Logic and Foundations of Mathematics Commons^™

Frege's Constraint And The Nature Of Frege's Foundational Program, Marco Panza, Andrea Sereni

Philosophy Faculty Articles and Research

Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ( ) or ‘Frege Constraint’ ( ), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how generalizes Frege’s views while comes closer to his original conceptions. Different authors diverge on the interpretation of and on whether it applies to definitions of both natural and real numbers. Our aim is to trace …

Asymptotic Quasi-Completeness And Zfc, Mirna Džamonja, Marco Panza

MPP Published Research

The axioms ZFC of first order set theory are one of the best and most widely accepted, if not perfect, foundations used in mathematics. Just as the axioms of first order Peano Arithmetic, ZFC axioms form a recursively enumerable list of axioms, and are, then, subject to Gödel’s Incompleteness Theorems. Hence, if they are assumed to be consistent, they are necessarily incomplete. This can be witnessed by various concrete statements, including the celebrated Continuum Hypothesis CH. The independence results about the infinite cardinals are so abundant that it often appears that ZFC can basically prove very little about such cardinals. …

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 …

Was Frege A Logicist For Arithmetic?, Marco Panza

MPP Published Research

The paper argues that Frege’s primary foundational purpose concerning arithmetic was neither that of making natural numbers logical objects, nor that of making arithmetic a part of logic, but rather that of assigning to it an appropriate place in the architectonics of mathematics and knowledge, by immersing it in a theory of numbers of concepts and making truths about natural numbers, and/or knowledge of them transparent to reason without the medium of senses and intuition.

Enthymemathical Proofs And Canonical Proofs In Euclid’S Plane Geometry, Abel Lassalle, Marco Panza

MPP Published Research

Since the application of Postulate I.2 in Euclid’s Elements is not uniform, one could wonder in what way should it be applied in Euclid’s plane geometry. Besides legitimizing questions like this from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.

The Logical Fallacies In Political Discourse, Zilin Cidre Zhou

Summer Research Program

I examined the use of logical fallacies in political discourse. Logical fallacies are fraudulent tricks people use in their argument to make it sound more credible while what they really do is to fool the audience. Out of more than 300 kinds of fallacies, I focused on 18 common ones by analyzing their use in debates about political issues. During conducting my research, I noted that being aware of my mental state is very important if I want to accurately detect the fallacies. Furthermore, while watching two sides debating, being impartial is as significant as staying calm. I also need …

Zero Textbook Cost Syllabus For Phi 1600 (Logic And Moral Reasoning), Andreea Prichea

Open Educational Resources

The goal of the course is for students to familiarize themselves with the structure of an argument. Identify and iterate the difference between inductive and deductive, valid and invalid arguments. Learn methods to determine if an argument is valid or invalid, and be able to identify faulty arguments based on the argument's structure. The core of the course will focus on deductive arguments as they relate to moral arguments. . The student throughout the course will be exposed to arguments in natural language, and try to analyze them through the methods learned.

Zero Textbook Cost Syllabus For Phi 1600 (Logic And Moral Reasoning), Alexander Steers-Mccrum

Open Educational Resources

The goal of this class is to familiarize students with formal and informal logic. Logic illustrates and explores the connections between ideas. It can help us evaluate our beliefs and make and understand arguments. Aside from its use in philosophy, logic is of particular importance in mathematics and law, and is foundational for computer science.

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.

Radical Social Ecology As Deep Pragmatism: A Call To The Abolition Of Systemic Dissonance And The Minimization Of Entropic Chaos, Arielle Brender

Student Theses 2015-Present

This paper aims to shed light on the dissonance caused by the superimposition of Dominant Human Systems on Natural Systems. I highlight the synthetic nature of Dominant Human Systems as egoic and linguistic phenomenon manufactured by a mere portion of the human population, which renders them inherently oppressive unto peoples and landscapes whose wisdom were barred from the design process. In pursuing a radical pragmatic approach to mending the simultaneous oppression and destruction of the human being and the earth, I highlight the necessity of minimizing entropic chaos caused by excess energy expenditure, an essential feature of systems that aim …

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 …

Review Of G. Israel, Meccanicismo. Trionfi E Miserie Della Visione Meccanica Del Mondo, Marco Panza

MPP Published Research

"This is Giorgio's Israel last book, which appeared only a few weeks after his untimely death, in September 2015. For many reasons, it can be considered as his intellectual legacy, since it comes back, in a new and organic way, to many of the research topics to which he devoted his life and his many publications, which include several papers in Historia Mathematica. One of these papers, co-authored with M. Menghini, appeared in vol. 25/4, 1998 and was devoted to Poincaré's and Enriques's opposite views on qualitative analysis, which is a theme also dealt with in this book (pp. 117–122)."

Classical Logic, Stewart Shapiro, Teresa Kouri Kissel

Philosophy Faculty Publications

[From introductory section]

Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language has components that correspond to a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record arguments that are valid for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions for at least part of the language.

The following sections provide the basics of a typical logic, sometimes called “classical elementary logic” or “classical first-order logic”....

Zero Textbook Cost Syllabus For Phi 1600 (Logic And Moral Reasoning), Jesse Rappaport

Open Educational Resources

This course examines the principles of clear and accurate thought, including sound and valid arguments and methods of scientific reasoning in moral and political argument.

Fictionalism, Semantics, And Ontology, Gordon Purves

Philosophy, Theology and Religious Studies Faculty Publications

This article expands upon the argument of a previous work which defended a variational account of scientific fictions. Specifically, I show that this understanding of scientific fictions can provide guidance for realist interpretations of scientific theories and models. Depending on a model's variational properties, different ontological commitments are appropriate, providing a principled way for a realist to moderate her views according to the structural properties of a given model. This reasoning is then applied the Lee-Yang theory and Kubo-Martin-Schwinger statistics, two foundational models in quantum statistical mechanics. The Lee-Yang theory is analyzed in a way that permits a robust realist …

Zero Textbook Cost Syllabus For Phi 1600 (Logic And Moral Reasoning), Gerrit Jan Kamperdyk

Open Educational Resources

This course examines the principles of clear and accurate thought, including sound and valid arguments and methods of scientific reasoning in moral and political argument.

On Defining 'Argument', Jeffrey C. Goodman

Faculty Scholarship

There is no concept more central to logic and critical thinking than the concept of an argument.  I here address extant definitions of ‘argument’ in the logical sense of the term and defend the claim that all are wanting: they are all extensionally inadequate, or at least problematically ambiguous.  An adequate definition of ‘argument’ will entail either an absolutist understanding or a relativistic one.  Current proposals, however, especially those typically found in logic and critical thinking textbooks, all seem to be objectionable no matter which sort of understanding one adopts.  I thus finish with a proposal for how to define …

Against The Intentional Definition Of Argument, G. C. Goddu

Philosophy Faculty Publications

Intentional definitions of argument, i.e. the conclusion being intended to follow from the premises, abound. Yet, there are numerous problem cases in which we appear to have arguments, but no intention. One way to try to avoid these problem cases is to appeal to acts, in which case one has to give up on the repeatability of arguments. One can keep repeatability and intentions if one resorts to act types, but then it appears that the problem cases re-emerge.

Seeing And Understanding Data: A Mini-Primary Source Project For Students Of Statistics, Charlotte Bolch, Beverly Wood

Publications

Mathematicians and scientists included drawings in their work before the first statistical graphs were invented. However, their illustrations were meant to depict quantifiable relationships rather than exposing statistical variability. The uses of pictorial representations or charts to convey trends among variable measurements dates back to at least the late 10th century. However, this means of communication did not gain the widespread appreciation it enjoys today until technological advances in the 21st century stimulated the growth of a new field, data visualization. The mini-Primary Source Project (PSP) Seeing and Understanding Data provides students the opportunity to explore the evolution of statistical …