Philosophy of Mathematics, Miscellaneous

The philosophy of superdeterminism is based on a single scientific fact about the universe, namely that cause and effect in physics are not real. In 2020, accomplished Swedish theoretical physicist, Dr. Johan Hansson published a physics proof using Albert Einstein’s Theory of Special Relativity that our universe is superdeterministic meaning a predetermined static block universe without cause and effect in physics. The unity of our universe originates from its creation from the same nothingness under the zero energy universe theory. However, (...) nothingness cannot actually cause a universe to exist on its own under superdeterminism due to the absence of cause and effect in physics. Moreover, nothingness is the absence of anything real and cannot logically be the reality underlying our universe. Consequently, our universe must exist from the reality of the logic or Logos underlying our physical reality resulting from the origination of our universe from the same nothingness or common creation. (shrink)

This article presents a fairness principle for evaluating decision-making based on predictions: a decision rule is unfair when the individuals directly impacted by the decisions who are equal with respect to the features that justify inequalities in outcomes do not have the same statistical prospects of being benefited or harmed by them, irrespective of their socially salient morally arbitrary traits. The principle can be used to evaluate prediction-based decision-making from the point of view of a wide range of antecedently specified (...) substantive views about justice in outcome distributions. (shrink)

This volume features new essays on the application and role of naturalism in philosophical inquiry. It serves as an important update on current controversies about naturalism. The contributors include leading figures who have written on naturalism and its relevance to a wide range of issues across philosophical subdisciplines. The chapter discuss how naturalism can be properly employed in different philosophical areas such as epistemology, metaphysics, philosophy of mind, philosophy of language, philosophy of religion, philosophy of time, philosophy of science, philosophy (...) of mathematics, philosophy of memory, cognitive science, ethics, meta-ethics, and normativity. Naturalism and Its Challenges will be of interest to scholars and advance students working in a wide range of philosophical disciplines. (shrink)

Brouwer's philosophy of mathematics is usually regarded as an intra-subjective, even solipsistic approach, an approach that also underlies his mathematical intuitionism, as he strived to create a mathematics that develops out of something inner and a-linguistic. Thus, points of connection between Brouwer's mathematical views and his views about and the social world seem improbable and are rarely mentioned in the literature. The current paper aims to challenge and change that. The paper employs a socially oriented prism to examine Brouwer's views (...) on the construction, use, and practice of mathematics. It focuses on Brouwer's views on language, his social interactions, and the importance of group context as they appear in the significs dialogues. It does so by exploring the establishment and dissolution of the significs movement, focusing on Gerrit Mannoury's influence and relationship with Brouwer and analyzing several fragments from the significs dialogues while emphasizing the role Brouwer ascribed to groups in forming and sharing new ideas. The paper concludes by raising two questions that challenge common historical and philosophical readings of intuitionism. (shrink)

Arithmetic is one of the foundations of our educational systems, but what exactly is it? Numbers are everywhere in our modern societies, but what is our knowledge of numbers really about? This book provides a philosophical account of arithmetical knowledge that is based on the state-of-the-art empirical studies of numerical cognition. It explains how humans have developed arithmetic from humble origins to its modern status as an almost universally possessed knowledge and skill. Central to the account is the realisation that, (...) while arithmetic is a human creation, the development of arithmetic is constrained by our evolutionarily developed cognitive architecture. Arithmetic is a sophisticated cultural development, but it is ultimately based on abilities with numerosities that we already possess as infants and share with many non-human animals. Therefore, arithmetic is not purely conventional, an arbitrary game akin to chess. Instead, arithmetic is deeply connected to our basic cognitive capacities. (shrink)

We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, we go (...) through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness. (shrink)

This volume contains 45 papers, written by the author alone or in collaboration with the following co-authors: Mumtaz Ali, Said Broumi, Sukanto Bhattacharya, Mamoni Dhar, Irfan Deli, Mincong Deng, Alexandru Gal, Valeri Kroumov, Pabitra Kumar Maji, Maikel Leyva-Vazquez, Feng Liu, Pinaki Majumdar, Munazza Naz, Karina Perez-Teruel, Rıdvan Sahin, A. A. Salama, Muhammad Shabir, Rajshekhar Sunderraman, Luige Vladareanu, Magdalena Vladila, Stefan Vladutescu, Haibin Wang, Hongnian Yu, Yan-Qing Zhang.

It is often claimed that the theory of function levels proposed by Frege in Grundgesetze der Arithmetik anticipates the hierarchy of types that underlies Church’s simple theory of types. This claim roughly states that Frege presupposes a type of functions in the sense of simple type theory in the expository language of Grundgesetze. However, this view makes it hard to accommodate function names of two arguments and view functions as incomplete entities. I propose and defend an alternative interpretation of first-level (...) function names in Grundgesetze into simple type-theoretic open terms rather than into closed terms of a function type. This interpretation offers a still unhistorical but more faithful type-theoretic approximation of Frege’s theory of levels and can be naturally extended to accommodate second-level functions. It is made possible by two key observations that Frege’s Roman markers behave essentially like open terms and that Frege lacks a clear criterion for distinguishing between Roman markers and function names. (shrink)

Does natural science give us reason to believe that mathematical statements are true? And does natural science give us reason to believe in some particular metaphysics of mathematics? These two questions should be firmly distinguished. My argument in this chapter is that a negative answer to the second question is compatible with an affirmative answer to the first. Loosely put, even if science settles the truth of mathematics, it does not settle its metaphysics.

This note identifies and comments on selected crucial traits of Robert Aumann’s philosophy of game theory. In doing so, it aims at carving out and expressing some notions tacitly held by many working game theorists and ideally even at triggering subsequent reflection on the philosophy of game theory in general. According to my reconstruction of Aumann’s position, sophisticated, relatively precise rules of language—an ultra-refined grammar for interactions—constitute the heart of game theory. Consequently, the heart of game theory is devoid, or (...) almost devoid, of empirical content. The final section proposes a nomenclature for lucidly discussing the relationships between theories, different types of models, and “the real world” and employs it to explicate and slightly amend Aumann’s remarks on the role of truth and fruitfulness in game theory. (shrink)

Although number sentences are ostensibly simple, familiar, and applicable, the justification for our arithmetical beliefs has been considered mysterious by the philosophical tradition. In this paper, I argue that such a mystery is due to a preconception of two realities, one mathematical and one nonmathematical, which are alien to each other. My proposal shows that the theory of numbers as properties entails a homogeneous domain in which arithmetical and nonmathematical truth occur. As a result, the possibility of arithmetical knowledge is (...) simply a consequence of the possibility of ordinary knowledge. (shrink)

The present paper concerns the Wittgenstein ontology project: an attempt to create a Semantic Web representation of Ludwig Wittgenstein’s philosophy. The project has been in development since 2006, and its current state enables users to search for information about Wittgenstein-related documents and the documents themselves. However, the developers have much more ambitious goals: they attempt to provide a philosophical subject matter knowledge base that would comprise the claims and concepts formulated by the philosopher. The current knowledge representation technology is not (...) well-suited for this task, and a non-standard approach is required. The creators of the Wittgenstein ontology project are aware of this fact; recently, they have been discussing conceptual devices adjusting the technology to their needs. The main goal of this paper is to present examples of a representation of philosophical content that make use of both the devices already proposed and some new inventions. The represented content comes from the Tractatus Logico-Philosophicus; more specifically, its theses concerning the problems in philosophy of mathematics. (shrink)

I defend the hypothesis that the semantic paradoxes, the paradoxes about collections, and the sorites paradoxes, are all paradoxes of reference fixing: they show that certain conventionally adopted and otherwise functional reference-fixing principles cannot provide consistent assignments of reference to certain relevant expressions in paradoxical cases. I note that the hypothesis has interesting implications concerning the idea of a unified account of the semantic, collection and sorites paradoxes, as well as about the explanation of their “recalcitrance”. I also note that (...) it does not necessarily imply that one should not expect the sometimes hoped for “unique” solution to a paradox of these kinds. (shrink)

Douglas Wadle celebrates the juxtaposition of art and science while examining music's influence on humanity's understanding of our place in the universe. Tracing the millennia-old love affair between music and science, Wadle chronicles the surprising ubiquity of musical training among history's greatest thinkers. He shines a spotlight on the intertwining stories of pattern and form and how they complement one another in our search for creativity and insight. Einstein's Violin relies on extensive research to tell the story of how music (...) impacts the pattern recognition software in our brains, facilitating more creative problem solving. Without digression into technical treatise, it focuses on the historical stories that best display music's beautiful interaction with mind and universe"--Back cover. (shrink)

In this paper, I present and discuss critically the main elements of Mario Bunge’s philosophy of mathematics. In particular, I explore how mathematical knowledge is accounted for in Bunge’s systemic emergent materialism.

The debate about the foundations of mathematical sciences traces back to Greek antiquity, with Euclid and the foundations of geometry. Through the flux of history, the debate has appeared in several shapes, places, and cultural contexts. Remarkably, it is a locus where logic, philosophy, and mathematics meet. In mathematical astronomy, Nicolaus Copernicus’s axiomatic approach toward a heliocentric theory of the universe has prompted questions about foundations among historians who have studied Copernican axioms in their terminological and logical aspects but never (...) examined them as a question of mathematical practice. Copernicus provides seven unproved assumptions in the introduction of the brief treatise entitled Nicolaus Copernicus’s draft on the models of celestial motions established by himself, better known as Commentariolus (ca. 1515), published circa 30 years before the final composition of his heliocentric theory (On the revolutions of the heavenly spheres, 1543). The assumptions deal with the renowned Copernican hypothesis of considering the Earth in motion and the Sun, not affected by motion, near the center of the universe. Although Copernicus decides to omit the proofs for the sake of brevity, the deductions in the Commentariolus are supposed to be drawn from the initial seven assumptions. Questions on the nature (are they postulates or axioms?) and the logic (is there an internal rigor?) of those assumptions have yet to be fully explored. By examining Copernicus’s seven assumptions as a question of mathematical practice, it is possible to hold historical, philosophical, and logical aspects of Copernican axiomatics together and understand them as part of Copernicus’s intuition and creativity. (shrink)

J. S. Silverberg, The Most Obscure and Inconvenient Tables ever Constructed.- D. J. Melville, Commercializing Arithmetic: The Case of Edward Hatton.- C. Baltus, Leading to Poncelet: A Story of Collinear Points.- R. Godard, Cauchy, Le Verrier et Jacobi sur le problème algébrique des valeurs propres et les inégalités séculaires des mouvements des planètes.- A. Ackerberg-Hastings, Mathematics in Astronomy at Harvard College Before 1839 as a Case Study for Teaching Historical Writing in Mathematics Courses.- J. J. Tattersall, S. L. McMurran, "Lectures (...) for Women" and the Founding of Newnham College, Cambridge.- D. Waszek, Are Euclid's Diagrams "Representations"? On an Argument by Ken Manders.- B. Buldt, Abstraction by Embedding and Constraint-Based Design.- W. Meyer, The Birth of Undergraduate Modern Algebra in the United States.- P. Liu, History as a Source of Mathematical Narrative in Developing Students' Interpretations of Mathematics.- F. Kamareddine, J. P. Seldin, Thoughts on Using the History of Mathematics to Teach the Foundations of Mathematical Analysis. (shrink)

In this multidisciplinary book, mathematician Matthew He provides integrative perspectives of algebraic biology, cognitive informatics, and poetic expressions of the human mind. Using classical Pythagorean Theorem and contemporary Category Theory, the proposed matrix models of the human mind connect three domains of the physical space of objective matters, mental space of subjective meanings, and emotional space of bijective modes; draws the connections between neural sparks and idea points, between synapses and idea lines, and between action potentials and frequency curves.

La presente investigación tiene como propósito general asumir un posicionamiento filosófico en clave histórico-social y de tipo anti-relativista para analizar el desarrollo histórico de la matemática, el cual aplicaremos a un caso en particular: la matemática del antiguo Egipto. Para ello se discutirán y criticarán, en primera instancia, determinadas posiciones filosóficas afines al cuasi-empirismo en matemática que, siendo relativistas, permitirán delinear nuestro propio posicionamiento en contraste: la existencia de una «matemática situada». Esta categoría filosófica tendrá como sustento teórico la noción (...) de conocimiento situado. Además, las implicaciones de esta son, según sostenemos, tanto filosóficas como historiográficas, ya que servirá para analizar las características del corpus de problemas matemáticos egipcios registrados en los diversos papiros matemáticos. En particular, haremos referencia a las expresiones lingüísticas egipcias para denotar las diversas operaciones aritméticas y el carácter algorítmico de los problemas, así como también la cuestión epistemológica de la empiria y cómo la perspectiva situada permite superar la dicotomía entre una matemática pura y otra aplicada, que consideramos no es ubicua para abordar la interpretación de la práctica matemática del antiguo país del Nilo. Palabras clave: Filosofía de la matemática, Anti-relativismo, Matemática situada, Extrañeza del pasado, Matemática egipcia. (shrink)

Es común escuchar que el mundo Occidental debe a los árabes el descubrimiento del álgebra. No obstante, el desarrollo de esta disciplina puede interpretarse como un crisol de distintas tradiciones científicas que fue posible gracias a la clasificación, traducción y crítica tanto de los clásicos como de las obras que los árabes obtuvieron de los pueblos que conquistaron. Entre estos trabajos se encontraba Los Elementos de Euclides. Los Elementos fueron cuidadosamente traducidos durante el califato de Al-Ma’mūn por el matemático Mohammed (...) ibn-Musa Al-Khwārizmī, autor de Al-jabr wa’l muqābalah, quien sentó los fundamentos de la disciplina que más tarde sería conocida como álgebra y quien, en la primera parte de su obra, nos proporciona tres métodos para resolver tres tipos de ecuaciones que llama “ecuaciones combinadas”. A lo largo de este artículo se ofrecen argumentos para sostener que los métodos para resolver estas ecuaciones constituyen una reinterpretación, en el terreno algebraico, de los teoremas 6, 7 y 8 del segundo libro de Los Elementos de Euclides. Mi objetivo es dar respuesta a la siguiente pregunta: ¿Qué lectura de Los elementos Euclides posibilitó la emergencia del álgebra en el mundo árabe? Para responderla necesario explorar el acercamiento que los árabes tuvieron con el Libro II de Los Elementos, con el fin de proponer una interpretación de los posibles factores que los llevaron a formular el álgebra. Esto último, en particular, se encuentra en la primera parte de la obra de Al-Khwārizmī. Palabras clave: Álgebra, Al-Khwārizmī, Euclides, Ecuación, Teorema. (shrink)

Mathematics has a long track record of refining the concepts by which we make sense of the world. For example, mathematics allows one to speak about different senses of "sameness", depending on the larger context. Phenomenology is the name of a philosophical discipline that tries to systematically investigate the first-personal perspective on reality and how it is constituted. Together, mathematics and phenomenology seem to be a good fit to derive statements about our experience that are, at the same time, well-defined, (...) precise, and significant to our inner lives. However, there are difficulties stemming from the fact that phenomenology deals with inherently subjective things. Phenomenological investigations seem to lose their appeal when trying to approach them in the clear-cut way afforded by mathematics. How to overcome this obstacle? We argue that the Arts play a special role in mediating between the precise statements of mathematics and the sometimes fuzzy nature of our experience. Mathematics and art are complementary ways to come to a comprehensive understanding of reality. (shrink)

This book is a collection of articles on research that attempts to connect logic and mathematics with empirical and cognition. There have been various such a...

Several natural languages such as English contain prima facie different kinds of referential and quantificational expressions. In particular, natural languages.

En este trabajo matemático-filosófico se estudian cuatro tópicos de la Lógica matemática: El método de construcción de modelos llamado Ultraproductos, la Propiedad de Interpolación de Craig, las Álgebras booleanas y los Órdenes parciales separativos. El objetivo principal del mismo es analizar la importancia que tienen dichos tópicos para el estudio de los fundamentos de la matemática, desde el punto de vista del platonismo matemático. Para cumplir con tal objetivo se trabajará en el ámbito de la Matemática, de la Metamatemática y (...) de la Filosofía de la matemática. El desarrollo de la investigación arrojó como resultado que tales tópicos son muy importantes para el estudio de los fundamentos de la matemática, desde el punto de vista del platonismo matemático, y en el trabajo se explica detalladamente con abundantes ejemplos el porqué (al final de cada sección y al final del mismo). (shrink)

En este artículo realizamos una reconstrucción del Programa original de Hilbert antes del surgimiento de los teoremas limitativos de la tercera década del siglo pasado. Para tal reconstrucción empezaremos por mostrar lo que Torretti llama los primeros titubeos formales de Hilbert, es decir, la defensa por el método axiomático como enfoque fundamentante. Seguidamente, mostraremos como estos titubeos formales se establecen como un verdadero programa de investigación lógico-matemático y como dentro de dicho programa la inquietud por la decidibilidad de los problemas (...) matemáticos y en específico la decidibilidad de la Lógica de primer orden cobra peso. Luego pasamos a analizar como la inquietud por la decibilidad toma lugar dentro del pensamiento filosófico-matemático de Hilbert presentándose como uno de los grandes problemas a los cuales la metamatemática debe encontrar una solución, esto lo hacemos mostrando un contraste con autores, como John von Neumann y Roberto Torretti, quienes de alguna u otra manera no interpretan el problema de la decidibilidad de la Lógica de primer orden como un problema de peso dentro del programa original de Hilbert. Finalmente argumentamos que el resultado meta-teórico de Church puede entenderse como una refutación del optimismo intelectual que permea a todo el programa original de Hilbert. (shrink)

From ancient times to 19th century geometryGeometry symbolized the essence of mathematical thinking and method, but modern philosophy of mathematics seems to have marginalized the philosophical status of geometryGeometry. The roots of this transformation will be sought in the ascendance of logical foundations in place of intuitive primacy as the cornerstone of mathematical certainty in the late 19th century. Nevertheless, geometry and geometrical thinking, in multiple manifestations, have continued to occupy a central place in the practice of mathematics proper. We (...) argue that this, together with advances in the neuroscience of mathematical processes, calls for an expansion of the present limited remit of the philosophy of mathematics. (shrink)

Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve multiple semantic functions? Whereas most philosophers deny that one of these uses is genuine, we instead argue that (...) number words, like many related expressions, are polymorphic, having multiple uses whose meanings are systematically related via type shifting. (shrink)

How is that when scientists need some piece of mathematics through which to frame their theory, it is there to hand? What has been called 'the unreasonable effectiveness of mathematics' sets a challenge for philosophers. Some have responded to that challenge by arguing that mathematics is essentially anthropocentric in character, whereas others have pointed to the range of structures that mathematics offers. Otavio Bueno and Steven French offer a middle way, which focuses on the moves that have to be made (...) in both the mathematics and the relevant physics in order to bring the two into appropriate relation. This relation can be captured via the inferential conception of the applicability of mathematics, which is formulated in terms of immersion, inference, and interpretation. In particular, the roles of idealisations and of surplus structure in science and mathematics respectively are brought to the fore and captured via an approach to models and theories that emphasize the partiality of the available information: the partial structures approach. The discussion as a whole is grounded in a number of case studies drawn from the history of quantum physics, and extended to contest recent claims that the explanatory role of certain mathematical structures in scientific practice supports a realist attitude towards them. The overall conclusion is that the effectiveness of mathematics does not seem unreasonable at all once close attention is paid to how it is actually applied in practice. (shrink)

I defend the thesis that Composition Entails Identity (CEI): that is, a whole is identical to all of its parts, taken together. CEI seems to be inconsistent, since it seems to require that the parts of a whole possess incompatible number properties (for instance, being one thing and being many things). I show that these number properties are, in fact, compatible.

Penelope Maddy has recently addressed the set-theoretic multiverseset-theoretic multiverse, and expressed reservations on its status and merits Foundations of mathematics. Essays in honor of W. Hugh Woodin’s 60th birthday. Contemporary mathematics. American Mathematical Society, Providence, pp. 289–322, 2017). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalismset-theoretic naturalism. I first distinguish three main forms of ‘multiversism’multiversism, and then I proceed to analyse MaddyMaddy’s concerns. Among other things, I take into account salient (...) aspects of multiverse-related mathematics, in particular, research programmes in set theory for which the use of the multiverse seems to be crucial, and show how one may provide responses to MaddyMaddy’s concerns based on a careful analysis of ‘multiverse practice’. (shrink)

Volume 2 contains both notebooks of "Time Management (Max) I and II" and thereby Gödel’s applied individual ethics, which he received among others through his teacher Heinrich Gomperz. Gödel thus incorporates the ethical ideal of self-perfection into his opus. The volume is prefaced by an introduction to relevant considerations from the ethics of the Stoics as well as ancient dietetics, which provide the philosophical background to understand Gödel’s approach. In addition, editor Eva-Maria Engelen presents how this fits into the context (...) of Gödel’s Philosophical Notebooks. (shrink)

The essential role of language in rational cognition is analysed. The approach is functional: only the results of the connection between language, reality, and thinking are considered. Scientific language is analysed as an extension and improvement of everyday language. The analysis gives a uniform view of language and rational cognition. The consequences for the nature of ontology, truth, logic, thinking, scientific theories, and mathematics are derived.

The book is a philosophical refection on the possibility of mathematical history. Are poosible models of historical phenomena so exact as those of physical ones? Mathematical models borrowed from quantum mechanics by the meditation of its interpretations are accomodated to history. The conjecture of many-variant history, alternative history, or counterfactual history is necessary for mathematical history. Conclusions about philosophy of history are inferred.

This article, written in Bengali ('Gonit Dorshon' means `philosophy of mathematics' ), briefly reviews a few of the major points of view toward mathematics and the world of mathematical entities, and interprets the philosophy of mathematics as an interaction between these. The existence of these different points of view is indicative that mathematics, in spite of being of universal validity, can nevertheless accommodate alternatives. In particular, I review the alternative viewpoints of Platonism and Intuitionism and present the case that in (...) spite of their great differences, they are not mutually exclusive - that both can be accommodated within the infinite edifice of mathematics. This, in turn, is argued to be consistent with the viewpoint of Category Theory that holds the promise of an entirely new interpretation of the world of mathematics and the relation of that world to the world of our concepts and ideas: mathematics is a human enterprise and mathematical logic is a reflection of how our ideas and concepts are formed and combined with one another. I venture that this, perhaps, is the view of mathematics that Ludwig Wittgenstein would espouse. (shrink)