these both deepen our understanding of the physical theories and induce further spread to mathematics and philosophy. (shrink)

This paper takes as a starting point certain notions from Peirce's writings and uses them to propose a picture of the part of mathematical practice that consists of hypothesis formation. In particular, three processes of hypothesis formation are considered: abstraction, generalisation, and an abductive-like inference. In addition Peirce's pragmatic conception of truth and existence in terms of higher-order concepts are used in order to obtain a kind of pragmatic realist picture of mathematics.

This article discusses the concept of mathematical metaphor as a tool for analyzing the formation of mathematical knowledge. It reflects on the work of Lakoff and Núñez as a reference point against which to rearticulate a richer notion of mathematical metaphor that can account for actual mathematical evolution. To reach its goal this article analyzes historical case studies, draws on cognitive research, and applies lessons from the history of metaphors in philosophy as analyzed by Derrida and de Man.

On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. The distinctions introduced aim at (...) clarifying discussions in philosophy of mathematics and contributing towards a more refined view of mathematical practice. (shrink)