Logic and Foundations of Mathematics Commons^™

Japanese-English Translation: Miki Kiyoshi —Thinking With Master Nishida (First Published In Fujin Kōron, August 1941) Complete Draft; Translated, Edited, And Revised By Christopher Southward, October 2022-September 2023 「西田先生のことども」、三木清著（初発 婦人公論、昭和十六年八月）, Christopher Southward

Comparative Literature Faculty Scholarship

Japanese-English Translation: Miki Kiyoshi —Thinking with Master Nishida (First Published in Fujin Kōron, August 1941) Complete Draft; Translated, Edited, and Revised by Christopher Southward, October 2022-September 2023「西田先生のことども」、三木清著（初発 婦人公論、昭和十六年八月）

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Deontic Meta-Rules, Francesco Olivieri, Guido Governatori, Matteo Cristani, Antonino Rotolo, Abdul Sattar

Centre for Computational Law

The use of meta-rules in logic, i.e., rules whose content includes other rules, has recently gained attention in the setting of non-monotonic reasoning: a first logical formalisation and efficient algorithms to compute the (meta)-extensions of such theories were proposed in Olivieri et al. (2021, Computing defeasible meta-logic. In JELIA 2021, LNCS, vol. 12678, pp. 69-84. Springer.). This work extends such a logical framework by considering the deontic aspect. The resulting logic will not just be able to model policies but also tackle well-known aspects that occur in numerous legal systems. The use of Defeasible Logic to model meta-rules in the …

Tarski And Bachmann In Regina: A Magical Connection, James T. Smith

Journal of Humanistic Mathematics

This is a personal account of an intersection of the schools of research in foundations of geometry founded by Alfred Tarski and Friedrich Bachmann. Their academic lineages and the origins of the schools are also described, as well as the mathematics that resulted from this intersection.

Moretheless, Abdelghani Alnahawi

Masters Theses

material investigations becoming questions with interjections

What Is A Number?, Nicholas Radley

HON499 projects

This essay is, in essence, an attempt to make a case for mathematical platonism. That is to say, that we argue for the existence of mathematical objects independent of our perception of them. The essay includes a somewhat informal construction of number systems ranging from the natural numbers to the complex numbers.

Three Essays On Substructural Approaches To Semantic Paradoxes, Brian C. Porter

Dissertations, Theses, and Capstone Projects

This thesis consists of three papers on substructural approaches to semantic paradoxes. The first paper introduces a formal system, based on a nontransitive substructural logic, which has exactly the valid and antivalid inferences of classical logic at every level of (meta)inference, but which I argue is still not classical logic. In the second essay, I introduce infinite-premise versions of several semantic paradoxes, and show that noncontractive substructural approaches do not solve these paradoxes. In the third essay, I introduce an infinite metainferential hierarchy of validity curry paradoxes, and argue that providing a uniform solution to the paradoxes in this hierarchy …

Zero, Śūnya And Pūrṇa: A Comparative Analysis, Animisha Tewari

Comparative Philosophy

Due to apparent duality in this world, one has to face a lot of difficulties while searching for the Truth. Our ego is the root cause for perception of duality and this in turn leads to suffering. This suffering can only be extinguished by attainment of the Truth, i.e, non-duality. However, in order to enable the finite intellect to comprehend the incomprehensible non-duality, this undifferentiated whole is sometimes denoted by nothingness (śūnya) or fullness (pūrṇa). Non-duality is usually understood by the numeral ‘1’ which stands for unity or oneness. The main aim of this paper is …

Ineffability, Emptiness And The Aesthetics Of Logic, Andreas Kapsner

Comparative Philosophy

In this essay, I explore the nature of the logical analysis of Buddhist thought that Graham Priest has offered in his book The Fifth Corner of Four (5of4). The paper traces the development of a logical value in- troduced in 5of4, which Priest has called e. The paper points out that certain criticisms I have made earlier still stand, but focuses on a recon- ceptualization of 5of4 in which these arguments carry less weight. This new perspective on the book, inspired by a response to my arguments by Priest himself, sees the logical analysis of Buddhism …

On Probabilistic Reasoning Of Actual Causation, Jingzhi Fang

Lingnan Theses and Dissertations

Probabilistic actual causation is a theory about actual causal relations in probabilistic scenarios. Compared with general (or type) causal connections, actual (or token, singular) causation involves specific and actual events occurring in a particular time and space. Halpern and Pearl proposed three mathematical definitions on actual causation via structural equation models (or causal models). Fenton-Glynn extended one of their definitions into a probabilistic version by following the probability-raising principle in the tradition of theorizing about probabilistic causation. The basic idea of this principle is that a cause shall raise the probability of its effect. He adopted interventional probabilities to analyse …

A Question Of Fundamental Methodology: Reply To Mikhail Katz And His Coauthors, Tom Archibald, Richard T. W. Arthur, Giovanni Ferraro, Jeremy Gray, Douglas Jesseph, Jesper Lützen, Marco Panza, David Rabouin, Gert Schubring

Philosophy Faculty Articles and Research

This paper is a response by several historians of mathematics to a series of papers published from 2012 onwards by Mikhail Katz and various co-authors, the latest of which was recently published in the Mathematical Intelligencer, “Two-Track Depictions of Leibniz’s Fictions” (Katz, Kuhlemann, Sherry, Ugaglia, and van Atten, 2021). At issue is a question of fundamental methodology. These authors take for granted that non-standard analysis provides the correct framework for historical interpretation of the calculus, and castigate rival interpretations as having had a deleterious effect on the philosophy, practice, and applications of mathematics. Rather than make this case by reasoned …

Necessity, Essence And Analyticity: Toward An Analytic Essentialist Account Of Necessity, Dongwoo Kim

Dissertations, Theses, and Capstone Projects

Some truths could not have failed to hold. Such are called metaphysically necessary truths. As Michael Dummett once aptly formulated, the philosophical problem about necessity is twofold: what makes necessary truths necessarily true and how do we recognize them as such? This dissertation aims to address these questions by developing and defending a novel account of necessity, which has the following three main theses: (1) the necessity of a statement about an entity is established as a consequence of a general principle implying that if the entity is a certain way then it is necessarily that way and the fact …

Foundational Mathematical Beliefs And Ethics In Mathematical Practice And Education, Richard Spindler

Journal of Humanistic Mathematics

Foundational philosophical beliefs about mathematics in the mathematical community may have an unappreciated yet profound impact on ethics in mathematical practice and mathematics education, which also affects practice. A philosophical and historical basis of the dominant platonic and formalist views of mathematics are described and evaluated, after which an alternative evidence-based foundation for mathematical thought is outlined. The dualistic nature of the platonic view based on intuition is then compared to parallel historical developments of universalizing ethics in Western thought. These background ideas set the stage for a discussion of the impact of traditional mathematical beliefs on ethics in the …

Ethics And Mathematics – Some Observations Fifty Years Later, Gregor Nickel

Journal of Humanistic Mathematics

Almost exactly fifty years ago, Friedrich Kambartel, in his classic essay “Ethics and Mathematics,” did pioneering work in an intellectual environment that almost self-evidently assumed a strict separation of the two fields. In our first section we summarize and discuss that classical paper. The following two sections are devoted to complement and contrast Kambartel’s picture. In particular, the second section is devoted to ethical aspects of the indirect and direct mathematization of modern societies. The final section gives a short categorization of various philosophical positions with respect to the rationality of ethics and the mutual relation between ethics and mathematics.

Unknowable Truths: The Incompleteness Theorems And The Rise Of Modernism, Caroline Tvardy

Honors Scholars Collaborative Projects

This thesis evaluates the function of the current history of mathematics methodologies and explores ways in which historiographical methodologies could be successfully implemented in the field. Traditional approaches to the history of mathematics often lack either an accurate portrayal of the social and cultural influences of the time, or they lack an effective usage of mathematics discussed. This paper applies a holistic methodology in a case study of Kurt Gödel’s influential work in logic during the Interwar period and the parallel rise of intellectual modernism. In doing so, the proofs for Gödel’s Completeness and Incompleteness theorems will be discussed as …

Semantic Completeness Of Intuitionistic Predicate Logic In A Fully Constructive Meta-Theory, Ian Ray

Masters Theses & Specialist Projects

A constructive proof of the semantic completeness of intuitionistic predicate logic is explored using set-generated complete Heyting Algebra. We work in a constructive set theory that avoids impredicative axioms; for this reason the result is not only intuitionistic but fully constructive. We provide background that makes the thesis accessible to the uninitiated.

Why Are They Called Real Numbers If They Aren’T Real, And Other Such Questions?, Rahmat Rashid

Honors Program Theses

This thesis studies the position of mathematical realism (the position that mathematical objects have ontological status) through history, starting with Pythagoras up until W.V.O Quine, and examining how these positions originate from each other. I hope to see how the position has changed and why, and provide an argument against the strongest of the realist positions, drawing on this extensive background. Finally, I advance my own argument against the strongest arguments for mathematical realism, and propose alternatives to a view of mathematical realism.

Russian Logics And The Culture Of Impossible: Part Ii: Reinterpreting Algorithmic Rationality, Ksenia Tatarchenko, Anya Yermakova, Liesbeth De Mol

Research Collection School of Social Sciences

This article reinterprets algorithmic rationality by looking at the interaction between mathematical logic, mechanized reasoning, and, later, computing in the Russian Imperial and Soviet contexts to offer a history of the algorithm as a mathematical object bridging the inner and outer worlds, a humanistic vision that we, following logician Vladimir Uspensky, call the “culture of the impossible.” We unfold the deep roots of this vision as embodied in scientific intelligentsia. In Part I, we examine continuities between the turn-of-the-twentieth-century discussions of poznaniye—an epistemic orientation towards the process of knowledge acquisition—and the postwar rise of the Soviet school of mathematical logic. …

Assessing Risk At The National Strategic Level: Visualization Tools For Military Planners, Wade A. Germann, Heather S. Gregg

The US Army War College Quarterly: Parameters

The reemergence of great power competition, conflict with near-peer competitor states below the level of armed conflict, and persisting threats from nonstate actors with transnational ambitions and global reach pose challenges for strategists planning, executing, and assessing military operations and strategy. Building on current visualization tools, two proposed models—the National Strategic Risk Abacus and the National Strategic Risk Radar Chart—address these challenges and better depict how the US military may inadvertently contribute to risk at the national strategic level.

The Agnostic Structure Of Data Science Methods, Domenico Napoletani, Marco Panza, Daniele Struppa

MPP Published Research

In this paper we argue that data science is a coherent and novel approach to empirical problems that, in its most general form, does not build understanding about phenomena. Within the new type of mathematization at work in data science, mathematical methods are not selected because of any relevance for a problem at hand; mathematical methods are applied to a specific problem only by `forcing’, i.e. on the basis of their ability to reorganize the data for further analysis and the intrinsic richness of their mathematical structure. In particular, we argue that deep learning neural networks are best understood within …

Patrick Aidan Heelan’S The Observable: Heisenberg’S Philosophy Of Quantum Mechanics, Paul Downes

Research Resources

The publication of Patrick Aidan Heelan’s The Observable, with forewords from Michel Bitbol, editor Babette Babich and the author himself, offers a timely invitation to reconsider the relation between quantum physics and continental philosophy.

Patrick Heelan does so, as a contemporary of and interlocutor with Werner Heisenberg on these issues, as a physicist himself who trained with leading figures of quantum mechanics (QM), Erwin Schrödinger and Eugene Wigner. Moreover, Heelan highlights Heisenberg’s interest in phenomenology as ‘a friend and frequent visitor of Martin Heidegger’ (55). Written originally in 1970 and unpublished then for reasons Babich explicates in her foreword, …

Dimentia: Footnotes Of Time, Zachary Hait

Senior Projects Spring 2021

Time from the physicist's perspective is not inclusive of our lived experience of time; time from the philosopher's perspective is not mathematically engaged, in fact Henri Bergson asserted explicitly that time could not be mathematically engaged whatsoever. What follows is a mathematical engagement of time that is inclusive of our lived experiences, requiring the tools of storytelling.

Analysis, Constructions And Diagrams In Classical Geometry, Marco Panza

MPP Published Research

Greek ancient and early modern geometry necessarily uses diagrams. Among other things, these enter geometrical analysis. The paper distinguishes two sorts of geometrical analysis and shows that in one of them, dubbed “intra-confgurational” analysis, some diagrams necessarily enter as outcomes of a purely material gesture, namely not as result of a codifed constructive procedure, but as result of a free-hand drawing.

Diagrams In Intra-Configurational Analysis, Marco Panza, Gianluca Longa

MPP Published Research

In this paper we would like to attempt to shed some light on the way in which diagrams enter into the practice of ancient Greek geometrical analysis. To this end, we will first distinguish two main forms of this practice, i.e., trans-configurational and intra-configurational. We will then argue that, while in the former diagrams enter in the proof essentially in the same way (mutatis mutandis) they enter in canonical synthetic demonstrations, in the latter, they take part in the analytic argument in a specific way, which has no correlation in other aspects of classical geometry. In intra-configurational analysis, diagrams represent …

Recognizing Mathematics Students As Creative: Mathematical Creativity As Community-Based And Possibility-Expanding, Meghan Riling

Journal of Humanistic Mathematics

Although much creativity research has suggested that creativity is influenced by cultural and social factors, these have been minimally explored in the context of mathematics and mathematics learning. This problematically limits who is seen as mathematically creative and who can enter the discipline of mathematics. This paper proposes a framework of creativity that is based in what it means to know or do mathematics and accepts that creativity is something that can be nurtured in all students. Prominent mathematical epistemologies held since the beginning of the twentieth century in the Western mathematics tradition have different implications for promoting creativity in …

Don’T Be So Fast With The Knife: A Reply To Kapsner, Graham Priest

Comparative Philosophy

The is a brief reply to the central objection against the construction of my The Fifth Corner of Four by Andi Kapsner in his “Cutting Corners: A Critical Note on Priest’s Five-Valued Catuṣkoṭi. This concerns the desirability of adding a fifth corner (ineffability) to the four of the catuṣkoṭi.

Cutting Corners: A Critical Note On Priest’S Five-Valued Catuṣkoṭi, Andreas Kapsner

Comparative Philosophy

Graham Priest has offered a rational reconstruction of Buddhist thought that involves, first, modeling the Catuṣkoṭi by a four valued logic, and then later adding a fifth value, read as “ineffability”. This note examines that fifth value and raises some concerns about it that seem grave enough to reject it. It then sketches an alternative to Priest’s account that has no need for the fifth value.

A Russellian Analysis Of Buddhist Catuskoti, Nicholaos Jones

Comparative Philosophy

Names name, but there are no individuals who are named by names. This is the key to an elegant and ideologically parsimonious strategy for analyzing the Buddhist catuṣkoṭi. The strategy is ideologically parsimonious, because it appeals to no analytic resources beyond those of standard predicate logic. The strategy is elegant, because it is, in effect, an application of Bertrand Russell's theory of definite descriptions to Buddhist contexts. The strategy imposes some minor adjustments upon Russell's theory. Attention to familiar catuṣkoṭi from Vacchagotta and Nagarjuna as well as more obscure catuṣkoṭi from Khema, Zhi Yi, and Fa Zang motivates the …

Connecting Ancient Philosophers’ Math Theory To Modern Fractal Mathematics, Colin Mccormack

Parnassus: Classical Journal

No abstract provided.

Between Evidence And Facts: An Argumentative Perspective Of Legal Evidence, Wenjing Du, Minghui Xiong

OSSA Conference Archive

In this paper, we will present an argumentative view of legal evidence. In an argumentation-based litigation game, the only purpose of the suitor (S) or the respondent (R) is to maximize their own legal rights while the purpose of the trier (T) is to maintain judicial fairness and justice. Different selections of evidence and different orders of presenting evidence will lead to different case-facts and even adjudicative results, the purpose of litigation is to reconcile a balance among the three parties - S, R, and T.

Real Possibility: Modality And Responsibility, Julia Gaul

Honors Scholar Theses

Imagine: someone is backing out of a parking space and does not look in their rear view mirror. They subsequently hit a car that was passing by. One could argue that they simply could have avoided the accident had they looked in their mirror. This non-actual possibility, that they could have looked in the mirror, seems legally and morally relevant. One could also argue that they could have avoided the accident had they stuck their feet out of their window and sung La Marseillaise.

My leading questions is: how do we distinguish possibilities that are legally and morally relevant from …