Logic and Foundations of Mathematics Commons^™

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, …

Magicicada And Mathematics: What The Parochiality Of Our Explanations Means For The Indispensability Argument, Jadyn Malone

Philosophy Senior Theses

This thesis examines whether the indispensability argument justifies belief in the existence of mathematical entities. I adopt ontological naturalism, arguing that we should believe in entities that feature in our best scientific theories, which are determined through inference to the best explanation. The indispensability argument contends that because mathematical entities are explanatorily indispensable to our best scientific theories – as demonstrated by examples like the prime number explanation of cicada life cycles – we should believe they exist just as we believe in electrons and stars.

However, I identify a challenge: our scientific explanations are inherently parochial, shaped by human …

On The Challenges Posed By Non-Machian Solutions To Relationalism In General Relativity, Zachary Matthew Zito

Undergraduate Honors Capstone Projects

The relationship between Mach’s Principle and solutions to Einstein’s field equations is examined, with special attention to Friedmann-Lematre-Robertson-Walker (FLRW) cosmology. Mach’s Principle is outlined, and the extent to which general relativity fulfills Machs vision is assessed. It is argued that several important solutions to Einsteins equations, particularly the FLRW cosmological models, embody key Machian features. In order to elucidate the FLRW cosmological model, the associated energy-momentum tensor for a perfect fluid under the assumptions of large-scale homogeneity and isotropy is derived and used to obtain the Friedmann equations that govern cosmic expansion. It is shown that FLRW cosmology can be …

01. Harry Stottlemeier's Discovery (Novel - Ebook), Matthew Lipman

Middle School Curriculum

One day Harry finds himself giving the wrong answer in science class and begins to wonder where he has gone wrong. This reflection soon involves his classmates, who begin to think together about the nature of thinking, inquiry and knowledge. With the help of their teacher, Harry and his classmates discover rules of formal and informal logic, relational logic and hypothetical thinking—-not as ends in themselves, but as tools in helping them understand themselves and their world. Some of the ideas they begin to explore this way include education, mind, rights, religion, art, cause and effect, causes and reasons, and …

Modeling Deductive Inference: A Historico-Philosophical Introduction To First-Order Logic, David J. Buller

Faculty Books & Book Chapters

This book is a companion text for lectures on first-order logic and its elementary metatheory (used in Intermediate Logic at Northern Illinois University). It covers the basic concepts of set theory necessary for a mathematical development of first-order logic; develops a formal language of first-order logic; presents a classical Tarskian semantics for the language and the “semantic” conception of logical consequence; presents a Gentzenian proof system and the “syntactic” conception of logical consequence; develops a partial decision procedure for logical consequence in the language; demonstrates applications of the formal system to modeling deductive inference expressed in natural language; and extends …

Unlocking The Power Of Logic: Thinking Smarter Every Day A Beginner's Guide To Logical Thinking, Jesus Diaz

A with Honors Projects

Power Point presented to an 11th and 12th grade high school audience, introducing them to the concept of logic.

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, …

The Grim Reaper Argument Is An Argument For No One, Daniel Linford, Alex Malpass

Philosophy Faculty Publications

According to a prominent philosophical argument for the beginning of time – the Grim Reaper Argument (GRA) – had it been possible for the past to be infinite, then a provably impossible scenario could have been constructed. Such a scenario is not possible, so, the GRA concludes, the past cannot be infinite. Here, we show that the GRA includes one premise acceptable only for Humeans and another premise acceptable only for anti-Humeans. Since, plausibly, everyone is either a Humean or an anti-Humean, the GRA is an argument for no one. Additionally, we argue that there may not be a way …

Polynomial Invariants For Classical Invariant Groups, John Diller

UNF Graduate Theses and Dissertations

The classical invariant theory of matrix groups plays a fundamental role in modern algebra and geometry, with origins in Hilbert’s Fourteenth Problem concerning the finite generation of rings of invariants. In this thesis, we study polynomial functions on spaces of matrices and investigate their invariants under natural actions of classical groups. After recalling the geometric and algebraic motivations behind Hilbert’s question, we focus on invariants for several complex Lie groups, including the general linear group GL(V), the symplectic group Sp(V), the orthogonal group O(V), and the special orthogonal group SO(V). Building on foundational work of Hermann Weyl and a modern …