18 Unconventional Essays on the Nature of Mathematics 18 Unconventional Essays on the Nature of Mathematics
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Jody Azzouni
ON “ON WHAT THERE IS”* BY JODY AZZOUNI Abstract: All sides in the recent debates over the Quine-Putnam Indispensability thesis presuppose Quine’s criterion for determining what a discourse is ontologically committed to. I subject the criterion to scrutiny, especially in regard to the available competitor-criteria, asking what means of evaluation there are for comparing alternative criteria against each other. Finding none, the paper concludes that ontological questions, in a certain sense, are philosophically indeterminate. (What is C. trying to pull?) marginalia on Quine’s copy of a letter from Carnap 1. A lot of philosophy of mathematics is motivated by considerations arising from what has come to be called the Quine-Putnam indispensability thesis;1 the claim, roughly, that if one’s best scientific (physical) theory requires existential quantification over certain entities, then one is onto- logically committed to such entities.2 Many books in this area, such as Chihara (1990), Field (1980), Hellman (1989), and Maddy (1990), draw their philosophical raison d’être from the view that scientific theories commit us to the existence of mathematical objects this way. The indispensability thesis, it seems, drives philosophers to hard choices: rewrite one’s science, rewrite one’s mathematics, or regretfully embrace extravagant ontologies. It’s quite unsurprising, therefore, that such a seminal claim has once again come under intense scrutiny; and equally unsurprising, I guess, to find philosophers on both sides of the philosophical fence. Maddy strongly Pacific Philosophical Quarterly 79 (1998) 1–18 0031–5621/98/0100–0000 © 1998 University of Southern California and Blackwell Publishers Ltd. -
The Walk of Life Vol
THE WALK OF LIFE VOL. 03 EDITED BY AMIR A. ALIABADI The Walk of Life Biographical Essays in Science and Engineering Volume 3 Edited by Amir A. Aliabadi Authored by Zimeng Wan, Mamoon Syed, Yunxi Jin, Jamie Stone, Jacob Murphy, Greg Johnstone, Thomas Jackson, Michael MacGregor, Ketan Suresh, Taylr Cawte, Rebecca Beutel, Jocob Van Wassenaer, Ryan Fox, Nikolaos Veriotes, Matthew Tam, Victor Huong, Hashim Al-Hashmi, Sean Usher, Daquan Bar- row, Luc Carney, Kyle Friesen, Victoria Golebiowski, Jeffrey Horbatuk, Alex Nauta, Jacob Karl, Brett Clarke, Maria Bovtenko, Margaret Jasek, Allissa Bartlett, Morgen Menig-McDonald, Kate- lyn Sysiuk, Shauna Armstrong, Laura Bender, Hannah May, Elli Shanen, Alana Valle, Charlotte Stoesser, Jasmine Biasi, Keegan Cleghorn, Zofia Holland, Stephan Iskander, Michael Baldaro, Rosalee Calogero, Ye Eun Chai, and Samuel Descrochers 2018 c 2018 Amir A. Aliabadi Publications All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. ISBN: 978-1-7751916-1-2 Atmospheric Innovations Research (AIR) Laboratory, Envi- ronmental Engineering, School of Engineering, RICH 2515, University of Guelph, Guelph, ON N1G 2W1, Canada It should be distinctly understood that this [quantum mechanics] cannot be a deduction in the mathematical sense of the word, since the equations to be obtained form themselves the postulates of the theory. Although they are made highly plausible by the following considerations, their ultimate justification lies in the agreement of their predictions with experiment. —Werner Heisenberg Dedication Jahangir Ansari Preface The essays in this volume result from the Fall 2018 offering of the course Control of Atmospheric Particulates (ENGG*4810) in the Environmental Engineering Program, University of Guelph, Canada. -
Once Upon a Party - an Anecdotal Investigation
Journal of Humanistic Mathematics Volume 11 | Issue 1 January 2021 Once Upon A Party - An Anecdotal Investigation Vijay Fafat Ariana Investment Management Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Arts and Humanities Commons, Other Mathematics Commons, and the Other Physics Commons Recommended Citation Fafat, V. "Once Upon A Party - An Anecdotal Investigation," Journal of Humanistic Mathematics, Volume 11 Issue 1 (January 2021), pages 485-492. DOI: 10.5642/jhummath.202101.30 . Available at: https://scholarship.claremont.edu/jhm/vol11/iss1/30 ©2021 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. The Party Problem: An Anecdotal Investigation Vijay Fafat SINGAPORE [email protected] Synopsis Mathematicians and physicists attending let-your-hair-down parties behave exactly like their own theories. They live by their theorems, they jive by their theorems. Life imi- tates their craft, and we must simply observe the deep truths hiding in their party-going behavior. Obligatory Preface It is a well-known maxim that revelries involving academics, intellectuals, and thought leaders are notoriously twisted affairs, a topologist's delight. -
Letters to the Editor
AI Magazine Volume 12 Number 3 (1991) (© AAAI) Letters q Editor: system-centered research. First, the the other does. No theorist is going Thank you for the opportunity to split between the neats and scruffies to spend his or her time attempting respond to the letters by Jim Hendler, is old and institutionalized, as to bring precision to a mess of hacks, James Herbsleb and Mike Wellman Hendler points out. Few researchers kludges and “knowledge,” and no regarding my survey of the Eighth are trained in both camps. Second, system builder is apt to find the National Conference on Artificial the pathologies that researchers in attempt informative. MAD does not Intelligence (AI Magazine, Volume 12, AAAI-90 themselves attributed to mean business as usual with occa- No. 1). The letters raise many inter- focusing on just one aspect did, in sional collaborative meetings. esting points that can be roughly fact, arise. Third, the expected advan- hfAn means assessing environmen- classified as questioning the validity tages of merging systems and models tal factors that affect behavior; of the survey and questioning the did, in fact, materialize (although the modelling the causal relationships proposed MAD methodology. Mike sample was very small). Wellman between a system’s design, its envi- Wellman says, “As Cohen acknowl- says that just because I didn’t see ronment, and its behavior; designing edges, a serious problem with the system-centered and model-centered or redesigning a system (or part of a survey is that the AAAI conference research reported together, doesn’t system); predicting how the system proceedings do not accurately repre- mean it wasn’t there; I say that I will behave; running experiments to sent the field.” Actually, I did not found pathoIogies (e.g., see my sec- test the predictions; explaining unex- acknowledge that AAAI is not repre- tions Models Without Systems and pected results and modifying modeis sentative; I just raised the possibility. -
All That Math Portraits of Mathematicians As Young Researchers
Downloaded from orbit.dtu.dk on: Oct 06, 2021 All that Math Portraits of mathematicians as young researchers Hansen, Vagn Lundsgaard Published in: EMS Newsletter Publication date: 2012 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Hansen, V. L. (2012). All that Math: Portraits of mathematicians as young researchers. EMS Newsletter, (85), 61-62. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. -
Stephen Cole Kleene
STEPHEN COLE KLEENE American mathematician and logician Stephen Cole Kleene (January 5, 1909 – January 25, 1994) helped lay the foundations for modern computer science. In the 1930s, he and Alonzo Church, developed the lambda calculus, considered to be the mathematical basis for programming language. It was an effort to express all computable functions in a language with a simple syntax and few grammatical restrictions. Kleene was born in Hartford, Connecticut, but his real home was at his paternal grandfather’s farm in Maine. After graduating summa cum laude from Amherst (1930), Kleene went to Princeton University where he received his PhD in 1934. His thesis, A Theory of Positive Integers in Formal Logic, was supervised by Church. Kleene joined the faculty of the University of Wisconsin in 1935. During the next several years he spent time at the Institute for Advanced Study, taught at Amherst College and served in the U.S. Navy, attaining the rank of Lieutenant Commander. In 1946, he returned to Wisconsin, where he stayed until his retirement in 1979. In 1964 he was named the Cyrus C. MacDuffee professor of mathematics. Kleene was one of the pioneers of 20th century mathematics. His research was devoted to the theory of algorithms and recursive functions (that is, functions defined in a finite sequence of combinatorial steps). Together with Church, Kurt Gödel, Alan Turing, and others, Kleene developed the branch of mathematical logic known as recursion theory, which helped provide the foundations of theoretical computer science. The Kleene Star, Kleene recursion theorem and the Ascending Kleene Chain are named for him. -
Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems*
Electronic Colloquium on Computational Complexity, Report No. 75 (2019) Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems* Lijie Chen Dylan M. McKay Cody D. Murray† R. Ryan Williams MIT MIT MIT Abstract A frontier open problem in circuit complexity is to prove PNP 6⊂ SIZE[nk] for all k; this is a neces- NP sary intermediate step towards NP 6⊂ P=poly. Previously, for several classes containing P , including NP NP NP , ZPP , and S2P, such lower bounds have been proved via Karp-Lipton-style Theorems: to prove k C 6⊂ SIZE[n ] for all k, we show that C ⊂ P=poly implies a “collapse” D = C for some larger class D, where we already know D 6⊂ SIZE[nk] for all k. It seems obvious that one could take a different approach to prove circuit lower bounds for PNP that does not require proving any Karp-Lipton-style theorems along the way. We show this intuition is wrong: (weak) Karp-Lipton-style theorems for PNP are equivalent to fixed-polynomial size circuit lower NP NP k NP bounds for P . That is, P 6⊂ SIZE[n ] for all k if and only if (NP ⊂ P=poly implies PH ⊂ i.o.-P=n ). Next, we present new consequences of the assumption NP ⊂ P=poly, towards proving similar re- sults for NP circuit lower bounds. We show that under the assumption, fixed-polynomial circuit lower bounds for NP, nondeterministic polynomial-time derandomizations, and various fixed-polynomial time simulations of NP are all equivalent. Applying this equivalence, we show that circuit lower bounds for NP imply better Karp-Lipton collapses. -
Academic Genealogy of the Oakland University Department Of
Basilios Bessarion Mystras 1436 Guarino da Verona Johannes Argyropoulos 1408 Università di Padova 1444 Academic Genealogy of the Oakland University Vittorino da Feltre Marsilio Ficino Cristoforo Landino Università di Padova 1416 Università di Firenze 1462 Theodoros Gazes Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo Angelo Poliziano Florens Florentius Radwyn Radewyns Geert Gerardus Magnus Groote Università di Mantova 1433 Università di Mantova Università di Firenze 1477 Constantinople 1433 DepartmentThe Mathematics Genealogy Project of is a serviceMathematics of North Dakota State University and and the American Statistics Mathematical Society. Demetrios Chalcocondyles http://www.mathgenealogy.org/ Heinrich von Langenstein Gaetano da Thiene Sigismondo Polcastro Leo Outers Moses Perez Scipione Fortiguerra Rudolf Agricola Thomas von Kempen à Kempis Jacob ben Jehiel Loans Accademia Romana 1452 Université de Paris 1363, 1375 Université Catholique de Louvain 1485 Università di Firenze 1493 Università degli Studi di Ferrara 1478 Mystras 1452 Jan Standonck Johann (Johannes Kapnion) Reuchlin Johannes von Gmunden Nicoletto Vernia Pietro Roccabonella Pelope Maarten (Martinus Dorpius) van Dorp Jean Tagault François Dubois Janus Lascaris Girolamo (Hieronymus Aleander) Aleandro Matthaeus Adrianus Alexander Hegius Johannes Stöffler Collège Sainte-Barbe 1474 Universität Basel 1477 Universität Wien 1406 Università di Padova Università di Padova Université Catholique de Louvain 1504, 1515 Université de Paris 1516 Università di Padova 1472 Università -
The Computational Complexity of Nash Equilibria in Concisely Represented Games∗
Electronic Colloquium on Computational Complexity, Report No. 52 (2005) The Computational Complexity of Nash Equilibria in Concisely Represented Games∗ Grant R. Schoenebeck y Salil P. Vadhanz May 4, 2005 Abstract Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payoffs are computed by a given boolean circuit, and graph games, where each agent's payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes. 1 Introduction In recent years, there has been a surge of interest at the interface between computer science and game theory. On one hand, game theory and its notions of equilibria provide a rich framework for modelling the behavior of selfish agents in the kinds of distributed or networked environments that often arise in computer science, and offer mechanisms to achieve efficient and desirable global outcomes in spite of the selfish behavior. -
Towards Non-Black-Box Lower Bounds in Cryptography
Towards Non-Black-Box Lower Bounds in Cryptography Rafael Pass?, Wei-Lung Dustin Tseng, and Muthuramakrishnan Venkitasubramaniam Cornell University, {rafael,wdtseng,vmuthu}@cs.cornell.edu Abstract. We consider average-case strengthenings of the traditional assumption that coNP is not contained in AM. Under these assumptions, we rule out generic and potentially non-black-box constructions of various cryptographic primitives (e.g., one-way permutations, collision-resistant hash-functions, constant-round statistically hiding commitments, and constant-round black-box zero-knowledge proofs for NP) from one-way functions, assuming the security reductions are black-box. 1 Introduction In the past four decades, many cryptographic tasks have been put under rigorous treatment in an eort to realize these tasks under minimal assumptions. In par- ticular, one-way functions are widely regarded as the most basic cryptographic primitive; their existence is implied by most other cryptographic tasks. Presently, one-way functions are known to imply schemes such as private-key encryp- tion [GM84,GGM86,HILL99], pseudo-random generators [HILL99], statistically- binding commitments [Nao91], statistically-hiding commitments [NOVY98,HR07] and zero-knowledge proofs [GMW91]. At the same time, some other tasks still have no known constructions based on one-way functions (e.g., key agreement schemes or collision-resistant hash functions). Following the seminal paper by Impagliazzo and Rudich [IR88], many works have addressed this phenomenon by demonstrating black-box separations, which rules out constructions of a cryptographic task using the underlying primitive as a black-box. For instance, Impagliazzo and Rudich rule out black-box con- structions of key-agreement protocols (and thus also trapdoor predicates) from one-way functions; Simon [Sim98] rules out black-box constructions of collision- resistant hash functions from one-way functions. -
Complex Numbers and Colors
Complex Numbers and Colors For the sixth year, “Complex Beauties” provides you with a look into the wonderful world of complex functions and the life and work of mathematicians who contributed to our understanding of this field. As always, we intend to reach a diverse audience: While most explanations require some mathemati- cal background on the part of the reader, we hope non-mathematicians will find our “phase portraits” exciting and will catch a glimpse of the richness and beauty of complex functions. We would particularly like to thank our guest authors: Jonathan Borwein and Armin Straub wrote on random walks and corresponding moment functions and Jorn¨ Steuding contributed two articles, one on polygamma functions and the second on almost periodic functions. The suggestion to present a Belyi function and the possibility for the numerical calculations came from Donald Marshall; the November title page would not have been possible without Hrothgar’s numerical solution of the Bla- sius equation. The construction of the phase portraits is based on the interpretation of complex numbers z as points in the Gaussian plane. The horizontal coordinate x of the point representing z is called the real part of z (Re z) and the vertical coordinate y of the point representing z is called the imaginary part of z (Im z); we write z = x + iy. Alternatively, the point representing z can also be given by its distance from the origin (jzj, the modulus of z) and an angle (arg z, the argument of z). The phase portrait of a complex function f (appearing in the picture on the left) arises when all points z of the domain of f are colored according to the argument (or “phase”) of the value w = f (z). -
PDF (Dissertation.Pdf)
Kind Theory Thesis by Joseph R. Kiniry In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2002 (Defended 10 May 2002) ii © 2002 Joseph R. Kiniry All Rights Reserved iii Preface This thesis describes a theory for representing, manipulating, and reasoning about structured pieces of knowledge in open collaborative systems. The theory's design is motivated by both its general model as well as its target user commu- nity. Its model is structured information, with emphasis on classification, relative structure, equivalence, and interpretation. Its user community is meant to be non-mathematicians and non-computer scientists that might use the theory via computational tool support once inte- grated with modern design and development tools. This thesis discusses a new logic called kind theory that meets these challenges. The core of the work is based in logic, type theory, and universal algebras. The theory is shown to be efficiently implementable, and several parts of a full realization have already been constructed and are reviewed. Additionally, several software engineering concepts, tools, and technologies have been con- structed that take advantage of this theoretical framework. These constructs are discussed as well, from the perspectives of general software engineering and applied formal methods. iv Acknowledgements I am grateful to my initial primary adviser, Prof. K. Mani Chandy, for bringing me to Caltech and his willingness to let me explore many unfamiliar research fields of my own choosing. I am also appreciative of my second adviser, Prof. Jason Hickey, for his support, encouragement, feedback, and patience through the later years of my work.