Recent discussions of practice in this Journal have appealed to what they describe as the classical concept of practice. In this paper, it is argued that if there is a single classical concept of practice, it has not been described with sufficient clarity for it to be of use in illuminating or correcting anything, even our ‘radically ambiguous’ common-sense understanding of educational practice; and that there are writers today whose understanding of practical wisdom is far superior to that of the (...) modern philosophers whose treatment of the topic has been relied upon. (shrink)

The relation between logic and philosophy of science, often taken for granted, is in fact problematic. Although current fashionable criticisms of the usefulness of logic are usually mistaken, there are indeed difficulties which should be taken seriously -- having to do, amongst other things, with different "scientific mentalities" in the two disciplines. Nevertheless, logic is, or should be, a vital part of the theory of science. To make this clear, the bulk of this paper is devoted to the key notion (...) of a "scientific theory" in a logical perspective. First, various formal explications of this notion are reviewed, then their further logical theory is discussed. In the absence of grand inspiring programs like those of Klein in mathematics or Hilbert in metamathematics, this preparatory ground-work is the best one can do here. The paper ends on a philosophical note, discussing applicability and merits of the formal approach to the study of science. (shrink)

Is rhetoric just a new and trendy way toépater les bourgeois?Unfortunately, I think that the newfound interest of some economists in rhetoric, and particularly Donald McCloskey in his new book and subsequent responses to critics, gives that impression. After economists have worked so hard for the past five decades to learn their sums, differential calculus, real analysis, and topology, it is a fair bet that one could easily hector them about their woeful ignorance of the conjugation of Latin verbs or (...) Aristotle's Six Elements of Tragedy. Moreover, it has certainly become an academic cliché that economists write as gracefully and felicitously as a hundred monkeys chained to broken typewriters. The fact that economists still trot out Keynes's prose in their defense is itself an index of the inarticulate desperation of an inarticulate profession. (shrink)

Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the unreasonable effectiveness of (...) mathematics and the relationship of mathematics to physics. (shrink)

This paper argues that in at least some cases, one proof of a given theorem is deeper than another by virtue of supplying a deeper explanation of the theorem — that is, a deeper account of why the theorem holds. There are cases of scientific depth that also involve a common abstract structure explaining a similarity between two otherwise unrelated phenomena, making their similarity no coincidence and purchasing depth by answering why questions that separate, dissimilar explanations of the two phenomena (...) cannot correctly answer. The connections between explanation, depth, unification, power, and coincidence in mathematics and science are compared. (shrink)

Bertrand Russell's contributions to last century's philosophy and, in particular, to the philosophy of mathematics cannot be overestimated.Russell, besides being, with Frege and G.E. Moore, one of the founding fathers of analytical philosophy, played a major rôle in the development of logicism, one of the oldest and most resilient1 programmes in the foundations of mathematics.Among his many achievements, we need to mention the discovery of the paradox that bears his name and the identification of its logical nature; the generalization to (...) the whole of mathematics of Frege's idea that it is not possible to draw a demarcation line between logic and arithmetic; the programme, carried out with Whitehead, of derivation of mathematics from the logical system of Principia Mathematica ; and the ramified theory of types, devised by Russell to protect the system of PM from the known paradoxes.Although there is an ample literature on these topics, it is quite important to reconsider Russell's contributions to the foundations of mathematics at a time when, as a consequence of the crisis of the classical programmes in the foundations of mathematics, new trends are beginning to develop within the philosophy of mathematics. These are trends which move in a very different direction from that of logicism, intuitionism, and Hilbert's programme.To see this we need to consider that, in spite of profound disagreements on the nature of mathematical activity, on the relationship existing between logic and mathematics, on the causes of and therapies for the paradoxes, etc., logicism, intuitionism, and Hilbert's programme share an important metaphor: the idea that mathematics is an edifice built on unshakable foundations,2 an edifice which makes possible only a cumulative growth of mathematical knowledge.Such a metaphor—which, together with more specific theses belonging to these schools of thought, remained unsupported …. (shrink)

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent (...) with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument. (shrink)

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

Résumé Dans cet article, nous discuterons de l’intégration du numérique à la sémiotique et proposerons qu’une modélisation conceptuelle puisse offrir un pont de dialogue entre ces deux domaines classiquement cloisonnés. Plus précisément, nous avancerons l’hypothèse que tout projet de recherche qui en appellera à l’informatique soit une démarche scientifique que s’il construit une théorie qui contient, en plus des modèles classiques que sont les modèles formel, computationnel et physique, un modèle conceptuel. Ce lieu, où les chercheur-es conceptualisent les multiples dimensions (...) de leur objet de recherche, sera alors défini en tant que socle d’une relation solide entre la sémiotique et l’informatique. Nous verrons que les différentes définitions du modèle conceptuel convergent vers la thèse soutenant que la connaissance scientifique met en œuvre une instance de conceptualisation qui doit identifier les diverses dimensions du problème en plus de devoir s’exprimer de manière à ce qu’elle soit communicable entre les membres des communautés épistémiques. Nous verrons, enfin, qu’un projet de recherché sémiotique exige la construction d’un tel modèle afin de décrire, généraliser et sélectionner des problèmes et que ce même modèle, conceptuel, sera aussi nécessaire si le ou la sémioticien-ne utilise l’informatique dans le cadre de sa recherche.fr. (shrink)

Mercier and Sperber illuminate many aspects of reasoning and rationality, providing refreshing and thoughtful analysis and elegant and well‐researched illustrations. They make a good case that reasoning should be viewed as a type of intuition, rather than a separate cognitive process or system. Yet questions remain. In what sense, if any, is reasoning a “module?” What is the link between rationality within an individual and rationality defined through the interaction between individuals? Formal theories of rationality, from logic, probability theory and (...) game theory, while not the focus of Mercier and Sperber's book, may help clarify this latter question. (shrink)

This paper investigates an important aspect of mathematical practice: that proof is required for a finished piece of mathematics. If follows that non-deductive arguments — however convincing — are never sufficient. I explore four aspects of mathematical research that have facilitated the impressive success of the discipline. These I call the Practical Virtues: Permanence, Reliability, Autonomy, and Consensus. I then argue that permitting results to become established on the basis of non-deductive evidence alone would lead to their deterioration. This furnishes (...) us with a partial rational justification for mathematicians strict insistence on proof. (shrink)

The view that a mathematical proof is a sketch of or recipe for a formal derivation requires the proof to function as an argument that there is a suitable derivation. This is a mathematical conclusion, and to avoid a regress we require some other account of how the proof can establish it.

Symbolic logics tend to be too mathematical for the philosophers and too philosophical for the mathematicians; and their history is too historical for most mathematicians, philosophers and logicians. This paper reflects upon these professional demarcations as they have developed during the century.

Jim Mackenzie; Evers & Walker and Forms of Knowledge, Journal of Philosophy of Education, Volume 19, Issue 2, 30 May 2006, Pages 199–209, https://doi.org/10.

This paper claims that the awareness of crucial philosophical questions and controversies, which have arisen during the historical evolution of fundamental concepts, ideas and processes in mathematics, should be an essential component of the professional knowledge of student teachers who intend to teach children mathematics.

According to a view going back to Plato, the aim of philosophy is to acquire knowledge and there is a method to acquire knowledge, namely a method of discovery. In the last century, however, this view has been completely abandoned, the attempt to give a rational account of discovery has been given up, and logic has been disconnected from discovery. This paper outlines a way of reconnecting logic with discovery.

Jim Mackenzie; Authority, Journal of Philosophy of Education, Volume 22, Issue 1, 30 May 2006, Pages 57–65, https://doi.org/10.1111/j.1467-9752.1988.tb00177.x.

Quine’s thesis of underdetermination is significantly weaker than it has been taken to be in the recent literature, for the following reasons: (i) it does not hold for all theories, but only for some global theories, (ii) it does not require the existence of empirically equivalent yet logically incompatible theories, (iii) it does not rule out the possibility that all perceived rivalry between empirically equivalent theories might be merely apparent and eliminable through translation, (iv) it is not a fundamental thesis (...) within Quine’s philosophy, and (v) it does not carry with it the anti-realistic consequences often associated with the thesis in recent debates. The paper analyzes Quine’s views on the matter and the changes they underwent over the years. A conjecture is put forth about why Quine’s thesis has been so widely misrepresented: Quine’s writings up to 1975 tackled primarily the formulation and justification of the thesis, but afterwards were concerned mostly with the question whether empirically equivalent rivals to the theory we hold are to be considered true also. When this latter discussion is read without bearing in mind Quine’s earlier formulation and justification of the thesis, his thesis seems to have stronger epistemic consequences than it actually does. A careful reading of his later writings shows, however, that the formulation of the thesis remained unchanged after 1975, and that his mature and considered views supported only a very mitigated version of the thesis. (shrink)

The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some (...) general unproven assumption. As an alternative, a bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach. (shrink)

The ArgumentStarting from the thesis that a science constructs the knowledge of the part of the world allotted to it, the present paper aims at bringing together all the various aspects of physics under a unified conceptual framework — that provided by the Marxian concept “mode of production.” After an introduction providing the initial plausibility grounds for the undertaking, the concept is analyzed into its conceptual elements in Part I of the paper. The analysis presents the reconstruction initiated by Louis (...) Althusser and developed by his followers. Part II starts from a characterization of physical problems. This offers a basis for “reading” physics in the terms introduced in Part I. The rest of Part II is devoted to the identification of all the aspects of physics with the conceptual elements in question. The paper aims at three things: to uncover the connections holding among seemingly disparate aspects of physics, usually discussed in almost total independence from each other; to take full account of the social dimensions of physics without vindicating social constructivism; to show that the boundaries separating the disciplines of philosophy, history and sociology of science are more arbitrary than usually considered. (shrink)

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, (...) the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered. (shrink)

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)

Attempts to lay a foundation for the sciences based on modern mathematics are questioned. In particular, it is not clear that computer science should be based on set-theoretic mathematics. Set-theoretic mathematics has difficulties with its own foundations, making it reasonable to explore alternative foundations for the sciences. The role of computation within an alternative framework may prove to be of great potential in establishing a direction for the new field of computer science.Whitehead''s theory of reality is re-examined as a foundation (...) for the sciences. His theory does not simply attempt to add formal rigor to the sciences, but instead relies on the methods of the biological and social sciences to construct his world-view. Whitehead''s theory is a rich source of notions that are intended to explain every element of experience. It is a product of Whitehead''s earlier attempt to provide a mathematical foundation for the physical sciences and is still consistent with modern physics. (shrink)

This paper seeks to determine the intuitive meaning of the concept of information by indicating its essential (definitional) features and relations with other concepts, such as that of knowledge. The term “information” – as with many other concepts, such as “process”, “force”, “energy” and “matter” – has a certain established meaning in natural languages, which allows it to be used, in science as well as in everyday life, without our possessing any somewhat stricter definition of it. The basic aim here (...) is thus to explicate what it amounts to in the context of its intuitive meaning as encountered in natural languages, what the subject of cognition implicitly presumes when using the term, and to which ontological situations it can be applied. I demonstrate that the essential features of the notion of information include the presence of a material medium, its transformation, the recording and reading of information encoded in the medium, and the grasp of what is recorded, coded and transmitted as an intentional object, where the latter is construed in terms broadly in line with the ontologies of Husserl and Ingarden. Along the way, a number of issues relating to the notion of information are also pointed out: the problem of informational identity, of the existence of virtual objects, and of the choice of an adequate information carrier, as well as formal-ontological problems, including those which concern relations between information carriers and intentional objects. (shrink)

What is the definition of 'type'? Having a clear and precise answer to this question would avoid many misunderstandings and prevent meaningless discussions that arise from them. But having such clear and precise answer to this question would also hurt science, "hamper the growth of knowledge" and "deflect the course of investigation into narrow channels of things already understood". In this essay, I argue that not everything we work with needs to be precisely defined. There are many definitions used by (...) different communities, but none of them applies universally. A brief excursion into philosophy of science shows that this is not just tolerable, but necessary for progress. Philosophy also suggests how we can think about this imprecise notion of type. (shrink)

The Argument Web is maturing as both a platform built upon a synthesis of many contemporary theories of argumentation in philosophy and also as an ecosystem in which various applications and application components are contributed by different research groups around the world. It already hosts the largest publicly accessible corpora of argumentation and has the largest number of interoperable and cross compatible tools for the analysis, navigation and evaluation of arguments across a broad range of domains, languages and activity types. (...) Such interoperability is key in allowing innovative combinations of tool and data reuse that can further catalyse the development of the field of computational argumentation. The aim of this paper is to summarise the key foundations, the recent advances and the goals of the Argument Web, with a particular focus on demonstrating the relevance to, and roots in, philosophical argumentation theory. (shrink)

Based on the material in the Lakatos Archive, this paper reconstructs, and then re-assesses, Lakatos’ epistemological project by placing it in the context of the debate on the role of reason in the history of science, and of the justification of rationality as a normative notion. It is claimed that Lakatos’ most fruitful ideas come from a peculiar philosophical combination of Hegelian historicism and Popperian fallibilism. The original tension, however, cannot be ultimately resolved. As a consequence, the problems that Lakatos (...) has to deal with in his attempt to justify a set of genuinely epistemological canons of scientific rationality that are not reducible to psychology or sociology of knowledge stand as a warning for any normative philosophy of science that takes history at its face value.Author Keywords: Lakatos; History; Method; Rationality. (shrink)

One important role of belief systems is to allow us to represent information about a certain domain of inquiry. This paper presents a formal framework to accommodate such information representation. Three cognitive models to represent information are discussed: conceptual spaces, state-spaces, and the problem spaces familiar from artificial intelligence. After indicating their weakness to deal with partial information, it is argued that an alternative, formulated in terms of partial structures, can be provided which not only captures the positive features of (...) these models, but also accommodates the partiality of information ubiquitous in science and mathematics. (shrink)

The original publication is available at JAIST Press http://www.jaist.ac.jp/library/jaist-press/index.html.

This article looks at recent work in cognitive science on mathematical cognition from the perspective of history and philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, it is argued that the focus of cognitive analyses of (...) historical developments of mathematics has been primarily on the former, even if they claim to be about the latter. (shrink)

We describe recent developments in research on mathematical practice and cognition and outline the nine contributions in this special issue of topiCS. We divide these contributions into those that address (a) mathematical reasoning: patterns, levels, and evaluation; (b) mathematical concepts: evolution and meaning; and (c) the number concept: representation and processing.

We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

In this article, extracted from his book Exploring Meinong's Jungle and Beyond, Sylvan argues that, contrary to widespread opinion, mathematics is not an extensional discipline and cannot be extensionalized without considerable damage. He argues that some of the insights of Meinong's theory of objects, and its modern development, item theory, should be applied to mathematics and that mathematical objects and structures should be treated as mind-independent, non-existent objects.

Kaposy (2010) argues that contemporary neuroscience cannot provide rational reasons for abandoning folk psychological concepts like self, personhood, or free will because these concepts are necessa...

There are two distinct meanings to ‘mathematical proof’. The connection between them is an unsolved problem. The first step in attacking it is noticing that it is an unsolved problem.

. Heuristic is a central concept of Lakatos' philosophy both in his early works and in his later work, the methodology of scientific research programs. The term itself, however, went through significant change of meaning. In this paper I study this change and the ‘metaphysical’ commitments behind it. In order to do so, I turn to his mathematical heuristic elaborated in Proofs and Refutations. I aim to show the dialogical character of mathematical knowledge in his account, which can open a (...) door to hermeneutic studies of mathematical practice. (shrink)

The article deals with Cantor’s diagonal argument and its alleged philosophical consequences such as that there are more reals than integers and, hence, that some of the reals must be independent of language because the totality of words and sentences is always count-able. My claim is that the main flaw of the argument for the existence of non-nameable objects or truths lies in a very superficial understanding of what a name or representation actually is.

Philosophers have tried to explain how science finds the truth by using new developments in logic to study scientific language and inference. R. G. Collingwood argued that only a logic of problems could take context into account. He was ignored, but the need to reconcile secure meanings with changes in context and meanings was seen by Karl Popper, W. v. O. Quine, and Mario Bunge. Jagdish Hattiangadi uses problems to reconcile the need for security with that for growth. But he (...) mistakenly insists that all problems are mere contradictions and artificially separates rigid from flexible aspects of meanings. In order to resolve the conflict we must (1) replace the quest for rigid terms with techniques for improvement, (2) use plausible arguments to uncover confused meanings, (3) use frameworks to choose problems and to regulate meanings, and (4) employ a bootstrap approach that uses frameworks to improve meanings and refined meanings to improve frameworks. (shrink)

Mathematics educators suggest that students of all ages need to participate in productive forms of mathematical argument (NCTM, 2000). Accordingly, we developed two complementary frameworks for analyzing the emergence of mathematical argumentation in one second‐grade classroom. Children attempted to resolve contesting claims about the “space covered” by three different‐looking rectangles of equal area measure. Our first analysis renders the topology of the semantic structure of the classroom conversation as a directed graph. The graph affords clear “at a glance” visualization of (...) how various senses of mathematics—as imagined, as performed, and as historically rooted—were interrelated. The graph represents the emergence and intercoordination between conceptual and procedural knowledge in the ebb and flow of classroom conversation. The graph has not just descriptive, but also predictive power: Interconnectedness among the nodes of the graph representing the first 40 min of conversation predicted the structure of recall in the final ten minutes by children who had played little overt role in the conversation up to that point. The second, complementary framework draws upon Goffman's expanded repertoire of roles in speech to demonstrate how the teacher orchestrated this classroom conversation to establish coherent argument. In this second analysis, we establish how the teacher mediated between the everyday talk of her students and the discourse of mathematics. Her revoicing of student talk created interstices for identity and participation in the formation of the argument. Together, the two forms of analysis illuminate the emergence of mathematical argument at two levels: as collective structure and concurrently, as individual activity. (shrink)

The concept of Mathematical Proof has been controversial for the past few decades. Different philosophers have offered different theories about the nature of Mathematical Proof, among which theories presented by Lakatos and Hersh have had significant similarities and differences with each other. It seems that a comparison and critical review of these two theories will lead to a better understanding of the concept of mathematical proof and will be a big step towards solving many related problems. Lakatos and Hersh argue (...) that, firstly, “mathematical proof” has two different meanings, formal and informal; and, secondly, informal proofs are affected by human factors, such as individual decisions and collective agreements. I call these two thesis, respectively, “proof dualism” and “humanism”. But on the other hand, their theories have significant dissimilarities and are by no means equivalent. Lakatos is committed to linear proof dualism and methodological humanism, while Hersh’s theory involves some sort of parallel proof dualism and sociological humanism. According to linear proof dualism, the two main types of proofs are provided in order to achieve a common goal: incarnation of mathematical concepts and methods and truth. However, according to the parallel proof dualism, two main types of proofs are provided in order to achieve two different types of purposes: production of a valid sequence of signs (the goal of the formal proof) and persuasion of the audience (the goal of the informal proof). Hersh’s humanism is informative and indicates pluralism; whereas, Lakatos’ version of humanism is normative and monistic. (shrink)