Philosophy of Mathematics
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Nonstandard Set Theories and Information Management
Nonstandard Set Theories and Information Management VAROL AKMAN akmantroycsbilkentedutr Department of Computer Engineering and Information Science Bilkent University Bilkent Ankara Turkey MUJDATPAKKAN pakkantrb ounbitnet Department of Computer Engineering Bosphorus University Bebek Istanbul Turkey Abstract The merits of set theory as a foundational to ol in mathematics stimulate its use in various areas of articial intelligence in particular intelligent information systems In this pap er a study of various nonstandard treatments of set theory from this p ersp ective is oered Applications of these alternative set theories to information or knowledge management are surveyed Keywords set theory knowledge representation information management commonsense rea soning nonwellfounded sets hyp ersets Intro duction Set theory is a branch of mo dern mathematics with a unique place b ecause various other branches can b e formally dened within it For example Book of the inuential works of N Bourbaki is devoted to the theory of sets and provides the framework for the remaining 1 volumes Bourbaki said in Goldblatt All mathematical theories may b e regarded as extensions of the general theory of sets On these foundations I can state that I can build up the whole of the mathematics of the present day This brings up the p ossibility of using set theory in foundational studies in AI Mc Carthy has emphasized the need for fundamental research in AI and claimed that AI needs mathematical and logical theory involving conceptual innovations In an -
New Foundations of Cost–Benefit Analysis
Regulation & Governance (2009) 3, 72–83 doi:10.1111/j.1748-5991.2009.01045.x RESEARCH FORUM New foundations of cost–benefit analysis A reply to Professors Sinden, Kysar, and Driesen Matthew Adler University of Pennsylvania Law School, Philadelphia, PA, USA Eric A. Posner University of Chicago Law School, Chicago, IL, USA Abstract This article responds to the criticisms of New Foundations of Cost–Benefit Analysis that appeared in a review by Amy Sinden, Douglas A. Kysar, and David M. Driesen. We argue that their criticisms are either based on misunderstandings of our approach or are too demanding, in the sense that no reasonable decision procedure would satisfy them. We illustrate this second argument by demon- strating that their preferred approach – feasibility analysis – has little to recommend it. Keywords: cost–benefit analysis, regulation, welfarism. Introduction The basic thrust of New Foundations of Cost–Benefit Analysis is to detach cost–benefit analysis (CBA) from Kaldor–Hicks efficiency,which we argue lacks moral relevance (Adler & Posner 2006, pp. 19–24), and instead to see CBA as a rough, administrable proxy for overall wellbeing. We defend the moral relevance of overall wellbeing by appealing, not to utilitarianism,but to“weak welfarism”– a much more catholic moral view that sees overall wellbeing as one among a possible plurality of foundational moral criteria, along with distributive considerations, deontological rights, and non-welfare values (Adler & Posner 2006, pp. 52–61). We also reject the simple equation of wellbeing and preference- satisfaction. An outcome benefits a person only if she prefers it and her preferences are self-interested and survive idealization (Adler & Posner 2006, pp. -
Arxiv:1204.2193V2 [Math.GM] 13 Jun 2012
Alternative Mathematics without Actual Infinity ∗ Toru Tsujishita 2012.6.12 Abstract An alternative mathematics based on qualitative plurality of finite- ness is developed to make non-standard mathematics independent of infinite set theory. The vague concept \accessibility" is used coherently within finite set theory whose separation axiom is restricted to defi- nite objective conditions. The weak equivalence relations are defined as binary relations with sorites phenomena. Continua are collection with weak equivalence relations called indistinguishability. The points of continua are the proper classes of mutually indistinguishable ele- ments and have identities with sorites paradox. Four continua formed by huge binary words are examined as a new type of continua. Ascoli- Arzela type theorem is given as an example indicating the feasibility of treating function spaces. The real numbers are defined to be points on linear continuum and have indefiniteness. Exponentiation is introduced by the Euler style and basic properties are established. Basic calculus is developed and the differentiability is captured by the behavior on a point. Main tools of Lebesgue measure theory is obtained in a similar way as Loeb measure. Differences from the current mathematics are examined, such as the indefiniteness of natural numbers, qualitative plurality of finiteness, mathematical usage of vague concepts, the continuum as a primary inexhaustible entity and the hitherto disregarded aspect of \internal measurement" in mathematics. arXiv:1204.2193v2 [math.GM] 13 Jun 2012 ∗Thanks to Ritsumeikan University for the sabbathical leave which allowed the author to concentrate on doing research on this theme. 1 2 Contents Abstract 1 Contents 2 0 Introdution 6 0.1 Nonstandard Approach as a Genuine Alternative . -
On the Brink of New Promise the FUTURE of U.S
On the Brink of New Promise THE FUTURE OF U.S. COMMUNITY FOUNDATIONS By Lucy Bernholz, Katherine Fulton, and Gabriel Kasper Blueprint Research & Design, Inc. and the Monitor Institute, a member of Monitor Group Created with funding support from the Charles Stewart Mott Foundation and the Ford Foundation © Copyright 2005 Blueprint Research & Design, Inc. and Monitor Company Group, LLP. We encourage readers to use and share the contents of this report, with the understanding that it is the intellectual property of Blueprint Research & Design and the Monitor Group, and that full attribution is required. An invitation from the Charles Stewart Mott and Ford foundations Our two foundations have been privileged to have worked with community foundations across the United States over the past quarter century. For both of us, this is an area of philanthropy we care about deeply. Th e Charles Stewart Mott and Ford foundations work at national and indeed international levels. Yet we both know how valuable it is for large private foundations to have strong, dynamic community founda- tions as partners working at the local level. Th is is why we embraced the opportunity to look deep into the future of the fi eld through the expert eyes of Lucy Bernholz, Katherine Fulton, and Gabriel Kasper. Over those years, Mott and Ford have made grants to community foundations to build general and administrative endowments, to provide peer-to-peer learning opportunities and technical assistance, and to strengthen programming expertise in areas such as low-income neighborhoods, the environment, and minority communities. We have also assisted the fi eld to develop abroad, and we have helped build the infrastructure for community foundations We recommend this report to community foundation staff, boards of directors, across the nation. -
The Problem of the Consistency of “New Foundations”
The Problem of the Consistency of \New Foundations" Randall Holmes September 10, 2013 1 Abstract: I will explain the nature of the long- standing problem of the consistency of the set theory New Foundations proposed by the philoso- pher W. v. O. Quine in 1937 both in the prior context of the development of set theory, its usefulness in mathematics, and the problem of the "paradoxes" of set theory, and in the pos- terior context of partial solutions to the consis- tency problem and related results. I do claim to have solved this problem (this is not generally agreed yet) but I am not going to talk about that on this occasion. The talk should be ac- cessible to a general audience of mathemati- cians; I hope that a graduate student or mature undergraduate would get something out of it too. Plan of the talk I'm going to talk about the problem of the consistency of Quine's set theory \New Foun- dations", which is the central issue of my tiny area of set theory. I currently believe that I have solved this prob- lem, but this has nothing to do with the present talk, or very little. What I propose to do is explain what the prob- lem is and put it in some kind of context. Why does one need a set theory? Why is there a problem of consistency of set theories? What is New Foundations anyway and why is there a problem with it in particular? What are the relevant related results and partial solutions to the problem? 2 What is a set? When I tell a layman with no maths that I work in set theory, they ask \What is a set?". -
Arxiv:1809.08676V1 [Math.LO] 23 Sep 2018 Hc Ahmtca O S.I Un U Htti Eednei N Dep Language Is Precise May Dependence Game
NAMING THE LARGEST NUMBER EXPLORING THE BOUNDARY BETWEEN MATHEMATICS AND THE PHILOSOPHY OF MATHEMATICS DAVID SIMMONS Abstract. What is the largest number accessible to the human imagination? The ques- tion is neither entirely mathematical nor entirely philosophical. Mathematical formula- tions of the problem fall into two classes: those that fail to fully capture the spirit of the problem, and those that turn it back into a philosophical problem. 1. Introduction “You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number–not an infinity–on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature. Are you ready? Get set. Go.” – Scott Aaronson, “Who can Name the Bigger Number?” 1 “The game of describing the largest integer, when played by experts, lapses into a hopeless argument over legitimacy.” – Joel Spencer, “Large numbers and unprovable theorems” [8] This paper addresses the question: what is the best strategy in a “name the largest number” contest such as the one described above? Of course, the answer depends on arXiv:1809.08676v1 [math.LO] 23 Sep 2018 what the rules of the contest are. Scott Aaronson’s idea of letting a “reasonable modern mathematician” judge the entries might sound fair at first, but the answer to the question of whether a given entry is a precise description of a number may depend on exactly which mathematician you ask. It turns out that this dependence is not incidental but fundamental, and slightly different standards for what counts as a valid entry give rise to radically different strategies for winning the game. -
Model Theory of Ultrafinitism I: Fuzzy Initial Segments of Arithmetic
Model Theory of Ultrafinitism I: Fuzzy Initial Segments of Arithmetic (Preliminary Draft) Mirco A. Mannucci Rose M. Cherubin February 1, 2008 Abstract This article is the first of an intended series of works on the model the- ory of Ultrafinitism. It is roughly divided into two parts. The first one addresses some of the issues related to ultrafinitistic programs, as well as some of the core ideas proposed thus far. The second part of the paper presents a model of ultrafinitistic arithmetics based on the notion of fuzzy initial segments of the standard natural numbers series. We also introduce a proof theory and a semantics for ultrafinitism through which feasibly consistent theories can be treated on the same footing as their classically consistent counterparts. We conclude with a brief sketch of a foundational program, that aims at reproducing the transfinite within the finite realm. 1 Preamble To the memory of our unforgettable friend Stanley ”Stan” Tennen- baum (1927 − 2005), Mathematician, Educator, Free Spirit As we have mentioned in the abstract, this article is the first one of a series dedicated to ultrafinitistic themes. arXiv:cs/0611100v1 [cs.LO] 21 Nov 2006 First papers often tend to take on the dress of manifestos, road maps, or both, and this one is no exception. It is the revised version of an invited conference talk, and was meant from the start for a quite large audience of philosophers, logicians, computer scientists, and mathematicians, who might have some in- terest in the ultrafinite. Therefore, neither the philosophico-historical, nor the mathematical side, are meant to be detailed investigations. -
A Finitist's Manifesto: Do We Need to Reformulate the Foundations Of
A Finitist’s Manifesto: Do we need to Reformulate the Foundations of Mathematics? Jonathan Lenchner 1 The Problem There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been sleep-walking in their infinitary mathematical paradise to take heed. Much of mathematics relies upon either (i) the “existence” of objects that contain an infinite number of elements, (ii) our ability, “in theory”, to compute with an arbitrary level of precision, or (iii) our ability, “in theory”, to compute for an arbitrarily large number of time steps. All of calculus relies on the notion of a limit. The monumental results of real and complex analysis rely on a seamless notion of the “continuum” of real numbers, which extends in the plane to the complex numbers and gives us, among other things, “rigorous” definitions of continuity, the derivative, various different integrals, as well as the fundamental theorems of calculus and of algebra – the former of which says that the derivative and integral can be viewed as inverse operations, and the latter of which says that every polynomial over C has a complex root. This essay is an inquiry into whether there is any way to assign meaning to the notions of “existence” and “in theory” in (i) to (iii) above. On the one hand, we know from quantum mechanics that making arbitrarily precise measure- ments of objects is impossible. By the Heisenberg Uncertainty Principle the moment we pin down an object, typically an elementary particle, in space, thereby bringing its velocity, and hence mo- mentum, down to near 0, there is a limit to how precisely we can measure its spatial coordinates. -
By WV QUINE Only the Truth Functions, Quantification, and Membership
538 MATHEMATICS: W. V. QUINE PROC. N. A. S. ON THE CONSISTENC Y OF "NEW FOUNDA TIONS" By W. V. QUINE HARVARD UNIVERSITY Communicated by Hassler Whitney, June 12, 1951 The question of the consistency of the system of set theory in my "New Foundations,"' briefly NF, has long been mooted.2 3 The question takes on added interest with Wang's new revision4 of the system of my Mathe- matical Logic;5 for Wang has proved that this revision, which is to be used in the forthcoming third printing of Mathematical Logic, is consistent if NF is consistent. The purpose of the present note is to point out that NF is consistent if and only if the system of Whitehead and Russell's Principia Mathematica,6 briefly PM, is consistent in a certain one of its natural ver- sions. PM may be conceived in a form in which the variables carry indices indi- cating logical type. Alternatively PM may be conceived under a conven- tion which Whitehead and Russell called typical ambiguity; under this con- vention the indices are suppressed but still only those formulae are ad- mitted as meaningful which are stratified, i.e., so constituted that type in- dices could be inserted everywhere without violating the theory of types. Another point on which we have a choice, in framing a version of PM, is the status of relations and relational predication: we may assume relational predication as primitive on a par with set-membership, and decree a sprawling annex to the theory of types to accommodate relations, as was done in effect by Whitehead and Russell; or alternatively we may, follow- ing Wiener,7 reduce relations to certain sets of sets of sets (at the cost of giving up relations between things of different types). -
Phenomenology, Logic and the Philosophy of Mathematics Richard Tieszen Index More Information
Cambridge University Press 0521837820 - Phenomenology, Logic and the Philosophy of Mathematics Richard Tieszen Index More information Index absolutism, 44, 332 see also Frege, G.; natural numbers; abstraction, 34, 36–37, 78, 84, 325, 327, Peano arithmetic (PA); Poincar´e, H.; 328 primitive recursive arithmetic (PRA); formal and material, 28 Weyl, H. see also founding/founded distinction artificial intelligence, 288 absurdity associationism, 210 formal a priori, 27 authentic/inauthentic distinction, 3, 42 material a priori, 27 axiom systems algorithmic methods, 32 definite formal, 4, 11–12, 29 alienation, 42 analysis, paradox of, 320 Becker, O., 8, 9, 14, 62, 83, 126, 237, 247, analytic a priori 254, 260, 268 judgments, 28 Bell, D., 329 see also analyticity; analytic/synthetic Benacerraf, P., 58, 64, 172 distinction Bernays, P., 108, 153, 245 Analytic and Continental philosophy, 1–2, Beth models, 289, 293 44–45, 66 BHK interpretation, 229, 232, 238, 242, analyticity, 185–190 245 rational intuition and, 188–190 biologism, 23 see also Frege, G.; G¨odel, K.; Poincar´e, Bishop, E., 228, 288 H.; Quine,W.V.O. Bolzano, B., 24, 51, 154, 318 analytic/synthetic distinction, 318 Boolos, G., 187 in Husserl, 27–28 Brentano, F., 1, 22 antinomies, 131 Brouwer, L. E. J., 7, 8, 118, 227, 228, 231, see also paradoxes 234, 235, 248, 249, 253, 254, 266, 278, antireductionism 283, 296 in Husserl, 33 see also intuitionism see also reductionism apophantic analytics, 28–29 calculation Aristotelian realism, 127 in science, 36, 40, 41–42 Aristotle, 127 Cantor, G., -
True History of Strict Finitism
A True(r) History of Strict Finitism Jean Paul Van Bendegem Vrije Universiteit Brussel Center for Logic and Philosophy of Science Universiteit Gent Confusion about what strict finitism is ° Both historical roots ° As what its meaning is Different names: ° Strict finitism ° Ultrafinitism ° Ultra-intuitionism Aim: to clarify matters (a bit) The “founding father” (usually mentioned as such) Alexander Yessenin-Volpin (sometimes Essenine-Volpin of Ésenine-Volpine) • ultra-intuitionism • articles quite cryptic • no direct interest in finitism • different aim (finitary consistency proof) YESSENIN-VOLPIN, A. S. : "Le programme ultra- intuitioniste des fondements des mathématiques". In: Infinitistic Methods, Proceedings Symposium on Foundations of Mathematics, Pergamon Press, Oxford, 1961, pp. 201-223. YESSENIN-VOLPIN, A. S. : "The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics". In: KINO, MYHILL & VESLEY (eds.), Intuitionism & proof theory. North-Holland, Amsterdam, 1970, pp. 3-45. YESSENIN-VOLPIN, A. S. : "About infinity, finiteness and finitization". In RICHMAN, F. (ed.), 1981, pp. 274-313. “Zenonian” sets Z: • if n belongs to Z, so does n+1 • Z is nevertheless finite in its entirety example : the collection of heartbeats in your youth But see: James R. Geiser: “A Formalization of Essenin-Volpin's Proof Theoretical Studies by Means of Nonstandard Analysis” (JSL, Vol. 39, No. 1, 1974, pp. 81-87) for an attempt at rigorous reconstruction “Take, for example, the unusual answer proposed by Alexander Yessenin-Volpin (Aleksandr Esenin-Volpin), a Russian logician of the ultra-finitist school who was imprisoned in a mental institution in Soviet Russia. Yessenin-Volpin was once asked how far one can take the geometric series of powers of 2, say (2 1, 2 2, 2 3, …, 2100 ). -
New Foundations of Transnational Private Regulation Author(S): Fabrizio Cafaggi Source: Journal of Law and Society, Vol
Cardiff University New Foundations of Transnational Private Regulation Author(s): Fabrizio Cafaggi Source: Journal of Law and Society, Vol. 38, No. 1, The Challenge of Transnational Private Regulation: Conceptual and Constitutional Debates (MARCH 2011), pp. 20-49 Published by: Wiley on behalf of Cardiff University Stable URL: https://www.jstor.org/stable/23030395 Accessed: 21-01-2019 16:18 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms Cardiff University, Wiley are collaborating with JSTOR to digitize, preserve and extend access to Journal of Law and Society This content downloaded from 154.59.124.94 on Mon, 21 Jan 2019 16:18:47 UTC All use subject to https://about.jstor.org/terms JOURNAL OF LAW AND SOCIETY VOLUME 38, NUMBER 1, MARCH 2011 ISSN: 0263-323X, pp. 20-49 New Foundations of Transnational Private Regulation Fabrizio Cafaggi* In section I oj this article, the factors driving towards the emergence of new transnational private regulation (TPR) are identified in com parison with, on the one hand, merchant law and, on the other, inter national public regimes. In section II, the focus is on the private sphere, looking at both the different conflicts of interests arising in the regulatory relationships and the need for governance responses.