Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.

The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...) existence of wide sets. Drawing upon Cantor’s notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”. (shrink)

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The categorical duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The hyperintensional profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal and hyperintensional profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is (...) genuinely logical. Second, the foregoing provides a hyperintensional account of the interpretation of mathematical and metamathematical vocabulary. (shrink)

CUPRINS CONTUR Re-Introducere, sau: Dincolo de „teoria şi practica” informării şi documentării – Spre o hermeneutică necesară Viorica Sâncrăian Atelier Philobiblon FOCUS Gheroghe Vais Biblioteca Universităţii din Cluj, 1906-1909 Dénes Győrfi Gyalui Farkas – fost director adjunct al bibliotecii universităţii din Cluj Vladimir F. Wertsman Seria filatelică multiculturală Librariana Meda-Diana Hotea „O scriere chineză în cifre arabe” Carmen Crişan Utilizarea bazelor de date ştiinţifice abonate de Biblioteca Centrala Universitara Lucian Blaga în anul 2005 Gabriela Morărescu Anul 2005 – o nouă (...) abordare a bibliotecilor filiale Mariana Falup Realizări şi perspective în automatizare şi modernizare Maria Petrescu Digitizarea documentelor culturale Alina Ioana Şuta De la schimbul internaţional de publicaţii la schimbul experienţei – un contact polonez Marcela-Georgeta Groza Biblioteca de Matematică şi Informatică între ieri şi mâine Luminiţa Tomuţa Remodelarea unei biblioteci – Remodelarea mentalităţilor Emilia-Maria Soporan O prietenie activă pe tărâmul multiculturalităţii: Andrei Pippidi - Adrian Marino Costel Dumitraşcu Informaţiile bibliografice sau: Pe urmele cărţii în bibliotecă ORIZONTURI Pavel Puşcaş Astrolabium Ioan Mihai Cochinescu Alchimia şi muzica Alin Mihai Gherman Muzică şi literatură – Aproximaţii şi însemnări ale unui meloman înrăit Monica Gheţ Muzica împotriva ciumei Florina Iliş Adrian Marino şi ideea de literatură din perspectivă hermeneutică Rodica Frenţiu Tatakau Hikaku Bungaku – Adrian Marino şi comparatismul militant în Japonia Mariana Soporan Fondul arhivistic Adrian Marino din Biblioteca Centrală Universitară „Lucian Blaga” Gábor Győrffy Adrian Marino: alternativa autohtonă a culturii libere Sidonia Grama Între spaţii ale amintirii şi locuri ale memoriei – Comemorarea a 15 ani de la revoluţie în Timişoara REFLEXII Ioana Robu- Sally Wood-Lamont Cea de-a 10-a Conferinţă a Asociaţiei Europene pentru Informare şi Biblioteci Medicale (EAHIL) Felix Ostrovschi Un alt fel de discurs – Adrian Marino Liana Iancu Radio, presă, carte – Iancu Tiberiu – şi după Kinga Tamás Váczy Leona (1913-1995) Biblioteca, spaţiu intelectual al omului sau: Trecut pentru viitor Adrian Grănescu Emil Pintea (16 decembrie 1944 - 6 ianuarie 2004) – polivalenţa unui destin Melinda Éva Szász Pasiune pentru carte şi artă – Dénes Gábor Meda-Diana Hotea „Lumină din lumină”: primăvară pascală – Expoziţie (07 aprilie - 03 mai 2005) Kolumbán Judit Expoziţia de manuscrise de sec. XVI-XVIII în Biblioteca Centrală Universitară „Lucian Blaga” – Homo scribens: genurile memoriei si tipologia scrierii în secolele XVI-XVIII-lea (Homo scribens: emlékezéskultúra és íráshasználati-szokások a XVI-XVIII században) Meda-Diana Hotea „Cartea cărţilor” – Memoria credinţei – Expoziţie (10.02 - 25.02.2005) Raluca Soare Clădirile clujene sau martorii tăcuţi Adrian Grănescu (Despre) Managementul construcţiilor de biblioteci – recenzie şi marginalii Maria-Stela Constantinescu-Matiţa Mircea Popa: Andrei VERESS – un bibliograf maghiar, prieten al românilor (Recenzie) István Király V. Activitatea Ştiinţifică a Universităţii „Babeş-Bolyai” – 2005, sau de la povara... la demnitatea istoriei Bodnár Róbert Pe urmele unei biblioteci pierdute Raluca Soare Un om, o carte, o bibliotecă – Traian Brad, un slujitor al cărţii Ildikó Bán Lidia Kulikovski: Accesul persoanelor dezavantajate la potenţialul bibliotecilor (Manual pentru bibliotecari) Chişinău, Editura Epigraf, 2006 (Recenzie) Iacob Mârza Meda-Diana Hotea: Catalogul cărţii rare din colecţiile B. C. U. „Lucian Blaga”– Donaţia Gh. Sion Vol. I (sec. XVI-XVIII) (Recenzie) Ruxandra Cesereanu Adrian Marino între unit-ideas şi Zeitgeist Iulia Grad Filozofia evreiască: între Ierusalim şi Atena Raluca Soare De-o parte şi de alta a cărţilor Adrian Grănescu (Recenzie) Boglárka Daróczi Literatura germană pentru copii apărută în România între 1944-1989 – O bibliografie. (shrink)

ABSTRACT Theories of sets such as Zermelo Fraenkel set theory are usually presented as the combination of two distinct kinds of principles: logical and set-theoretic principles. The set-theoretic principles are imposed ‘on top’ of first-order logic. This is in agreement with a traditional view of logic as universally applicable and topic neutral. Such a view of logic has been rejected by the intuitionists, on the ground that quantification over infinite domains requires the use of intuitionistic rather than classical logic. (...) In the following, I consider constructive set theories, which use intuitionistic rather than classical logic, and argue that they manifest a distinctive interdependence or an entanglement between sets and logic. In fact, Martin-Löf type theory identifies fundamental logical and set-theoretic notions. Remarkably, one of the motivations for this identification is the thought that classical quantification over infinite domains is problematic, while intuitionistic quantification is not. The approach to quantification adopted in Martin-Löf’s type theory is subtly interconnected with its predicativity. I conclude by recalling key aspects of an approach to predicativity inspired by Poincaré, which focuses on the issue of correct quantification over infinite domains and relate it back to Martin-Löf type theory. (shrink)

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated by Herbrand’s Induction-Axiom (...) Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

In this paper, I consider the use of mathematical results in philosophical arguments arriving at conclusions with non-mathematical content, with the view that in general such usage requires additional justification. As a cautionary example, I examine Kreisel’s arguments that the Continuum Hypothesis is determined by the axioms of Zermelo-Fraenkel set theory, and interpret Weston’s 1976 reply as showing that Kreisel fails to provide sufficient justification for the use of his main technical result. If we take the perspective that mathematical (...) results are used in the context of a modelling of something not necessarily mathematical, then the situation is clarified somewhat, and the procedure for arriving at justification for the use of such results becomes clear. I give an example of a particularly strong form this justification might take, using the idea of formalism independence due to Gödel and Kennedy. (shrink)

This paper introduces the theory QST of quasets as a formal basis for the Nmatrices. The main aim is to construct a system of Nmatrices by substituting standard sets by quasets. Since QST is a conservative extension of ZFA (the Zermelo-Fraenkel set theory with Atoms), it is possible to obtain generalized Nmatrices (Q-Nmatrices). Since the original formulation of QST is not completely adequate for the developments we advance here, some possible amendments to the theory are also considered. One of (...) the most interesting traits of such an extension is the existence of complementary quasets which admit elements with undetermined membership. Such elements can be interpreted as quantum systems in superposed states. We also present a relationship of QST with the theory of Rough Sets RST, which grants the existence of models for SQT formed by rough sets. Some consequences of the given formalism for the relation of logical consequence are also analysed. (shrink)

Cuprins CONTUR Re-Introducere sau: Dincolo de „teoria şi practica” informării şi documentării – Spre o hermeneutică posibilă şi necesară Proiectul şi Programul PHILOBIBLON( în noua formulare) FOCUS Dana Stana, Omonimia şi paronimia în bibliologie Victoria Frâncu, Profesia de bibliotecar la graniţa dintre spaţiul bibliotecii şi ciberspaţiu Olimpia Curta, Laboratorul de informatică şi profesioniştii săi Ionel Enache, Fundamentele teoretice ale marketingului de bibliotecă Maria Petrescu, Bibliotecile digitale şi impactul lor asupra tinerilor Adriana Szekely, Liana Grigore, Bibliorev – în continuă schimbare (...) István Király V., Proiect în vederea Acreditării ISI al revistei PHILOBIBLON Valeria Salánki, Cultura organizaţională. Propunere de studiu asupra culturii organizaţionale – octombrie 2008 Marian Petcu, Originile faptului divers în presa română Tudor-George Pereverza Expresia bibliografică a Iconografiei eminesciene (Fotografie şi artă plastică, 1939-1989) Vlad A. Codrea, Gabriela-Rodica Morărescu, Forray Erzsébet, Catalogus Raritatum et Benefactorum, un manuscris reprezentativ din perioada de început a Muzeului de Ştiinţele Naturii din Aiud 4 Alin Mihai Gherman, Pornind de la o Bucoavnă necunoscută Roxana Bălăucă, Tipărituri franceze în colecţiile BCU „Lucian Blaga”: secolul al XVIII-lea Maria-Stela Constantinescu-Matiţa, Szabó Károly – bibliotecar, bibliograf şi istoric Roxana Bălăucă, Theodor Aman în Donaţia Sion Margareta Berchez, Îmbogăţirea colecţiilor Bibliotecii Universităţii de Ştiinţe Agricole şi Medicină Veterinară Cluj-Napoca prin schimbul de publicaţii de-a lungul timpului Ioana Rotund, O oază francofonă la Cluj Ilona Okos-Rigó, Biblioteca şi Grădina Botanică Clujeană Gabriela Pop, Institutul de Iudaistică şi Istorie Evreiască „Dr. Moshe Carmilly” şi Biblioteca de Studii Iudaice Mariana Falup, Împrumutul interbibliotecar intern şi internaţional şi livrarea de documente la B.C.U. Cluj-Napoca ORIZONTURI Doru Radosav, Viaţa ca alterego. Petrea Icoanei: travesti şi clandestinitate în mişcarea de rezistenţă anticomunistă Ionuţ Costea, Mitbiografia între propagandă şi memorie Gabriela Morărescu ;Vlad Codrea, Preocupări pentru cunoaşterea şi protecţia naturii în scrierile unui entuziast naturalist al secolului XIX: Basiliu Basiota Alin Mihai Gherman, Vechi lexic bibliotecăresc Raluca Betea, Influenţa artei Renaşterii asupra decoraţiei icoanelor din secolele XVI-XVIII din Transilvania şi Maramureş 5 Monica Mureşan, Căsătoria civilă ca aspect al modernizării societăţii transilvănene la sfârşitul secolului al XIX –lea. – Discurs oficial şi receptare socială reflectate în presa vremii Ancuţa-Lăcrimioara Chiş, Diferenţa şi discriminarea socio-politică a femeii Irina Petraş, Casa, locul vieţii (fragmente) Monica Mureşan Pentru o istorie a morţii în peisajul istoriografic românesc - prezentarea unor contribuţii colective recente: Religiozitate şi atitudini în faţa morţii în spaţiul transilvan din premodernitate până în secolul XX, coord. Mihaela Grancea, 2005 şi Discursuri despre moarte în Transilvania secolelor XVI-XX, coord Mihaela Grancea şi Ana Dumitran, 2006. REFLEXII Olimpia Curta, Reviste electronice. Baze şi perspective de Alice Keller – Recenzie Adrian Grănescu, Arhitectura Clinicilor Universitare din Cluj 1886-1903 Dorina Buia, Itinerar de suflet la Tăul Muced Raluca Soare, De la teoriile ataşamentului la tehnicile de intervenţie în psihoterapie. (Školka Enikő – Teorii explicative, Modele şi Tehnici de intervenţie în psihologie clinică şi psihoterapie) Sidonia Nedeianu Grama, Mitbiografia politică în istoria orală Raluca Soare, Opera bibliothecariorum 1990-2007 – Recenzie Ionuţ Costea, Kovács Mária, A kolozsvári „Lucian Blaga” Központi Egyetemi Könyvtár 19. századi magyar nyelvű kéziratainak katalógusa. Első kötet: történelmi és földrajzi kéziratok/ Catalogul manuscriselor maghiare din secolul al 19-lea din colecţiile Bibliotecii Centrale Universitare „Lucian Blaga” Volumul I: Manuscrise de istorie şi geografie, Editura Argonaut, Cluj-Napoca, 2007, 218 p. – Recenzie 6 Ana Maria Căpâlneanu, Lola Maria Petrescu – Biblioteca şi provocările secolului XXI, Cluj-Napoca: Risoprint, 2007 István Király V, Bibliografia ca instrument al clarificării de sine Kinga Tamás, Viorica Sâncrăian - un coleg şi un bibliotecar de excepţie În colecţia BIBLIOTHECA BIBLIOLOGICA au apărut Revista PHILOBIBLON – Volumele apărute. (shrink)

A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds, Watson (2010) suggests that Bohn‘s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of (...) junky worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo-Fraenkel set theory ZF, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: Topology (and possibly other "algebraic" mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds. (shrink)

Jakob Friedrich Fries (1773-1843): A Philosophy of the Exact Sciences -/- Shortened version of the article of the same name in: Tabula Rasa. Jenenser magazine for critical thinking. 6th of November 1994 edition -/- 1. Biography -/- Jakob Friedrich Fries was born on the 23rd of August, 1773 in Barby on the Elbe. Because Fries' father had little time, on account of his journeying, he gave up both his sons, of whom Jakob Friedrich was the elder, to the Herrnhut Teaching (...) Institution in Niesky in 1778. Fries attended the theological seminar in Niesky in autumn 1792, which lasted for three years. There he (secretly) began to study Kant. The reading of Kant's works led Fries, for the first time, to a deep philosophical satisfaction. His enthusiasm for Kant is to be understood against the background that a considerable measure of Kant's philosophy is based on a firm foundation of what happens in an analogous and similar manner in mathematics. -/- During this period he also read Heinrich Jacobi's novels, as well as works of the awakening classic German literature; in particular Friedrich Schiller's works. In 1795, Fries arrived at Leipzig University to study law. During his time in Leipzig he became acquainted with Fichte's philosophy. In autumn of the same year he moved to Jena to hear Fichte at first hand, but was soon disappointed. -/- During his first sojourn in Jenaer (1796), Fries got to know the chemist A. N. Scherer who was very influenced by the work of the chemist A. L. Lavoisier. Fries discovered, at Scherer's suggestion, the law of stoichiometric composition. Because he felt that his work still need some time before completion, he withdrew as a private tutor to Zofingen (in Switzerland). There Fries worked on his main critical work, and studied Newton's "Philosophiae naturalis principia mathematica". He remained a lifelong admirer of Newton, whom he praised as a perfectionist of astronomy. Fries saw the final aim of his mathematical natural philosophy in the union of Newton's Principia with Kant's philosophy. -/- With the aim of qualifying as a lecturer, he returned to Jena in 1800. Now Fries was known from his independent writings, such as "Reinhold, Fichte and Schelling" (1st edition in 1803), and "Systems of Philosophy as an Evident Science" (1804). The relationship between G. W. F. Hegel and Fries did not develop favourably. Hegel speaks of "the leader of the superficial army", and at other places he expresses: "he is an extremely narrow-minded bragger". On the other hand, Fries also has an unfavourable take on Hegel. He writes of the "Redundancy of the Hegelistic dialectic" (1828). In his History of Philosophy (1837/40) he writes of Hegel, amongst other things: "Your way of philosophising seems just to give expression to nonsense in the shortest possible way". In this work, Fries appears to argue with Hegel in an objective manner, and expresses a positive attitude to his work. -/- In 1805, Fries was appointed professor for philosophy in Heidelberg. In his time spent in Heidelberg, he married Caroline Erdmann. He also sealed his friendships with W. M. L. de Wette and F. H. Jacobi. Jacobi was amongst the contemporaries who most impressed Fries during this period. In Heidelberg, Fries wrote, amongst other things, his three-volume main work New Critique of Reason (1807). -/- In 1816 Fries returned to Jena. When in 1817 the Wartburg festival took place, Fries was among the guests, and made a small speech. 1819 was the so-called "Great Year" for Fries: His wife Caroline died, and Karl Sand, a member of a student fraternity, and one of Fries' former students stabbed the author August von Kotzebue to death. Fries was punished with a philosophy teaching ban but still received a professorship for physics and mathematics. Only after a period of years, and under restrictions, he was again allowed to read philosophy. From now on, Fries was excluded from political influence. The rest of his life he devoted himself once again to philosophical and natural studies. During this period, he wrote "Mathematical Natural Philosophy" (1822) and the "History of Philosophy" (1837/40). -/- Fries suffered from a stroke on New Year's Day 1843, and a second stroke, on the 10th of August 1843 ended his life. -/- 2. Fries' Work Fries left an extensive body of work. A look at the subject areas he worked on makes us aware of the universality of his thinking. Amongst these subjects are: Psychic anthropology, psychology, pure philosophy, logic, metaphysics, ethics, politics, religious philosophy, aesthetics, natural philosophy, mathematics, physics and medical subjects, to which, e.g., the text "Regarding the optical centre in the eye together with general remarks about the theory of seeing" (1839) bear witness. With popular philosophical writings like the novel "Julius and Evagoras" (1822), or the arabesque "Longing, and a Trip to the Middle of Nowhere" (1820), he tried to make his philosophy accessible to a broader public. Anthropological considerations are shown in the methodical basis of his philosophy, and to this end, he provides the following didactic instruction for the study of his work: "If somebody wishes to study philosophy on the basis of this guide, I would recommend that after studying natural philosophy, a strict study of logic should follow in order to peruse metaphysics and its applied teachings more rapidly, followed by a strict study of criticism, followed once again by a return to an even closer study of metaphysics and its applied teachings." -/- 3. Continuation of Fries' work through the Friesian School -/- Fries' ideas found general acceptance amongst scientists and mathematicians. A large part of the followers of the "Fries School of Thought" had a scientific or mathematical background. Amongst them were biologist Matthias Jakob Schleiden, mathematics and science specialist philosopher Ernst Friedrich Apelt, the zoologist Oscar Schmidt, and the mathematician Oscar Xavier Schlömilch. Between the years 1847 and 1849, the treatises of the "Fries School of Thought", with which the publishers aimed to pursue philosophy according to the model of the natural sciences appeared. In the Kant-Fries philosophy, they saw the realisation of this ideal. The history of the "New Fries School of Thought" began in 1903. It was in this year that the philosopher Leonard Nelson gathered together a small discussion circle in Goettingen. Amongst the founding members of this circle were: A. Rüstow, C. Brinkmann and H. Goesch. In 1904 L. Nelson, A. Rüstow, H. Goesch and the student W. Mecklenburg travelled to Thuringia to find the missing Fries writings. In the same year, G. Hessenberg, K. Kaiser and Nelson published the first pamphlet from their first volume of the "Treatises of the Fries School of Thought, New Edition". -/- The school set out with the aim of searching for the missing Fries' texts, and re-publishing them with a view to re-opening discussion of Fries' brand of philosophy. The members of the circle met regularly for discussions. Additionally, larger conferences took place, mostly during the holidays. Featuring as speakers were: Otto Apelt, Otto Berg, Paul Bernays, G. Fraenkel, K. Grelling, G. Hessenberg, A. Kronfeld, O. Meyerhof, L. Nelson and R. Otto. On the 1st of March 1913, the Jakob-Friedrich-Fries society was founded. Whilst the Fries' school of thought dealt in continuum with the advancement of the Kant-Fries philosophy, the members of the Jakob-Friedrich-Fries society's main task was the dissemination of the Fries' school publications. In May/June, 1914, the organisations took part in their last common conference before the gulf created by the outbreak of the First World War. Several members died during the war. Others returned disabled. The next conference took place in 1919. A second conference followed in 1921. Nevertheless, such intensive work as had been undertaken between 1903 and 1914 was no longer possible. -/- Leonard Nelson died in October 1927. In the 1930's, the 6th and final volume of "Treatises of the Fries School of Thought, New Edition" was published. Franz Oppenheimer, Otto Meyerhof, Minna Specht and Grete Hermann were involved in their publication. -/- 4. About Mathematical Natural Philosophy -/- In 1822, Fries' "Mathematical Natural Philosophy" appeared. Fries rejects the speculative natural philosophy of his time - above all Schelling's natural philosophy. A natural study, founded on speculative philosophy, ceases with its collection, arrangement and order of well-known facts. Only a mathematical natural philosophy can deliver the necessary explanatory reasoning. The basic dictum of his mathematical natural philosophy is: "All natural theories must be definable using purely mathematically determinable reasons of explanation." Fries is of the opinion that science can attain completeness only by the subordination of the empirical facts to the metaphysical categories and mathematical laws. -/- The crux of Fries' natural philosophy is the thought that mathematics must be made fertile for use by the natural sciences. However, pure mathematics displays solely empty abstraction. To be able to apply them to the sensory world, an intermediatory connection is required. Mathematics must be connected to metaphysics. The pure mechanics, consisting of three parts are these: a) A study of geometrical movement, which considers solely the direction of the movement, b) A study of kinematics, which considers velocity in Addition, c) A study of dynamic movement, which also incorporates mass and power, as well as direction and velocity. -/- Of great interest is Fries' natural philosophy in view of its methodology, particularly with regard to the doctrine "leading maxims". Fries calls these "leading maxims" "heuristic", "because they are principal rules for scientific invention". -/- Fries' philosophy found great recognition with Carl Friedrich Gauss, amongst others. Fries asked for Gauss's opinion on his work "An Attempt at a Criticism based on the Principles of the Probability Calculus" (1842). Gauss also provided his opinions on "Mathematical Natural Philosophy" (1822) and on Fries' "History of Philosophy". Gauss acknowledged Fries' philosophy and wrote in a letter to Fries: "I have always had a great predilection for philosophical speculation, and now I am all the more happy to have a reliable teacher in you in the study of the destinies of science, from the most ancient up to the latest times, as I have not always found the desired satisfaction in my own reading of the writings of some of the philosophers. In particular, the writings of several famous (maybe better, so-called famous) philosophers who have appeared since Kant have reminded me of the sieve of a goat-milker, or to use a modern image instead of an old-fashioned one, of Münchhausen's plait, with which he pulled himself from out of the water. These amateurs would not dare make such a confession before their Masters; it would not happen were they were to consider the case upon its merits. I have often regretted not living in your locality, so as to be able to glean much pleasurable entertainment from philosophical verbal discourse." -/- The starting point of the new adoption of Fries was Nelson's article "The critical method and the relation of psychology to philosophy" (1904). Nelson dedicates special attention to Fries' re-interpretation of Kant's deduction concept. Fries awards Kant's criticism the rationale of anthropological idiom, in that he is guided by the idea that one can examine in a psychological way which knowledge we have "a priori", and how this is created, so that we can therefore recognise our own knowledge "a priori" in an empirical way. Fries understands deduction to mean an "awareness residing darkly in us is, and only open to basic metaphysical principles through conscious reflection.". -/- Nelson has pointed to an analogy between Fries' deduction and modern metamathematics. In the same manner, as with the anthropological deduction of the content of the critical investigation into the metaphysical object show, the content of mathematics become, in David Hilbert's view, the object of metamathematics. -/-. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

In this paper a class of languages which are formal enough for mathematical reasoning is introduced. Its languages are called mathematically agreeable. Languages containing a given MA language L, and being sublanguages of L augmented by a monadic predicate, are constructed. A mathematical theory of truth (shortly MTT) is formulated for some of those languages. MTT makes them fully interpreted MA languages which posses their own truth predicates. MTT is shown to conform well with the eight norms formulated for theories (...) of truth in the paper 'What Theories of Truth Should be Like (but Cannot be)', by Hannes Leitgeb. MTT is also free from infinite regress, providing a proper framework to study the regress problem. Main tools used in proofs are Zermelo-Fraenkel (ZF) set theory and classical logic. (shrink)

Para matemáticos interesados en problemas de fundamentos, lógico-matemáticos y filósofos de la matemática, el axioma de elección es centro obligado de reflexión, pues ha sido considerado esencial en el debate dentro de las posiciones consideradas clásicas en filosofía de la matemática (intuicionismo, formalismo, logicismo, platonismo), pero también ha tenido una presencia fundamental para el desarrollo de la matemática y metamatemática contemporánea. Desde una posición que privilegia el quehacer matemático, nos proponemos mostrar los aportes que ha tenido el axioma en varias (...) áreas fundamentales de la matemática, su aplicación en la lógica de primer orden, así como una breve descripción de las pruebas de consistencia relativa debidas a Gödel y Cohen, las cuales establecieron su independencia del sistema axiomático Zermelo-Fraenkel (ZF). Con todo lo anterior mostraremos cómo el quehacer matemático contemporáneo se adscribe al platonismo matemático en los términos de Bernays y Ferreirós. Revisaremos también los argumentos de Zermelo y Cantor para permitir el uso de asunciones en la matemática, los cuales se acercan a los planteamientos de la investigación científica y esbozan relaciones con la filosofía de la práctica matemática. Finalmente, justificamos el uso del axioma de elección en la contemporaneidad, abogando por unas relaciones de equidad entre la matemática y la filosofía, presentando además su plena vigencia, a través de la referencia a algunos problemas abiertos en la actualidad que vinculan el axioma de elección con la teoría de Ramsey. (shrink)

In this paprer a class of so called mathematically acceptable (shortly MA) languages is introduced First-order formal languages containing natural numbers and numerals belong to that class. MA languages which are contained in a given fully interpreted MA language augmented by a monadic predicate are constructed. A mathematical theory of truth (shortly MTT) is formulated for some of these languages. MTT makes them fully interpreted MA languages which posses their own truth predicates, yielding consequences to philosophy of mathematics. MTT is (...) shown to conform well with the eight norms presented for theories of truth in the paper 'What Theories of Truth Should be Like (but Cannot be)' by Hannes Leitgeb. MTT is also free from infinite regress, providing a proper framework to study the regress problem. (shrink)