Philosophy of Mathematics

U članku se razmatra grčki ideal mjere, najčešće izražen pojmom sredine. To nije samo svegrčki ideal nego drevna i svevremena baština, koja je konvergirala odgoju u ujednosti ‘dobrote i ljepote’. On je življen u polisu, kao što je bio i personalna, unutarduševna težnja. U Aristotelovoj politici počelo mjere prvenstveno je vezano za balansiranje političkih poredaka, ali postoji i druga perspektiva u kojoj je više riječ o vrlini i moralnim kvalitetama aristokracije, gdje se identificira dobrog čovjeka i građanina. Taj pristup srodniji (...) je istraživanjama u Aristotelovim etičkim tekstovima. Pažljiva analiza ovih distinkcija iz knjige Politika zaključuje se s uvidima o odgoju i vrlini kao komplementarnim aspektima praktičke filozofije. The article considers the Greek ideal of moderation, most often expressed through the notion of the Mean. It is not merely a pangreek ideal, but an ancient and everlasting inheritance, which converged to the education in the unity of ‘goodness and beauty’. It is lived in the polis, as it was also a personal aspiration of the inner soul. In Aristotle’s politics, the source of moderation is primarily connected to the balance of political orders, however there is another perspective in which it is more about the virtue and moral qualities of aristocracy, which identifies the good man and citizen. That approach is more akin to research conducted in Aristotle’s texts on ethics. Careful analysis of these distinctions from the book Politics is concluded with insights on education and virtue as complementary aspects of practical philosophy. (shrink)

Reverse mathematics is a new take on an old idea: asking which axioms are necessary to prove a given theorem. This question was first asked about the parallel axiom in Euclid’s geometry and later about the axiom of choice in set theory. Obviously, such questions can be asked in many fields of mathematics, but in recent decades, it has proved fruitful to focus on subsystems of second-order arithmetic, where much of mainstream mathematics resides. It has been found that many basic (...) theorems of analysis and topology, as well as certain parts of infinite algebra and combinatorics, can be proved in such systems. And, remarkably, almost all the basic theorems fall into one of five particular systems: a base system RCA0 of “constructive mathematics” and four others defined by certain axioms about real numbers. Moreover, many of the theorems not provable in RCA0 turn out to be equivalent to one of these defining axioms, so we know precisely which axiom is needed to prove them. Thus, after some motivational remarks about the parallel axiom and the axiom of choice, we will concentrate on the study of RCA0 and its extensions, which is what “reverse mathematics” is generally taken to mean today. (shrink)

In the last couple of years, a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the questions what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences (i.e., explanations that don’t cite causes in the explanans) sometimes take a form of the question of what makes mathematical models explanatory, especially whether highly idealized models in science can be explanatory and in virtue of (...) what they are explanatory. These questions raise further issues about counterfactuals, modality, and explanatory asymmetries: i.e., do mathematical and non-causal explanations support counterfactuals, and how ought we to understand explanatory asymmetries in non-causal explanations? Even though these are very common issues in the philosophy of physics and mathematics, they can be found in different guises in the philosophy of biology where there is the statistical interpretation of the Modern Synthesis theory of evolution, according to which the post-Darwinian theory of natural selection explains evolutionary change by citing statistical properties of populations and not the causes of changes. These questions also arise in philosophy of ecology or neuroscience in regard to the nature of topological explanations. The question here is can the mathematical (or more precisely topological) properties in network models in biology, ecology, neuroscience, and computer science be explanatory of physical phenomena, or are they just different ways to represent causal structures. The aim of this special issue is to unify all these debates around several overlapping questions. These questions are: are there genuinely or distinctively mathematical and non-causal explanations?; are all distinctively mathematical explanations also non-causal; in virtue of what they are explanatory; does the instantiation, implementation, or in general, applicability of mathematical structures to a variety of phenomena and systems play any explanatory role? This special issue provides a platform for unifying the debates around several key issues and thus opens up avenues for better understanding of mathematical and non-causal explanations in general, but also, it will enable even better understanding of key issues within each of the debates. (shrink)

El análisis de las propiedades geométricas de las configuraciones finitas ha sido uno de los objetivos fundamentales del estudio de las diversas geometrías discretas y de la geometría combinatoria. Este artículo propone plantear la posibilidad de una elucidación de lo matemático desde una perspectiva derivada de las ‘matemáticas en acción’ y no desde una concepción ‘analítico gramatical’ de sus fundamentos, y establecer, al menos, un mínimo umbral de validez, que articule una interpretación semiótica de los signos matemáticos, a través del (...) análisis de un par de ejemplos elementales de geometría configuracional, cuyas construcciones son propuestas por el autor. (shrink)

El presupuesto según el cual el contenido de una manifestación compleja está en función de los contenidos de sus partes componentes, expresa claramente una intuición que solemos tener sobre lo múltiple; implica una reflexión sobre la relación entre el todo y las partes que lo componen; involucra una teoría de las multiplicidades que entraña atributos de naturaleza matemática; presenta el problema de cómo los seres humanos nos relacionamos con los entornos del mundo para generar unidad de sentido. La significación es (...) un proceso de síntesis. Y desentrañar los mecanismos de funcionamiento de dicho proceso es un enigma de carácter eminentemente fenomenológico. La matemática misma, como acto de significación, como conjunto de actos de contenidos intencionales, comparte este carácter y es susceptible de tratamiento fenomenológico. Noesis y semiosis se conjugan para dotar al saber matemático de un contenido, de articulación compleja, que trasciende las perspectivas meramente logicistas, formalistas o intuicionistas. Una primera aproximación a esta problemática es el propósito del presente trabajo. (shrink)

This article develops a criticism of Alain Badiou’s assertion that “mathematics is ontology.” I argue that despite appearances to the contrary, Badiou’s case for bringing set theory and ontology together is problematic. To arrive at this judgment, I explore how a case for the identification of mathematics and ontology could work. In short, ontology would have to be characterised to make it evident that set theory can contribute to it fundamentally. This is indeed how Badiou proceeds in Being and Event. (...) I review his descriptions of the ontological problematic at some length here, only to argue that set theory is a poor fit. Although philosophers working on questions of being were certainly occupied with matters of oneness and the part-whole relationship, I argue that Badiou’s discussion of philosophical sources points towards a mereological treatment, not a set-theoretic one. Finally, I suggest that Badiou’s philosophical interpretations of key set-theoretic results are better understood as some sort of analogising between mathematics, ontology, and philosophical anthropology. (shrink)

The latest draft (posted 05/14/22) of this short, concise work of proof, theory, and metatheory provides summary meta-proofs and verification of the work and results presented in the Theory and Metatheory of Atemporal Primacy and Riemann, Metatheory, and Proof. In this version, several new and revised definitions of terms were added to subsection SS.1; and many corrected equations, theorems, metatheorems, proofs, and explanations are included in the main text. The body of the text is approximately 18 pages, with 3 sections; (...) 1, an Introduction (with a 108 page listing of key terms & definitions), sect. 2, the Results, sect. 3, Discussion (commentary & predictions), and sect. 4, Works Cited. As much as possible, the style is intended for readability and understanding by very bright children (with some interest & knowledge of maths, etc.) and very interested nonprofessionals. The results of this project also enable upgrades of number theory, set theory, proof theory, metamathematics, the foundations of science, and quantum mechanics theory (etc.). (shrink)

In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah ̃a, working with only three numerals (one, two, many) can help us to change our perception (...) of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive light. (shrink)

The objective of this paper is to examine the thought of John Philoponus contra Aristotle, as it pertains to consciousness and its emergence, in light of both contemporary cosmology and philosophy. It will be argued that in an eternal universe the emergence of consciousness is an impossibility. The inspiration for this line of reasoning is found in Philoponus’ sixth century arguments against Aristotle on the eternity of the world. It will be shown that much of Philoponus’ argumentation is corroborated by (...) contemporary cosmology and philosophy. (shrink)

Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.

In a career that spans 60 years so far, W.W. Tait has made many highly influential contributions to logic, the philosophy of mathematics, and their history. The present collection of new essays - contributed by former students, colleagues, and friends - is a Festschrift, i.e., a celebration of his life and work. The essays address a variety of themes prominent in his work or related to it. The collection starts with an introduction in which Tait's contributions are sketched and put (...) into context. The eleven essays that follow are arranged in three parts: Part I. Proof Theory and its History; Part II. Logic and Philosophy of Mathematics; and Part III. History of Logic and Philosophy of Mathematics. Each of the essays contributes substantially to one or several of these areas. The authors included are: Steve Awodey, Solomon Feferman, Michael Friedman, Warren Goldfarb, Geoffrey Hellman, William Howard, Stephen Menn, Rebecca Morris, Charles Parsons, Erich Reck, Thomas Ricketts, and Wilfried Sieg. The editor, Erich H. Reck is Professor of Philosophy at the University of California at Riverside. (shrink)

The text of this interview is based on a conversation between Bagryana Popov and Juliane Römhild on 1 September 2021. In this interview, Bagryana discusses two works which unite her research into political trauma and site-specific performance in the context of political repression under the communist regime in Bulgaria. For her choreography He is not here and the performance event Traces Bagryana returned to Sofia, the city of her birth, to explore her own family history and her grandfather’s incarceration as (...) a political prisoner. He is not here is a meditation on the body at the receiving end of political power developed in collaboration with dancer Simon Ellis. Traces takes the audience first to the courthouse in Sofia, where Bagryana’s grandfather was trialled, and then to her family’s flat where Bagryana’s daughter reads from the diaries of her grandmother. Both works were a part of Bagryana’s PhD project at Melbourne University. (shrink)

Well-known for his work on absolutism, divine right theory, and his contextual reading of Hobbes’ ideas, Sommerville also published successful critical editions of Sir Robert Filmer and King James vi and I’s political writing. Sommerville’s engagement in key historiographical debates on early- modern British history, involving “opposing camps” of revisionists and post-revisionists, is less explored. Here, I focus on the question whether pre-Civil War England was immune to ideological conflict or, instead, featured a confrontation between King and Parliament based on (...) ideas of power, liberty and obedience. I also highlight the continued relevance of Sommerville’s innovative account of English political thought as deeply shaped by European theories. His work reminds us of the role of the history of political thought in rectifying false claims and unchecked opinions, so commonly expounded in our divided world. In conclusion, I advance my own critical interpretation of Sommerville’s views on absolutism and patriarchalism. (shrink)

This paper recounts my encounter with the ideological context of Hobbes’s system as a graduate student in political theory through the teaching and scholarship of Professor Johann Sommerville. This encounter made me recognize that political theorists should study not only systems of political philosophy but also their ideological contexts, whose primary components are not “languages” but ideas and arguments deployed in debates concerning issues of political legitimacy of a particular time. Specifically, I realized that incorporating ideological contexts into the study (...) of political theory could render the quest for legitimate political orders in different ages and cultures accessible to modern readers of diverse backgrounds, enable a more accurate grasp of the originality of a philosophical system, and uncover new resources for political theorizing. (shrink)

Economic methodology has been dominated by developments in the philosophy of science. This book's central thesis is that a great deal can be gained by refocusing attention on developments in the philosophy of mathematics, in particular those that took place over the course of the twentieth century. In this book the authors argue that a close examination of the major developments in the philosophy of mathematics both deepens and enriches our understanding of the formalisation of economics, while also offering novel (...) methods of critically evaluating the strengths and limitations of formalism in economics. This book represents an innovative attempt to more fully understand the complexity of the interaction between developments in the philosophy of mathematics and the process of formalisation in economics. (shrink)