ΦNEWS Editorial

ΦNEWS Editorial ........................................................................ 3 ConstructionandSchematainMathematics ..................................... 4 MassesofFormalPhilosophy ................................................... 27 Towardsa TheoryofConvention ............................................... 30 ReturntoSelf-Reference ....................................................... 63 Announcements ................................................................ 65 Conferences 65 New Publications 92 AiML — Advances in Modal Logic 122 ASL — Association for Symbolic Logic 127 FoLLI — Foundation of Logic, Language and Information 131 Master in Pure and Applied Logic 132 Automated Reasoning, Web Ontologies and Multi-Agent Systems 135 FunnyFragments ............................................................. 137 ΦNEWS ...................................................................... 140 IndexofNames ............................................................... 143 ΦNEWS Volume 9 April 2006 • • Published by ΦLOG and Springer Edited by Vincent F. Hendricks, Pelle Guldborg Hansen, Stig Andur Pedersen and Dov M. Gabbay VOLUME 9 1 2006 ΦNEWS is published by ΦLOG The Network for Philosophical Logic and Its Applications Department of Philosophy and Science Studies Roskilde University, P6 P.O. Box 260 DK4000 Roskilde, Denmark Phone: (+45) 4674 2343 Fax: (+45) 4674 3012 Website: http://www.phinews.ruc.dk Website: http://www.philog.ruc.dk and Springer P.O. Box 17 3300 AA Dordrecht, The Netherlands Website: http://www.springeronline.com Edited by Vincent F. Hendricks (Editor-in-chief, [email protected]) Pelle Guldborg Hansen (Editor-in-chief, [email protected]) Stig Andur Pedersen (Editor, [email protected]) Dov M. Gabbay (Editor, [email protected]) Volume 9, April 2006 Visit http://www.phinews.ruc.dk for a free subscription Concept, LATEX2ε-Layout, Graphics and Sleeve Designs by Vincent F. Hendricks Typeset in LATEX2ε Printed by Roskilde University Press, Denmark 2006 Sleeve printed by Grønager’s — Grafisk Produktion AS, Denmark 2006 ISSN 1602-1444 c 2006 by Vincent F. Hendricks, Pelle Guldborg Hansen, Stig Andur Pedersen, Dov M. Gabbay, ° ΦLOG and Springer ΦLOG is sponsored by The Danish Research Council for the Humanities VOLUME 9 2 2006 EDITORIAL ΦNEWS Welcome to the ninth volume of ΦNEWS–The Free Newsletter for Philosophical Logic and Its Applications, published jointly by ΦLOG –The Network for Philosophical Logic and Its Applications and Springer. ΦLOG is sponsored by The Danish Research Council for the Humanities. ΦNEWS is published biannually. The newsletter is freely distributed to subscribers and an on-line copy may be obtained from the ΦNEWS website at http://www.phinews.ruc.dk Volume 9 includes the papers ‘Construction and Schemata in Mathematics’ by Klaus Frovin Jørgensen and ‘Towards a Theory of Convention’ by Pelle Guldborg Hansen, information about the Masses of Formal Philosophy project, announcements of upcoming conferences, new publications, initiatives, programs and announcements from AiML — Advances in Modal Logic, ASL — Association for Symbolic Logic, FoLLI — Foundation of Logic, Language and Information, TARK — Theoretical Aspects of Reasoning about Knowledge. The next volume of ΦNEWS is scheduled for October 2006. ΦNEWS publishes contributions in terms of expositional papers, announcements of workshops, seminars, conferences, forthcoming publications, new initiatives and other material within the aim and scope of the newsletter. Send your written contribution (preferably in TEX, LATEX, LATEX2ε) to either one of the ΦNEWS editors. Contact information, and additional information on how to submit material, is available on page 140. April, 2006 Vincent F. Hendricks Pelle Guldborg Hansen Stig Andur Pedersen Dov M. Gabbay Editors VOLUME 9 3 2006 CONSTRUCTION AND SCHEMATA IN MATHEMATICS KLAUS FROVIN JØRGENSEN Department of Philosophy and Science Studies Roskilde University, Denmark [email protected] ΦNEWS Recently I read Patrick Suppes claiming that “the most important open scandal in philosophy is the problem of free will” [17, p. 205]. I very much agree with Suppes thattheproblemofthefreewillisamajorpuzzle,whichweshouldtrytogeta better understanding of by examining the deeper issues connected with the free will. This essay, however, does not treat the problem of the free will. It concerns the problems of the ontology and epistemology of mathematics. In general, the problems of the philosophy of mathematics are just as old and–if it makes sense to talk about solvability of such problems–perhaps just as unsolved as the problem of the free will. Mathematics is a very important ingredient of knowledge. In its most simple form mathematics plays a necessary role in our understanding of the surrounding world and is necessary for solving simple problems of ordinary life. At the other end of the simplicity-scale we find the mathematics as used in science. Also here mathematics has a necessary role in our descriptions of nature and the way in which we are involved with it and each other. It truly amazes me that there seems to be very little consensus with respect to the ontology and epistemology of mathematics. There can be no doubt that the natural numbers exit; I mean, we use them all the time. The question is only: In which sense? This question I have treated this question elsewhere [13], thus I will here be more interested in the existence of mathematical objects whose nature seems more abstract. Platonism is an important part of the philosophy of mathematics. The upshot of Platonism is its simplicity with respect to existence and truth: Mathematical objects are existing non-temporally and non-spatially in a universe which we can access only by rational thinking. They exist eternally and independently of human beings and are perfectly real. Mathematical statements are therefore objectively true or false and their truth-value is also independent from humans.1 Within Platonism one introduces in this way a dualism2 and from which it follows that all kinds of constructivism are incompatible with Platonism. As Plato himself expresses it: 1 Kurt Gödel, for instance, expressed views in this directions [8]. 2 It is well-known what the general problem of dualisms is: How are the dual worlds connected? VOLUME 9 4 2006 ΦNEWS [...] noonewhohasevena slightacquaintancewithgeometrywilldenythat the nature of this science is in flat contradiction with the absurd language used by mathematicians, for want of better terms. They constantly talk of “opera- tions” like “squaring”, “applying”, “adding”, and so on, as if the object were to do something, whereas the true purpose of the whole subject is knowledge — knowledge, moreover, of what eternally exists, not of anything that comes to be this or that at some time and ceases to be. (Republic 527a) I find this element of Platonism rather problematic and in the following I would like to outline a philosophy of mathematics which on the one hand takes the language used by mathematicians as serious as possible: When mathematicians talk, for instance, about constructing the real numbers what can they possibly mean? On the other hand I have no intensions towards ‘taking the fists from the boxer’. On the contrary, I think to much energy has been put into producing normative philosophy of mathematics in discussing what is ‘good’ or what is ‘bad’ mathematics. I want to do a certain kind of naturalized philosophy of mathematics. We see mathematics is being done in a great many aspects of the human life. I would like to know what is going on–the interest is not in the direction of what should be going on. Equivalence classes and quotient structures Mathematicians often talk about constructionsinconnectionwithquotientstructures. Let X be some set and suppose is an equivalence relation on X.3 Given any element x of X we now construct another∼ object [x] which is the collection of all elements which are equivalent to x, i.e., [x]= y X x y . { ∈ | ∼ } [x] is called the equivalence class generated by x under and the question now is, in which sense the equivalence class is constructed andinwhichsenseit∼ exists.Given such an X and we quite often collect all the equivalence classes and form a new set X/ consisting∼ of all the equivalence classes, in other words: ∼ a X/ if and only if, a is an equivalence class generated ∈ ∼ by some element of X under . ∼ By extensionality of sets we can lift the equivalence relation on X to an identity relation on X/ . ∼ ∼ [x]=[y] if and only if x y. ∼ 3 That is an equivalence relation on X means that tree conditions obtain for with respect to elements of∼ X. Firstly, any element x is related to itself, i.e., x x;secondly,x ∼y implies y x; and thirdly, x y and y z implies x z. ∼ ∼ ∼ ∼ ∼ ∼ VOLUME 9 5 2006 ΦNEWS For at very simple example let X be the set of all cars. If x y obtains just in case x has the same color as does y,then is an equivalence relation.∼ 4 We can then form X/ which consists of all car-colors,∼ so to speak. Philosophers would tend to think of ∼X/ as a collection of concepts, whereas mathematicians would think of it as a collection∼ of new objects. We can also construct Q as the collection of equivalence classes of ordered pairs of whole numbers (a, b),withb =0. The equivalence relation is defined by 6 (a, b) (c, d) if and only if ad = bc. ∼ The equivalence class [(a, b)] is then understood as the rational number a/b. More generally, we can construct equivalence classes in the following way. Let f be a function form X to Y .Thenf defines an equivalence relation on X by x y if and only if f(x)=f(y). ∼ Here [x] is the set of all elements y, which are mapped to f(x). The equivalence relation is the kernel of f: ker f = (x, y) X X f(x)=f(y) . { ∈ × | } Now suppose that we are given even more. Let us look at the case where we are given a homomorphism ϕ : X Y where X and Y are structures, e.g., groups or rings, of the same type.5 Now,→ a homomorphism ϕ : X Y is a certain map- ping which preservers the structure of X to Y , in the sense→ that within Y there is a sub-structure which in a certain sense imitates X.

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