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Mathematical Practice
Edited by Silvia De Toffoli (University School of Advanced Studies IUSS Pavia)
About this topic
| Summary | Philosophy of mathematical practice is a branch of philosophy of mathematics starting with the assumption that mathematics is not only a body of eternal truths, but also a human activity with its specific dynamics of change and history. By observing a wide range of mathematical practices, including advanced practices, questions beyond foundations and access to abstract objects arise. Examples of such questions are: Why do mathematicians reprove a theorem which already has an accepted proof? When is a proof explanatory? What is mathematical understanding? What are the epistemic roles of diagrams and visualization in mathematics? How did certain mathematical concepts evolve over time? These questions tend to admit localized answers, specific to certain contexts, rather than applying for all mathematics. In recent years, researchers have been accumulating detailed case studies to answer them. Moreover, interdisciplinary endeavours have been pursued, bringing together not only philosophers and historians, but also cognitive scientists, sociologist, anthropologists, mathematics education researchers, and computer scientists. |
| Key works | The collection Mancosu 2008 contains an important sample of articles on the philosophy of mathematical practice. For a more interdisciplinary collection, in which issues in sociology of mathematics and mathematical education are also included, see Van Bendegem & van Kerkhove 2007. Book length studies are Corfield 2003, Ferreirós 2015, and Wagner 2017. |
| Introductions | The introduction of Mancosu 2008 is very informative and clarifies the position of philosophy of mathematical practice in the landscape of philosophy of mathematics. For general descriptions of different approaches on mathematical practice, see Van Bendegem 2014. A recent survey article is Carter 2019. |
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Related
Siblings
- Explanation in Mathematics (110)
- The Infinite (512)
- The Application of Mathematics (319)
- History of Mathematics (766)
- Philosophy of Mathematics, General Works (486)
- Mathematical Explanation (173)
- Philosophy of Mathematics, Misc (664)
See also
- Explanation in Mathematics (110)
- Visualization in Mathematics (111)
- Mathematical Cognition (618 | 111)
- History of Mathematics (766)
- Areas of Mathematics (10,680 | 6,270)
Contents
380 found
Order:
| 1 — 50 / 380 |
- A general framework for a Second Philosophy analysis of set-theoretic methodology.Carolin Antos & Deborah Kant - manuscriptPenelope Maddy’s Second Philosophy is one of the most well-known ap- proaches in recent philosophy of mathematics. She applies her second-philosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of means-ends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows us to clarify the procedure and (...)
- Mathematicians against the myth of genius: beyond the envy interpretation.Terence Rajivan Edward - manuscriptThis paper examines Timothy Gowers’ attempt to counter a mythology of genius in mathematics: that to be a mathematician one has to be a mathematical genius. Someone might take such attacks on the myth of genius as expressions of envy, but I propose that there is another reason for cautioning against placing a high value on genius, by turning to research in the humanities.
- Can AI Abstract the Architecture of Mathematics?Posina Rayudu - manuscriptThe irrational exuberance associated with contemporary artificial intelligence (AI) reminds me of Charles Dickens: "it was the age of foolishness, it was the epoch of belief" (cf. Nature Editorial, 2016; to get a feel for the vanity fair that is AI, see Mitchell and Krakauer, 2023; Stilgoe, 2023). It is particularly distressing—feels like yet another rerun of Seinfeld, which is all about nothing (pun intended); we have seen it in the 60s and again in the 90s. AI might have had (...)
- John W. Dawson, Jr. Why Prove it Again: Alternative Proofs in Mathematical Practice. Basel: Birkhäuser, 2015. ISBN: 978-3-319-17367-2 ; 978-3-319-17368-9 . Pp. xii + 204. [REVIEW]Jessica Carter - forthcoming - Philosophia Mathematica:nkw003.
- Mature Intuition and Mathematical Understanding.William D'Alessandro & Irma Stevens - forthcoming - Journal of Mathematical Behavior.Mathematicians often describe the importance of well-developed intuition to productive research and successful learning. But neither education researchers nor philosophers interested in epistemic dimensions of mathematical practice have yet given the topic the sustained attention it deserves. The trouble is partly that intuition in the relevant sense lacks a usefully clear characterization, so we begin by offering one: mature intuition, we say, is the capacity for fast, fluent, reliable and insightful inference with respect to some subject matter. We illustrate the (...)
- Mathematical Justification without Proof.Silvia De Toffoli - forthcoming - In Giovanni Merlo, Giacomo Melis & Crispin Wright, Self-knowledge and Knowledge A Priori. Oxford University Press.According to a widely held view in the philosophy of mathematics, direct inferential justification for mathematical propositions (that are not axioms) requires proof. I challenge this view while accepting that mathematical justification requires arguments that are put forward as proofs. I argue that certain fallacious putative proofs considered by the relevant subjects to be correct can confer mathematical justification. But mathematical justification doesn’t come for cheap: not just any argument will do. I suggest that to successfully transmit justification an argument (...)
- The Epistemic Roles of Diagrams.Silvia De Toffoli - forthcoming - In Kurt Sylvan, Ernest Sosa, Jonathan Dancy & Matthias Steup, The Blackwell Companion to Epistemology, 3rd edition. Wiley Blackwell.
- On the Value of Reformulating.Josh Hunt - forthcoming - Journal of Philosophy.Throughout science and mathematics, expert inquirers often reformulate existing problem-solving procedures and theories. But what value is there to reformulating, particularly when one already knows how to solve a given problem? Is reformulating merely instrumentally valuable for other practical or epistemic aims, or does it constitute a distinctive kind of epistemic achievement? I argue that by changing what we need to know to solve a problem, significant reformulations constitute a kind of intellectual value. Whereas some reformulations are trivial notational variants, (...)
- The Multiverse View and Set-Theoretic Practice.Deborah Kant - forthcoming - Kriterion – Journal of Philosophy.Hamkins’ multiverse view is a prominent position on the nature of set theory. It is posited against the universe view and proposed as a philosophical theory explaining current set-theoretic practice. This paper confronts the multiverse view with the results of an interview study investigating current set-theoretic practice. The study reveals a heterogeneity of set-theoretic research practices. The multiverse view is found to align well with pluralist research practices but not with absolutist practices. The generalisation claim of the multiverse view fails (...)No categories
- Critical Math Kinds: A Framework for the Philosophy of Alternative Mathematics.Franci Mangraviti - forthcoming - Erkenntnis:1-21.Mathematics, even more than the other sciences, is often presented as essentially unique, as if it could not be any other way. And yet, prima facie alternative mathematics are all over the place, from non-Western mathematics to mathematics based on nonclassical logics. Taking inspiration from Robin Dembroff’s analysis of critical gender kinds, and from Andrew Aberdein and Stephen Read’s analysis of alternative logics, in this paper I will introduce a practice-centered framework for the study of alternative mathematics based on the (...)
- What mathematical explanation need not be.Elijah Chudnoff & Silvia De Toffoli - 2025 - Journal of Mathematical Behavior 79 (101255):1-12.Recent works in the philosophy of mathematical practice and mathematical education have challenged orthodox views of mathematical explanation by developing Understanding-first accounts according to which mathematical explanation should be cashed out in terms of understanding. In this article, we explore two arguments that might have motivated this move, (i) the context-sensitivity argument and (ii) the inadequacy of knowing why argument. We show that although these arguments are derived from compelling observations, they ultimately rest on a misunderstanding of what Explanation-first accounts (...)
- Pragmatic Insights into Set-Theoretic Practice: Exploring Disagreement and Agreement among Practitioners.Deborah Kant - 2025 - Frankfurt am Main: Vittorio Klostermann.Many believe mathematical truth is indisputable. However, the set-theoretic independence phenomenon challenges this idea. Certain statements about infinite sets, like the continuum hypothesis, are neither true nor false according to the standard axioms. While philosophers have offered various diagnoses of this problem, this book posits that the set-theoretic community is key to solving the issue, proposing a pragmatic approach. It presents the first extensive empirical study, featuring interviews with 28 set theorists from varied backgrounds. It explores the spectrum of disagreement (...)
- Notations for neurodiverse learners.Sophie Marchand & Dirk Schlimm - 2025 - Journal of Mathematical Behavior 79.Notations are essential for mathematics, mathematical logic, and many other disciplines. In order for them to be used, they have to be learned and understood, which is relative to the perceptual and cognitive resources of their users. However, most reflections about the design of notations have not taken into consideration the diversity of possible users. In recent years, various groups of people have been identified who exhibit specific strengths and challenges with regard to the reading and processing of written information. (...)
- Infinite Practices, One Mathematics: Challenging Mathematical Pluralism.Melisa Vivanco - 2025 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 56 (1):1-11.Theories about the foundations of mathematics often encounter a problem similar to the traditional demarcation problem in science. In this context, it is pertinent to examine the first candidate for the identifying property of mathematical pluralism: reduction within a structure. As I argue here, this notion is insufficient for a coherent definition of structure within the plurality. In the end, demarcating a plurality of mathematics can be as problematic as demarcating a unitary mathematics. -/- .
- (1 other version)Introduction to Views from Other Domains.Andrew Aberdein - 2024 - In Bharath Sriraman, Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 2589-2596.The study of mathematical practice has always been an interdisciplinary enterprise and is not confined to history and philosophy. Important contributions have been made by scholars from many domains, including sociology, education, argumentation theory, rhetoric, formal epistemology, and theology. The chapters in this section provide overviews and introductions into the insights that scholars of mathematical practice may glean from these disciplines.
- Argumentation in Mathematical Practice.Andrew Aberdein & Zoe Ashton - 2024 - In Bharath Sriraman, Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 2665-2687.Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting meanings (...)
- Cultures of Mathematical Practice in Alexandria in Egypt: Claudius Ptolemy and His Commentators (Second–Fourth Century CE).Alberto Bardi - 2024 - In Bharath Sriraman, Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 1727-1744.Claudius Ptolemy’s mathematical astronomy originated in Alexandria in Egypt under Roman rule in the second century CE and held for more than a millennium, even beyond the Copernican theories (sixteenth century). To trace the flourishing of such mathematical creativity requires an understanding of Ptolemy’s philosophy of mathematical practice, the ancient commentators of Ptolemaic works, and the historical context of Alexandria in Egypt, a multicultural city which became a cradle of cultures of mathematical practices and blossomed into the Ptolemaic system and (...)
- Beyond semantic pollution: Towards a practice-based philosophical analysis of labelled calculi.Fabio De Martin Polo - 2024 - Erkenntnis:1-30.This paper challenges the negative attitudes towards labelled proof systems, usually referred to as semantic pollution, by arguing that such critiques overlook the full potential of labelled calculi. The overarching objective is to develop a practice-based philosophical analysis of labelled calculi to provide insightful considerations regarding their proof-theoretic and philosophical value. To achieve this, successful applications of labelled calculi and related results will be showcased, and comparisons with other relevant works will be discussed. The paper ends by advocating for a (...)
- Proofs for a price: Tomorrow’s ultra-rigorous mathematical culture.Silvia De Toffoli - 2024 - Bulletin (New Series) of the American Mathematical Society 61 (3):395–410.Computational tools might tempt us to renounce complete cer- tainty. By forgoing of rigorous proof, we could get (very) probable results for a fraction of the cost. But is it really true that proofs (as we know and love them) can lead us to certainty? Maybe not. Proofs do not wear their correct- ness on their sleeve, and we are not infallible in checking them. This suggests that we need help to check our results. When our fellow mathematicians will be (...)
- The Epistemological Subject(s) of Mathematics.Silvia De Toffoli - 2024 - In Bharath Sriraman, Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 2880-2904.Paying attention to the inner workings of mathematicians has led to a proliferation of new themes in the philosophy of mathematics. Several of these have to do with epistemology. Philosophers of mathematical practice, however, have not (yet) systematically engaged with general (analytic) epistemology. To be sure, there are some exceptions, but they are few and far between. In this chapter, I offer an explanation of why this might be the case and show how the situation could be remedied. I contend (...)
- Husserl's Philosophy of Mathematical Practice.Mirja Hartimo - 2024 - Cambridge University Press.Husserl’s Philosophy of Mathematical Practice explores the applicability of the phenomenological method to philosophy of mathematical practice. The first section elaborates on Husserl’s own understanding of the method of radical sense-investigation (Besinnung), with which he thought the mathematics ofhis time should be approached. The second section shows how Husserl himself practiced it in tracking both constructive and platonistic features in mathematical practice. Finally, the third section situates Husserlian phenomenology within the contemporary philosophy of mathematical practice, where the examined styles are (...)
- The Epistemology of the Infinite.Patrick J. Ryan - 2024 - Dissertation, University of California, BerkeleyThe great mathematician, physicist, and philosopher, Hermann Weyl, once called mathematics the “science of the infinite.” This is a fitting title: contemporary mathematics—especially Cantorian set theory—provides us with marvelous ways of taming and clarifying the infinite. Nonetheless, I believe that the epistemic significance of mathematical infinity remains poorly understood. This dissertation investigates the role of the infinite in three diverse areas of study: number theory, cosmology, and probability theory. A discovery that emerges from my work is that the epistemic role (...)Remove from this listExport citation
Bookmark - Reverse Mathematics.John Stillwell - 2024 - In Bharath Sriraman, Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 1963-1988.Reverse mathematics is a new take on an old idea: asking which axioms are necessary to prove a given theorem. This question was first asked about the parallel axiom in Euclid’s geometry and later about the axiom of choice in set theory. Obviously, such questions can be asked in many fields of mathematics, but in recent decades, it has proved fruitful to focus on subsystems of second-order arithmetic, where much of mainstream mathematics resides. It has been found that many basic (...)
- A Taxonomy for Set-Theoretic Potentialism.Davide Sutto - 2024 - Philosophia Mathematica:1-28.Set-theoretic potentialism is one of the most lively trends in the philosophy of mathematics. Modal accounts of sets have been developed in two different ways. The first, initiated by Charles Parsons, focuses on sets as objects. The second, dating back to Hilary Putnam and Geoffrey Hellman, investigates set-theoretic structures. The paper identifies two strands of open issues, technical and conceptual, to clarify these two different, yet often conflated, views and categorize the potentialist approaches that have emerged in the contemporary debate. (...)
- Signs as a Theme in the Philosophy of Mathematical Practice.David Waszek - 2024 - In Bharath Sriraman, Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer.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 (...)
- Value Judgments in Mathematics: G. H. Hardy and the (Non-)seriousness of Mathematical Theorems.Simon Weisgerber - 2024 - Global Philosophy 34 (1):1-24.One of the general criteria G. H. Hardy identifies and discusses in his famous essay A Mathematician’s Apology (Cambridge University Press, Cambridge, 1940) by which a mathematician’s patterns must be judged is seriousness. This article focuses on one of Hardy’s examples of a non-serious theorem, namely that 8712 and 9801 are the only numbers below 10000 which are integral multiples of their reversals, in the sense that 8712 = 4·2178, and 9801 = 9·1089. In the context of a discussion of (...)
- A hub-and-spoke model of geometric concepts.Mario Bacelar Valente - 2023 - Theoria : An International Journal for Theory, History and Fundations of Science 38 (1):25-44.The cognitive basis of geometry is still poorly understood, even the ‘simpler’ issue of what kind of representation of geometric objects we have. In this work, we set forward a tentative model of the neural representation of geometric objects for the case of the pure geometry of Euclid. To arrive at a coherent model, we found it necessary to consider earlier forms of geometry. We start by developing models of the neural representation of the geometric figures of ancient Greek practical (...)
- Σ01 soundness isn’t enough: Number theoretic indeterminacy’s unsavory physical commitments.Sharon Berry - 2023 - British Journal for the Philosophy of Science 74 (2):469-484.It’s sometimes suggested that we can (in a sense) settle the truth-value of some statements in the language of number theory by stipulation, adopting either φ or ¬φ as an additional axiom. For example, in Clarke-Doane (2020b) and a series of recent APA presentations, Clarke-Doane suggests that any Σ01 sound expansion of our current arithmetical practice would express a truth. In this paper, I’ll argue that (given a certain popular assumption about the model-theoretic representability of languages like ours) we can’t (...)
- Formal Ontology and Mathematics. A Case Study on the Identity of Proofs.Matteo Bianchetti & Giorgio Venturi - 2023 - Topoi 42 (1):307-321.We propose a novel, ontological approach to studying mathematical propositions and proofs. By “ontological approach” we refer to the study of the categories of beings or concepts that, in their practice, mathematicians isolate as fruitful for the advancement of their scientific activity (like discovering and proving theorems, formulating conjectures, and providing explanations). We do so by developing what we call a “formal ontology” of proofs using semantic modeling tools (like RDF and OWL) developed by the computer science community. In this (...)
- Unrealistic Models in Mathematics.William D'Alessandro - 2023 - Philosophers' Imprint 23 (#27).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 (...)
- Who's Afraid of Mathematical Diagrams?Silvia De Toffoli - 2023 - Philosophers' Imprint 23 (1).Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...)
- (1 other version)Recalcitrant Disagreement in Mathematics: An “Endless and Depressing Controversy” in the History of Italian Algebraic Geometry.Silvia De Toffoli & Claudio Fontanari - 2023 - Global Philosophy 33 (38):1-29.If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...)
- From a Doodle to a Theorem: A Case Study in Mathematical Discovery.Juan Fernández González & Dirk Schlimm - 2023 - Journal of Humanistic Mathematics 13 (1):4-35.We present some aspects of the genesis of a geometric construction, which can be carried out with compass and straightedge, from the original idea to the published version (Fernández González 2016). The Midpoint Path Construction makes it possible to multiply the length of a line segment by a rational number between 0 and 1 by constructing only midpoints and a straight line. In the form of an interview, we explore the context and narrative behind the discovery, with first-hand insights by (...)
- Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups.Joshua Habgood-Coote & Fenner Stanley Tanswell - 2023 - Episteme 20 (2):281-307.In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little (...)
- The hidden use of new axioms.Deborah Kant - 2023 - In Carolin Antos, Neil Barton & Giorgio Venturi, The Palgrave Companion to the Philosophy of Set Theory. Palgrave.This paper analyses the hidden use of new axioms in set-theoretic practice with a focus on large cardinal axioms and presents a general overview of set-theoretic practices using large cardinal axioms. The hidden use of a new axiom provides extrinsic reasons in support of this axiom via the idea of verifiable consequences, which is especially relevant for set-theoretic practitioners with an absolutist view. Besides that, the hidden use has pragmatic significance for further important sub-groups of the set-theoretic community---set-theoretic practitioners with (...)$159.99 new (collection) View on Amazon.comRemove from this list Direct download (3 more)Export citation Bookmark
- Rethinking inconsistent mathematics.Franci Mangraviti - 2023 - Dissertation, Ruhr University BochumThis dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for a genuine revolution in (...)
- Szemerédi’s theorem: An exploration of impurity, explanation, and content.Patrick J. Ryan - 2023 - Review of Symbolic Logic 16 (3):700-739.In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (...)
- Mathematical Progress — On Maddy and Beyond.Simon Weisgerber - 2023 - Philosophia Mathematica 31 (1):1-28.A key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of ‘mathematical importance’, broadly (...)
- Idéaux de preuve : explication et pureté.Andrew Arana - 2022 - In Andrew Arana & Marco Panza, Précis de philosophie de la logique et des mathématiques, Volume 2, philosophie des mathématiques. Paris: Editions de la Sorbonne. pp. 387-425.Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.$50.12 new (collection) View on Amazon.comRemove from this list Direct downloadExport citation Bookmark
- 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.
- Predicativity and constructive mathematics.Laura Crosilla - 2022 - In Gianluigi Oliveri, Claudio Ternullo & Stefano Boscolo, Objects, Structures, and Logics. Cham (Switzerland): Springer.In this article I present a disagreement between classical and constructive approaches to predicativity regarding the predicative status of so-called generalised inductive definitions. I begin by offering some motivation for an enquiry in the predicative foundations of constructive mathematics, by looking at contemporary work at the intersection between mathematics and computer science. I then review the background notions and spell out the above-mentioned disagreement between classical and constructive approaches to predicativity. Finally, I look at possible ways of defending the constructive (...)(collection) View on Amazon.comRemove from this list Direct download (2 more)Export citation Bookmark 4 citations
- 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 for a (...)
- Objectivity and Rigor in Classical Italian Algebraic Geometry.Silvia De Toffoli & Claudio Fontanari - 2022 - Noesis 38:195-212.The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of (...)
- Simulation of hybrid systems under Zeno behavior using numerical infinitesimals.Alberto Falcone, Alfredo Garro, Marat Mukhametzhanov & Yaroslav Sergeyev - 2022 - Communications in Nonlinear Science and Numerical Simulation 111:article number 106443.This paper considers hybrid systems — dynamical systems that exhibit both continuous and discrete behavior. Usually, in these systems, interactions between the continuous and discrete dynamics occur when a pre-defined function becomes equal to zero, i.e., in the system occurs a zero-crossing (the situation where the function only “touches” zero is considered as the zero-crossing, as well). Determination of zero-crossings plays a crucial role in the correct simulation of the system in this case. However, for models of many real-life hybrid (...)
- The Philosophy of the Concept and the Specificity of Mathematics.Matt Hare - 2022 - In Peter Osborne, Afterlives: transcendentals, universals, others. London: CRMEP Books. pp. 101-129.
- Symmetry and Reformulation: On Intellectual Progress in Science and Mathematics.Josh Hunt - 2022 - Dissertation, University of MichiganScience and mathematics continually change in their tools, methods, and concepts. Many of these changes are not just modifications but progress---steps to be admired. But what constitutes progress? This dissertation addresses one central source of intellectual advancement in both disciplines: reformulating a problem-solving plan into a new, logically compatible one. For short, I call these cases of compatible problem-solving plans "reformulations." Two aspects of reformulations are puzzling. First, reformulating is often unnecessary. Given that we could already solve a problem using (...)
- A Hippocratic Oath for mathematicians? Mapping the landscape of ethics in mathematics.Dennis Müller, Maurice Chiodo & James Franklin - 2022 - Science and Engineering Ethics 28 (5):1-30.While the consequences of mathematically-based software, algorithms and strategies have become ever wider and better appreciated, ethical reflection on mathematics has remained primitive. We review the somewhat disconnected suggestions of commentators in recent decades with a view to piecing together a coherent approach to ethics in mathematics. Calls for a Hippocratic Oath for mathematicians are examined and it is concluded that while lessons can be learned from the medical profession, the relation of mathematicians to those affected by their work is (...)
- Tables as powerful representational tools.Dirk Schlimm - 2022 - In Valeria Giardino, Sven Linker, Tony Burns, Francesco Bellucci, J. M. Boucheix & Diego Viana, Diagrammatic Representation and Inference. 13th International Conference, Diagrams 2022, Rome, Italy, September 14–16, 2022, Proceedings. Springer. pp. 185-201.Tables are widely used for storing, retrieving, communicating, and processing information, but in the literature on the study of representations they are still somewhat neglected. The strong structural constraints on tables allow for a clear identification of their characteristic features and the roles these play in the use of tables as representational and cognitive tools. After introducing syntactic, spatial, and semantic features of tables, we give an account of how these affect our perception and cognition on the basis of fundamental (...)
- Some paradoxes of infinity revisited.Yaroslav Sergeyev - 2022 - Mediterranian Journal of Mathematics 19:143.In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah ̃a, working with only three numerals (one, two, many) can help us to change our perception (...)
- Visual Proofs as Counterexamples to the Standard View of Informal Mathematical Proofs?Simon Weisgerber - 2022 - In Giardino V., Linker S., Burns R., Bellucci F., Boucheix J.-M. & Viana P., Diagrammatic Representation and Inference. 13th International Conference, Diagrams 2022, Rome, Italy, September 14–16, 2022, Proceedings. Springer, Cham. pp. 37-53.A passage from Jody Azzouni’s article “The Algorithmic-Device View of Informal Rigorous Mathematical Proof” in which he argues against Hamami and Avigad’s standard view of informal mathematical proof with the help of a specific visual proof of 1/2+1/4+1/8+1/16+⋯=1 is critically examined. By reference to mathematicians’ judgments about visual proofs in general, it is argued that Azzouni’s critique of Hamami and Avigad’s account is not valid. Nevertheless, by identifying a necessary condition for the visual proof to be considered a proper proof (...)
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