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	Paul Benacerraf - Wikipedia, the free encyclopedia
		Paul Benacerraf is an American philosopher of mathematics based at Princeton University.
		His brother is Nobel Prize-winning immunologist Baruj Benacerraf.
		Benacerraf is perhaps best know for his paper "what numbers could not be" and for his highly succussful anthology on the philosophy of mathematics, co-edited with Hilary Putnam.
	en.wikipedia.org /wiki/Paul_Benacerraf (114 words)

	SATAN STULTIFIED
		Benacerraf is therefore precluded from using Gödel's first theorem, and turns instead to its corollary, Gödel's second theorem, that in any formal system rich enough for elementary arithmetic, the consistency of that system cannot be proved-in-that-system, unless the system is in fact inconsistent.
		Benacerraf's mechanism, which avoids contradiction only by never specifying what sort of machine a human being is alleged to be, is, from a common sense point of view, a position too empty to be worth holding.
		Benacerraf is claiming that the man is a machine, although for every particular machine he could be we can show that he is not that one.
	www.univ.trieste.it /~etica/2003_1/4_monographica.htm (4059 words)

	HH - What 'What Numbers Could Not Be', by Paul Benacerraf, Is (Site not responding. Last check: 2007-10-29)
		Benacerraf's essay is structured into three sections: the first is an exposition of some of the current accounts of number; the second section discusses the particular difficulties associated with such accounts, concluding that they are fundamentally wrong, and the third section is an attempt to justify the conclusion of the previous section.
		Benacerraf is claiming no such thing himself here: although it does not become apparent until later on, these are not his views, he is merely presenting them as a subject for later discussion.
		Benacerraf explains that to characterise the numbers is only to describe the structure, without any identification of the individual elements, and that this is why numbers are not objects at all.
	hilton.org.uk /what_numbers_are_not.phtml (3452 words)

	The Platonist theory of the forms must stand as one of the most frustrating in the history of philosophy (Site not responding. Last check: 2007-10-29)
		Benacerraf believes that this similarity does not exist and having dutifully acknowledged Platonism’s indisputable semantic appeal, he moves on to reject the theory on an epistemological basis using his notion of causal knowledge.
		Benacerraf believes that Gödel and other mathematical realists must inevitably encounter major difficulties precisely at this point when challenged to provide the connection between the mathematical truths that one claims to know and the grounds on which one can come to know them.
		Since Benacerraf’s assertion that Platonism is epistemologically unsatisfactory is based on the notion of causal knowledge, it seems reasonable to focus on that theory of knowledge when determining whether this particular challenge will be fatal for Platonism or not.
	www.student.cs.uwaterloo.ca /~smckenna/PHIL359/Paper-Nov28.htm (1572 words)

	Baruj Benacerraf - Wikipedia, the free encyclopedia
		Baruj Benacerraf, M.D. Baruj Benacerraf (born 29 October 1920) is a Venezuelan-American immunologist who shared the 1980 Nobel Prize in Physiology or Medicine for the "discovery of the Major histocompatibility complex genes which encode cell surface molecules important for the immune system's distinction between self and non-self".
		Benacerraf moved to Paris from Venezuela with his family in 1925.
		In 1940 he emigrated to the USA and in 1943, he became a naturalized citizen.
	en.wikipedia.org /wiki/Baruj_Benacerraf (213 words)

	Structuralism, Category Theory and Philosophy of Mathematics
		Paul Benacerraf's paper, "What Numbers Could Not Be", is often credited with initiating the philosophical thesis of structuralism, although the structuralist attitude had been prevalent among working mathematicians since the 1930s.
		Benacerraf considers Frege's solution.[282] Frege chose as the number 3 the extension of the concept "equivalent with some 3-members set"; that is, for Frege a number was an equivalence class - the class of all classes equivalent with a given class.
		Benacerraf concludes that there is no one account which conclusively establishes which sets are the "real" numbers, and he doesn't believe that there could be such an argument.
	www.mmsysgrp.com /strctcat.htm (7237 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		I shall thus be considering Benacerraf's reasons for proposing to extend the Tarskian analysis to mathematics as well as for choosing the causal theory as a desideratum for mathematical knowledge.
		Benacerraf is clearly in favour of the extension, for which he offers the following argument.
		Such a paradigm cannot simply be assumed, as Benacerraf seems to do, because it has yet not been shown that the differences between the two kinds of knowledge--empirical and mathematical--could be accounted for without appeal to a non-causalist mathematical epistemology.
	www.ifcs.ufrj.br /cefm/textos/SPINTOP.DOC (9031 words)

	THE MATH ENQUIRER (Site not responding. Last check: 2007-10-29)
		Benacerraf’s argument hinges on the fact that the Peano Axioms are satisfied by all three accounts, (and, indeed, are satisfied by a potentially infinite number of different progressions).
		The Peano Axioms serve as a basis for deriving the theorems of elementary arithmetic, and since the Axioms are satisfied by all three accounts, they do not provide a reason for picking one account over the other.
		But, Benacerraf argues, there are no arguments or reasons for choosing one account over the other.
	www.humboldt.edu /~math/currentevents/colloquim/archives/fall_04/sep23_04.shtml (513 words)

	AAA000 BIBLIOGRAPHY -- LOCKE LECTURES
		Benacerraf, Paul "What numbers could not be" (1965); reprinted in Benacerrraf and Putnam, 272-294.
		Benacerraf, Paul "What mathematical truth could not be", in Morton and Stich, pp.
		Benacerraf, Paul "Mathematical truth" (1973) ; reprinted in Benacerrraf and Putnam, 404-420.
	webware.princeton.edu /vanfraas/seminarS2002/Locke-BIBLIOGR.htm (1686 words)

	USSY 206
		Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics, Englewood Cliffs, Prentice-Hall, Inc., 1964.
		Paul Bernays, On Platonism in Mathematics: 274 - 286.
		Paul Carus,The Philosophical Basis of Mathematics, The Foundations of Mathematics, Chicago, the Open Court Publishing Company, 1908.
	www.cwru.edu /artsci/math/singer/courses/Quest/references.htm (329 words)

	Structuralism, Category Theory and Philosophy of Mathematics (Site not responding. Last check: 2007-10-29)
		Benacerraf argues for the structuralist position by first presenting an example in which the sons of two militant logicists, Ernie and Johnny, first learn logic and set theory instead of elementary number theory.
		Benacerraf concludes that numbers could not be sets at all on the grounds that there are no good reasons to say that any particular number is some particular set, for any system of objects that forms a recursive progression would be adequate.
		White's first argument in support of his thesis relies on the assumption that since Benacerraf continues to use number-words and numerals, they must be interpreted as singular terms, and therefore have referents in the "world".
	cs.wwc.edu /~aabyan/CII/strctcat.htm (8116 words)

	A Priori and A Posteriori [Internet Encyclopedia of Philosophy] (Site not responding. Last check: 2007-10-29)
		It would seem, for instance, to require that the objects of rational insight be eternal, abstract, Platonistic entities existing in all possible worlds.
		As a result of this and related concerns, many contemporary philosophers have either denied that there is any a priori justification, or have attempted to offer an account of a priori justification that does not appeal to rational insight.
		Paul and Anscombe (New York: Harper and Row).
	www.iep.utm.edu /a/apriori.htm (5580 words)

	Is Platonism Dead? (Site not responding. Last check: 2007-10-29)
		Both theories, however, have been found wanting, particularly by Paul Benacerraf in his 1973 paper, "Mathematical Truth." In Platonism and Anti-Platonism in Mathematics, Mark Balaguer claims that both views are defensible, but also that there is no fact of the matter as to which of the two views is correct.
		The variety of platonism that he argues for is referred to as FBP, which stands for 'full-blooded platonism', and it is, essentially, the view that all logically possible mathematical objects actually do exist.
		The book begins with Benacerraf's epistemological problem for mathematics, as presented in "Mathematical Truth." Since Benacerraf relies on a causal theory of knowledge to highlight the epistemological difficulties of platonism, platonists are left with a relatively easy way out, namely the rejection of the causal theory.
	www.chass.utoronto.ca /pcu/old_pcu/noesis/issue_v/noesis_v_5.html (3030 words)

	Oliver Pooley, Oxford Philosophy
		Paul Benacerraf, ``Mathematical Truth'', Journal of Philosophy 70 (1973): 661--679.
		Bob Hale and Crispin Wright, ``Benacerraf’s Dilemma Revisited'', European Journal of Philosophy 10 (2002): 101--129.
		Paul Benacerraf, ``What Numbers Could not Be'', Philosophical Review 74 (1965): 47--73.
	users.ox.ac.uk /~ball0402/teaching/frege.html (237 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		Paul Benacerraf's (1973) seminal paper "Mathematical Truth," which underscored the difficulty of providing an account of mathematical knowledge compatible with a Platonist theory of mathematical truth, spurred a series of more general investigations into the question of knowledge of abstract entities.
		Benacerraf (1973) maintains that our best theory of mathematical truth commits us to the view that the truth conditions for such statements involve reference to abstract entities and that our best account of knowledge requires a causal relation between knowers and the entities referred to by the truth conditions of the statements that they know.
		Scientific naturalists, however, cannot assume that Benacerraf's problem will appear within the science that replaces philosophy as opposed to being one of the problems that is discarded in the transition.
	www.unl.edu /philosop/people/faculty/casullo/ac_IRLP-Essay.htm (12247 words)

	Are Minds Like Numbers? - Readings in Philosophy of Mathematics
		PAUL BENACERRAF, "WHAT NUMBERS COULD NOT BE." Philosophical Review.
		Paul Benacerraf, "What Mathematical Truth Could Not Be -I" in Benacerraf and His Critics, Morton, Adam (ed) Blackwell, 1996 [see below]
		In these, as in the better understood cases regarding the Godel incompleteness results, the desired philosophical results are claimed to depend on a significant prior injection of philosophical presuppositions.
	www4.ncsu.edu /~n51ls801/PHI340mirror/phimath.html (811 words)

	Oxford Scholarship Online: The Indispensability of Mathematics (Site not responding. Last check: 2007-10-29)
		It is shown that the argument is silent on many issues such as whether numbers are sets, and whether mathematical entities are causally inert or not.
		Paul Benacerraf's two objections to mathematical realism are also considered.
		Finally, the use of indispensability arguments outside the philosophy of mathematics is considered, and, in particular, an indispensability argument for modal realism is discussed.
	www.oxfordscholarship.com /oso/public/content/philosophy/019513754X/acprof-019513754X-chapter-7.html (125 words)

	Green,I
		B lymphocyte-enriched cell populations cultured with mitogens in initial suspension cultures formed colonies in soft agar when the same mitogenic agent was present in the lower layer of a two-layer soft agar system.
		L H Glimcher, A M Kruisbeek, W E Paul, I Green
		W J Martin, L Ellman, I Green, B Benacerraf
	bioinfo.pl /auth:Green,I (7352 words)

	Oxford Scholarship Online: Ways a World Might Be (Site not responding. Last check: 2007-10-29)
		This paper explores the analogy between mathematical Platonism and modal realism, and between Benacerraf's dilemma and the epistemological objection.
		It is argued that the parallels and contrasts may clarify both modal realism and the general problem of model epistemology.
		The paper begins with a sketch of Benacerraf's reasons for thinking that there is a prima facie conflict between a straightforward account of mathematical truth and a reasonable account of mathematical knowledge.
	www.oxfordscholarship.com /oso/public/content/philosophy/0199251487/acprof-0199251487-chapter-3.html (210 words)

	Philosophy of Mathematics (Site not responding. Last check: 2007-10-29)
		Though some of the texts are unfortunately rather technical, no knowledge of mathematics is assumed: almost every Friday (see the detailed calendar below) will be devoted to discussion, largely to help you get the point of some of the texts without (or with, depending on class interest) all the technical details.
		Paul Benacerraf and Hilary Putnam, eds., Philosophy of Mathematics: Selected Readings, 2nd edition (Cambridge University Press, New York, 1983).
		A wiki is a web page that can be edited through your browser, that is, one that you can not only read, but change and add to.
	zillion.philosophy.arizona.edu /~PhilosophyofMathematics/syll4.html (1249 words)

	Philosophy of Mind Bibliography, Part 4: Philosophy of Artificial Intelligence (Site not responding. Last check: 2007-10-29)
		Benacerraf's "paradox" is illusory; there are no strong consequences of Godel's theorem for mechanism.
		Reply to Good 1967: a human can trump any given machine, so the human is not the machine, whether or not the human is superior across the board.
		Lucas, J. Satan stultified: A rejoinder to Paul Benacerraf.
	consc.net /biblio/4.html (3554 words)

	Princeton - PWB 11/23/98 - Page one
		Paul Benacerraf came to Princeton as an undergraduate and stayed for 50 years so far
		In the autumn of 1948, Paul Benacerraf entered Princeton University as a freshman.
		Today, in this autumn of 1998, the Philosophy Department chair and former provost can boast 50 years of study (undergraduate and graduate), teaching and administrative service, all on this campus.
	www.princeton.edu /~compub/pwb/98/1123 (931 words)

	[No title]
		Paul Benacerraf is the classic expositor of two famous problems in the philosophy of mathematics.
		Paul Benacerraf has famously argued that nothing in the practice of mathematicians singles out a particular series of mathematical objects as the referents of number-terms.
		Topic to be determined in accordance with the interests of the group.
	www.st-andrews.ac.uk /~ar29/PY5306.html (692 words)

	Language, Mind, and Art : Essays in Appreciation and Analysis, In Honor of Paul Ziff (Synthese Library) by D. Jamieson ...
		This volume is a collection of essays in appreciation, analysis and honor of Paul Ziff, one of the leading American philosophers of the post-World War II period.
		The volume begins with a reminiscence by Paul Benacerraf, and ends with selections from an unpublished volume of plays by Paul Ziff.
		The volume should appeal to anyone whose work has been influenced by Ziff, or is interested in central philosophical problems concerning language, mind, and art.
	www.gettextbooks.com /isbn_0792328108.html (238 words)

	Hilary Putnam Bibliography
		Hilary Putnam and Paul Benacerraf (New York: Prentice-Hall, 1964), pp.
		Paul A. Schilpp (La Salle, Ill.: Open Court, 1974), vol.
		Lewis E. Hahn and Paul A. Schilpp (La Salle, Ill.: Open Court, 1986), pp.
	www.pragmatism.org /putnam (6164 words)

	Table of contents for Library of Congress control number 82025257 (Site not responding. Last check: 2007-10-29)
		Table of contents for Philosophy of mathematics : selected readings / edited by Paul Benacerraf, Hilary Putnam.
		Remarks on the definition and nature of mathematics Haskell B. Curry 12.
		What numbers could not be Paul Benacerraf 16.
	www.loc.gov /catdir/toc/cam025/82025257.html (162 words)

	Leiter Reports: A Group Blog (Jan. 23-May 31 2006): Summary of Major Philosophy Faculty Moves for 2004-05
		New York University: Added Paul Horwich (philosophy of language, philosophy of science) from CUNY and James Pryor (epistemology) from Princeton University.
		Paul Benacerraf (philosophy of math, logic) and Bas van Fraassen (philosophy of science, logic) are both phasing into retirement, and so will not be part of the faculty lists for the fall 2006 PGR surveys.
		Princeton is trying to recruit Paul Boghossian (philosophy of language and mind) from NYU and Alex Byrne (philosophy of mind, metaphysics) from MIT.
	leiterreports.typepad.com /blog/2005/09/summary_of_majo.html (2578 words)

	Memo to Paul Ernest
		Paul Ernest, School of Education, University of Exeter, UK FR: Kirby Urner, 4D Solutions, Portland, Oregon, USA RE: Your Social Constructivism as a Philosophy of Mathematics etc. CC:
		Synergetics-L (e-list and archives) Dear Sir, I was just now perusing your book, while sipping coffee, at our local Powell's Books on Hawthorne Blvd (while getting the oil changed in the Subaru at Jiffy Lube).
		I invite you to explore these exhibits and share any feedback: On Ludwig Wittgenstein's Contribution to a Pragmatic Philosophy http://www.teleport.com/~pdx4d/lw.html Investigations into the Linear Algebra Concepts used in the XYZ and Quadray Language Games
	www.grunch.net /synergetics/ernest.html (1154 words)

	PS842.htm
		Paul W. Humphreys, "Quantitative Probabilistic Causality and Structural Scientific Realism"
		Paul M. Churchland, "Subjective Qualia from a Materialist Point of View"
		Robert Cummins, "The Mind of the Matter: Comments on Paul Churchland"
	www.msu.edu /unit/phl/PSA/PS842.htm (412 words)

	John Simon Guggenheim Memorial Foundation 1967 Fellows Page
		Paul H. Avrich, Distinguished Professor Emeritus of History, Queens College, City University of NY: 1967.
		Paul Benacerraf, Stuart Professor of Philosophy & Provost, Princeton University: 1967.
		George Paul Georghiou, Professor of Entomology, University of California, Riverside: 1967.
	www.gf.org /67fellow.html (2649 words)

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