From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, 2017 BarIlan University
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 epsilondelta ones.
The Proscriptive Principle And Logics Of Analytic Implication, 2017 The Graduate Center, City University of New York
The Proscriptive Principle And Logics Of Analytic Implication, Thomas M. Ferguson
All Graduate Works by Year: Dissertations, Theses, and Capstone Projects
The analogy between inference and mereological containment goes at least back to Aristotle, whose discussion in the Prior Analytics motivates the validity of the syllogism by way of talk of parts and wholes. On this picture, the application of syllogistic is merely the analysis of concepts, a term that presupposes—through the root ἀνά + λύω —a mereological background.
In the 1930s, such considerations led William T. Parry to attempt to codify this notion of logical containment in his system of analytic implication AI. Parry’s original system AI was later expanded to the system PAI. The hallmark of Parry’s ...
Complexity Reduction In StateBased Modeling, 2016 Portland State University
Complexity Reduction In StateBased Modeling, Martin Zwick
Martin Zwick
For a system described by a relation among qualitative variables (or quantitative variables "binned" into symbolic states), expressed either settheoretically or as a multivariate joint probability distribution, complexity reduction (compression of representation) is normally achieved by modeling the system with projections of the overall relation. To illustrate, if ABCD is a four variable relation, then models ABC:BCD or AB:BC:CD:DA, specified by two triadic or four dyadic relations, respectively, represent simplifications of the ABCD relation. Simplifications which are lossless are always preferred over the original full relation, while simplifications which lose constraint are still preferred if the ...
Enhancements To Crisp Possibilistic Reconstructability Analysis, 2016 Portland State University
Enhancements To Crisp Possibilistic Reconstructability Analysis, Anas AlRabadi, Martin Zwick
Martin Zwick
Modified Reconstructibility Analysis (MRA), a novel decomposition within the framework of settheoretic (crisp possibilistic) Reconstructibility Analysis, is presented. It is shown that in some cases while 3variable NPNclassified Boolean functions are not decomposable using Conventional Reconstructibility Analysis (CRA), they are decomposable using Modified Reconstructibility Analysis (MRA). Also, it is shown that whenever a decomposition of 3variable NPNclassified Boolean functions exists in both MRA and CRA, MRA yields simpler or equal complexity decompositions. A comparison of the corresponding complexities for AshenhurstCurtis decompositions, and Modified Reconstructibility Analysis (MRA) is also presented. While both AC and MRA decompose some but not all NPNclasses ...
Prestructuring Neural Networks Via Extended Dependency Analysis With Application To Pattern Classification, 2016 Portland State University
Prestructuring Neural Networks Via Extended Dependency Analysis With Application To Pattern Classification, George G. Lendaris, Thaddeus T. Shannon, Martin Zwick
Martin Zwick
We consider the problem of matching domainspecific statistical structure to neuralnetwork (NN) architecture. In past work we have considered this problem in the function approximation context; here we consider the pattern classification context. General Systems Methodology tools for finding problemdomain structure suffer exponential scaling of computation with respect to the number of variables considered. Therefore we introduce the use of Extended Dependency Analysis (EDA), which scales only polynomially in the number of variables, for the desired analysis. Based on EDA, we demonstrate a number of NN prestructuring techniques applicable for building neural classifiers. An example is provided in which EDA ...
Complexity And Decomposability Of Relations, 2016 Portland State University
Complexity And Decomposability Of Relations, Martin Zwick
Martin Zwick
A discrete multivariate relation, defined settheoretically, is a subset of a cartesian product of sets which specify the possible values of a number of variables. Where three or more variables are involved, the highest order relation, namely the relation between all the variables, may or may not be decomposable without loss into sets of lower order relations which involve subsets of the variables. In a completely parallel manner, the highest order relation defined informationtheoretically, namely the joint probability distribution involving all the variables, may or may not be decomposed without loss into lowerorder distributions involving subsets of the variables. Decomposability ...
Exploring Mathematical Strategies For Finding Hidden Features In MultiDimensional Big Datasets, 2016 University of Houston
Exploring Mathematical Strategies For Finding Hidden Features In MultiDimensional Big Datasets, Tri Duong, Fang Ren, Apurva Mehta
STAR (STEM Teacher and Researcher) Presentations
With advances in technology in brighter sources and larger and faster detectors, the amount of data generated at national user facilities such as SLAC is increasing exponentially. Humans have a superb ability to recognize patterns in complex and noisy data and therefore, data is still curated and analyzed by humans. However, a human brain is unable to keep up with the accelerated pace of data generation, and as a consequence, the rate of new discoveries hasn't kept pace with the rate of data creation. Therefore, new procedures to quickly assess and analyze the data are needed. Machine learning approaches ...
Mathematical Practice And Human Cognition, 2016 Indiana University  Purdue University Fort Wayne
Mathematical Practice And Human Cognition, Bernd Buldt
Philosophy Faculty Presentations
Frank Quinn (of JaffeQuinn fame, see [1]) worked out the basics of his own account of mathematical practice, an account that is informed by an analysis of contemporary mathematics and its pedagogy (see [2]). Taking this account as our starting point, we can characterize the current mathematical practice to acquire and work with new concepts as a cognitive adaptation strategy that, first, emerged to meet the challenges posed by the growing abstractness of its objects and which, second, proceeds according to the following threepronged approach:

(i) sever as many ties to ordinary language as possible and limit ordinary language explanations ...
On Fixed Points, Diagonalization, And SelfReference, 2016 Indiana University  Purdue University Fort Wayne
On Fixed Points, Diagonalization, And SelfReference, Bernd Buldt
Philosophy Faculty Presentations
We clarify the respective role fixed points, diagonalization, and self reference play in proofs of G ̈odel’s first incompleteness theorem. We first show that the usual fixedpoint construction can be reduced to a double diagonalization; this is done to address widely held views such as that fixedpoint are “paradoxical” (Priest), or work by “black magic” (Soare), or that their construction is “intuitively unclear” (Kotlarski). We then discuss three notions of selfreference; this can be seen an extension of a recent study by Halbach and Visser and is meant to show that we do not (yet?) have a robust theory ...
Mathematical Practice And Human Cognition, 2016 Indiana University  Purdue University Fort Wayne
Mathematical Practice And Human Cognition, Bernd Buldt
Philosophy Faculty Presentations
Frank Quinn (of JaffeQuinn fame, see [1]) worked out the basics of his own account of mathematical practice, an account that is informed by an analysis of contemporary mathematics and its pedagogy (see [2]). Taking this account as our starting point, we can characterize the current mathematical practice to acquire and work with new concepts as a cognitive adaptation strategy that, first, emerged to meet the challenges posed by the growing abstractness of its objects and which, second, proceeds according to the following threepronged approach:

(i) sever as many ties to ordinary language as possible and limit ordinary language explanations ...
Sentential Logic, 2016 CSU, San Bernardino
Sentential Logic, Tony Roy
Excerpted from the longer Roy, Symbolic Logic, including chapter 1 and just the first parts of chapters 2  7.
From the preface: There is, I think, a gap between what many students learn in their first course in formal logic, and what they are expected to know for their second. While courses in mathematical logic with metalogical components often cast only the barest glance at mathematical induction or even the very idea of reasoning from definitions, a first course may also leave these untreated, and fail explicitly to lay down the definitions upon which the second course is based. The ...
Symbolic Logic, 2016 CSU, San Bernardino
Symbolic Logic, Tony Roy
From the preface: There is, I think, a gap between what many students learn in their first course in formal logic, and what they are expected to know for their second. While courses in mathematical logic with metalogical components often cast only the barest glance at mathematical induction or even the very idea of reasoning from definitions, a first course may also leave these untreated, and fail explicitly to lay down the definitions upon which the second course is based. The aim of this text is to integrate material from these courses and, in particular, to make serious mathematical logic ...
On Fixed Points, Diagonalization, And SelfReference, 2016 Indiana University  Purdue University Fort Wayne
On Fixed Points, Diagonalization, And SelfReference, Bernd Buldt
Philosophy Faculty Publications
We clarify the respective roles fixed points, diagonalization, and self reference play in proofs of Gödel’s first incompleteness theorem.
The Logic Of Uncertain Justifications, 2016 Kenyon College
The Logic Of Uncertain Justifications, Robert Milnikel
Robert Milnikel
No abstract provided.
The Logic Of Uncertain Justifications, 2016 Kenyon College
The Logic Of Uncertain Justifications, Robert Milnikel
Robert Milnikel
No abstract provided.
Conservativity For Logics Of Justified Belief: Two Approaches, 2016 Kenyon College
Conservativity For Logics Of Justified Belief: Two Approaches, Robert Milnikel
Robert Milnikel
No abstract provided.
Exploring Argumentation, Objectivity, And Bias: The Case Of Mathematical Infinity, 2016 University of Ontario Institute of Technology
Exploring Argumentation, Objectivity, And Bias: The Case Of Mathematical Infinity, Ami Mamolo
OSSA Conference Archive
This paper presents an overview of several years of my research into individuals’ reasoning, argumentation, and bias when addressing problems, scenarios, and symbols related to mathematical infinity. There is a long history of debate around what constitutes “objective truth” in the realm of mathematical infinity, dating back to ancient Greece (e.g., Dubinsky et al., 2005). Modes of argumentation, hindrances, and intuitions have been largely consistent over the years and across levels of expertise (e.g., Brown et al., 2010; Fischbein et al., 1979, Tsamir, 1999). This presentation examines the interrelated complexities of notions of objectivity, bias, and argumentation as ...
On The Conjugacy Problem For Automorphisms Of Trees, 2016 Boise State University
On The Conjugacy Problem For Automorphisms Of Trees, Kyle Douglas Beserra
Boise State University Theses and Dissertations
In this thesis we identify the complexity of the conjugacy problem of automorphisms of regular trees. We expand on the results of Kechris, Louveau, and Friedman on the complexities of the isomorphism problem of classes of countable trees. We see in nearly all cases that the complexity of isomorphism of subtrees of a given regular countable tree is the same as the complexity of conjugacy of automorphisms of the same tree, though we present an example for which this does not hold.
On A MultipleChoice Guessing Game, 2016 Bethel College  Mishawaka
On A MultipleChoice Guessing Game, Ryan Cushman, Adam J. Hammett
The Research and Scholarship Symposium
We consider the following game (a generalization of a binary version explored by Hammett and Oman): the first player (“Ann”) chooses a (uniformly) random integer from the first n positive integers, which is not revealed to the second player (“Gus”). Then, Gus presents Ann with a koption multiple choice question concerning the number she chose, to which Ann truthfully replies. After a predetermined number m of these questions have been asked, Gus attempts to guess the number chosen by Ann. Gus wins if he guesses Ann’s number. Our goal is to determine every mquestion algorithm which maximizes the probability ...
Epistemic Considerations On ExtensiveForm Games, 2016 Graduate Center, City University of New York
Epistemic Considerations On ExtensiveForm Games, Cagil Tasdemir
All Graduate Works by Year: Dissertations, Theses, and Capstone Projects
In this thesis, we study several topics in extensiveform games. First, we consider perfect information games with belief revision with players who are tolerant of each other’s hypothetical errors. We bound the number of hypothetical nonrational moves of a player that will be tolerated by other players without revising the belief on that player’s rationality on future moves, and investigate which games yield the backward induction solution.
Second, we consider players who have no way of assigning probabilities to various possible outcomes, and deﬁne players as conservative, moderate and aggressive depending on the way they choose, and show ...