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Articles 1 - 30 of 31
Full-Text Articles in Logic and Foundations
Opening The Lantern: The Leiden Declaration, Mark Huber
Opening The Lantern: The Leiden Declaration, Mark Huber
Journal of Humanistic Mathematics
The Leiden Declaration on Artificial Intelligence and Mathematics provides insight into how mathematicians view their discipline. However, when positioning mathematics against AI, the declaration falls short in recognizing recent advances. This column introduces the reader who might be unfamiliar with these new methods through the formal language Lean and discusses how these new abilities might change the way journals operate.
Riding The Rails Of Reason: A Dialogue On Truth, Logic, And Proof, Surinder Pal Singh Kainth
Riding The Rails Of Reason: A Dialogue On Truth, Logic, And Proof, Surinder Pal Singh Kainth
Journal of Humanistic Mathematics
On a quiet train ride, I found myself in conversation with Noor, an inquisitive teenager with sharp questions about truth, logic, and mathematical proof. As we talked, I used the train itself as a metaphor to explain how mathematical proofs provide certainty, far beyond what repetitive verification alone can offer. Our discussion ranged from common misconceptions about the foundations of logic to the need for clear definitions and axioms. We also touched on fundamental ideas such as the challenges posed by the Axiom of Choice and the limitations revealed by Gödel’s incompleteness theorem.
A Nonstandard Exploration Of Approximate Identity And Unitization, Tong (Nicole) Wu
A Nonstandard Exploration Of Approximate Identity And Unitization, Tong (Nicole) Wu
HMC Senior Theses
The goal of this senior thesis is to explore general nonstandard analysis and some possible applications to 𝐶*-algebras in functional analysis. More specifically, we shall define an approximate identity of a 𝐶*-algebra using nonstandard analysis and study nonstandard hulls of internal 𝐶*-algebra in the context of different unitizations. We shall also prove a few results for ideals in 𝐶*-algebra using nonstandard definitions of approximate identities. We shall also briefly discuss the history and developments of nonstandard analysis.
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, …
Belonging, Lawrence M. Lesser
The Making Of Mathematics: An Interview With Carlo Cellucci, Marshall Gordon
The Making Of Mathematics: An Interview With Carlo Cellucci, Marshall Gordon
Journal of Humanistic Mathematics
Carlo Cellucci is professor emeritus of the Sapienza University of Rome, and has written books on logic, mathematics, and philosophy. In the following interview conducted via back-and-forth email correspondence, he discusses his 2022 book, The Making of Mathematics: Heuristic Philosophy of Mathematics [3], which presents a new paradigm for thinking about, doing, and teaching mathematics.
Mathematics And Determinism: Chaos, Quantum Mechanics, And The Limits Of Predictive Structure, Jackson T. Salumbides
Mathematics And Determinism: Chaos, Quantum Mechanics, And The Limits Of Predictive Structure, Jackson T. Salumbides
CMC Senior Theses
This thesis examines the relationship between mathematics and determinism by analyzing how chaos theory, quantum mechanics, and formal mathematical limits challenge traditional conceptions of predictability and causal structure. Chaos theory shows that deterministic systems can exhibit practical unpredictability due to sensitivity to initial conditions. Quantum mechanics introduces probabilistic outcomes that complicate deterministic interpretation, though alternative frameworks such as Bohmian mechanics and superdeterminism attempt to restore determinism at conceptual cost. Additionally, results from mathematical logic, including Gödel’s incompleteness theorems and Turing’s undecidability, demonstrate intrinsic limitations on what can be deduced or computed, even in fully deterministic systems. By synthesizing these areas, …
Discordium Mathematica - A Symphony In Aleph Minor, Vijay Fafat
Discordium Mathematica - A Symphony In Aleph Minor, Vijay Fafat
Journal of Humanistic Mathematics
How did Mathematics arise? Who created it? Why is it subject to Godel’s Incompleteness Theorems? And what does all this have to do with Coleridge’s poem, “Kubla Khan”, and “The Person from Porlock”? Here is a complete mythology of Mathematics set in an epic poetry format, fusing thoughts and verses from Western religions and Eastern mysticism… Those with immense patience and careful reading shall reap the fruit… (best read on a large screen or in printed form)
Self-Reference And Diagonalisation, Joël A. Doat
Self-Reference And Diagonalisation, Joël A. Doat
Journal of Humanistic Mathematics
This poem is an exercise on self-reference and diagonalisation in mathematics featuring Turing’s proof of the undecidability of the halting problem, Cantor’s cardinality argument, the Burali-Forti paradox, and Epimenides' liar paradox.
Using Bloom's Taxonomy For Math Outreach Within And Outside The Classroom, Manmohan Kaur
Using Bloom's Taxonomy For Math Outreach Within And Outside The Classroom, Manmohan Kaur
Journal of Humanistic Mathematics
Not everyone is a great artist, but we don’t often hear, “I dislike art.” Most people are able to appreciate visual arts, music and sports, without necessarily excelling in it themselves. On the other hand, the phrase “I dislike math” is widely prevalent. This is especially ironic in our current society, where mathematics affects our day-to-day activities in essential ways such as e-commerce and e-mail. This paper describes the opportunity to popularize mathematics by focusing on its fun and creative aspects, and illustrates this opportunity through a brief discussion of interdisciplinary topics that expose the beauty, elegance and value of …
A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo
A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo
HMC Senior Theses
Mac Lane’s Coherence Theorem is a subtle, foundational characterization of monoidal categories, a categorical concept which is now an important and popular tool in areas of pure mathematics and theoretical physics. Mac Lane’s original proof, while extremely clever, is written somewhat confusingly. Many years later, there still does not exist a fully complete and clearly written version of Mac Lane’s proof anywhere, which is unfortunate as Mac Lane’s proof provides very deep insight into the nature of monoidal categories. In this thesis, we provide brief introductions to category theory and monoidal categories, and we offer a precise, clear development of …
The Rubbish Researchers Puzzle, Michael W. Lucht
The Rubbish Researchers Puzzle, Michael W. Lucht
Journal of Humanistic Mathematics
The Rubbish Researchers Puzzle is a humorous short story about the Blue-Eyed Islanders Puzzle, cultural insensitivity in logic problems, and the quality of research.
What Makes A Theory Of Infinitesimals Useful? A View By Klein And Fraenkel, Vladimir Kanovei, Karin Katz, Mikhail Katz, Thomas Mormann
What Makes A Theory Of Infinitesimals Useful? A View By Klein And Fraenkel, Vladimir Kanovei, Karin Katz, Mikhail Katz, Thomas Mormann
Journal of Humanistic Mathematics
Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.
From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, Mikhail Katz, Luie Polev
From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, Mikhail Katz, Luie Polev
Journal of Humanistic Mathematics
In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting “the definition” to the students as a monolithic absolute. We hope that our experiences could be useful to other instructors wishing to follow this method of instruction. A poll run at the conclusion of the course indicates that students tend to favor infinitesimal definitions over epsilon-delta ones.
Sudoku Variants On The Torus, Kira A. Wyld
Sudoku Variants On The Torus, Kira A. Wyld
HMC Senior Theses
This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.
Four Years With Russell, Gödel, And Erdős: An Undergraduate's Reflection On His Mathematical Education, Michael H. Boggess
Four Years With Russell, Gödel, And Erdős: An Undergraduate's Reflection On His Mathematical Education, Michael H. Boggess
CMC Senior Theses
Senior Thesis at CMC is often described institutionally as the capstone of one’s undergraduate education. As such, I wanted my own to accurately capture and reflect how I’ve grown as a student and mathematician these past four years. What follows is my attempt to distill lessons I learned in mathematics outside the curriculum, written for incoming undergraduates and anyone with just a little bit of mathematical curiosity. In it, I attempt to dispel some common preconceptions about mathematics, namely that it’s uninteresting, formulaic, acultural, or completely objective, in favor of a dynamic historical and cultural perspective, with particular attention paid …
On The Occasion Of Your Graduation, Robert Dawson
On The Occasion Of Your Graduation, Robert Dawson
Journal of Humanistic Mathematics
A letter from an absent supervisor to a doctoral student about to graduate reveals a terrible secret.
Prove It!, Kenny W. Moran
Prove It!, Kenny W. Moran
Journal of Humanistic Mathematics
A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.
The Mathematical Landscape, Antonio Collazo
The Mathematical Landscape, Antonio Collazo
CMC Senior Theses
The intent of this paper is to present the reader will enough information to spark a curiosity in to the subject. By no means is the following a complete formulation of any of the topics covered. I want to give the reader a tour of the mathematical landscape. There are plenty of further details to explore in each section, I have just touched the tip the iceberg. The work is basically in four sections: Numbers, Geometry, Functions, Sets and Logic, which are the basic building blocks of Math. The first sections are a exposition into the mathematical objects and their …
Tesselland: A Mathematical Oddment, Martin Glover
Tesselland: A Mathematical Oddment, Martin Glover
Humanistic Mathematics Network Journal
No abstract provided.
Bridging To Infinity, Mike Pinter
Bridging To Infinity, Mike Pinter
Humanistic Mathematics Network Journal
The author's own experiences as a mathematics student and teacher have influenced how he thinks about the infinite. Author Madeleine L'Engle has also shaped his thinking with her writing. The author offers some thoughts that connect some of L'Engle's writing with his experience.
Mathematics, The Liberal Arts, And Slavish Devotions, J. D. Phillips
Mathematics, The Liberal Arts, And Slavish Devotions, J. D. Phillips
Humanistic Mathematics Network Journal
No abstract provided.
Does A Mathematical/Scientific Worldview Lead To A Clearer Or More Distorted View Of Reality?: Purposive Musings Inspired From Readings In The Urantia Book, The Cosmic Family, Volume I, And Elsewhere, Jeru
Humanistic Mathematics Network Journal
No abstract provided.
Notes On Formal Constructivism, D. Joyner, P. Lejarraga
Notes On Formal Constructivism, D. Joyner, P. Lejarraga
Humanistic Mathematics Network Journal
Our aim is to sketch some ideas related to how we (as in, we two) think we (as in, we humans) think. "That theory is useless. It isn't even wrong." - Wolfgang Pauli. Our hope in this paper is to provide a theory, admittedly somewhat vague, of how we think about mathematics. We also hope our ideas do not cause the reader to be reminded of Pauli's quote above. These notes were motivated by the interesting book by Changeaux and Connes.
Fivefolded Asymmetrical Hand: A Poetic Essay, S. Robert Wilson
Fivefolded Asymmetrical Hand: A Poetic Essay, S. Robert Wilson
Humanistic Mathematics Network Journal
No abstract provided.
Journal Review: Third International Anthology On Paradoxism, Joanne Growney
Journal Review: Third International Anthology On Paradoxism, Joanne Growney
Humanistic Mathematics Network Journal
No abstract provided.
What "Is" Mathematics?: In Memoriam Of Gian-Carlo Rota, Gian-Carlo Rota
What "Is" Mathematics?: In Memoriam Of Gian-Carlo Rota, Gian-Carlo Rota
Humanistic Mathematics Network Journal
No abstract provided.
Number, Infinity And Truth: Reflections On The Spiritual In Mathematics, James V. Rauff
Number, Infinity And Truth: Reflections On The Spiritual In Mathematics, James V. Rauff
Humanistic Mathematics Network Journal
No abstract provided.
A Reflection On The Word: Remembering The Word “Word” Is Reflexive, Paul Fjelstad, Ivan Ginchev
A Reflection On The Word: Remembering The Word “Word” Is Reflexive, Paul Fjelstad, Ivan Ginchev
Humanistic Mathematics Network Journal
No abstract provided.
An Informal History Of Classical Rhetoric For Mathematicians (Plato And Aristotle), Phillip Keith, Sandra Z. Keith
An Informal History Of Classical Rhetoric For Mathematicians (Plato And Aristotle), Phillip Keith, Sandra Z. Keith
Humanistic Mathematics Network Journal
No abstract provided.