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Articles 1 - 30 of 128
Full-Text Articles in Logic and Foundations of Mathematics
The Limits Of Universal Logic: A Cross-Cultural Archaeology Of Logical Critique, Murat Kelikli
The Limits Of Universal Logic: A Cross-Cultural Archaeology Of Logical Critique, Murat Kelikli
Comparative Philosophy
This study examines critiques of logic articulated across diverse historical and cultural traditions within a comparative framework. The critiques developed in Islamic, Buddhist, Chinese, Indian, Zen, and modern Western philosophy do not, in the main, reject logic or reasoning as such; rather, they target the conceptualization of logic as a universal, context-independent, and singular normative standard. The analysis shows that objections raised at ontological, epistemic, linguistic, methodological, and political levels converge on a set of shared concerns rooted in the structural features of logic itself, most notably categorical determinacy, universalist assumptions, and claims of context-independence. At the same time, these …
The Modal Catuṣkoṭi: Formalizing Emptiness In Classical Modal Logic, Chance Chapman
The Modal Catuṣkoṭi: Formalizing Emptiness In Classical Modal Logic, Chance Chapman
Comparative Philosophy
Nāgārjuna’s arguments in the Mūlamadhyamakakārikā rely extensively on modus tollens, yet this inference is invalid in the paraconsistent frameworks typically used to formalize the catuṣkoṭi. We develop a modal extension of Belnap-Dunn semantics to address this, expressed as: ¬□(v(P) = t) ∧ ¬□(v(P) = f) ∧ ¬□(v(P) = b) ∧ ¬□(v(P) = n). This formula applies self-reflexively, resolves longstanding interpretive debates regarding verses within Nāgārjuna’s Mūlamadhyamakakārikā and Vigrahavyāvartanī such as his “no thesis” defense, and bridges interpretive gaps between traditionally designated Eastern and Western logical traditions, providing a Western analytical translation of Indian dialectic into symbolic modal logic that …
Why Read The Classics (Of Mathematical Proofs)?, Cosimo Perini Brogi
Why Read The Classics (Of Mathematical Proofs)?, Cosimo Perini Brogi
Journal of Humanistic Mathematics
Inspired by Italo Calvino’s (almost) homonymous writing, this short essay suggests a formal perspective to complement the typical categorisation of classical results in mathematics on aesthetic bases. The traditional proof that √2 is not a rational number provides a simple example of proof mining. When studied through the tools of logic, that proof has also led to designing new systems for formal proving based on cyclic proofs. This example thus supports the central thesis of this essay that reading the classics of mathematical proofs enables us to acquire a taste for mathematical beauty and can open new directions in mathematics …
A Cross-System Analysis Of Structural Commensurability Between Axiomatic Foundations Of Set Theory (Zfc) And The Principles Of Transcendent Wisdom, Mohammadjavad Maarefvand
A Cross-System Analysis Of Structural Commensurability Between Axiomatic Foundations Of Set Theory (Zfc) And The Principles Of Transcendent Wisdom, Mohammadjavad Maarefvand
Comparative Philosophy
Building on observed similarities in uniqueness proofs for the empty set and the Necessary Existent (Wājib al-Wujūd), this study engages the meta-methodological issue of cross-system commensurability. It identifies four instances of structural commensurability between Zermelo-Fraenkel set theory with Choice (ZFC) and Mulla Sadra's Transcendent Wisdom (Ḥikmat al-Muta'āliyah). First, identity principles: the Axiom of Extensionality provides a formal operational criterion analogous to the philosophical principle of identity through absence of distinguishing features. Second, multiplicity mechanisms: ZFC's constructor axioms (bottom-up construction) and emanation principles (top-down explanation) address analogous problems with inverse orientations. Third, transcending limitation: the Axiom of Infinity ensuring quantitative extensional …
Challenging The Double-Negation Principle: An Analysis Of Strategic Theories Of Sun Zi, Carnot, And Clausewitz Through Intuitionist Logic, Antonino Drago
Challenging The Double-Negation Principle: An Analysis Of Strategic Theories Of Sun Zi, Carnot, And Clausewitz Through Intuitionist Logic, Antonino Drago
Comparative Philosophy
To study a subject similarly represented in Eastern and Western cultures, I examine some classical theories of war strategy according to a new analytical method. This method is based on the recognition that the most important strategic theories are organized as theories essentially not deduced from axioms according to classical logic. Such theories reason through doubly negated propositions (DNPs), whose meanings differ from those of their affirmative counterparts. This corresponds to the use of intuitionist logic instead of classical logic. An inspection of the works of the texts of Sun Zi, Lazare Carnot, and Clausewitz reveals many such propositions, and …
A Dualistic Interpretation Of Mathematical Creation Through Art And Argumentation, Sofia Almpani, Petros Stefaneas, Mihir Chakraborty
A Dualistic Interpretation Of Mathematical Creation Through Art And Argumentation, Sofia Almpani, Petros Stefaneas, Mihir Chakraborty
Journal of Humanistic Mathematics
Creativity, informal reasoning and dynamic exchange of ideas form the pulsating heart of mathematical creation. In this approach, the concept of a mathematical object surpasses conventional boundaries of formal presentation, as they also encompass the intention to prove, significant creative stages within the proving, and the overall experience of prover's journey, which may involve arguments, debates, discovery insights, aesthetic visualizations, and narrative elements. In this paper we present two conceptual frameworks, namely Argumentation-based Proof-Events Calculus (APEC) and Mathematical RUPAs, in order to provide distinct yet interconnected perspectives on informal thinking, knowledge creation, and proving in mathematics. Following the two perspectives, …
Tarski And Bachmann In Regina: A Magical Connection, James T. Smith
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.
Zero, Śūnya And Pūrṇa: A Comparative Analysis, Animisha Tewari
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
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 …
Foundational Mathematical Beliefs And Ethics In Mathematical Practice And Education, Richard Spindler
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
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.
Assessing Risk At The National Strategic Level: Visualization Tools For Military Planners, Wade A. Germann, Heather S. Gregg
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.
Recognizing Mathematics Students As Creative: Mathematical Creativity As Community-Based And Possibility-Expanding, Meghan Riling
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
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
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
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
Connecting Ancient Philosophers’ Math Theory To Modern Fractal Mathematics, Colin Mccormack
Parnassus: Classical Journal
No abstract provided.
The Conceptions Of Self-Evidence In The Finnis Reconstruction Of Natural Law, Kevin P. Lee
The Conceptions Of Self-Evidence In The Finnis Reconstruction Of Natural Law, Kevin P. Lee
St. Mary's Law Journal
Finnis claims that his theory proceeds from seven basic principles of practical reason that are self-evidently true. While much has been written about the claim of self-evidence, this article considers it in relation to the rigorous claims of logic and mathematics. It argues that when considered in this light, Finnis equivocates in his use of the concept of self-evidence between the realist Thomistic conception and a purely formal, modern symbolic conception. Given his respect for the modern positivist separation of fact and value, the realism of the Thomistic conception cannot be the foundation for the natural law as Finnis would …
Are Logic And Math Relevant To Social Debates?, Michael A. Lewis
Are Logic And Math Relevant To Social Debates?, Michael A. Lewis
Journal of Humanistic Mathematics
Social debates, as well as discussions about certain highly charged issues, such as racism, gender identity, and sexuality, usually turn on the uses or mentions of key words. That is, the conclusions we can draw from such discussions depend on how certain terms are used or mentioned in them. Yet participants in social debates may often fail to precisely define their terms or fail to make important distinctions in terms uttered by others. Both logic and mathematics pay attention to the importance of precise definitions when it comes to engaging in discussions, arguments, or proofs. Logic also makes an important …
Engaging The Paradoxical: Zeno's Paradoxes In Three Works Of Interactive Fiction, Michael Z. Spivey
Engaging The Paradoxical: Zeno's Paradoxes In Three Works Of Interactive Fiction, Michael Z. Spivey
Journal of Humanistic Mathematics
For over two millennia thinkers have wrestled with Zeno's paradoxes on space, time, motion, and the nature of infinity. In this article we compare and contrast representations of Zeno's paradoxes in three works of interactive fiction, Beyond Zork, The Chinese Room, and A Beauty Cold and Austere. Each of these works incorporates one of Zeno's paradoxes as part of a puzzle that the player must solve in order to advance and ultimately complete the story. As such, the reader must engage more deeply with the paradoxes than he or she would in a static work of fiction. …
Maths Living In Social Arenas, From Practice To Foundations, Nigel Vinckier
Maths Living In Social Arenas, From Practice To Foundations, Nigel Vinckier
Journal of Humanistic Mathematics
Maths comes to life in human interaction. This has consequences for the mathematics itself. This paper discusses how this ``coming to life'' of mathematics in different social arenas influences the foundations of maths. We will argue that this influence is profound, to the extent that it is hard to upkeep the idea that there is or should be one foundation on which all mathematics can be built.
Fatal Attractions, Elective Affinities, And Deadly Epistemologies, Ibpp Editor
Fatal Attractions, Elective Affinities, And Deadly Epistemologies, Ibpp Editor
International Bulletin of Political Psychology
This article cites film, the novel, and news report to underline the deadly seriousness of the quest for knowledge.
Pagan Winter, Samm Willard
Pagan Winter, Samm Willard
Sophia and Philosophia
Isn’t this a lovely place to pick apart your lover’s face
Some say the river bank’s a sacred place
Others think that’s such a silly thing to say
But I would never try to prove them wrong on such a blissful day
The colors of the leaves will soon have changed
The yellows and the greens will fade to gray
But I will lose a quiet hour to the darkest day
A pagan winter’s on its way
I will see the death of God before it’s Christmas day
A pagan winter’s on its way
Well isn’t this some lovely clay …
Haunted By A Memory I Never Lived, Carlos Hiraldo
Haunted By A Memory I Never Lived, Carlos Hiraldo
Sophia and Philosophia
I am haunted by a memory I never lived. My mother and father are sitting in their house in Brooklyn with my baby sister watching the 1969 moon landing. Born in 1971, I wasn’t there. But I spent my toddler years in the waning residue of excitement about the landing and listening to adults talk about where they had watched it. As a child, I was baffled by how vivid this event that occurred without me was to people of my parents’ age. Except for some surviving pictures of the living room, I never knew the house in which they …
Symmetry And Measuring: Ways To Teach The Foundations Of Mathematics Inspired By Yupiaq Elders, Jerry Lipka, Barbara Adams, Monica Wong, David Koester, Karen Francois
Symmetry And Measuring: Ways To Teach The Foundations Of Mathematics Inspired By Yupiaq Elders, Jerry Lipka, Barbara Adams, Monica Wong, David Koester, Karen Francois
Journal of Humanistic Mathematics
Evident in human prehistory and across immense cultural variation in human activities, symmetry has been perceived and utilized as an integrative and guiding principle. In our long-term collaborative work with Indigenous Knowledge holders, particularly Yupiaq Eskimos of Alaska and Carolinian Islanders in Micronesia, we were struck by the centrality of symmetry and measuring as a comparison-of-quantities, and the practical and conceptual role of qukaq [center] and ayagneq [a place to begin]. They applied fundamental mathematical principles associated with symmetry and measuring in their everyday activities and in making artifacts. Inspired by their example, this paper explores the question: Could symmetry …
From Solvability To Formal Decidability: Revisiting Hilbert’S “Non-Ignorabimus”, Andrea Reichenberger
From Solvability To Formal Decidability: Revisiting Hilbert’S “Non-Ignorabimus”, Andrea Reichenberger
Journal of Humanistic Mathematics
The topic of this article is Hilbert’s axiom of solvability, that is, his conviction of the solvability of every mathematical problem by means of a finite number of operations. The question of solvability is commonly identified with the decision problem. Given this identification, there is not the slightest doubt that Hilbert’s conviction was falsified by Gödel’s proof and by the negative results for the decision problem. On the other hand, Gödel’s theorems do offer a solution, albeit a negative one, in the form of an impossibility proof. In this sense, Hilbert’s optimism may still be justified. Here I argue that …
Recapture, Transparency, Negation And A Logic For The Catuṣkoṭi, Adrian Kreutz
Recapture, Transparency, Negation And A Logic For The Catuṣkoṭi, Adrian Kreutz
Comparative Philosophy
The recent literature on Nāgārjuna’s catuṣkoṭi centres around Jay Garfield’s (2009) and Graham Priest’s (2010) interpretation. It is an open discussion to what extent their interpretation is an adequate model of the logic for the catuskoti, and the Mūla-madhyamaka-kārikā. Priest and Garfield try to make sense of the contradictions within the catuskoti by appeal to a series of lattices – orderings of truth-values, supposed to model the path to enlightenment. They use Anderson & Belnaps's (1975) framework of First Degree Entailment. Cotnoir (2015) has argued that the lattices of Priest and Garfield …
Call Thee Ishmael, Mark Backus
Call Thee Ishmael, Mark Backus
Sophia and Philosophia
“Moby-Dick is a strangely compelling book.”[1] Scholarship and commentary help the reader understand why Ishmael’s tale is so compelling, but not always why it is strangely so. The perennial search for a master key to unlock the strangeness of Moby-Dick beneath its infinite layers has added more mesmerizing layers, but if many of the proposed keys fit into the lock of Moby-Dick, why is there yet a sense that none have completely opened “the great flood-gates?” (Moby-Dick 22, hereafter “MD”). Is it because none of them are right, or that they are only partly right, or that …
Dialetheism, Paradox, And Nāgārjuna’S Way Of Thinking, Richard H. Jones
Dialetheism, Paradox, And Nāgārjuna’S Way Of Thinking, Richard H. Jones
Comparative Philosophy
Nāgārjuna’s doctrine of emptiness, his ideas on “two truths” and language, and his general method of arguing are presented clearly by him and can be stated without paradox. That the dialetheists today can restate his beliefs in paradoxical ways does not mean that Nāgārjuna argued that way; in fact, their restatements misrepresent and undercut his arguments.
Platonic Agonism: A Dialogical Addendum To Plato’S Sophist, Bennett Foster
Platonic Agonism: A Dialogical Addendum To Plato’S Sophist, Bennett Foster
Sophia and Philosophia
The following addendum to Plato’s Sophist was fabricated as a kind of experimental answer to a specific contextual question: What is the relation of Plato’s conception of philosophy to the practice of the agōn in Ancient Greece? For the “contest-system,”[1] to adopt Gouldner's phrase, has long been recognized as one of the salient features of Greek culture in the centuries leading up to Plato’s time.[2] Yet in the dialogues Plato never gives an explicit critique of the agōn the way he does other cultural phenomena, such as politics, poetry, rhetoric, education, etc. Many scholars have therefore concluded that Plato is …