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Psychometrics conceptualizes a person's proficiency (or ability, or competence), in a cognitive or educational domain, as a latent numerical quantity. Yet both conceptual and empirical studies have shown that the assumption of quantitative structure for such phenomena is unlikely to be tenable. A reason why most applications of psychometrics nevertheless continue to treat them as if they were numerical quantities may be that quantification is thought to be necessary to enable measurement. This is indeed true if one regards the task of measurement as the location of a measurand at a point on the real number line (the viewpoint adopted by, for example, the representational theory of measurement, the realist theory of measurement as the discovery of ratios, and Rasch measurement theory). But this is not the only philosophically respectable way of defining the notion of measurement. This paper suggests that van Fraassen's more expansive view of measurement as, in general, location in a logical space (which could be the real continuum, as in metrological applications in the physical sciences, but could be a different mathematical structure), provides a more appropriate conceptual framework for psychometrics. Taking educational measurement as a case study, it explores what that could look like in practice, drawing on fuzzy logic and mathematical order theory. It suggests that applying this approach to the assessment of intersubjectively constructed phenomena, such as a learner's proficiency in an inherently fuzzily-defined subject area, entails recognizing the theory-dependent nature of valid representations of such phenomena, which need not be conceived of structurally as values of quantities. Finally, some connections are made between this "qualitative mathematical" theorization of educational assessment, and the application of techniques from machine learning and artificial intelligence in this area. [ABSTRACT FROM AUTHOR]

Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as implicit definitions of the objects in the range of their operators. In second-order logic (SOL) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (Proceedings of the Aristotelian Society87, 137–151, 1986) and Zalta (Abstract Objects, vol. 160 of Synthese Library, 1983), examines explicit definitions of abstract objects. These axioms state that there is a unique abstract encoding all concepts satisfying a given formula ϕ (F) , with F a concept variable. Such a system is inconsistent in full SOL. It can be made consistent with several intricate tweaks, as Zalta has shown. Our approach in this article is simpler: we use a novel method to establish consistency in a restrictive version of predicative SOL. The resulting system, RPEAO, interprets first-order PA in extensional contexts, and has a natural extension delivering a peculiar interpretation of PA 2 . [ABSTRACT FROM AUTHOR]

This article introduces a special issue of Philosophical Investigations focused on Ludwig Wittgenstein's philosophy of mathematics and its connection to the philosophy of mathematical practice (PMP). The author acknowledges that while Wittgenstein is a highly influential philosopher, his work in the philosophy of mathematics is often overshadowed by his other writings. The special issue includes nine articles that cover various topics related to Wittgenstein's philosophy of mathematics and PMP. The articles explore Wittgenstein's practice-based approach to meaning, his views on set theory, his struggles with arithmetic and higher mathematics, and his critique of Cantorian set theory. The volume aims to deepen our understanding of mathematical practice, contribute to the discourse on Wittgenstein's philosophy of mathematics, and introduce Wittgensteinian ideas to the wider community of philosophers of mathematical practice. [Extracted from the article]

The status of mathematical facts has long been taken to be unclear in Wittgenstein's philosophy of mathematics, and often, it seems that he wants to eliminate mathematical facts in favour of facts about our beliefs or behaviour. In this paper, I argue that by reading Wittgenstein as a radical conventionalist, we can give a reading of the relevant passages according to which Wittgenstein doesn't deny that there are mathematical facts, but rather denies that one needs a metaphysical account of what mathematical facts are and how they relate to the world that goes beyond the minimal claims of radical conventionalism—that empirical facts about how we would find natural to project our training into new cases and the constitution of concepts by our agreement are enough. At the end of the paper, I discuss the implications of this reading on how to understand a rule's determination of its own application. [ABSTRACT FROM AUTHOR]

This article is a preface to a special issue of the Journal of Difference Equations and Applications dedicated to Saber Elaydi on his 80th birthday. Elaydi is recognized for his significant contributions to the field of difference equations and discrete dynamical systems. He has had a successful academic career, serving as a professor and chair at Trinity University in Texas. Elaydi has also been a valuable teacher and mentor, inspiring students and attracting disciples from around the world. He has written influential books on difference equations and has been involved in organizing conferences and founding journals in the field. Elaydi's research interests cover a wide range of topics, including mathematical biology, neuroscience, and economics. He has made substantial contributions to these areas and has been recognized with prestigious awards. The article expresses appreciation for Elaydi's achievements and wishes him continued success in his mathematical and personal life. [Extracted from the article]

Showing that a problem is hard for a model of computation is one of the most challenging tasks in theoretical computer science, logic and mathematics. For example, it remains beyond reach to find an explicit problem that cannot be computed by polynomial size propositional formulas (PF). As a model of computation, logic programs (LP) under answer set semantics are as expressive as PF and also \(\mathtt{NP}\) -complete for satisfiability checking. In this article, we show that the PAR problem is hard for LP, i.e., deciding whether a binary string contains an odd number of \(1\) 's requires exponential size LP. The proof idea is first to transform logic programs into equivalent boolean circuits and then apply a probabilistic method known as random restriction to obtain an exponential lower bound. Based on the main result, we generalize a sufficient condition for identifying hard problems for LP and give a separation map for an LP family from a computational point of view, whose members are all equally expressive and share the same reasoning complexity. [ABSTRACT FROM AUTHOR]

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. [ABSTRACT FROM AUTHOR]

The aim of the 17th century philosophers, in general, was to show that, by taking as an example the success of the human mind embodied in science, definitive knowledge could be achieved in philosophy, just as in science. To this end, firstly, they described in detail the method to be followed in order to obtain the knowledge. In science, the efforts of scientists such as Kepler, Galileo, and Newton to explain the universe based on mathematics led mathematics to stand out as a way of thinking. In this period, while mathematical research methods enabled the examination of physical phenomena, philosophy, on the other hand, was seen as a model to reach exact knowledge. Rationalist philosophers such as Descartes, Spinoza, and Leibniz all agree that people should use their own minds, not authorities, and moving from this idea, mathematics is the method that will lead us to the truth. This paper aims to show how these three great philosophers applied mathematics to their own philosophies, based on their differences and similarities. [ABSTRACT FROM AUTHOR]

We aim to show from a new perspective that Quine's ontological relativity, based largely on his so-called "proxy-function argument", falls short of being a rigorously coherent philosophical conception, as it exhibits significant formal defects. This new perspective enables exposing the shortcomings of Quine's position and suggests a possible reformulation of the original position. Moreover, we argue that his ontological relativity is inconsistent with the empirical data associated with some of our best physical theories, such as quantum mechanics. We refer to fundamental concepts of philosophy and the foundations of mathematics in order to clarify our critique of Quine's position concerning the relation between formalized theories and both what we can know about the real world and how we come to know it. [ABSTRACT FROM AUTHOR]

Copyright of Beytulhikme: An International Journal of Philosophy is the property of Beytulhikme: An International Journal of Philosophy 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.)

The research on the interpretation of mathematics as part of the sciences of mind, introduced as 'Geisteswissenschaften' in the German philosophy, reflects the relationship between mathematics and arts, including architecture. This study focuses on the early works of the philosopher Max Bense where he described the contours of an intellectual history of mathematics related to philosophy. He pointed out strong relationships between mathematics and all areas of human creation abilities. In particular, his investigations on the relationship between arts, including architecture, and mathematics in important historical phases reveal that the mathematics of an age is reflected aesthetically in the artistic styles or theories of art. His thesis posits that a high point in mathematics corresponds to a high point in art. Later on, Bense developed an aesthetic theory on the basis of mathematics, semiotics and information theory. [ABSTRACT FROM AUTHOR]

Gödel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those implicit assumptions. This notion of implicit commitment motivates directly or indirectly several research programmes in logic and the foundations of mathematics; yet there hasn't been a direct logical analysis of the notion of implicit commitment itself. In a recent paper, we carried out an initial assessment of this project by studying necessary conditions for implicit commitments; from seemingly weak assumptions on implicit commitments of an arithmetical system |$S$|⁠ , it can be derived that a uniform reflection principle for |$S$| —stating that all numerical instances of theorems of |$S$| are true—must be contained in |$S$| 's implicit commitments. This study gave rise to unexplored research avenues and open questions. This paper addresses the main ones. We generalize this basic framework for implicit commitments along two dimensions: in terms of iterations of the basic implicit commitment operator, and via a study of implicit commitments of theories in arbitrary first-order languages, not only couched in an arithmetical language. [ABSTRACT FROM AUTHOR]

The interests of the Chilean engineer and philosopher Manuel Atria Ramírez cover a variety of subjects and disciplines, but most of his published work focuses on philosophy of mathematics and physics. After a first phase of youth, of evident Thomistic inclination, his mature thought shows the influence of the great epistemological schools that thrive during the first half of the 20th century. Beyond illustrating the reception that was given in those years in the Latin American academic scene to the new European philosophical tendencies, we think that the concise expository style of the author and his deep understanding of mathematical physics and its history, have the merit to pose with freshness and originality the question referring to the ultimate foundation of mathematics, and its connection with reality. In this article we present and analyze the fundamental lines of Professor Atria`s philosophy of mathematics and physics, as well as those questions open for further research.

The paper is devoted to the problem of describing reality in the language of mathematics and logic in connection with intellectual intuition. The question raised is how the basic requirements of mathematical theory and logic will change if some of the multiverse models of modern physics are taken as the basis. Mathematics is considered in the context of various historical approaches. It is shown that some of the well-known requirements of a formal theory (such as consistency) may begin to play a different role if the multiverse hypothesis is accepted. In the framework of theories based on the idea of multiple worlds, the logical consequence, the natural law of Duns Scotus, the law of excluded middle, and other well-known facts of classical logic which in some cases cause controversy due to their intuitive unacceptability are resolved. The paper discusses an approach based on paraconsistent logics: such logics can be considered the first to correspond to multiverse theories. [ABSTRACT FROM AUTHOR]

Formal methods (FMs) have been cited as a viable methodology for developing high-quality software. However, the steep learning curve in efficiently using the underlying discrete mathematics and logic has hindered FMs' adoption, leading to a decline in their initial interest in the 1980s. Traditionally, technical approaches have been pursued to address the FMs challenge. Having taken cognisance of a similar pre-4IR decline in AI, the researcher views FMs as technology and considers solutions at intersections of 4IR-5IR technology adoption, leveraged by the support of governors, termed leadership-management in this work. Following a qualitative research choice, scholarly literature is reviewed, and sets of qualitative propositions are defined to develop a conceptual framework for a 4IR-5IR leadership-management adoption of FMs. Aspects that emerged and are incorporated into the framework are cross-functional and executive levels of leadership, transformative, adaptive, and servant leadership styles, using FM tools that embed a high level of user experience, and 4IR technologies, augmented with 5IR human aspects. The framework is hoped to motivate a company's leadership to contribute to technology and technical ICT-based decision-making increasingly. Future work in this area would involve securing input from practitioners and exercising the framework in an industrial setting. [ABSTRACT FROM AUTHOR]

Hermann's Weyl Das Kontinuum has inspired several studies in logic and foundations of mathematics over the last century. The book provides a remarkable reconstruction of a large portion of classical mathematics on a predicative basis. However, diverging interpretations of the predicative system formulated by Weyl have been proposed in the literature. In the present work, I analyze an early formalization of Weyl's ideas proposed by [Casari, E. 1964. Questioni di Filosofia Della Matematica, Milano: Feltrinelli] and compare it with other, more well-known, accounts, such as those proposed respectively by Feferman and Avron. In this way, I fill a gap in the literature on Weyl's predicative mathematics by shedding light on an interpretation that has never been studied. Moreover, Casari's work is plausibly the first systematic reconstruction of Weyl's system. In particular, I highlight the different insights about predicativity that can be found in Weyl's work. Casari's reconstruction focuses on the logical and mathematical processes described by Weyl in the first part of his book. Through the analysis of this unexplored perspective, I shed light on fundamental aspects of Weyl's work and elucidate the alleged ambiguities it presents. In particular, in line with Avron's most recent reconstruction of Weyl's ideas, this analysis supports a stronger conception of predicativity than what is commonly attributed to Weyl in the literature. As a result, this work not only provides an analysis of a neglected interpretation of Das Kontinuum, but also contributes to elucidating Weyl's classical conception of predicativity. [ABSTRACT FROM AUTHOR]

The article titled "Discordium Mathematica" explores the concept of mathematics as a cosmic symphony and the creation of perfection by Mathamos, the God of Mathematics. The text incorporates references to various mythologies and philosophical ideas, presenting mathematics as a divine art form. It delves into the intricate nature of mathematics, its connection to harmony and rhythm, and its ability to reveal profound truths. The article is written in a poetic and metaphorical style, inviting readers to contemplate the beauty and significance of mathematics. [Extracted from the article]

The idea that some objects are metaphysically "cheap" has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume's Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one‐to‐one: (HP)#xx=#yyiffxx≈yy$\#xx=\#yy\textrm { iff }xx\approx yy$. The principal aim of this article is to use the notion of grounding to develop this sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of metaphysical "cheapness." Second, we must rebut the "bad company" objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege's Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures "cheap" abstracta and permissible abstraction. [ABSTRACT FROM AUTHOR]

Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein's approach had roots in Poncelet's principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked some of the central ideas in Riemann's theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker's work had a strongly qualitative character that imbued Klein's early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet's earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie). [ABSTRACT FROM AUTHOR]

Most people, evidence suggests, have a hard time thinking straight. Symbolic logic is a tool that can help remedy this problem. Unfortunately, it is highly abstract and uses symbols whose meanings rely on unintuitive arbitrary conventions. Without sacrificing rigor, iconic logic is more concrete and uses icons that resemble what they stand for and whose meanings are thus easier to picture, process, and remember. Here I review and critique iconic existential graphs and concept diagrams--the former link iconic logic to iconic mathematics; the latter expand popular Euler or Venn diagrams and have, to some degree, been empirically investigated for user-friendliness. I lay out how expertise in perception, cognition, and genetics can inform and improve such empirical research to help make iconic logic more ergonomic. After all, logic is a tool, and tools should not only suit their use but also their user. [ABSTRACT FROM AUTHOR]

With respect to the metaphysics of infinity, the tendency of standard debates is to either endorse or to deny the reality of ‘the infinite’. But how should we understand the notion of ‘reality’ employed in stating these options? Wittgenstein’s critical strategy shows that the notion is grounded in a confusion: talk of infinity naturally takes hold of one’s imagination due to the sway of verbal pictures and analogies suggested by our words. This is the source of various philosophical pictures that in turn give rise to the standard metaphysical debates: that the mathematics of infinity corresponds to a special realm of infinite objects, that the infinite is profoundly huge or vast, or that the ability to think about infinity reveals mysterious powers in human beings. First, I explain Wittgenstein’s general strategy for undermining philosophical pictures of ‘the infinite’ – as he describes it in Zettel; and then show how that critical strategy is applied to Cantor’s diagonalization proof in Remarks on the Foundations of Mathematics II. [ABSTRACT FROM AUTHOR]

Eric Snyder's book, "Semantics and the Ontology of Number," is a valuable resource for those interested in the philosophy of mathematics. The book explores the relationship between language and the existence of numbers, focusing on the "easy argument" for their existence. Snyder argues that linguistic considerations are crucial in understanding the ontology of numbers and supports a cautious form of referentialism. The book also discusses the use of type-shifting operations in semantics to analyze the meaning of number expressions. Overall, the book offers insights into the philosophy of mathematics and the semantics of number expressions. [Extracted from the article]

This article discusses the historical approach to the philosophy of mathematics, with a focus on the influence of philosopher Imre Lakatos. The author, Donald Gillies, argues that Lakatos played a significant role in introducing historical approaches to the philosophy of mathematics, although he was not the first to do so. Gillies examines various works that cite Lakatos and explores the impact of the historical approach on the field. He concludes that historical approaches to the philosophy of mathematics have produced valuable insights and should be considered a progressive research program. [Extracted from the article]

Jean W. Rioux's book, "Thomas Aquinas' Mathematical Realism," explores Aquinas' philosophy of mathematics and its relevance to contemporary problems in the field. Rioux contrasts Aquinas' views with those of Aristotle, highlighting their complementarity. Aquinas' philosophy emphasizes the conditions that mathematical objects must fulfill to be considered legitimate, rather than simply questioning their existence. Rioux also examines Aquinas' views on the construction of mathematical objects and their relationship to proof construction. The book provides insights into the nature of mathematics and its various aspects, making it a valuable resource for scholars in medieval philosophy and the philosophy of mathematics. [Extracted from the article]

The article "Indispensability" in the journal Philosophia Mathematica discusses the development of Indispensability Arguments in the philosophy of mathematics. These arguments support mathematical realism by asserting the indispensability of mathematics to our best scientific theories and explanations. The article explores the traditional Quine-Putnam version of the argument and addresses objections to each premise. It also introduces the enhanced indispensability argument, which focuses on the indispensability of mathematics for scientific explanations. The authors provide strategies for defending Indispensability Arguments against objections and offer a comprehensive overview of the topic. Overall, the article presents a concise and convincing defense of Indispensability Arguments, making it a valuable resource for those interested in the philosophy of mathematics. [Extracted from the article]

The online version of this paper has been temporarily withdrawn due to problems with the display of certain elements. A corrected version will be made available as soon as these issues have been resolved. The correct PDF version of the paper remains available via the PDF button under the paper's title. The publisher apologizes to readers and to the author for these problems.By Neil TennantReported by Author [Extracted from the article]

Mathematical pluralism can take one of three forms: (1) every consistent mathematical theory consists of truths about its own domain of individuals and relations; (2) every mathematical theory, consistent or inconsistent, consists of truths about its own (possibly uninteresting) domain of individuals and relations; and (3) the principal philosophies of mathematics are each based upon an insight or truth about the nature of mathematics that can be validated. (1) includes the multiverse approach to set theory. (2) helps us to understand the significance of the distinguished non‐logical individual and relation terms of even inconsistent theories. (3) is a metaphilosophical form of mathematical pluralism and hasn't been discussed in the literature. In what follows, I show how the analysis of theoretical mathematics in object theory exhibits all three forms of mathematical pluralism. [ABSTRACT FROM AUTHOR]

We discuss the mathematical modelling of two of the main mechanisms that pushed forward the emergence of multicellularity: phenotype divergence in cell differentiation and between-cell cooperation. In line with the atavistic theory of cancer, this disease being specific of multicellular animals, we set special emphasis on how both mechanisms appear to be reversed, however not totally impaired, rather hijacked, in tumour cell populations. Two settings are considered: the completely innovating, tinkering, situation of the emergence of multicellularity in the evolution of species, which we assume to be constrained by external pressure on the cell populations, and the completely planned—in the body plan —situation of the physiological construction of a developing multicellular animal from the zygote, or of bet hedging in tumours, assumed to be of clonal formation, although the body plan is largely—but not completely—lost in its constituting cells. We show how cancer impacts these two settings and we sketch mathematical models for them. We present here our contribution to the question at stake with a background from biology, from mathematics and from philosophy of science. [ABSTRACT FROM AUTHOR]

In mathematics, philosophy, cosmology, and theology, the notion of infinity has generated ample debate. Much less discussion has been generated by the notion of finiteness. However, when we consider finitely supported sets, the notion of finiteness becomes more interesting and richer. We present several independent definitions of finite sets within the framework of finitely supported structures, emphasizing the differences between these definitions. [ABSTRACT FROM AUTHOR]

We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$ , as $\kappa $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin's axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert's axiom of completeness. [ABSTRACT FROM AUTHOR]

The article is a review of two books by John Stillwell: "Algebraic Number Theory for Beginners: Following a Path From Euclid to Noether" and "The Story of Proof: Logic and the History of Mathematics." The reviewer acknowledges that Stillwell's works may not be directly relevant to modeling and applied mathematics, but praises his writing skills. The review discusses the content and structure of "Algebraic Number Theory for Beginners," noting that it may not be suitable for beginners due to unclear prerequisites and a lack of depth in certain topics. The reviewer also briefly mentions the overlap between algebraic number theory and elementary number theory. The review concludes with a brief mention of "The Story of Proof," describing it as a senior-level book on logic and the history of mathematics. [Extracted from the article]

In order to promote greater integration between Literature and research in Mathematical Education, we present in this study an analysis of the text "Some Popular Fallacies about Vivisection", a pamphlet published in 1875 by Lewis Carroll. In this work, Carroll employs principles of Classical Logic to question opinions about the practice of vivisection. We discuss reasons justifying the relevance of literature in research in the field of Mathematical Education and provide a general overview of the content and structure of Carroll's text, also offering a brief historical contextualization of the pamphlet's theme, vivisection. Finally, we exemplify how Classical Logic can be employed as an alternative approach to interpreting Carroll's work, thus demonstrating the intersection between Mathematics and Literature in this study. [ABSTRACT FROM AUTHOR]

The article discusses the field of "picturistics" and the challenges in understanding the nature of pictures and picture perception. It highlights the complexity of depicting space and the different types of space involved in the visual experience. The lack of a general theory of pictures is attributed to the multidisciplinary knowledge required and the dominance of linear perspective in representing visual space. The article also explores the paradoxical nature of pictures, which are both flat and deep, and emphasizes the importance of the artistic perspective in understanding picture perception. The paper suggests that an integrated approach combining art, philosophy, and science could pave the way for a future discipline of picturistics. [Extracted from the article]

This document is an editors' introduction to a special issue of the Journal of Spectral Theory dedicated to honoring Edward Brian Davies on his 80th birthday. It provides a brief biography of Davies, highlighting his academic achievements and contributions to the field of mathematics, particularly in spectral theory and the study of non-self-adjoint operators. The document also mentions Davies' role in attracting talented mathematicians from the former Soviet Union to the UK and his influence on the development of the theory of linear partial differential equations in the UK. The introduction concludes by listing the topics covered in the special issue and expressing the editors' hope that readers will find joy in reading the volume. [Extracted from the article]

"Visionen neuer Wissenschaft: Zur Dialogischen Dichtung von Dante Alighieri und Johannes Kepler" by Laetitia Rimpau is a book that explores the similarities in poetry and cosmology between Dante Alighieri and Johannes Kepler. It delves into the literary tradition of Platonic world models and the astronomical aspects of Dante's Convivio, as well as Kepler's poetry and the frontispiece to the Rudolphine Tables. Both Dante and Kepler were critical of the church's monopoly on learning and worked in the pagan tradition of cosmology. The book also examines the relationship between poetry and astronomy throughout history, focusing on the representation of constellations in ancient Greek poetry and the connections between Dante's cosmology and astronomy, as well as Kepler's scientific poetry. Overall, it provides a comprehensive exploration of the intersection between poetry and astronomy from antiquity to the Romantic period. [Extracted from the article]

This paper focuses on Kojève's account of history and philosophy of science. Kojève's understanding of science can be characterized as internalism, which is evident in his holistic view of philosophy, theology, quantum physics, and the history of classical Newtonian mechanics. It precipitates the facilitation of a further inquiry into the Christian genesis, secular evolution, and subsequent de-Christianization of scientific thought. The paper includes a critical scrutiny of Kojève's philosophical tenets, followed by a comparative analysis of the views of Hegel, Koyré, and Kojève. The primary objective of this research is to juxtapose Kojève's doctrines with Hegel's contemplations on the history and philosophy of science. In addition to identifying affinities, notably the emphasis on the Christian concept of God's Incarnation for the advancement of science, I draw the distinctions between the positions of Hegel, Kojève, and Koyré, specifically concerning the valuation of mathematical knowledge. [ABSTRACT FROM AUTHOR]

In response to Felix Mühlhölzer's critical discussion of my book Wittgenstein on Mathematics (2021), I argue against his main objections, in particular his view that one shouldn't even look for coherence in Wittgenstein's philosophy of mathematics, and his claim that Wittgenstein's insistence on the empirical applicability of mathematical concepts need not be taken seriously because in one passage it is prefaced by the words 'I want to say'. I also correct a number of irksome misunderstandings in Mühlhölzer's review. Repeatedly he ascribes to me a position that I merely present for discussion – and reject or qualify in the end. In particular, he wrongly ascribes to me the empiricist view that mathematical propositions are derived from experience. Some of Wittgenstein's remarks seem to express such a view, so I discuss it but conclude that, some exceptions apart, it is untenable and also inconsistent with Wittgenstein's key ideas about mathematical proof. [ABSTRACT FROM AUTHOR]

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature , rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones , i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality. [ABSTRACT FROM AUTHOR]

Copyright of Beytulhikme: An International Journal of Philosophy is the property of Beytulhikme: An International Journal of Philosophy 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.)

Computer assisted theorem proving is an increasingly important part of mathematical methodology, as well as a long-standing topic in artificial intelligence (AI) research. However, the current generation of theorem proving software have limited functioning in terms of providing new proofs. Importantly, they are not able to discriminate interesting theorems and proofs from trivial ones. In order for computers to develop further in theorem proving, there would need to be a radical change in how the software functions. Recently, machine learning results in solving mathematical tasks have shown early promise that deep artificial neural networks could learn symbolic mathematical processing. In this paper, I analyze the theoretical prospects of such neural networks in proving mathematical theorems. In particular, I focus on the question how such AI systems could be incorporated in practice to theorem proving and what consequences that could have. In the most optimistic scenario, this includes the possibility of autonomous automated theorem provers (AATP). Here I discuss whether such AI systems could, or should, become accepted as active agents in mathematical communities. [ABSTRACT FROM AUTHOR]