Kanterian, Edward (2007) Ludwig Wittgenstein, Reaktion Books
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Nominalism, Trivialism, Logicism
Nominalism, Trivialism, Logicism Agustín Rayo∗ May 1, 2014 This paper is an effort to extract some of the main theses in the philosophy of mathematics from my book, The Construction of Logical Space. I show that there are important limits to the availability of nominalistic paraphrase-functions for mathematical languages, and sug- gest a way around the problem by developing a method for specifying nominalistic contents without corresponding nominalistic paraphrases. Although much of the material in this paper is drawn from the book—and from an earlier paper (Rayo 2008)—I hope the present discussion will earn its keep by motivating the ideas in a new way, and by suggesting further applications. 1 Nominalism Mathematical Nominalism is the view that there are no mathematical objets. A standard problem for nominalists is that it is not obvious that they can explain what the point of a mathematical assertion would be. For it is natural to think that mathematical sentences like ‘the number of the dinosaurs is zero’ or ‘1 + 1 = 2’ can only be true if mathematical objects exist. But if this is right, the nominalist is committed to the view that such sentences are untrue. And if the sentences are untrue, it not immediately obvious why they would be worth asserting. ∗For their many helpful comments, I am indebted to Vann McGee, Kevin Richardson, Bernhard Salow and two anonymous referees for Philosophia Mathematica. I would also like to thank audiences at Smith College, the Università Vita-Salute San Raffaele, and MIT’s Logic, Langauge, Metaphysics and Mind Reading Group. Most of all, I would like to thank Steve Yablo. -
Biography Paper – Georg Cantor
Mike Garkie Math 4010 – History of Math UCD Denver 4/1/08 Biography Paper – Georg Cantor Few mathematicians are house-hold names; perhaps only Newton and Euclid would qualify. But there is a second tier of mathematicians, those whose names might not be familiar, but whose discoveries are part of everyday math. Examples here are Napier with logarithms, Cauchy with limits and Georg Cantor (1845 – 1918) with sets. In fact, those who superficially familier with Georg Cantor probably have two impressions of the man: First, as a consequence of thinking about sets, Cantor developed a theory of the actual infinite. And second, that Cantor was a troubled genius, crippled by Freudian conflict and mental illness. The first impression is fundamentally true. Cantor almost single-handedly overturned the Aristotle’s concept of the potential infinite by developing the concept of transfinite numbers. And, even though Bolzano and Frege made significant contributions, “Set theory … is the creation of one person, Georg Cantor.” [4] The second impression is mostly false. Cantor certainly did suffer from mental illness later in his life, but the other emotional baggage assigned to him is mostly due his early biographers, particularly the infamous E.T. Bell in Men Of Mathematics [7]. In the racially charged atmosphere of 1930’s Europe, the sensational story mathematician who turned the idea of infinity on its head and went crazy in the process, probably make for good reading. The drama of the controversy over Cantor’s ideas only added spice. 1 Fortunately, modern scholars have corrected the errors and biases in older biographies. -
BWS Newsletter No 18 24/11/2015, 20:45
BWS Newsletter no 18 24/11/2015, 20:45 From: British Wittgenstein Society [[email protected]] Subject: Newsletter no.18 BWS website home August 2013 BWS Newsletter Issue no 18 Contents Nota Bene The Wittgenstein-Skinner Archive Pdf version (216 kb) Nota Bene Arthur Gibson on Wittgenstein’s Conferences rediscovered archive Around the world Wittgenstein postings In early October 1941 German bombers Lecture series attacked Oakington RAF base. Victims Pastures new were rushed to hospital in Cambridge. Housekeeping The only slightly later admission of a Executive Committee polio patient was unnoticed, by-passed, being left untreated for very many hours in a corridor. This is how Wittgenstein’s closest friend Francis Skinner came to die at the age of 29. About BWS Within the week of Skinner’s funeral, in a state of trauma, Ludwig attempted to resign his Philosophy Chair; arranged to leave Cambridge for Guy’s Hospital working to fulfil his, now memorial, plan with Francis; attended Francis’ funeral; reclaimed from Skinner’s family the Wittgenstein- BWS is a British focal point for research and exchange Skinner Archive, and posted them to Skinner’s school of ideas among Wittgenstein scholars and students friend, Reuben Goodstein. Eventually Goodstein gave the throughout the world. Archive to the Mathematical Association. It was a much- appreciated invitation from the Association (and full This Newsletter will be sent exclusively to members of acknowledgement to it in references here), with the the BWS, on a regular basis, in order to draw attention support of Trinity College, for me to research and prepare to updates on the website, or to share as yet unpublished this unpublished Archive for book publication. -
Wittgenstein in Exile
Wittgenstein in Exile “My thoughts are one hundred per cent Hebraic.” -Wittgenstein to Drury, 19491 Wittgenstein was born in 1889 into one of the richest families in Central Europe. He lived and learned at home, in Vienna, until 1903, when he was 14. We have no record of his thoughts about the turn of the last century, but it is unlikely that it seemed very significant to him. The Viennese of the time had little inclination to consider the possibilities of change, and the over-ripe era in which Wittgenstein grew up did not really end until Austria-Hungary’s defeat, in World War I, and subsequent dismantling. But the family in which Wittgenstein grew up apparently felt that European culture had already come to an end in the 1840’s. And Wittgenstein himself felt he belonged to an era that had vanished with the death of the composer Robert Schumann (1810-1856).2 Somewhere in the middle of the Nineteenth Century there was an important change into the contemporary era, of which Wittgenstein did not feel a part. Wittgenstein’s understanding of history, and his consequent self-understanding in relation to his times, was deeply influenced by Oswald Spengler, who in 1918 published The Decline of the West [Der Untergang des Abenlandes]. This book, expanded to a second volume in 1922, and revised in 1923, became a best-seller in post-war Europe. Wittgenstein made numerous references to it in 1930-1931, and acknowledged Spengler as one of his ten noteworthy influences.3 According to Spengler, cultures grow, flower, and deteriorate naturally, according to their own internal form, much as a human being does. -
Our Conceptual Understanding of a Phenomenon, While the Logic of Induction Adds Quantitative Details to the Conceptual Knowledge
DOCUMENT RESUME ED 376 173 TM 021 982 AUTHOR Ho, Yu Chong TITLE Abduction? Deduction? Induction? Is There a Logic of Exploratory Data Analysis? PUB DATE Apr 94 NOTE 28p.; Paper presented at the Annual Meeting of the American Educational Research Association (New Orleans, LA, April 4-8, 1994). PUB TYPE Reports Descriptive (141) Speeches/Conference Papers (150) EDRS PRICE MF01/PCO2 Plus Postage. DESCRIPTORS *Comprehension; *Deduction; Hypothesis Testing; *Induction; *Logic IDENTIFIERS *Abductive Reasoning; *Exploratory Data Analysis; Peirce (Charles S) ABSTRACT The philosophical notions introduced by Charles Sanders Peirce (1839-1914) are helpfu: for researchers in understanding the nature of knowledge and reality. In the Peircean logical system, the logic of abduction and deduction contribute to our conceptual understanding of a phenomenon, while the logic of induction adds quantitative details to the conceptual knowledge. Although Peirce justified the validity of induction as a self-corrective process, he asserted that neither induction nor deduction can help us to unveil the internal structure of meaning. As exploratory data analysis performs the function of a model builder for confirmatory data analysis, abduction plays the role of explorer of viable paths to further inquiry. Thus, the logic of abduction fits well into exploratory data analysis. At the stage of abduction, the goal is to explore the data, find out a pattern, and suggest a plausible hypothesis; deduction is to refine the hypothesis based upon other plausible premises; and induction is the empirical substantiation. (Contains 55 references.) (Author) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. is *********************************************************************** Abduction? Deduction? Induction? Is there a Logic of Exploratory Data Analysis? Yu Chong Ho University of Oklahoma Internet: [email protected] April 4, 1994 U S. -
Explanations
© Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. CHAPTER 1 Explanations 1.1 SALLIES Language is an instrument of Logic, but not an indispensable instrument. Boole 1847a, 118 We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic; the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two. De Morgan 1868a,71 That which is provable, ought not to be believed in science without proof. Dedekind 1888a, preface If I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems whilst the root drives into the depthswx . Frege 1893a, xiii Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians. Russell 1903a, 451 1.2 SCOPE AND LIMITS OF THE BOOK 1.2.1 An outline history. The story told here from §3 onwards is re- garded as well known. It begins with the emergence of set theory in the 1870s under the inspiration of Georg Cantor, and the contemporary development of mathematical logic by Gottlob Frege andŽ. especially Giuseppe Peano. A cumulation of these and some related movements was achieved in the 1900s with the philosophy of mathematics proposed by Alfred North Whitehead and Bertrand Russell. -
Wittgenstein, Ludwig (1889-1951) by Craig Kaczorowski
Wittgenstein, Ludwig (1889-1951) by Craig Kaczorowski Encyclopedia Copyright © 2015, glbtq, Inc. Entry Copyright © 2004, glbtq, inc. Reprinted from http://www.glbtq.com The Austrian-British philosopher Ludwig Wittgenstein is considered one of the most significant thinkers of the twentieth century, especially for his valuable contributions within the field of Linguistic Philosophy. His major works on language and linguistics include Tractatus Logico-Philosophicus (1922), and Philosophical Investigations (1953), which was published posthumously. Early Life and Education Born Ludwig Josef Johann Wittgenstein in Vienna on April 26, 1889, he was the youngest of eight children raised in a wealthy and cultured Jewish family. Initially educated privately at home, at the age of 14 he began attending a school in Linz, Austria that specialized in mathematics and natural science. In 1906, he went to Berlin to study mechanical engineering and two years later registered at the University of Manchester in England to study for his doctorate in engineering. Over time, however, Wittgenstein developed a greater interest in mathematics and mathematical logic, and he transferred to Trinity College, University of Cambridge, where he took courses from 1912 to 1913 given by the distinguished philosopher and mathematician Bertrand Russell. In November 1912, on the recommendation of fellow student John Maynard Keynes, Wittgenstein was elected to the elite Cambridge society known as the Apostles, which at that time had an aura of homoeroticism. This homoerotic atmosphere made Wittgenstein uncomfortable, however, and he stopped attending meetings. Finding Cambridge a less than ideal place to work, since he felt that his fellow academics lacked depth, Wittgenstein went to Skjolden in Norway and continued his mathematical and philosophical investigations in seclusion. -
What Is Neologicism?∗
What is Neologicism? 2 by Zermelo-Fraenkel set theory (ZF). Mathematics, on this view, is just applied set theory. Recently, ‘neologicism’ has emerged, claiming to be a successor to the ∗ What is Neologicism? original project. It was shown to be (relatively) consistent this time and is claimed to be based on logic, or at least logic with analytic truths added. Bernard Linsky Edward N. Zalta However, we argue that there are a variety of positions that might prop- erly be called ‘neologicism’, all of which are in the vicinity of logicism. University of Alberta Stanford University Our project in this paper is to chart this terrain and judge which forms of neologicism succeed and which come closest to the original logicist goals. As we look back at logicism, we shall see that its failure is no longer such a clear-cut matter, nor is it clear-cut that the view which replaced it (that 1. Introduction mathematics is applied set theory) is the proper way to conceive of math- ematics. We shall be arguing for a new version of neologicism, which is Logicism is a thesis about the foundations of mathematics, roughly, that embodied by what we call third-order non-modal object theory. We hope mathematics is derivable from logic alone. It is now widely accepted that to show that this theory offers a version of neologicism that most closely the thesis is false and that the logicist program of the early 20th cen- approximates the main goals of the original logicist program. tury was unsuccessful. Frege’s (1893/1903) system was inconsistent and In the positive view we put forward in what follows, we adopt the dis- the Whitehead and Russell (1910–13) system was not thought to be logic, tinctions drawn in Shapiro 2004, between metaphysical foundations for given its axioms of infinity, reducibility, and choice. -
Charles Sanders Peirce - Wikipedia, the Free Encyclopedia 9/2/10 4:55 PM
Charles Sanders Peirce - Wikipedia, the free encyclopedia 9/2/10 4:55 PM Charles Sanders Peirce From Wikipedia, the free encyclopedia Charles Sanders Peirce (pronounced /ˈpɜrs/ purse[1]) Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician, and scientist, born in Cambridge, Massachusetts. Peirce was educated as a chemist and employed as a scientist for 30 years. It is largely his contributions to logic, mathematics, philosophy, and semiotics (and his founding of pragmatism) that are appreciated today. In 1934, the philosopher Paul Weiss called Peirce "the most original and versatile of American philosophers and America's greatest logician".[2] An innovator in many fields (including philosophy of science, epistemology, metaphysics, mathematics, statistics, research methodology, and the design of experiments in astronomy, geophysics, and psychology) Peirce considered himself a logician first and foremost. He made major contributions to logic, but logic for him encompassed much of that which is now called epistemology and philosophy of science. He saw logic as the Charles Sanders Peirce formal branch of semiotics, of which he is a founder. As early as 1886 he saw that logical operations could be carried out by Born September 10, 1839 electrical switching circuits, an idea used decades later to Cambridge, Massachusetts produce digital computers.[3] Died April 19, 1914 (aged 74) Milford, Pennsylvania Contents Nationality American 1 Life Fields Logic, Mathematics, 1.1 United States Coast Survey Statistics, Philosophy, 1.2 Johns Hopkins University Metrology, Chemistry 1.3 Poverty Religious Episcopal but 2 Reception 3 Works stance unconventional 4 Mathematics 4.1 Mathematics of logic C. -
Logicism and Mathematical Practices – Russell’S Theory of Metrical Geometry in the Principles of Mathematics (1903) Sébastien Gandon
Logicism and mathematical practices – Russell’s theory of metrical geometry in The Principles of Mathematics (1903) Sébastien Gandon To cite this version: Sébastien Gandon. Logicism and mathematical practices – Russell’s theory of metrical geometry in The Principles of Mathematics (1903). 2008. halshs-00348291 HAL Id: halshs-00348291 https://halshs.archives-ouvertes.fr/halshs-00348291 Preprint submitted on 18 Dec 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Logicism and mathematical practices – Russell’s theory of metrical geometry in The Principles of Mathematics (1903) Sébastien GANDON IUF / PHIER, Clermont Université 1- In a letter to the French historian of mathematics P. Dugac, dated 12/05/1984, the great mathematician J. Dieudonné wrote: The controversy between Poincaré and Russell is very enlightening; it reveals quite obviously how completely invalid the reasonings of the alleged “mathematician” Russell are about everything connected to mathematics; he could have been wholly self-taught on the subject, since what he says shows he -
Wittgenstein's Philosophical Development: Phenomenology, Grammar, Method, and the Anthropological View
Nordic Wittgenstein Review 5 (2) 2016 INVITED PAPER Juliet Floyd jfloyd @ bu.edu Chains of Life: Turing, Lebensform, and the Emergence of Wittgenstein’s Later Style Abstract This essay accounts for the notion of Lebensform by assigning it a logical role in Wittgenstein’s later philosophy. Wittgenstein’s additions of the notion to his manuscripts of the PI occurred during the initial drafting of the book 1936-7, after he abandoned his effort to revise The Brown Book. It is argued that this constituted a substantive step forward in his attitude toward the notion of simplicity as it figures within the notion of logical analysis. Next, a reconstruction of his later remarks on Lebensformen is offered which factors in his reading of Alan Turing’s “On computable numbers, with an application to the Entscheidungsproblem“ (1936/7), as well as his discussions with Turing 1937-1939. An interpretation of the five occurrences of Lebensform in the PI is then given in terms of a logical “regression” to Lebensform as a fundamental notion. This regression characterizes Wittgenstein’s mature answer to the question, “What is the nature of the logical?” 1. Introduction Lebensform is an important gauge of the development of Wittgenstein’s thought. It represents, from a regressive point of view, the place he finally chased down to. In foundational matters such as these, it is often the simplest, naïve perspective that comes last in a thinker’s (or a tradition’s) evolution, the working through 7 Juliet Floyd CC-BY of how to get back home to the primordial basis, the extraction of fundamentals from a jungle of technical accretions. -
Cambridge UP. Ancestral, Axiomatic Method, Borderline
Audi, R. Ed. 1999. Cambridge Dictionary of Philosophy. Cambridge: Cambridge UP. ancestral, axiomatic method, borderline case, categoricity, Church (Alonzo), conditional, convention T, converse (outer and inner), corresponding conditional, degenerate case, domain, De Morgan, ellipsis, laws of thought, limiting case, logical form, logical subject, material adequacy mathematical analysis, omega, proof by recursion, recursive function theory, scheme, scope, Tarski (Alfred), tautology, universe of discourse. CITE AS: Corcoran, J. 1999. “Ancestral”, Cambridge Dictionary of Philosophy. R.Audi, Ed. Cambridge: Cambridge UP. p. 65. ancestral (of a given relation R), the relation (also called the transitive closure of R) that relates one given individual to a second if and only if the first can be “reached” from the second by repeated “applications” of the given relation R. The ancestor relation is the ancestral of the parent relation since one person is an ancestor of a second if the first is a parent of the second or the first is a parent of a parent of the second or the first is a parent of a parent of a parent of the second, and so on. Frege discovered a simple method of giving a materially adequate and formally correct definition of the ancestral of a given relation in terms of the relation itself (plus logical concepts). This method is informally illustrated as follows: in order for one person A to be an ancestor of a second person B it is necessary and sufficient for A to have every property that belongs to every parent of B and that belongs to every parent of any person to whom it belongs.