Results for 'Mathematical practice'

1000+ found
Order:
  1. Virtue Theory of Mathematical Practices: An Introduction.Andrew Aberdein, Colin Jakob Rittberg & Fenner Stanley Tanswell - 2021 - Synthese 199 (3-4):10167-10180.
    Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, ready to (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  2. Bayesian Perspectives on Mathematical Practice.James Franklin - 2020 - Handbook of the History and Philosophy of Mathematical Practice.
    Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and (...)
    Download  
     
    Export citation  
     
    Bookmark  
  3. Proof, Explanation, and Justification in Mathematical Practice.Moti Mizrahi - 2020 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 51 (4):551-568.
    In this paper, I propose that applying the methods of data science to “the problem of whether mathematical explanations occur within mathematics itself” (Mancosu 2018) might be a fruitful way to shed new light on the problem. By carefully selecting indicator words for explanation and justification, and then systematically searching for these indicators in databases of scholarly works in mathematics, we can get an idea of how mathematicians use these terms in mathematical practice and with what frequency. (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  4.  91
    Word Choice in Mathematical Practice: A Case Study in Polyhedra.Lowell Abrams & Landon D. C. Elkind - 2019 - Synthese (4):1-29.
    We examine the influence of word choices on mathematical practice, i.e. in developing definitions, theorems, and proofs. As a case study, we consider Euclid’s and Euler’s word choices in their influential developments of geometry and, in particular, their use of the term ‘polyhedron’. Then, jumping to the twentieth century, we look at word choices surrounding the use of the term ‘polyhedron’ in the work of Coxeter and of Grünbaum. We also consider a recent and explicit conflict of approach (...)
    Download  
     
    Export citation  
     
    Bookmark  
  5. Explanation in Mathematics: Proofs and Practice.William D'Alessandro - 2019 - Philosophy Compass 14 (11).
    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 (...)
    Download  
     
    Export citation  
     
    Bookmark   5 citations  
  6. The Interplay Between Mathematical Practices and Results.Mélissa Arneton, Amirouche Moktefi & Catherine Allamel-Raffin - 2014 - In Léna Soler, Sjoerd Zwart, Michael Lynch & Vincent Israel-Jost (eds.), Science After the Practice Turn in the Philosophy, History, and Social Studies of Science. New York - London: Routledge. pp. 269-276.
    Download  
     
    Export citation  
     
    Bookmark  
  7. Dialogue Types, Argumentation Schemes, and Mathematical Practice: Douglas Walton and Mathematics.Andrew Aberdein - 2021 - Journal of Applied Logics 8 (1):159-182.
    Douglas Walton’s multitudinous contributions to the study of argumentation seldom, if ever, directly engage with argumentation in mathematics. Nonetheless, several of the innovations with which he is most closely associated lend themselves to improving our understanding of mathematical arguments. I concentrate on two such innovations: dialogue types (§1) and argumentation schemes (§2). I argue that both devices are much more applicable to mathematical reasoning than may be commonly supposed.
    Download  
     
    Export citation  
     
    Bookmark  
  8. Mathematizing as a Virtuous Practice: Different Narratives and Their Consequences for Mathematics Education and Society.Deborah Kant & Deniz Sarikaya - 2020 - Synthese 199 (1-2):3405-3429.
    There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people’s willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative considerations. Our strategy (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  9. Univalent Foundations as a Foundation for Mathematical Practice.Harry Crane - 2018
    I prove that invoking the univalence axiom is equivalent to arguing 'without loss of generality' (WLOG) within Propositional Univalent Foundations (PropUF), the fragment of Univalent Foundations (UF) in which all homotopy types are mere propositions. As a consequence, I argue that practicing mathematicians, in accepting WLOG as a valid form of argument, implicitly accept the univalence axiom and that UF rightly serves as a Foundation for Mathematical Practice. By contrast, ZFC is inconsistent with WLOG as it is applied, (...)
    Download  
     
    Export citation  
     
    Bookmark  
  10. Teaching and Learning Guide For: Explanation in Mathematics: Proofs and Practice.William D'Alessandro - 2019 - Philosophy Compass 14 (11).
    This is a teaching and learning guide to accompany "Explanation in Mathematics: Proofs and Practice".
    Download  
     
    Export citation  
     
    Bookmark  
  11.  41
    Lakatos' Undone Work: The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science_ - Introduction to the Special Issue on _Lakatos’ Undone Work.Sophie Nagler, Hannah Pillin & Deniz Sarikaya - 2022 - Kriterion - Journal of Philosophy 36:1-10.
    We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, which gave rise to this special issue. Lastly, (...)
    Download  
     
    Export citation  
     
    Bookmark  
  12.  58
    Lisa A. Shabel, Mathematics in Kant’s Critical Philosophy-Reflections on Mathematical Practice. Routledge New York & London 2003, pp.192 $ 65.00 (hbk) ISBN 0-415-93955-0. Recensione di Francesco Tampoia 3/06/2004. [REVIEW]Francesco Tampoia - manuscript
    Scopo di questo agile ma denso volume è approfondire “The part played by the mathematical construction in the context of a full investigation of Kant’s theory of sensibility, that to say the Transcendental Aesthetic”. Si tratta della ripresentazione della tesi di dottorato della Shabel, da cui la stessa ha riportato ampi squarci per un articolo award-winning 1998 dal titolo ”Kant on the Symbolic Construction of Mathematical Concepts” (Studies in the History and the Philosophy of Science). Non si tratta (...)
    Download  
    Translate
     
     
    Export citation  
     
    Bookmark  
  13. Crunchy Methods in Practical Mathematics.Michael Wood - 2001 - Philosophy of Mathematics Education Journal 14.
    This paper focuses on the distinction between methods which are mathematically "clever", and those which are simply crude, typically repetitive and computer intensive, approaches for "crunching" out answers to problems. Examples of the latter include simulated probability distributions and resampling methods in statistics, and iterative methods for solving equations or optimisation problems. Most of these methods require software support, but this is easily provided by a PC. The paper argues that the crunchier methods often have substantial advantages from the perspectives (...)
    Download  
     
    Export citation  
     
    Bookmark  
  14. Mathematics and Argumentation.Andrew Aberdein - 2009 - Foundations of Science 14 (1-2):1-8.
    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.
    Download  
     
    Export citation  
     
    Bookmark   9 citations  
  15. Reconstructing the Unity of Mathematics Circa 1900.David J. Stump - 1997 - Perspectives on Science 5 (3):383-417.
    Standard histories of mathematics and of analytic philosophy contend that work on the foundations of mathematics was motivated by a crisis such as the discovery of paradoxes in set theory or the discovery of non-Euclidean geometries. Recent scholarship, however, casts doubt on the standard histories, opening the way for consideration of an alternative motive for the study of the foundations of mathematics—unification. Work on foundations has shown that diverse mathematical practices could be integrated into a single framework of axiomatic (...)
    Download  
     
    Export citation  
     
    Bookmark  
  16.  3
    Unrealistic Models in Mathematics.William D'Alessandro - forthcoming - Philosophers’ Imprint.
    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 (...)
    Download  
     
    Export citation  
     
    Bookmark  
  17. Mathematical Wit and Mathematical Cognition.Andrew Aberdein - 2013 - Topics in Cognitive Science 5 (2):231-250.
    The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, (...)
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  18. Envisioning Transformations – The Practice of Topology.Silvia De Toffoli & Valeria Giardino - 2016 - In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012--2014. Zurich, Switzerland: Birkhäuser. pp. 25-50.
    The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities (...)
    Download  
     
    Export citation  
     
    Bookmark   10 citations  
  19.  94
    The Changing Practices of Proof in Mathematics: Gilles Dowek: Computation, Proof, Machine. Cambridge: Cambridge University Press, 2015. Translation of Les Métamorphoses du Calcul, Paris: Le Pommier, 2007. Translation From the French by Pierre Guillot and Marion Roman, $124.00HB, $40.99PB. [REVIEW]Andrew Arana - 2017 - Metascience 26 (1):131-135.
    Review of Dowek, Gilles, Computation, Proof, Machine, Cambridge University Press, Cambridge, 2015. Translation of Les Métamorphoses du calcul, Le Pommier, Paris, 2007. Translation from the French by Pierre Guillot and Marion Roman.
    Download  
     
    Export citation  
     
    Bookmark  
  20. The Great Gibberish - Mathematics in Western Popular Culture.Markus Pantsar - 2016 - In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012--2014. Springer International Publishing. pp. 409-437.
    In this paper, I study how mathematicians are presented in western popular culture. I identify five stereotypes that I test on the best-known modern movies and television shows containing a significant amount of mathematics or important mathematician characters: (1) Mathematics is highly valued as an intellectual pursuit. (2) Little attention is given to the mathematical content. (3) Mathematical practice is portrayed in an unrealistic way. (4) Mathematicians are asocial and unable to enjoy normal life. (5) Higher mathematics (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  21. Mathematical Gettier Cases and Their Implications.Neil Barton - manuscript
    Let mathematical justification be the kind of justification obtained when a mathematician provides a proof of a theorem. Are Gettier cases possible for this kind of justification? At first sight we might think not: The standard for mathematical justification is proof and, since proof is bound at the hip with truth, there is no possibility of having an epistemically lucky justification of a true mathematical proposition. In this paper, I argue that Gettier cases are possible (and indeed (...)
    Download  
     
    Export citation  
     
    Bookmark  
  22. Mathematical Representation: Playing a Role.Kate Hodesdon - 2014 - Philosophical Studies 168 (3):769-782.
    The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features which instead (...)
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  23. Marriages of Mathematics and Physics: A Challenge for Biology.Arezoo Islami & Giuseppe Longo - 2017 - Progress in Biophysics and Molecular Biology 131:179-192.
    The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the (...)
    Download  
     
    Export citation  
     
    Bookmark   6 citations  
  24. What Are Mathematical Diagrams?Silvia De Toffoli - 2022 - Synthese 200 (2):1-29.
    Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  25. Groundwork for a Fallibilist Account of Mathematics.Silvia De Toffoli - 2021 - Philosophical Quarterly 7 (4):823-844.
    According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too (...)
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  26.  48
    On Faith in the Practice of Mathematics.Silvere Gangloff - manuscript
    Download  
     
    Export citation  
     
    Bookmark  
  27. Mathematical Monsters.Andrew Aberdein - 2019 - In Diego Compagna & Stefanie Steinhart (eds.), Monsters, Monstrosities, and the Monstrous in Culture and Society. Wilmington, DE, USA: pp. 391-412.
    Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The (...)
    Download  
     
    Export citation  
     
    Bookmark  
  28. The Structuralist Mathematical Style: Bourbaki as a Case Study.Jean-Pierre Marquis - 2022 - In Claudio Ternullo Gianluigi Oliveri (ed.), Boston Studies in the Philosophy and the History of Science. New York, État de New York, États-Unis: pp. 199-231.
    In this paper, we look at Bourbaki’s work as a case study for the notion of mathematical style. We argue that indeed Bourbaki exemplifies a mathematical style, namely the structuralist style.
    Download  
     
    Export citation  
     
    Bookmark  
  29. What is Mathematical Rigor?John Burgess & Silvia De Toffoli - 2022 - Aphex 25:1-17.
    Rigorous proof is supposed to guarantee that the premises invoked imply the conclusion reached, and the problem of rigor may be described as that of bringing together the perspectives of formal logic and mathematical practice on how this is to be achieved. This problem has recently raised a lot of discussion among philosophers of mathematics. We survey some possible solutions and argue that failure to understand its terms properly has led to misunderstandings in the literature.
    Download  
     
    Export citation  
     
    Bookmark  
  30. Observations on Sick Mathematics.Andrew Aberdein - 2010 - In Bart van Kerkhove, Jean Paul van Bendegem & Jonas de Vuyst (eds.), Philosophical Perspectives on Mathematical Practice. College Publications. pp. 269--300.
    This paper argues that new light may be shed on mathematical reasoning in its non-pathological forms by careful observation of its pathologies. The first section explores the application to mathematics of recent work on fallacy theory, specifically the concept of an ‘argumentation scheme’: a characteristic pattern under which many similar inferential steps may be subsumed. Fallacies may then be understood as argumentation schemes used inappropriately. The next section demonstrates how some specific mathematical fallacies may be characterized in terms (...)
    Download  
     
    Export citation  
     
    Bookmark   9 citations  
  31. A New Role for Mathematics in Empirical Sciences.Atoosa Kasirzadeh - 2021 - Philosophy of Science 88 (4):686-706.
    Mathematics is often taken to play one of two roles in the empirical sciences: either it represents empirical phenomena or it explains these phenomena by imposing constraints on them. This article identifies a third and distinct role that has not been fully appreciated in the literature on applicability of mathematics and may be pervasive in scientific practice. I call this the “bridging” role of mathematics, according to which mathematics acts as a connecting scheme in our explanatory reasoning about why (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  32. Nietzsche’s Philosophy of Mathematics.Eric Steinhart - 1999 - International Studies in Philosophy 31 (3):19-27.
    Nietzsche has a surprisingly significant and strikingly positive assessment of mathematics. I discuss Nietzsche's theory of the origin of mathematical practice in the division of the continuum of force, his theory of numbers, his conception of the finite and the infinite, and the relations between Nietzschean mathematics and formalism and intuitionism. I talk about the relations between math, illusion, life, and the will to truth. I distinguish life and world affirming mathematical practice from its ascetic perversion. (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  33. Wisdom Mathematics.Nicholas Maxwell - 2010 - Friends of Wisdom Newsletter (6):1-6.
    For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...)
    Download  
    Translate
     
     
    Export citation  
     
    Bookmark   1 citation  
  34. Nominalism and Mathematical Intuition.Otávio Bueno - 2008 - ProtoSociology 25:89-107.
    As part of the development of an epistemology for mathematics, some Platonists have defended the view that we have intuition that certain mathematical principles hold, and intuition of the properties of some mathematical objects. In this paper, I discuss some difficulties that this view faces to accommodate some salient features of mathematical practice. I then offer an alternative, agnostic nominalist proposal in which, despite the role played by mathematical intuition, these difficulties do not emerge.
    Download  
     
    Export citation  
     
    Bookmark   5 citations  
  35. A Mathematical Model of Quantum Computer by Both Arithmetic and Set Theory.Vasil Penchev - 2020 - Information Theory and Research eJournal 1 (15):1-13.
    A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section (...)
    Download  
     
    Export citation  
     
    Bookmark  
  36. Many-Valued Logics. A Mathematical and Computational Introduction.Luis M. Augusto - 2020 - London: College Publications.
    2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive modeling, and (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  37. Mathematics as Make-Believe: A Constructive Empiricist Account.Sarah Elizabeth Hoffman - 1999 - Dissertation, University of Alberta (Canada)
    Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of mathematics are considered and rejected. Constructive (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  38. Using Corpus Linguistics to Investigate Mathematical Explanation.Juan Pablo Mejía Ramos, Lara Alcock, Kristen Lew, Paolo Rago, Chris Sangwin & Matthew Inglis - 2019 - In Eugen Fischer & Mark Curtis (eds.), Methodological Advances in Experimental Philosophy. London: Bloomsbury Academic. pp. 239–263.
    In this chapter we use methods of corpus linguistics to investigate the ways in which mathematicians describe their work as explanatory in their research papers. We analyse use of the words explain/explanation (and various related words and expressions) in a large corpus of texts containing research papers in mathematics and in physical sciences, comparing this with their use in corpora of general, day-to-day English. We find that although mathematicians do use this family of words, such use is considerably less prevalent (...)
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  39.  90
    Computing, Modelling, and Scientific Practice: Foundational Analyses and Limitations.Philippos Papayannopoulos - 2018 - Dissertation,
    This dissertation examines aspects of the interplay between computing and scientific practice. The appropriate foundational framework for such an endeavour is rather real computability than the classical computability theory. This is so because physical sciences, engineering, and applied mathematics mostly employ functions defined in continuous domains. But, contrary to the case of computation over natural numbers, there is no universally accepted framework for real computation; rather, there are two incompatible approaches --computable analysis and BSS model--, both claiming to formalise (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  40.  83
    Where Opposites Meet: Mathematics Between Science And Humanities.Ivano Zanzarella - 2019 - Scienza E Filosofia 22:302-321.
    The connection between science and mathematics is often considered necessary and insoluble. Therefore, a relationship between mathematics and humanities or arts is deemed exceptional or sometimes unnatural. Nevertheless, on the basis of historical, ontological and epistemological researches it can be noted that it’s impossible to warrant the immediate identification between mathematics and sciences on a deeper level than the practical one. Given the instrumentality and then the unnecessity of this connection, the relationship between mathematics and not-scientific disciplines is undeniable, even (...)
    Download  
    Translate
     
     
    Export citation  
     
    Bookmark   1 citation  
  41. Computational Reverse Mathematics and Foundational Analysis.Benedict Eastaugh - manuscript
    Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in (...)
    Download  
     
    Export citation  
     
    Bookmark  
  42. The "Artificial Mathematician" Objection: Exploring the (Im)Possibility of Automating Mathematical Understanding.Sven Delarivière & Bart Van Kerkhove - 2017 - In B. Sriraman (ed.), Humanizing Mathematics and its Philosophy. Cham: Birkhäuser. pp. 173-198.
    Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  43. An Inquiry Into the Practice of Proving in Low-Dimensional Topology.Silvia De Toffoli & Valeria Giardino - 2015 - In Gabriele Lolli, Giorgio Venturi & Marco Panza (eds.), From Logic to Practice. Zurich, Switzerland: Springer International Publishing. pp. 315-336.
    The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, (...)
    Download  
     
    Export citation  
     
    Bookmark   14 citations  
  44.  53
    From Practical to Pure Geometry and Back.Mario Bacelar Valente - 2020 - Revista Brasileira de História da Matemática 20 (39):13-33.
    The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical geometry, some of which are basically (...)
    Download  
     
    Export citation  
     
    Bookmark  
  45. Holobiont Evolution: Mathematical Model with Vertical Vs. Horizontal Microbiome Transmission.Joan Roughgarden - 2020 - Philosophy, Theory, and Practice in Biology 12 (2).
    A holobiont is a composite organism consisting of a host together with its microbiome, such as a coral with its zooxanthellae. To explain the often intimate integration between hosts and their microbiomes, some investigators contend that selection operates on holobionts as a unit and view the microbiome’s genes as extending the host’s nuclear genome to jointly comprise a hologenome. Because vertical transmission of microbiomes is uncommon, other investigators contend that holobiont selection cannot be effective because a holobiont’s microbiome is an (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  46. Mathematical Deduction by Induction.Christy Ailman - 2013 - Gratia Eruditionis:4-12.
    In attempt to provide an answer to the question of origin of deductive proofs, I argue that Aristotle’s philosophy of math is more accurate opposed to a Platonic philosophy of math, given the evidence of how mathematics began. Aristotle says that mathematical knowledge is a posteriori, known through induction; but once knowledge has become unqualified it can grow into deduction. Two pieces of recent scholarship on Greek mathematics propose new ways of thinking about how mathematics began in the Greek (...)
    Download  
    Translate
     
     
    Export citation  
     
    Bookmark  
  47.  95
    Reconstruction in Philosophy of Mathematics.Davide Rizza - 2018 - Dewey Studies 2 (2):31-53.
    Throughout his work, John Dewey seeks to emancipate philosophical reflection from the influence of the classical tradition he traces back to Plato and Aristotle. For Dewey, this tradition rests upon a conception of knowledge based on the separation between theory and practice, which is incompatible with the structure of scientific inquiry. Philosophical work can make progress only if it is freed from its traditional heritage, i.e. only if it undergoes reconstruction. In this study I show that implicit appeals to (...)
    Download  
     
    Export citation  
     
    Bookmark  
  48. Normativity and Mathematics: A Wittgensteinian Approach to the Study of Number.J. Robert Loftis - 1999 - Dissertation, Northwestern University
    I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view (...)
    Download  
     
    Export citation  
     
    Bookmark  
  49. Truth, Proof and Gödelian Arguments: A Defence of Tarskian Truth in Mathematics.Markus Pantsar - 2009 - Dissertation, University of Helsinki
    One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established (...)
    Download  
     
    Export citation  
     
    Bookmark   7 citations  
  50. Chasing Individuation: Mathematical Description of Physical Systems.Zalamea Federico - 2016 - Dissertation, Paris Diderot University
    This work is a conceptual analysis of certain recent developments in the mathematical foundations of Classical and Quantum Mechanics which have allowed to formulate both theories in a common language. From the algebraic point of view, the set of observables of a physical system, be it classical or quantum, is described by a Jordan-Lie algebra. From the geometric point of view, the space of states of any system is described by a uniform Poisson space with transition probability. Both these (...)
    Download  
     
    Export citation  
     
    Bookmark  
1 — 50 / 1000