Philosophy Of Mathematics Research Papers (original) (raw)

This was a talk at "Theology in Mathematics?" (Kraków, Poland, June 8-10, 2014).
Abstract: A deep conviction of the majority of mathematicians on the brink of the 20th century was that mathematics is or at least must be infallible, consistent, rigorous, certain, necessary and universal, as well as free and applicable to the world without restriction. This common belief needs to be explained. This very conviction or belief was responsible for a heated argument provoked by set-theoretic paradoxes which were interpreted as an indication of the foundational crisis of mathematics (Grundlagenkrise der Mathematik). The crisis caused the emergence of diverse programs for the foundations of mathematics. Those programs gave birth, on the one hand, to the contemporary philosophy of mathematics and, on the other, to the formation of a new research field within mathematics: mathematical logic and the foundations of mathematics. This paper proposes the hypothesis that mathematics, seen from the foundationalist perspective, served at the time as a substitute for theology. According to this approach, philosophy of mathematics mediates an impact between theology and mathematics. To confirm this hypothesis I consider the prehistory of such an absolutist account of mathematics and pay a special attention to theological and quasi-theological ideas of the key figures of the three main foundationalist programs (logicism, intuitionism and formalism).

The thesis is an investigation into the logical pluralism debate, aiming to understand how the philosophical commitments sustaining each side to the debate connects to more general issues connected to the foundations of logic. My investigation centers on the following three notions: (1) Epistemic justification, (2) The metaphysical "ground" for logical truth, and (3) Normativity. Chapter 1 traces the monistic and pluralistic conception of logic back to its philosophical/mathematical roots, which we find in the writings of Rudolf Carnap and Gottlob Frege. I argue that logical pluralism - in its more plausible, epistemic (rather than ontic) form - was enabled by the semantic shift which Carnap seems to have anticipated and that, from a conventionalist perspective, his 'Principle of Tolerance' follows as a consequence of that shift. Chapter 2 concerns the issues ensuing from Willard V. O Quine’s critique of Carnap's conventionalism, which had a devastating effect for his foundationalist project. The aim is in particular to address the issue of meaning-variance, a crucial assumption for the conventionalist approach to pluralism. In chapter 3, I present another framework for pluralism, due to Stewart Shapiro’s [2014] ‘modelling’ conception of logic, according which logic is conceived as a mathematical model of natural language. Shapiro argues that our concept of logical consequence is vague and in need of a sharpening to attain a fixed meaning. Pluralism follows from there being two or more equally "correct" such sharpenings; i.e., relative to our theoretical aims. I argue that the modelling-conception is the best way to approach a justification of basic logical laws. However, since that conception also grounds Timothy Williamson’s [2017] argument for monism, I argue that the conception ultimately fails to establish logical pluralism. Since both Williamson and Shapiro take a pragmatic approach to justification, I conclude that the question of pluralism does not turn on epistemological commitments (i.e., on (1)), and suggest instead that it is a matter of (2), i.e., of one's conception of the "ground" for logical truth.

- by Julie Lauvsland
- •
- Model Theory, Pragmatism, Philosophy Of Mathematics, Philosophy of Logic

"Michel Weber et Pierfrancesco Basile (sous la direction de), Chromatikon III. Annuaire de la philosophie en procès — Yearbook of Philosophy in Process, Louvain-la-Neuve, Presses universitaires de Louvain, 2007. (300 p. ; ISBN... more

- by Michel Weber
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- Gnosticism, Metaphysics, Epistemology, Theology

This paper discusses results that arise in specific configurations pertaining to invariance under isoconjugation. The results lead to revolutionary theorems and crucial properties in both Euclidean and Projective geometry. After discussion of important theorems and properties of associated configurations, the authors present and prove an important, new result and its application in difficult geometrical configurations.

- by Ivan Zelich and +1
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- Mathematics, Applied Mathematics, Algebra, Geometry And Topology

The paper deals with the aesthetic and religious dimensions of mathematics. These dimensions are considered as closely connected, though reciprocally non-reducible. “Mathematical beauty” is already firmly established as a term in the philosophy of mathematics. Here, an attempt is made to bring forward two additional candidates: “mathematical sublime” and “numinous mathematics”. The last one is meant to designate the recognition of some mathematical practices as inspiring anticipation of the meeting with the divine reality or producing a feeling of its presence. The first one is used here to designate the related feelings in disguise, i.e., being reinterpreted or transferred from the straightforwardly religious to the aesthetic sphere. Taking Kant’s theory of the sublime as a starting point, the paper introduces a related account of it that treats mathematical beauty through mathematical sublimity as a more fundamental category. Within this account, religious experience, the aesthetics of the sublime and mathematical practice are closely interlinked through an appropriate interpretation of the idea of the infinite. Both mathematical and art symbolism are seen as an endeavour to represent the infinite within the finite, which correlates well with the definition of mathematics as “the science of the infinite” (Hermann Weyl).Keywords: philosophy of mathematics, philosophy of mathematical practice, mathematical aims, mathematical beauty, the sublime, numinous, the infinite, symbolism.

Here a preprint has been uploaded.

An Otherwise Non-distinct Geneaology:"Mathematical Logic" as seen from the Perspective of Alain Badiou's Philosophical System: The aim of this essay is to map eleven transformations of mathematical logic in Badiou's philosophical system. These transformations otherwise organize the development of logic and mathematics within the context of French epistemology. We argue that to grasp Badiou's ontological argument, especially as set out in Being and Event, it is necessary to adopt a perspective critical of logicism and logical positivism. In other words, as naive set theory is created by Georg Cantor, it is independent of the project of logicism stemming from Russell and Wittgenstein's work on Frege. Indeed, set theory should not be reduced to the field of logic, but be understood as mathematical per se. This is the line of development philosophy of mathematics, epistemology and logic took within the structuralist background of French philosophy.

- by Norman R Madarasz
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- Philosophy Of Mathematics, Alain Badiou

This is a rough and ready mix of finished sections and notes for a master-class on Alain Badiou's set theory ontology focusing in particular on the initial Meditations of Being and Event and attempting to identify different options open to Badiou in his construction of an ontology of multiplicity. It was used as the basis for a workshop presentation along with a powerpoint - which, incidentally, Badiou himself attended!

- by Oliver Feltham
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- Ontology, Philosophy Of Mathematics, Alain Badiou

We live in an age of movement. More than at any other time in history , people and things move longer distances, more frequently, and faster than ever before. All that was solid melted into air long ago and is now in full circulation around the world like dandelion seeds adrift on turbulent winds. We find ourselves, in the early twenty-first century, in a world where every major domain of human activity has become increasingly defined by motion. 1 We have entered a new historical era defined in large part by movement and mobility and are now in need of a new historical on-tology appropriate to our time. The observation that the end of the twentieth century and the beginning of the twenty-first was marked by an increasingly " liquid " and " mobile modernity " is now something widely recognized in the scholarly literature at the turn of the century. 2 Today, however, our orientation to this event is quite different. Almost twenty years into the twenty-first century we now find ourselves situated on the other side of this heralded transition. The question that confronts us today is thus a new one: how to fold all that has melted back up into new solids. 3

- by Thomas Nail
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- Critical Theory, Religion, History, Social Movements

Hilbert's program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862-1943), and was pursued by him and his collaborators at the University of Gottingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert's proposal called for a new foundation of mathematics based on two pillars: the axiomatic method and finitary proof theory.

- by Richard Zach
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- Philosophy Of Mathematics, David Hilbert
- by İbrahim Halil Kayacan
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- Mathematics, Philosophy Of Mathematics, Matematika

L'apport des connaissances dans la science est soumis à l’expérience. Dans la vision de Pascal les anciens les ont trouvées seulement ébauchées, et nous les laisserons à ceux qui viendront après nous en un état plus accompli que nous ne les avons reçues se voyant ainsi infinies en l’étendue de leurs recherches. Cet écrit propose d’explorer la tension philosophique autour de l’expérience de simulation et de l’expérience de pensée qui créent et explorent tous deux des mondes hypothétiques.

- by Matthieu Verry
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- Sociology, Computer Science, Materials Science, Philosophy

This paper will utilize advanced computational mathematics and Quantum statistics for an algorithmic view on our universe. We will look at the theoretical nature of wave propagation, different Quantum states and propose a mathematical analysis for how it ties all together. Sequential equations applicable to nature and a utilization of Quantum similarity as proposed in my previous papers will also be used. The purpose of this paper is to provide a new approach to algorithmic Quantization in terms of describing the biggest complexities in Quantum Mechanics.

- by Armando Marques-Guedes
- •
- History, Cultural History, Sociology, Political Sociology

The systematic implications of irrational numbers, for those interested.

- by Nathan Coppedge
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- Mathematics, Number Theory, Set Theory, Proof Theory
- by Federico Demaria
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- Marketing, Engineering, Environmental Engineering, Religion

STRONG INDUCTION and its underlying definitions are presented in this white paper (knowledge base).

- by Matheus Lobo
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- Mathematics, Proof Theory, Philosophy Of Mathematics, Mathematics Education

In the article, the philosophical significance of quantum computation theory for philosophy of mathematics is discussed. In particular, I examine the notion of "quantum-assisted proof" (QAP); the discussion sheds light on the problem of the nature of mathematical proof; the potential empirical aspects of mathematics and the realism-antirealism debate (in the context of the indispensability argument). I present a quasi-empiricist account of QAP's, and discuss the possible impact on the discussions centered around the Enhanced Indispensabity Argument (EIA).

Atomic numbers of Hydrogen, Helium, Boron, Neon, Calcium, Technetium, and Protactinium are members of a cryptic numerical progression series. Contextually, this research paper involves postulation and hypothesis formulation around some... more

- by Aidan Lyon
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- Philosophy, Philosophy of Science, Philosophy Of Mathematics, Australasian Philosophy

Im Kontext mit allen benannten Phänomenen in diesem Post, ergibt Alles ein schlüssiges Bild zu
„United States INC“ Konkurs?
Wieviel davon stimmt kann ich nicht sagen. Da ich die Wahrscheinlichkeit als eher hoch erachte, dass da im Hintergrund in diese Richtung schon was passiert ist und laufend "gewerkelt" wird, möchte ich diese interessante Meldung bringen. Interessant auch vor dem Hintergrund des buchstäblichen
Impfwahnsinns
und
was in Afghanistan passiert
und
der neuen Goldenen Bulle der Erlösung Proklamation
und
im Kontext mit der Papstrede im Deutschen Bundestag 2011,
sowie
neues von Prof. Franz Hörmann zu Zukunft der Wissenschaften und dem Gelsystem
und
Botschaften vom Botschafter der physisch lebendigen Erde zu Recht und Kräfte
über
der Wiederanbindung zum ein 1 einzig All-Bewusstsein
mit
der Wiederanbindung verloren gegangener Wahrnehmung der 12 Sinnesorgane.
und der physischen Lebenordnung kosmischer Gesetze mit den Naturgesetzen.
zuerst Video ansehen
https://www.academia.edu/video/kAwrO1

In the framework of the philosophy of contemporary mathematics, Hellman and Awodey both hold an interesting discussion on the role of Zermelo-Fraenkel Set Theory and Category Theory in the perspective of a good foundation for mathematics. For Hellman, neither Set Theory nor Category Theory constitutes a good foundational framework for mathematics and, in addition, Categories does not achieve a strong autonomy regarding Sets. Awodey’s claim is that Category Theory is a best option in the frame of a new way of understanding what a foundation of mathematics means. In this sense, the aim of this paper is to highlight the philosophical main features of this discussion, to establish some related positions and to show some interesting consequences for the philosophy of mathematics.

Indicates that everything standing on top of naive set theory is also proven.

The article outlines a thesis that, from the cognitive point of view, natural numbers share common structure with the physical experiences encountered by students during their early childhood years.
This common structure displays conceptual duality which is described in the context of numbers as magnitude and multitude. The issue is examined by using tangible examples of walking, cutting bread and pouring water into beakers. The article presents a cognitive view of learning.
Key words: mathematical thinking, mathematical practices, cognition of mathematics, conceptual development

- by Wes Raykowski
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- Mathematics, Education, Philosophy Of Mathematics, Mathematics Education
- by Mark Balaguer
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- Philosophy, Philosophy Of Mathematics
- by James Schwartz
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- Philosophy Of Mathematics, Structuralism (Philosophy), Nominalism

Mathematics education includes beliefs about the nature of mathematics and the ways about mathematics teaching and
learning. Many thoughts brings as a result of looking at the philosophy of mathematics, which is necessary for understanding
the nature of mathematics. These thoughts today might make mathematics fallible rather than absolute correct information.
Beliefs about the nature of mathematics also are leading learning and teaching activities in mathematics classrooms. Reuben
Hersh, a mathematician, while explaining mathematics as a social-cultural-historical reality with mental, physical aspects and
as a part of the humanities that achieving a compromise similar to science and constructing reproducible results; reveals the
philosophy of his own mathematical experience in his studies of philosophy of mathematics. Reuben Hersh's mathematician
with his academic roles reflected in his studies of philosophy of mathematics that he emphasized experience. At this point,
the experiences of Reuben Hersh as a person who does mathematics and come from within mathematics set an example for
mathematics teachers. This study aims to provide an approach about the nature of mathematics based on Reuben Hersh and to
give suggestions for mathematics teachers.

- by Carlo Veronesi
- •
- Epistemology, Philosophy of Science, Philosophy Of Mathematics, Galileo Galilei

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. The crucial idea of a continuum is used to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results.

- by José Ferreirós
- •
- Philosophy of Science, Philosophy Of Mathematics, Epistemología

An investigative attempt at explaining Euclid's first definition in the "Elements," and answering the question "what is a point"? I assert that a point is an indefinitely small quantity of space, and thus can "have no part" because it is... more

A century and a half ago, a revolution in human thought began that has gone largely unrecognized by modern scholars: A system of non-Euclidean geometries was developed that literally changed the way that we view our world. At first, some thought that space itself was non-Euclidean and four-dimensional, but Einstein ended their 'speculations' when he declared that time was the fourth dimension. Yet our commonly perceived space is truly four-dimensional. Einstein unwittingly circumvented that particular revolution in thought and delayed its completion for a later day, although his work was also necessary for the completion of that revolution. That later day is now approaching. The natural progress of science has brought us back to the point where science again needs to consider the physical reality of a higher-dimensional space. Science must acknowledge the truth that space is four-dimensional and space-time is five-dimensional, as required by accepted physical theories and observations, before it can move forward with a new unified fundamental theory of physical reality.

We give a pictorial, and absurdly simple, proof that transparently illustrates why four colours suffice to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal... more

- by Alexandre Borovik
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- Philosophy Of Mathematics, Mathematics Education, Gifted Education

The present paper deals with the ontological status of numbers and considers Frege´s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path that, departing from Frege's initial premises, drives to a conception of numbers as synthetic a priori in a more Kantian sense. On this basis, it tentatively explores a possible derivation of basic logical rules on their behalf, suggesting a more rudimentary basis to inferential thinking, which supports reconsidering the difference between logical thinking and AI. Finally, it reflects upon the contributions of this approach to the problem of the a priori.

- by Olga Ramírez Calle
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- Philosophy Of Mathematics, A Priori Knowledge, Logical Empiricism, Frege