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This document is a book review of "Generations of Reason: A Family's Search for Meaning in Post-Newtonian England" by Joan L. Richards. The book explores the struggles of mathematicians in post-Newtonian England as they grappled with the power of reason and its relationship to Anglican doctrine. It focuses on the experiences of individuals such as William Frend, Theophilus Lindsey, and Augustus de Morgan, who were drawn to heterodox beliefs and sought to reconcile reason with their religious convictions. The review highlights the intersections between mathematics, religion, and social reform during this period, and discusses the influence of figures like Franz Joseph Gall and John Elliotson on the family's exploration of phrenology and mesmerism. The author suggests that Charles Darwin's "Origin of Species" can be seen as a representation of the challenges faced by this family of reason. [Extracted from the article]

"The Story of Proof: Logic and the History of Mathematics" by John Stillwell is a book that explores the development of mathematical proofs throughout history. The author examines the emergence of proofs in antiquity and the long-standing dominance of Euclid's foundations of geometry. Stillwell presents a historical narrative of mathematical achievements and subfields, highlighting the ways in which different areas of mathematics are interconnected. The book also discusses the influence of linguistic and conceptual developments on the evolution of mathematics, including the role of infinity and the impact of notation. While the text assumes some prior knowledge of mathematical proofs, it provides numerous examples and explanations to help readers understand the subject. The book concludes with a discussion of modern foundations of mathematics and the potential for computer-assisted proof and automated proof verification. [Extracted from the article]

Distinctively mathematical explanations (DMEs) explain natural phenomena primarily by appeal to mathematical facts. One important question is whether there can be an ontic account of DME. An ontic account of DME would treat the explananda and explanantia of DMEs as ontic items (ontic objects, properties, structures, etc.) and the explanatory relation between them as an ontic relation (e.g., Pincock, 2015; Povich, 2021). Here I present a conventionalist account of DME, defend it against objections, and argue that it should be considered ontic. Notably, if indeed it is ontic, the conventionalist account seems to avoid a convincing objection to other ontic accounts (Kuorikoski, 2021).

In a series of articles, Mauricio Suárez defends the neutrality of fictionalism with respect to the scientific realism-anti-realism debate. Suárez understands fictionalism from a strictly methodological point of view, linked to the practice of model building in the context of the philosophy of science. He moves away from the type of fictionalism analysed in other areas of philosophy such as metaphysics, the philosophy of language, aesthetics or the philosophy of mathematics. Following Vaihinger's position, he emphasizes the inferential role of fiction in scientific modelling and argues that scientific fictionalism is not incompatible with scientific realism, as is often believed. We argue against Suárez's position and reject the ubiquitous character assigned to fictions in scientific discourse, as well as the deflationary view of scientific realism defended by Suárez. We conclude that when the semantic, epistemic, and metaphysical aspects at stake in the realism-antirealism debate are taken into account, the alleged compatibility between scientific realism and fictionalism starts to generate some tension. [ABSTRACT FROM AUTHOR]

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the 'width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the 'Leibniz biconditionals' stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. [ABSTRACT FROM AUTHOR]

Dominique Pradelle's book, "Être et genèse des idéalités: Un ciel sans éternité," explores the nature of mathematical knowledge and the paradox of mathematical objects being both ideal and timeless. Pradelle examines the views of philosophers such as Thomas Aquinas, Bernard Bolzano, and Edmund Husserl, who argue that idealities exist independently of subjective mental acts and have intersubjective and omnitemporal validity. The book also discusses the phenomenological constitution of mathematical objects and their dependence on a horizon structure, as well as the role of axiomatization and formalization in changing the nature of mathematical objects. Pradelle's work challenges naive realism and supports a nuanced understanding of mathematical objects. [Extracted from the article]

"Joel D. Hamkins.Lectures on the Philosophy of Mathematics" is an introductory textbook written by a prominent set theorist. The book covers both mathematical and philosophical material, combining interesting mathematical material with discussions of philosophical issues. While the book may cover too much mathematical material for some readers, it could be of great interest to students studying a combination of math and philosophy. The book is divided into chapters on Number, Rigor, Infinity, Geometry, Proof, Computability, Incompleteness, and Set theory, with the chapters on Number and Set theory being the longest. The author often adopts a lecturing attitude, introducing key ideas but leaving the issues open for further exploration. The book also discusses the ontological controversy between the universe and multiverse views in set theory, with the author presenting arguments for the multiverse view. Overall, the book provides a wide and ambitious coverage of the philosophy of mathematics, although some readers may find certain topics could have been presented in a different light or more carefully. [Extracted from the article]

The text discusses two books on the philosophy of mathematics education. The first book, "Connecting Humans to Equations," explores the human dimensions of mathematics and covers topics such as the ontological and epistemological aspects of mathematics. The second book, "Critical Mathematics Education," presents the Critical Mathematics Education (CME) paradigm, which aims to empower students to think critically about mathematics and its societal implications. Both books offer valuable insights into the philosophy of mathematics education and are recommended for philosophers and mathematics teachers. [Extracted from the article]

The article provides a review of the book "Critical Mathematics Education" by Ole Skovsmose, which offers a unique perspective on mathematics education. Skovsmose emphasizes the importance of mathematics as a practice and explores the concepts of hope and social justice in critical mathematics education. The book also discusses the role of real-world problem areas as learning contexts and the idea of risk in mathematical modeling. It delves into the reasons behind low achievement in mathematics and students' lack of interest, arguing that learning should be an active process influenced by social, political, and economic factors. The book introduces new concepts and analyses, drawing on philosophers to offer a philosophy of critical mathematics education that considers power, positioning, and pedagogy. It also explores the ethics of mathematics and the impact of mathematics on society. The book concludes with a personal story from the author's experiences, highlighting the global need for critical mathematics education. [Extracted from the article]

Carlo Cellucci's book, "The Making of Mathematics - Heuristic/Philosophy of Mathematics," challenges the traditional view of mathematics as theorem proving through the axiomatic method. Cellucci argues that mathematics is best understood as problem solving through the analytic method. He emphasizes the importance of reconnecting the context of discovery with that of justification, allowing for a more complete understanding of mathematical arguments. Cellucci also critiques textbooks that omit the real mathematical process, advocating for the inclusion of heuristics in problem solving. Overall, his work has the potential to reshape the way mathematics is presented and understood in the future. [Extracted from the article]

This paper aims to discuss the revitalization of education for sustainable development (ESD) in mathematics education, particularly in relation to mathematics curricula for grades 1–10, using the example of the new Norwegian curriculum, LK20, which came into effect at the beginning of the 2020 school year. Several studies in the past two decades have identified disengagement of sustainability learning (SL) within mathematics education and called for a change in the philosophy of mathematics education to integrate sustainability into the teaching and learning of mathematics. Using the qualitative content analysis method, we examined three types of documents: the core curriculum, the mathematics curriculum, and one Norwegian mathematics textbook series called Matemagisk. We find that sustainable development (SD) is one of the interdisciplinary issues addressed in LK20. Even though the mathematics curriculum does not explicitly incorporate terms such as 'sustainability' or 'sustainable development', indicating that mathematics and sustainability are unconnected, its six core elements—exploration and problem-solving, modeling and applications, reasoning and argumentation, representation and communication, abstraction and generalization, and mathematical fields of knowledge—provide opportunities for integrating sustainability learning (SL). On the other hand, looking at one of the mathematics textbook series, Matemagisk, for grades 4, 7, and 10, it appears that tasks that use sustainability contexts were included implicitly or explicitly, indicating another opportunity to facilitate sustainable learning. By including sustainability contexts in tasks and mathematical activities, it is possible to boost the process of embedding ESD in mathematics education without compromising content. [ABSTRACT FROM AUTHOR]

In this paper, I study the kind of questions we can ask about the existence of numbers. In addition to asking whether numbers exist, and how, I argue that there is also a third relevant question: why numbers exist. In platonist and nominalist accounts this question may not make sense, but in the psychologist account I develop, it is as well-placed as the other two questions. In fact, there are two such why-questions: the causal why-question asks what causes numbers to exist and the teleological why-question asks for what purpose numbers exist. I argue that in a psychologist constructivist account, in which numbers are understood to exist as referents of a particular type of culturally shared concepts, both why-questions can get plausible answers. [ABSTRACT FROM AUTHOR]

I show that Wittgenstein's critique of G.H. Hardy's mathematical realism naturally extends to Paul Benacerraf's influential paper, 'Mathematical Truth'. Wittgenstein accuses Hardy of hastily analogizing mathematical and empirical propositions, thus leading to a picture of mathematical reality that is somehow akin to empirical reality despite the many puzzles this creates. Since Benacerraf relies on that very same analogy to raise problems about mathematical 'truth' and the alleged 'reality' to which it corresponds, his major argument falls prey to the same critique. The problematic pictures of mathematical reality suggested by Hardy and Benacerraf can be avoided, according to Wittgenstein, by disrupting the analogy that gives rise to them. I show why Tarskian updates to our conception of 'truth' discussed by Benacerraf do not answer Wittgenstein's concerns. That is, because they merely presuppose what Wittgenstein puts into question, namely, the essential uniformity of 'truth' and 'proposition' in ordinary discourse. [ABSTRACT FROM AUTHOR]

The cell-cell signaling mechanisms that are the basis for all of physiology have been used to trace evolution back to the unicellular state, and beyond, to the "First Principles of Physiology". And since our physiology derives from the Cosmos based on Symbiogenesis, it has been hypothesized that the cell behaves like a functional Mobius Strip, having no 'inside or outside' cell membrane surface - it is continuous with the Cosmos, its history being codified from Quantum Entanglement to Newtonian Mechanics, affording the cell consciousness and unconsciousness/subconsciousness as a continuum for the first time. Similarly, Klein and Maimon have concluded that their 'Soft Logic' mathematics also constitutes a Mobius Strip, using both a real number axis, combined with a zero axis, numerically representing cognition. This is congruent with the cell as 'two-tiered' consciousness, the first tier being the real-time interface between the cell membrane and its environment; the second tier constituting integrated physiology, referencing the consciousness of the Cosmos. Thus, there is coherence between physiology, consciousness and mathematics for the first time. [ABSTRACT FROM AUTHOR]

This article aims to carry out an analysis and assessment of the relationships between philosophy and science, which reveals the most diverse conceptions about whether the first is science or not and how these have been visualized in the course of time; all of which supports the need to address an important concept for science that is "crisis". Accordingly, it offers the results of a survey applied to teachers and students of the Mathematics-Fiscal specialty of the Secular University "Eloy Alfaro" of Manabí (ULEAM); and later, reveal and argue the existing relationships between both sciences. As a general objective, it is proposed to base the role played by Philosophy as an essential science for the development of Physics and vice versa. In it, the theoretical-methodological design was presented based on the determination of the research problem, the scientific questions, the determination of the object and field of investigation, the tasks to be undertaken; and finally the theoretical, empirical and statistical mathematical methods were presented, making evident their importance and the reasons for their use in the thesis that is being undertaken. Likewise, as a conclusion, through examples, considerations are provided about how Philosophy has influenced Physics and vice versa, which shows the lack of training in the first in such a way that explicit and implicit phenomena can be adequately addressed in the field. Physics learning. The research is part of the project "Improvement of Pedagogical Practices in Educational Institutions in the South Zone of Manabí.". [ABSTRACT FROM AUTHOR]

Severin Schroeder's book, "Wittgenstein and Mathematics," provides a comprehensive and accessible study of Ludwig Wittgenstein's later philosophy of mathematics. The book is divided into two parts, with the first part exploring Wittgenstein's early thoughts on mathematics and the criticisms of logicism, and the second part covering a wide range of topics discussed in Wittgenstein's later writings, such as rule-following, proof, inconsistency, and completeness. Schroeder also addresses the distinction between Wittgenstein's "grammar view" and "calculus view" of mathematics, highlighting the normative aspect of mathematics. The book offers a clear understanding of Wittgenstein's ideas on major problems in the philosophy of mathematics and is highly recommended for those interested in this field. [Extracted from the article]

Research has indicated a need for mathematics-specific affective measures and reported a lack of research exploring how these beliefs relate to one another, specifically with identity measures. Thus, the aims of this study are: (1) to validate measurement items for a mathematics teacher identity scale and a mathematics teaching philosophy scale, and (2) to explore correlations that may exist between these and other mathematics-specific beliefs. Exploratory factor analysis was conducted to provide evidence to support the inclusion of instrument items measuring both the mathematics teacher identity and mathematics teaching philosophy scales. Additionally, results indicated strong or moderate correlations among many of the identity and beliefs measures, such as mathematics identity and mathematics teacher identity. Mathematical mindsets, however, were either weakly correlated or not significantly correlated with all other measures. This study extends previous research on the understanding of the relationships among mathematics specific affective measures. [ABSTRACT FROM AUTHOR]

There are two distinct but interrelated questions concerning Aristotle's account of infinity that have been the subject of recurring debate. The first of these, what I call here the interpretative question, asks for a charitable and internally coherent interpretation of the limited pieces of text where Aristotle outlines his view of the 'potential' (and not 'actual') infinite. The second, what I call here the philosophical question, asks whether there is a way to make Aristotle's notion of the potential infinite coherent and rigorous with modern tools that can stand as a rival to the widely-accepted view of the infinite as characterized in a mathematical theory of sets. In this paper, I argue that the theoretical roles that Aristotle intends his account of the potential infinite to fulfill lead to a deep and irresoluble tension that can help explain the persistence of debates on both of these questions. I do so by turning to the places where Aristotle attempts to argue for or against the existence of particular infinite processes to show that he slides between different underlying notions of when changes are possible. Making these underlying notions clear can help us better understand the role of Aristotle's account in the history of philosophy, the possible pitfalls for a contemporary theory of the potential infinite, and what each of these debates might learn from each other. [ABSTRACT FROM AUTHOR]

The book "Evolutionary Debunking Arguments: Ethics, Philosophy of Religion, Philosophy of Mathematics, Metaphysics, and Epistemology" edited by Diego E. Machuca explores the topic of debunking arguments, which seek to attack beliefs by showing their problematic origins. The book covers various philosophical disciplines, including ethics, philosophy of religion, philosophy of mathematics, metaphysics, and epistemology. While some chapters in the book are critical of debunking arguments, others offer optimistic responses and defenses against them. The book provides a valuable resource for scholars interested in debunking arguments and related topics. [Extracted from the article]

Conventionalism about mathematics has much in common with two other views: fictionalism and the multiverse view (aka plenitudinous platonism). The three views may differ over the existence of mathematical objects, but they agree in rejecting a certain kind of objectivity claim about mathematics, advocating instead an extreme pluralism. The early parts of the paper will try to elucidate this anti‐objectivist position, and question whether conventionalism really offers a third form of it distinct from fictionalism and the multiverse view. The paper then turns to anti‐objectivism about logic, and suggests that here too conventionalism offers no distinct alternative, and that there are limits on the extent of pluralism/anti‐objectivism. It also argues that these limits can't be used to argue for more extensive objectivity within mathematics, and also that they do allow for more limited pluralism within logic, e.g. as regards resolution of semantic and property‐theoretic paradoxes. [ABSTRACT FROM AUTHOR]

This essay raises the question of material hermeneutics in Heelan's philosophy of techno-science. For Heelan, a continental philosophy of technoscience, referring to Husserl and Heidegger and especially to Merleau-Ponty, features hermeneutic contexts of mathematics and measurement as well as laboratory observation, including what the later Heelan spoke of as "portable laboratories," for the sake of objectivity and "meaning making." For Paul Feyerabend, this material practice corresponded to the use of both techniques of observation and instrumentation, and not less "propaganda" in the case of Galileo which practice for Heelan included the ontological status of measures and numbers as well as apprenticeship in what Heelan called "contingent local practical cultural milieus." The essay includes a discussion of Heidegger on mathematics and Bruno Latour on pasteurization. [ABSTRACT FROM AUTHOR]

In his later works, the great logician and mathematician Kurt Gödel concentrated his focus on philosophical problems such as the implications of set theory, the grammar and philosophy of language, objectivity and relativity, the ontological proof of God's existence, and phenomenology as an exact method. This essay explores how Gödel read the philosophy (of logic and mathematics) of his time and why he turned his attention to Husserl's phenomenology to describe the foundations of mathematics. To begin with, Gödel employs Husserl's significant distinction between Weltanschauung (worldview) philosophy and philosophy as rigorous science: According to the Weltanschauung philosophy, the spirit of time constantly changes so that the ideas discussed and goals attempted are meant to be temporal, and not for the sake of eternal truths, but for that of their own perfection; philosophy as rigorous science, on the other hand, is supratemporal so that its aim is to discover absolute and timeless values. As for the worldview of his time, Gödel saw the development of philosophy, and mathematics leaned toward skepticism, pessimism, and positivism. For instance, the antinomies of set theory have shaken the grounds on which mathematics and logic are founded. Gödel also used these paradoxes in his incompleteness theorems to prove that there are some statements that can neither be proven nor disproved within a system. This also means that arithmetic is not sufficient to prove consistency. From this, however, Gödel does not come to a conclusion for a nihilism in mathematics and logic: These mere antinomies of set theory do not "necessarily" lead us to logical positivism, and neither to such a materialism, nor to any kind of pessimistic theory of knowledge. The incompleteness theorems assert that there are arithmetical propositions that are true but neither provable nor unprovable within their own calculus, so that arithmetic is intrinsically incomplete. However, instead of Alfred Tarski's pathological view of examining the detections within the faulty system and then reforming the system all together, Gödel holds that we need to change our methods to find new patterns that describe the antinomies pointing to the unrecoverable reality of the mathematical world. Thus, Gödel does not follow any variation of the Weltanschauung philosophy of his time, either attempting to reduce mathematical realities to mathematical proofs in order to get rid of antinomies, or endeavoring to rescue a complete system of truths by a closed formal system, both Weltanschauung philosophies fail to set forth a realistic method. In this context, Gödel finds the task of phenomenology analogous to what he pursues, in terms of a systematic framework for the foundations of mathematics. Husserl's phenomenology, in Gödel's account, proliferates the intuition of (mathematical) essences and clarifies the meaning of undefinable concepts such as the antinomies of set theory. Applying phenomenological reduction to the objective reality of the mathematical world, Gödel believes that one obtains a clear experiential reality of the essential characteristics of (mathematical and logical) concepts. Briefly, what Gödel finds in Husserl's phenomenology that corresponds to his way of mathematical realism is a thoroughly designated method that gives us mathematical essences back again. [ABSTRACT FROM AUTHOR]

Copyright of Erdem is the property of Ataturk Kultur Merkezi and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A lot of feminist and decolonial critiques of knowledge get confronted with the reference to »hard sciences« or »mathematical reasoning« as examples that supposably showcase that there are epistemic spheres of no political dimensions and complete universality. This paper commits to a deconstruction of exactly such a reasoning in directly arguing to view Mathematics not only as also being knowledge production bearing political dimensions but even claiming it to play a particular role in securing hegemonic structures of knowing. In putting forth this argument I have three main claims: a) What is commonly referred to as »mathematics « is actually a specifically Western way of conceptualizing what it means to »think mathematically« (here referred to as Mathematics). b) The Western concept of Mathematics is conceptually entangled with Western concepts of Rationality and Humanism. c) These three concepts and their entanglement can be understood as a re-establishing epistemically violent structures of knowledge. [ABSTRACT FROM AUTHOR]

Mathematical proofs are not sequences of arbitrary deductive steps--each deductive step is, to some extent, rational. This paper aims to identify and characterize the particular form of rationality at play in mathematical proofs. The approach adopted consists in viewing mathematical proofs as reports of proof activities--that is, sequences of deductive inferences--and in characterizing the rationality of the former in terms of that of the latter. It is argued that proof activities are governed by specific norms of rational planning agency, and that a deductive step in a mathematical proof qualifies as rational whenever the corresponding deductive inference in the associated proof activity figures in a plan that has been constructed rationally. It is then shown that mathematical proofs whose associated proof activities violate these norms are likely to be judged as defective by mathematical agents, thereby providing evidence that these norms are indeed present in mathematical practice. We conclude that, if mathematical proofs are not mere sequences of deductive steps, if they possess a rational structure, it is because they are the product of rational planning agents. [ABSTRACT FROM AUTHOR]

This document is a collection of personal recollections and reflections on the author's experiences in the fields of mathematics and philosophy. The author discusses their mentors, colleagues, and interactions with renowned logicians and philosophers. They also mention their involvement in conferences, collaborations on books, and visits to various locations related to their work. The text concludes with heartfelt eulogies for three individuals who had a significant impact on the author's life. Overall, the document provides a glimpse into the author's academic and personal journey in these fields and pays tribute to influential figures in the discipline. [Extracted from the article]

The field of numerical cognition provides a fairly clear picture of the processes through which we learn basic arithmetical facts. This scientific picture, however, is rarely taken as providing a response to a much‐debated philosophical question, namely, the question of how we obtain number knowledge, since numbers are usually thought to be abstract entities located outside of space and time. In this paper, I take the scientific evidence on how we learn arithmetic as providing a response to the philosophical question of how we obtain number knowledge. I reject the view that numbers are abstract entities located outside of space and time and, alternatively, derive from the scientific evidence a novel account of the nature of numbers. In this account, numbers are reifications of the counting procedure and arithmetic statements are seen as describing the functioning of counting and calculation techniques. [ABSTRACT FROM AUTHOR]

Copyright of Kant-Studien is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

It is proved that the basic framework of the premises and reasoning of Wittgenstein's “Tractatus Logico-philosophicus” corresponds quite well to the transcendental method (as formulated by H. Cohen). Whereas Kant’s philosophy proceeds from the fact of existence of mathematics and mathematised natural science and investigates their conditions of possibility, Wittgenstein proceeds from the fact that propositions of language describe reality and reveals the conditions of possibility of such descriptions. Kant, answering the question about the conditions of possibility of the named sciences, comes to the idea of the transcendental subject and the distinction between the world of phenomena and the thing in itself. Wittgenstein's investigation of the conditions of possibility that the world is described by propositions leads to the assertion that both the world and language are together in logical space. The latter constitutes the a priori and transcendental condition of the possibility that language “reaches out” to reality. For both the theories - in the “Critique of Pure Reason” and in the “Tractatus Logico-Philosophicus” - the idea of the boundary is important. In the “Critique of Pure Reason” it is the boundary of possible experience and cognition, while in the “Tractatus Logico-Philosophicus” it is the boundary of what can be thought and expressed by meaningful propositions. Related to the different definitions of the boundary is the difference in the treatment of mathematised natural science. For “The Critique of Pure Reason” was created in the era of unconditional acceptance of Newtonian mechanics. And the “Tractatus Logico-Philosophicus” was created at the time of the crisis of the Newtonian paradigm and its replacement by other notions of time and space. However, the idea of boundary, which is present in both doctrines, determines closeness in the attitude towards metaphysics between the author of “The Critique of Pure Reason” and the author of “The Tractatus Logico-Philosophicus”. The study also shows that Wittgenstein did not follow logicism in his philosophy of mathematics. For him, both mathematical objects and propositions of logic are constructions. The conviction about the constructive character of mathematical and logical objects shows an affinity with the Kantian tradition in the philosophy of mathematics.

Wrinch managed only second-class honors on Part I. Senechal speculates that this result served as a wake-up call as, two years later, Wrinch finished Part II among the Wranglers. Senechal sees in Wrinch's devotion to the revelatory power of simplification and beauty a culture clash with biologists and chemists. Senechal recounts an apposite conversation between Bertrand Russell and the philosopher Jean Nicod, one of the students who studied I Principia Mathematica i with Russell and Wrinch in 1917. "For Beauty", I replied - "And I - for Truth - Themself are One - We Brethren, are", He said - And so, as Kinsmen, met a Night - We talked between the Rooms - Until the Moss had reached our lips - And covered up - Our names - - Emily Dickinson I I Died for Beauty i is Marjorie Senechal's enthralling biography of little-known mathematician Dorothy Wrinch (1894-1976). [Extracted from the article]

In this paper we discuss the importance of the symbolization in mathematics through the analysis of four aspects we judge to be the most important ones. Our discussion will be centralized in some syntactical aspects of mathematics and the four aspects we are going to discuss are: i. precision and concision; ii. the possibility of extension to other problems; iii. heuristics; iv. structure revelation. [ABSTRACT FROM AUTHOR]

The paper discusses a possible way of justifying the existence of a multiplicity of cryptomorphic axiomatic theories. It is argued that what renders this multiplicity inevitable are the applications of the theory. They give rise to difficulties whose solution necessitates the application of different conceptual tools. The paper is organized as follows: (§1) introduces the concept of cryptomorphism and the canonic example for the phenomenon: matroid theory; (§2) discusses five cryptomorphic approaches to the definition of rationality in the framework of decision theory: through utility functions, weak orders, choice operators, layered permutations, and pop-stack sortability; (§3) shows that each of these different approaches can serve as a basis for the introduction of a different modification of the original theory. This establishes the heuristic value of cryptomorphic approaches in the domain of decision theory. [ABSTRACT FROM AUTHOR]

In this paper, the connection between logicism and the principle of tolerance in Carnap's philosophy of logic and mathematics is to be presented in terms of the history of its development. Such development is conditioned by two lines of criticism to Carnap's attempt to combine Logicism and Conventionalism, the first of which comes from Gödel, the second from Alfred Tarski. The presentation will take place in three steps. First, the Logicism of Carnap before the publication of The Logical Syntax of Language will be analyzed in its main features. Second, the relationship between Logicism and the Principle of Tolerance at the time of the publication of Logical Syntax, and thirdly, in the so-called semantical phase of Carnap's thought will be examined. On the basis of the two preceding points, we will then critically analyze two interpretations of the possibility to maintain a philosophical position like that of Carnap nowadays. [ABSTRACT FROM AUTHOR]

The link between logic and mathematics has been recognized by theorists from different fields, but empirical investigations on this link remain scarce, and even less is known about the mechanisms underlying such a link. The current study aimed to address this issue by investigating the relations between transitive reasoning and children's mathematical competence, and exploring the potential mechanisms involved. A sample of 101 sixth graders were assessed on their transitive reasoning skills, their mathematics achievement, various potential mediators, as well as general cognitive factors (serving as control variables). The results showed that children's transitive reasoning performance was significantly related to their mathematics achievement beyond the effects of general cognitive factors, and this relation was mediated by fraction knowledge and Relation to Operands understanding. The findings not only confirmed the relation between transitive reasoning and mathematical competence, but also shed light on the mechanisms underlying such a relation. [ABSTRACT FROM AUTHOR]

In this paper, we connect two research directions concerning numeral symbol systems and their epistemological significance. The first direction concerns the cognitive processes involved in acquiring and applying different numeral symbols, e.g. the Indo-Arabic or Roman numeral systems. The second direction is a semiotic one, with focus on Charles Peirce's Philosophy of Notation. Peirce's work on logical formalism is well known, but he also wrote extensively on numeral systems. Here we take Peirce's considerations on central notions like iconicity and simplicity and examine their relevance for comparing different numeral symbol systems. We argue that simplicity and iconicity, for example, cannot be understood as single notions. Instead, they should be connected to different aims of numeral symbols that different systems fulfill to different degrees. Consequently, we focus on the kind of trade-offs that different symbol systems imply in acquiring and applying numeral symbol systems. [ABSTRACT FROM AUTHOR]

The goal of this paper is to show – by way of a case study – how the contents of Proclus' Commentary on the First Book of Euclid's Elements were incorporated into university teaching in the sixteenth century. I analyse the impact of Proclus on the works of the Spanish mathematician and university professor Jerónimo Muñoz (ca. 1520–1591). In order to do so, I examine two manuscripts: Adnotationes in commentaria Procli super Euclidem (MS Vat. Lat. 6996), and Astrologicarum et geographicarum institutionum libri sex (MS Vat. Lat. 6998). I show that the contents of Proclus' commentary pervade Muñoz's mathematical writings and influence his mathematical ontology, his classification of mathematical disciplines, and the history and terminology of geometry that he adopts. Moreover, I expound on how Proclus' text inspired Muñoz to maintain that the fifth postulate was a theorem, leading him to attempt a demonstration of it which pre-dates knowledge in the Latin West of Naṣīr ad-Dīn's (1201–1274) previous attempt. [ABSTRACT FROM AUTHOR]

Some recent scholars have argued that Spinoza's conception of causation should be understood in terms of the Aristotelian notion of a formal cause. I argue that while they are right to identify causation in Spinoza as a relation of entailment from an essence, they are mistaken about its philosophical pedigree. I examine three suggested lines of influence: (a) the late scholastic conception of emanation; (b) early modern philosophy of mathematics; and (c) Descartes's notion of the causa sui. In each case, the evidence indicates that causation in Spinoza should be categorized in Aristotelian terms as efficient and not formal. [ABSTRACT FROM AUTHOR]

A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Button and Walsh have introduced a view they call 'internalism', according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while internalism arguably explains how we pick out unique mathematical structures, this does not suffice to account for the determinacy of mathematical discourse. [ABSTRACT FROM AUTHOR]

Even the claim that mathematicians do not examine their hypotheses, which Plato's Socrates makes explicitly and repeatedly as a contrast between mathematics and dialectic, is valuable only when you face both its falsity and its truth. She is less interested in Plato knowing Eudoxus's completed theory or other advanced mathematical facts, than she is in Plato's awareness of mathematicians striving - in some ways like dialecticians but in other ways unlike them - along their way to creating such theories. That is, Plato believed mathematicians establish that their hypotheses solve problems and that is all the mathematicians seek, and Plato approved of that. (p. 7, compare the retrospective summary on p. 40) 2 PLATO IN ENGLISH: TEXT AND TRANSLATION Landry's default translation is [8] (Landry p. 1n4). [Extracted from the article]