Mathematics has always been a core part of western education, from the medieval quadrivium to the large amount of arithmetic and algebra still compulsory in high schools. It is an essential part. Its commitment to exactitude and to rigid demonstration balances humanist subjects devoted to appreciation and rhetoric as well as giving the lie to postmodernist insinuations that all “truths” are subject to political negotiation. In recent decades, the character of mathematics has changed – or rather broadened: it has (...) become the enabling science behind the complexity of contemporary knowledge, from gene interpretation to bank risk. Mathematical understanding is all the more necessary for future jobs, as well as remaining, as ever, a prophylactic against the more corrosive philosophical views emanating from the humanities. (shrink)

We compare the medieval projects of commentaries and disputations with the modern projects of formal ontology and of mathematics.

The city of Kruja (Albania)contains three types of dwellings that date back to different periods of time: the historic ones, the socialist ones, the modern ones. This paper has to deal only with the historic building's typology. The questionnaire that is applied will be considered for the development of mathematical regression based on specific data for this category. Variation between the relevant variables of the questionnaire is fairly or inverse-linked with a certain percentage of influence. The aim of this study (...) is to have better insights into the important role of building occupancy, people's lifestyle, economic level, and the quality of life of the residents in the inner medieval citadel, the physics characteristics, and the energy efficiency of the historical dwellings. The variables such as the number of residents, quality of life, the surface of the dwellings, heating instruments and time spent in the dwelling, current living conditions, monthly bills, level of satisfaction, restoration, and building improvements, interact with each other with probabilities with different positive or negative percentages. (shrink)

This article seeks the origin, in the theories of Ibn al-Haytham (Alhazen), Descartes, and Berkeley, of two-stage theories of spatial perception, which hold that visual perception involves both an immediate representation of the proximal stimulus in a two-dimensional ‘‘sensory core’’ and also a subsequent perception of the three dimensional world. The works of Ibn al-Haytham, Descartes, and Berkeley already frame the major theoretical options that guided visual theory into the twentieth century. The field of visual perception was the first area (...) of what we now call psychology to apply mathematics, through geometrical models as used by Euclid, Ptolemy, Ibn al-Haytham, and Descartes (among others). The article shows that Kepler’s discovery of the retinal image, which revolutionized visual anatomy and entailed fundamental changes in visual physiology, did not alter the basic structure of theories of spatial vision. These changes in visual physiology are advanced especially in Descartes' Dioptrics and his L'Homme. Berkeley develops a radically empirist theory vision, according to which visual perception of depth is learned through associative processes that rely on the sense of touch. But Descartes and Berkeley share the assertion that there is a two-dimensional sensory core that is in principle available to consciousness. They also share the observation that we don't usually perceived this core, but find depth and distance to be phenomenally immediate, a point they struggle to accommodate theoretically. If our interpretation is correct, it was not a change in the theory of the psychology of vision that engendered the idea of a sensory core, but rather the introduction of the theory into a new metaphysical context. (shrink)

Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are devoted. (...) At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

Il contributo consiste in una analisi e commento della prima sezione del De docta ignorantia di Nicola Cusano (capp. I-XVI), dedicata ai concetti di precisione matematica e di uguaglianza. Il saggio offre la possibilità di ripercorrere la teoria della conoscenza di Cusano, laddove l'impossibilità per la ragione di giungere ad una mens-ura precisa dell'oggetto da conoscere non si trasforma in una mera cultura del limite, bensì la filosofia negativa diviene base per la mistica. In altri termini l'obiettivo specifico del saggio (...) è rilevare come il metodo della dotta ignoranza apra alla conoscenza dell'infinito proprio nel momento in cui essa nega ogni conoscenza razionale precisa. La matematica diviene dunque il fondamento su cui far leva per innalzare la propria mente (transferre) verso la conoscenza simbolica di Dio, nel medesimo solco tracciato da Dionigi e dalla tradizione della mistica speculativa. (shrink)

The present Yearbook (which is the fourth in the series) is subtitled Trends & Cycles. Already ancient historians (see, e.g., the second Chapter of Book VI of Polybius' Histories) described rather well the cyclical component of historical dynamics, whereas new interesting analyses of such dynamics also appeared in the Medieval and Early Modern periods (see, e.g., Ibn Khaldūn 1958 [1377], or Machiavelli 1996 [1531] 1). This is not surprising as the cyclical dynamics was dominant in the agrarian social systems. (...) With modernization, the trend dynamics became much more pronounced and these are trends to which the students of modern societies pay more attention. Note that the term trend – as regards its contents and application – is tightly connected with a formal mathematical analysis. Trends may be described by various equations – linear, exponential, power-law, etc. On the other hand, the cliodynamic research has demonstrated that the cyclical historical dynamics can be also modeled mathematically in a rather effective way (see, e.g., Usher 1989; Chu and Lee 1994; Turchin 2003, 2005a, 2005b; Turchin and Korotayev 2006; Turchin and Nefedov 2009; Nefedov 2004; Korotayev and Komarova 2004; Korotayev, Malkov, and Khaltourina 2006; Korotayev and Khaltourina 2006; Korotayev 2007; Grinin 2007), whereas the trend and cycle components of historical dynamics turn out to be of equal importance. (shrink)

Aristotle divided arguments that persuade into the rhetorical (which happen to persuade), the dialectical (which are strong so ought to persuade to some degree) and the demonstrative (which must persuade if rightly understood). Dialectical arguments were long neglected, partly because Aristotle did not write a book about them. But in the sixteenth and seventeenth century late scholastic authors such as Medina, Cano and Soto developed a sound theory of probable arguments, those that have logical and not merely psychological force but (...) fall short of demonstration. Informed by late medieval treatments of the law of evidence and problems in moral theology and aleatory contracts, they considered the reasons that could render legal, moral, theological, commercial and historical arguments strong though not demonstrative. At the same time, demonstrative arguments became better understood as Galileo and other figures of the Scientific Revolution used mathematical proof in arguments in physics. Galileo moved both dialectical and demonstrative arguments into mathematical territory. (shrink)

This self-contained one page paper produces one valid two-premise premise-conclusion argument that is a counterexample to the entire three traditional rules of distribution. These three rules were previously thought to be generally applicable criteria for invalidity of premise-conclusion arguments. No longer can a three-term argument be dismissed as invalid simply on the ground that its middle is undistributed, for example. The following question seems never to have been raised: how does having an undistributed middle show that an argument's conclusion does (...) not follow from its premises? This result does nothing to vitiate the theories of distribution developed over the period beginning in medieval times. What it does vitiate is many if not all attempts to use distribution in tests of invalidity outside of the standard two-premise categorical arguments—where they were verified on a case-by-case basis without further theoretical grounding. In addition it shows that there was no theoretical basis for many if not all claims of fundamental status of rules of distribution. These results are further support for approaching historical texts using mathematical archeology. (shrink)

The symbolism of nature as a book in which one reads is of ancient origin. This study focuses on the question of its mathematical and theological language in the biblical context and on the background of changes in natural philosophy, especially in the Renaissance period. The biblical context is associated with the paradigm shift in the Renaissance period, because all the researched authors addressed the questions of meaning and methods of research of nature in connection with the hermeneutics of biblical (...) texts. The change in attitude towards the method and results of nature research is connected with two concepts of the relationship of mathematics to the real world. As this world is considered to be created by God, the relationship of mathematics and matter is linked to the question of God's relationship to mathematics. In relation to nature, it was initially significantly limited to geometry, largely under the influence of Euclid and Pythagoras. In the Renaissance period, the preconditions for analytical geometry and infinitesimal calculus were already being created. Thanks to them, Newton was able to build a new physics that combined ancient and medieval approaches to astronomy and terrestrial nature, originally diametrically different, into one whole. (shrink)

-/- FILOSOFIA: CRÍTICA À METAFÍSICA -/- PHILOSOPHY: CRITICISM TO METAPHYSICS -/- Por: Emanuel Isaque Cordeiro da Silva - UFRPE Alana Thaís Mayza da Silva - CAP-UFPE RESUMO: A Metafísica (do grego: Μεταφυσική) é uma área inerente à Filosofia, dito isto, é uma esfera que compreende o mundo e os seres humanos sob uma fundamentação suprassensível da realidade, bem como goza de fundamentação ontológica e teológica para explicação dos dilemas do nosso mundo. Logo, não goza da experiência e explicação científica com (...) base na matemática, ciências, observação, análise, etc. e sim da explicação apenas teorética sem a análise empírica. Por fim, filósofos de divergentes escolas e eras da história da filosofia fundamentaram suas críticas quanto a Metafísica; e este breve trabalho tem como objetivo analisar as críticas dos filósofos e trazer uma conclusão clara e objetiva para os leitores, sejam leigos ou não. Palavras-chave: Metafísica. Crítica. Teoria. Suprassensível. Empírico. Experiência. ABSTRACT Metaphysics is an area inherent to philosophy, that is, it is a sphere that comprises the world and human beings under a supersensitive foundation of reality, as well as enjoying ontological and theological foundations for explaining the dilemmas of our world. Therefore, it does not enjoy the scientific experience and explanation based on mathematics, science, observation, analysis, etc., but only the theoretical explanation without empirical analysis. Finally, philosophers from divergent schools and eras in the history of philosophy based their criticism of Metaphysics, and this brief work aims to analyze the criticism of philosophers and bring a clear and objective conclusion to readers, whether lay or not. Keywords: Metaphysics. Criticism. Theory. Supersensitive. Empirical. Experience. INTRODUÇÃO Para uma fundamentação alicerçada na observação, análise e conclusões pragmáticas, é imprescindível aclarar o conceito de Metafísica como esfera inerente da Filosofia, bem como disciplina primordial para o currículo do Filósofo formado nas Universidades globais. Para isso, discorro acerca do conceito de Metafísica desde a alcunha aristotélica de 'Filosofia primeira', passando à análise modernista, em especial à kantiana e seu criticismo (Crítica da Metafísica é a alma desse trabalho), sob análise de autores didáticos e, por fim, sob a elucidação da esfera Metafísica mediante os dicionários de Filosofia de Japiassú e Marcondes, e de Nicola Abbagnano. No conjunto de obras aristotélicas entituladas de Metafísica, Aristóteles buscou elucidar 'o ser enquanto ser', isto é, buscar a objetividade das coisas e do mundo mediante a subjetividade e a realidade suprassensível dessas coisas. Logo, a explicação dos fenômenos que cercavam a Grécia clássica eram explicados além do alicerce das Ciências tradicionais da época (Física, Química, Biologia, etc.). Todavia, para Aristóteles, a Metafísica consiste na 'ciência primeira' no que se refere o fornecimento de um fundamento único para todas as demais ciências, ou seja, dar-lhes o objeto ao qual elas se referem e os princípios dos quais todas elas dependem. Por fim, a Metafísica implica ser uma enciclopédia das ciências um inventário completo e exaustivo de todas as ciências, em suas relações de coordenação e subordinação, nas tarefas e nos limites atribuídos a cada uma, de modo definitivo. Na modernidade, a Metafísica perde a centralidade do mundo da Filosofia, quem é o precursor de tal declínio é Immanuel Kant e seu criticismo. Tal descendência é elucidada e aclarada posteriormente no presente trabalho. A Metafísica moderna ganha uma nova interpretação, sendo alcunhada e levada por analogia à Ontologia, expressada, segundo Kant, como conceito de gnosiologia. Para Kant, Metafísica é a fonte inerente do estudo e explicação das formas alicerçadas na razão, bem como fundamento de toda realidade suprassensível que se deve basear para elucidar e aclarar o mundo moderno. Ainda segundo o filósofo, a Metafísica é a fonte de todos os princípios reais para a explicação da realidade. Sendo ela, por fim um 'sistema filosófico' alicerçado sobre uma perspectiva ontológica, teológica e/ou suprassensível da realidade. No nosso finalismo, a inerência da Metafísica à filosofia transcende a explicação de todo o universo e sua totalidade (matéria e forma como alcunha Aristóteles). Logo, na Filosofia Clássica com Platão e, especialmente em Aristóteles, o Mito passou a ser ciência e essa ciência era determinada de Metafísica. Vale salientar que a alcunha 'Metafísica' foi, primeiramente, usada por Andrônico de Rodes . Por fim, a Metafísica primordial para fundamentar as explicações necessárias para os fenômenos que se decorrera na Grécia antiga, foi perdendo centralidade ao passo da história da filosofia e, na modernidade, com Hume e Kant passa por uma grande crise, discorrida agora no trabalho. -/- DA CRÍTICA À METAFÍSICA Platão criou um sistema metafísico de explicação da realidade e do mundo que influenciou muitos pensadores e religiosos. Boa parte da história da filosofia foi composta de sistemas metafísicos. Mas, se, por um lado, o pensamento metafísico tem uma vida longa e profícua, por outro, é alvo de críticas constantes, principalmente por filósofos modernos e contemporâneos, que questionam a validade das teses metafísicas, seja porque tratam de coisas que não podem ser conhecidas, como alma, Deus e formas inteligíveis, seja porque suas conclusões são consideradas enganosas. ARISTÓTELES E A FILOSOFIA PRIMEIRA As ciências teóricas ou teoréticas são aquelas que produzem um saber universal, válido em qualquer situação, e necessário, isto é, que não pode ser de outro modo. Essas ciências estão voltadas para a contemplação da verdade. O sábio que se dedica a elas encontra um fim em si mesmo, pois o resultado de sua investigação não gera nenhum objeto exterior, como uma construção ou uma escultura, mas beneficia sua própria alma. Elas estão organizadas em Metafísica, Matemática e Física, que inclui a Psicologia ou a ciência da alma. As principais obras teoréticas escritas por Aristóteles são Física, De anima e Metafísica. Filosofia Primeira, que chegou até nós com o nome de Metafísica, também chamada muitas vezes por Aristóteles de Teologia .com a ressalva de que Teologia aqui não tem o sentido que é dado a ela contemporaneamente), é a expressão que designava a mais elevada das ciências teoréticas, diferenciando-se da chamada Filosofia Segunda, ou Física. Aristóteles abre o Livro I da Metafísica com a célebre passagem: Todos os homens têm, por natureza, desejo de conhecer: uma prova disso é o prazer das sensações, pois, fora até da sua utilidade, elas nos agradam por si mesmas e, mais que todas as outras, as visuais [...l. (ARISTÓTELES, 1969). A busca pelo conhecimento encontra-se na própria natureza humana, desde o nascimento, mesmo que se manifeste em seu grau mais rudimentar. Prova disso é o conhecimento por meio das sensações ou dos sentidos, entre os quais se destaca o da visão. É o chamado conhecimento por empiria ou saber empírico, que não pode ser ensinado porque é adquirido imediata e concretamente quando percebemos as coisas sensíveis. Também há um saber que vem da técnica. O técnico é aquele que tem o conhecimento dos meios para chegar a um fim. Como ensina o historiador da Filosofia, Julián Marias, apesar de o saber técnico ser superior ao saber empírico, ambos são necessários em nossa vida: 1..l Portanto, a tékhne [técnica) é superior à empeiria; mas esta também é necessária, por exemplo, para curar, porque o médico não tem de curar o homem, e sim Sócrates, e o homem apenas de modo mediato. (MARIAS, J. 2015). O conhecimento metafísico é menos necessário em nossa vida cotidiana, mas nenhum outro lhe é superior. A Metafísica é o conhecimento pelas causas e, como vimos, o conhecimento é científico quando ele nos dá os princípios e as causas das coisas. Para Aristóteles, tudo o que existe é efeito de uma causa e ela é a responsável por esse algo ser necessariamente de um jeito e não de outro. De acordo com Aristóteles, conhecer algo é conhecer pela causa. Dessa forma, a explicação de algo ou o seu conhecimento deve sempre dizer por que algo é necessariamente de determinado modo. Sem esse tipo de explicaçào, não pode haver propriamente conhecimento. Daí se constata o caráter rigoroso da explicação científica: ela não apenas sabe que "algo é isto", mas é capaz de explicar por que algo é necessariamente isto. A definição aristotélica de Metafísica Chauí indaga em seu livro Iniciação à Filosofia a questão central do que Aristóteles entende por sua alcunha ou Metafísica. Chauí explana: Embora Aristóteles admita que para cada tipo de Ser e suas essências existe uma ciência teorética própria (física, biologia, psicologia, matemática, astronomia), ele também defende a necessidade de uma ciência geral, mais ampla, mais universal, anterior a todas essas, cujo objeto não seja esse ou aquele tipo de Ser, essa ou aquela modalidade de essência, mas o Ser em geral, a essência em geral. Essa ciência, para Aristóteles, é a Filosofia Primeira ou metafísica, que investiga o que é a essência e aquilo que faz com que haja essências particulares diferenciadas, que estuda o Ser enquanto Ser. (CHAUÍ, 2010). O objeto investigado pela Metafísica também é muito diferente do objeto conhecido pelo saber empírico ou pelo saber técnico. O saber empírico nos faz conhecer as coisas particulares, como Sócrates, o Oráculo de Delfos e a caneta esferográfica; o saber técnico nos faz conhecer as diversas técnicas, como os procedimentos médicos e a engenharia de construir templos. O que o saber metafísico, então, nos permite conhecer? Segundo Aristóteles, a Metafisica estuda o ser enquanto ser. Não considera o ser de modo particular como fazem as demais ciências — por exemplo, ao investigar Sócrates, os procedimentos médicos etc. mas busca o que é universal, o que envolve tanto Sócrates quanto os procedimentos médicos, tanto o Oráculo de Delfos quanto a arte de construir templos. Segundo Aristóteles, a Metafísica trata das causas primeiras, que são as quatro seguintes: • a causa material, que é a matéria de que é feita alguma coisa. Em uma escultura, por exemplo, a causa material pode ser o bronze ou o mármore; • a causa formal, que é a forma ou a essência das coisas. No exemplo da escultura, é a forma ou a aparência que possibilita que a reconheçamos como uma escultura (e não como um poste, por exemplo); • a causa eficiente, que é o agente que produz a coisa. Uma escultura é produzida por um artista; • a causa final, que é a razão ou a finalidade das coisas, A finalidade da escultura é o prazer estético. A Metafísica também se ocupa da substância. A substância responde pelos significados do ser. Mas o que é a substância? Trata-se de uma questão complexa no pensamento aristotélico. Aristóteles recusa-se a entender a substância como sendo a forma platónica. Ela nào é o suprassensível. Seria entao a substância o sensível individual? As coisas são matéria e forma; amatéria é o substrato da coisa (por exemplo, o mármore é a matéria da estátua). Sem sua forma, ela é apenas potencialidade, mas sem a matéria qualquer realidade sensível se desvaneceria. Assim, a matéria pode ser dita substância em sentido impróprio. A forma, por seu turno, é aquilo que determina a matéria, é a sua essência; ela é substancia em sentido próprio. Mas a matéria é também matéria enformada. As coisas sensíveis são, a um só tempo, matéria e forma. E esse composto também pode ser legitimamente chamado de substância. Aristóteles apresenta três gêneros de substâncias: • as substâncias sensíveis, que nascem, morrem e, por isso, passam por todo tipo de mudança; • as substancias sensíveis e incorruptíveis, que não passam por nenhum tipo de mudança, como os planetas e as estrelas; • a substância suprassensível, que é superior às outras duas: o Primeiro Motor Imóvel, o deus aristotélico, causa de todo movimento, mas sendo ele mesmo imóvel. (Ele precisa ser imóvel porque se ele também se movesse precisaria haver uma causa para esse movimento, o que levaria a uma espécie de regressão ao infinito). Assim como o demiurgo de Platão, o deus aristotélico nao é um deus criador. Não criou o Universo nem gerou o movimento dos corpos. Ao contrário, o deus de Aristóteles exerce uma atraçáo sobre o Universo, motivando sua existência, e sobre os corpos, gerando seu movimento, ou seja, atrai tudo para si como objeto de amor. Não é, portanto, causa eficiente do mundo e do movimento do mundo, mas causa final. O deus aristotélico nao cria o mundo do nada por um ato de vontade. Ele nao é causa eficiente do mundo, como indicamos ser o artista causa eficiente de sua escultura. A METAFÍSICA CLÁSSICA OU MODERNA Grandes revoluções no pensamento do século XVII deram estopim à uma grande crise na esfera metafísica. De modo resumido a Metafísica sofreu uma grande transformação e uma nova elaboração, sem a fundamentação embasada no pensamento platônico, aristotélico ou neoplatônico. Sendo assim a nova metafísica é caracterizada por: [...] afirmação da incompatibilidade entre fé e razão, acarretando a separação de ambas, de sorte que a religião e a filosofia possam seguir caminhos próprios, mesmo que a segunda não esteja publicamente autorizada a expor ideias que contradigam as verdades ou dogmas da fé [...] [...] redefinição do conceito de Ser ou substância. Os modernos conservam a definição tradicional da substância como o Ser que existe em si e por si mesmo, que subsiste em si e por si mesmo. Porém, em lugar de considerar que a substância se define por gênero e espécie, havendo tantos tipos de substâncias quantos gêneros e espécies houver, passa-se a definir a substância levando em consideração seus predicados essenciais ou seus atributos essenciais, isto é, aquelas propriedades ou atributos sem os quais uma substância não é o que ela é. Após o supracitado, os cartesianos e Descartes dirão que há somente três substâncias essenciais, divergindo totalmente do pensamento aristotélico, as substâncias cartesianas são a Alma, a matéria dos corpos e Deus. Para empiristas, só se é possível conhecer a substância corpórea, logo não se é possível elaborar uma metafísica, e sim uma geometria ou uma física que se ocupa do estudo dessas matérias corpóreas. Baruch Spinoza (1632-1677) explanou a primordialidade de se elaborar uma definição genérica e universal, comumente aceita por toda a comunidade de filósofos de Aristóteles a modernidade. Podemos reduzir o conceito proposto da seguinte maneira: “substância é aquilo que existe em si e por si e não depende de outros para existir”.Logo, diz Espinosa, que há apenas uma substância no Universo que não depende de outra para existir, tal substância é o próprio Deus ou a natureza. Nessa revolução do pensamento filosófico, em peculiar à metafísica, houve também a redefinição do conceito de causa e causalidade. Em Aristóteles haviam quatro tipos de causas, agora, na modernidade, se propunha apenas dois tipos: a eficiente e a final. Chauí explana que a Causa eficiente é aquela na qual uma ação anterior determina como consequência necessária a produção de um efeito, e que seu alcance é universal na natureza. Causa final é aquela que determina, para os seres pensantes, a escolha da realização ou não realização de uma ação, e que só opera na ação de Deus e nas ações humanas. Houve também, nessa revolução, uma quebra do paradigma metafísico, isto é, a metafísica não se dividia mais entre a teologia, a psicologia racional e a cosmologia racional. Agora, a metafísica ganha um novo rumo, um rumo próprio de estudo embasado em suas próprias fundamentações. Logo, a metafísica se apropria de três e apenas três ideias de substância. A substância infinita, ou o próprio Deus. A substância pensante que é a consciência como faculdade de reflexão e de representação da realidade alicerçada na razão. E, por fim, a substância extensa que é a natureza embasada nos princípios e leis regidas pela matemática e a mecânica. DAVID HUME E A CRISE DA METAFÍSICA O filósofo escocês David Hume (1711-1776) nos ajuda a entender em parte o teor das críticas ao pensamento metafísico. "Se tomarmos em nossas mãos um volume qualquer, de teologia ou metafísica escolástica, por exemplo, façamos a pergunta: contém ele qualquer raciocínio abstrato referente a números e quantidades? Não. Contém qualquer raciocínio experimental referente a questões de fato e de existência? Não. Às chamas com ele, então, pois não pode conter senão sofismas e ilusão." (HUME, 2004) Hume critica a teologia e a metafísica escolástica, que foi a metafísica praticada principalmente nas escolas cristãs durante a Idade Média, porque suas concepções não tratam nem de conceitos matemáticos, que são formais e podem levar a conclusões seguras - por exemplo, "2 + 2 = 4" -, nem de afirmações sobre fatos, que podem ser verificados empiricamente, isto é, pelos órgãos dos sentidos —como "a aceleração de um corpo em queda livre é de 9,8 metros por segundo", Assim, Hume aconselha o leitor a jogar as afirmações metafísicas na fogueira, pois são consideradas ilusões, fantasias ou produto de erros de raciocínio e nada têm a acrescentar ao conhecimento humano. A partir de Hume, a metafísica, tal como existira desde os gregos, tornara-se impossível. O CRITICISMO KANTIANO E O FIM DA METAFÍSICA CLÁSSICA Entre a Dissertação de 1770 e a Crítica da razão pura, mais de dez anos se passaram. Apesar de a obra ser volumosa, a redaçào da Crítica da razão pura não chegou a cinco meses de trabalho. De início, Kant pretendia apenas revisar sua dissertação, mas o trabalho acabou por levá-lo a figurar entre os grandes nomes do pensamento filosófico, como Aristóteles e Descartes. Kant fez um diagnóstico bastante negativo da situação da Filosofia de sua época, em particular da Metafísica, que, embora fosse considerada a Filosofia Primeira, patinava em questões impossíveis de serem resolvidas. Ao mesmo tempo, buscou inserir a Filosofia no rumo seguido pelas Ciências Naturais, pela Lógica e pela Matemática, que já tinham encontrado o caminho da Ciência. Em alguns pontos, o projeto kantiano se aproximou de algumas características do projeto cartesiano. Descartes duvidou de todo o saber conhecido, pois queria encontrar um fundamento seguro que sustentasse a edificação da Filosofia e, mais particularmente, da Ciência. Diferentemente de Descartes, contudo, Kant nào duvidou de todo o conhecimento que havia aprendido. Para ele, bastava estabelecer os limites do conhecimento racional, ou seja, o que a razão podia conhecer e o que era impossível de ser conhecido. No fim do século XVIII, de acordo com a proposta de Wolff, a Metafísica era dividida em duas partes: Metafísica geral e Metafísica especial. A primeira tratava da Ontologia (ciência do ser enquanto ser, como definido por Aristóteles), e a segunda se ocupava do ser humano (Psicologia), do mundo (Cosmologia) e de Deus (Teologia racional), Os principais temas in-vestigados eram a imortalidade da alma, a liberdade a finitude ou infinitude do mundo e a existência de Deus. No entanto, de acordo com Kant, as questões metafísicas clássicas extrapolavam as capacidades de conhecimento do ser humano, pois estavam tão afastadas da experiência que a razão só podia pensá-las, sem conhecê-las realmente. As questões postas pela Metafísica clássica extrapolavam o terreno de qualquer experiência possível e, no entanto, a razão, por sua própria natureza, apresenta questões que não pode evitar, mas que também náo pode responder por não terem nenhuma pedra de toque na experiência. Disso decorre a pergunta fundamental sobre a possibilidade de haver um conhecimento metafísico. Publicada em 1781, a Crítica da razão pura foi uma tentativa de responder a essa pergunta. A obra instituiu um "tribunal da razao" para examinar a própria razão, e não para tomar partido no conflito entre racionalistas, como Descartes e Leibniz, que elaboraram sistemas metafísicos, e empiristas, como Locke e Hume, que basearam o conhecimento unicamente nos sentidos. Criticar a razão é determinar as possibilidades, os limites e o alcance do próprio conhecimento. A transformação foi tao grande para a Filosofia que essa obra nao pôde ser ignorada. A situação da Metafísica No texto a seguir, retirado do célebre prefácio à primeira edição de Crítica da razão pura, podemos ver o diagnóstico de Kant sobre a situaçao da Metafísica no período em que vivia. A razão humana, num determinado dominio dos seus conhecimentos, possui o singular destino de se ver atormentada por questões, que náo pode evitar, pois lhe são impostas pela sua natureza, mas às quais também não pode dar resposta por ultrapassarem completamente as suas possibilidades. Não é por culpa sua que cai nessa perplexidade. Parte de princípios, cujo uso é inevitável no decorrer da experiência e, ao mesmo tempo, suficientemente garantido por esta. Ajudada por estes princípios eleva-se cada vez mais alto (como de resto lho consente a natureza) para condições mais remotas. Porém, logo se apercebe de que, desta maneira, a sua tarefa há de ficar sempre inacabada, porque as questões nunca se esgotam; vê-se obrigada, por conseguinte, a refugiar-se em princípios, que ultrapassam todo o uso possível da experiência e, não obstante, estão ao abrigo de qualquer suspeita, pois o senso comum está de acordo com eles. Assim, a razão humana cai em obscuridades e contradições, que a autorizam a concluir dever ter-se apoiado em erros, ocultos algures, sem contudo os poder descobrir. Na verdade, os princípios de que se serve, uma vez que ultrapassam os limites de toda experiência, já não reconhecem nesta qualquer pedra de toque. O teatro destas disputas infindáveis chama-se Metafísica. (KANT, I. 2010). O fim da Metafísica clássica Podemos agora retomar o questionamento sobre os motivos pelos quais a física newtoniana é possível como ciência e entender também por que a metafísica racionalista tradicional não é conhecimento científico, segundo a teoria kantiana. A física trata de regras e leis dos fenômenos, das representações conformadas pela sensibilidade e pelo entendimento, ou seja, refere-se a objetos do conhecimento humano. A metafísica clássica, por sua vez, trata de coisas que não fazem parte da experiência sensível. Deus e a alma, por exemplo, não são apreendidos pela sensibilidade, não são impressões conformadas ou fenômenos e, portanto, não podem ser conhecidos pelo ser humano. Ideias ou conceitos dessa espécie podem ser pensados pela razão, mas é impossível conhecê-los, pois estão além da capacidade humana de conhecimento. Por esse motivo, a física é possível como ciência e a metafísica tradicional não. Com base nisso, Kant provocou uma revolução na teoria do conhecimento, que ele mesmo designou como "virada copernicana na filosofia", Se Copémico mudou a forma de entender o mundo ao defender a ideia de que o Sol está no centro do sistema solar, Kant mudou a forma de o ser humano compreender o mundo e a si próprio ao estabelecer a ideia de que o ser humano, e não os objetos externos, é o centro do conhecimento, O ser humano não é um agente passivo, que só recebe informações, mas atua decisivamente na constituição do conhecimento e do que se compreende como realidade. Os elementos a priori que determinam essa constituição do conhecimento são o escopo da filosofia kantiana, também conhecida como transcendental. Logo, o pensamento kantiano alterou profundamente a Metafísica. A existência de Deus ou a imortalidade da alma, por exemplo, passaram a ser questões que já não podiam ser respondidas. Se percebemos apenas as coisas que ocorrem em um tempo específico (antes, agora, depois), como seremos capazes de perceber algo que está na eternidade (Deus)? Como afirmar que a alma é imortal se não conseguimos percebê-la com os nossos sentidos? O conhecimento a partir de Kant, então, é sempre conhecimento de objetos, isto é, de conteúdos elaborados e modificados pelo sujeito que conhece. O conhecimento passou a se ocupar, assim, de fenómenos, e não mais da própria coisa, como pretendiam os metafísicos clássicos. Com a crítica kantiana, tornou-se impossível conhecer efetivamente os temas da Metafísica clássica e emitir qualquer opinião segura sobre assuntos que não podem ser assentados na experiência. Podemos pensar sobre Deus, podemos querer que nossa a seja imortal, mas jamais teremos a possibilidade de verificar esses assuntos em nossa experiência ou mesmo ter sobre eles verdadeiro conhecimento. Nada nos impede de acreditar em Deus ou na imortalidade da alma, porém não podemos exigir que a Filosofia se pronuncie a respeito desses temas. Na verdade, a crítica kantiana não diz que Deus existe nem que não existe, mas apenas que não é capaz de saben Deus, não obstante sua importância para a Filosofia do próprio Kant, é uma questão de fé. AUGUSTE COMTE E A METAFÍSICA NO SÉCULO XIX O filósofo francês Auguste Comte (1798-1857), considerado o pai do positivismo, fundamenta sua filosofia também em oposição às concepções metafísicas. Em seu entendimento, a busca por princípios gerais ou essências, entidades que estão além do que podemos observar ou perceber pelos órgãos dos sentidos, é algo infrutífero. Comte explana: "[...] não é supérfluo assinalar agora, de modo direto, a preponderância contínua da observação sobre a imaginação, como o principal caráter lógico da sã filosofia moderna, dirigindo nossas pesquisas, não para causas essenciais, mas para leis efetivas, dos diversos fenômenos naturais. Sem ser doravante imediatamente contestado, permanece este princípio fundamental muitas vezes desconhecido nos trabalhos especiais. Embora as diferentes ordens de especulações concedam, sem dúvida, à imaginação uma alta participação, isto é, constan temente empregada para criar ou aperfeiçoar os meios da vinculação entre os fatos constatados, mas o ponto de partida e a sua direçào não lhe poderiam pertencer em nenhum caso. Ainda quando procedemos verdadeiramente a priori, é claro que as considerações gerais que nos guiam foram inicialmente fundadas, quer na ciência correspondente, quer em outra, na simples observação, única fonte de sua realidade e também de sua fecundidade. Ver para prever: tal é o caráter permanente da verdadeira ciência. Tudo prever sem ter nada visto constitui somente uma absurda utopia metafísica, ainda muito seguida." (COMTE, 1978). A observação é considerada a única fonte de realidade. Abandonando a observação e apoiando-se única e exclusivamente na imaginação, como seria característico das especulaçôes metafísicas, o pensamento pode levar não para o conhecimento das coisas, mas para a fantasia. A ciência, então, deve ter como base as coisas que podem ser observadas. Todas as especulações e as teorias científicas, em última instância, devem ter como fundamento a realidade observável. LUDWIG WITTGENSTEIN E A METAFÍSICA CONTEMPORÂNEA O filósofo austríaco Ludwig Wittgenstein (1889-1951) também se opõe às afirmações metafísicas por considerar que estas tratam de problemas (ou pseudoproblemas) que não podem ser formulados claramente e, portanto, não podem ser respondidos. “O método correto da filosofia seria o seguinte: só dizer o que pode ser dito, ou seja, as proposições das ciências naturais [...] e depois, quando alguém quisesse dizer algo metafísico, mostrar-lhe que nas suas proposições existem sinais aos quais não foi dada uma denotação.” (WITTGENSTEIN, 1995). Quer dizer, as sentenças ou os termos metafísicos se referem a coisas, objetos ou seres enigmáticos ou misteriosos, que não podem ser tratados com clareza, isto é, não se sabe exatamente o que são, o que impossibilitaria a formulação de problemas e soluções reais. "Deus", "alma" ou "vida após a morte", por exemplo, são termos que se referem especificamente a quê? Os "problemas" metafísicos estariam no campo do mistério e não no da investigação racional. Por isso, como afirma Wittgenstein, "acerca daquilo de que se não pode falar, tem que se ficar em silêncio". CONCLUSÃO (ÕES) Ora, a Metafísica na Grécia antiga foi imprescindível para o entendimento dos mais variados dilemas do mundo antigo, sejam perguntas simples, ou complexas. Iniciada como um sistema filosófico platônico, passando a ser filosofia primeira em Aristóteles, a metafísica incorporou elementos teológicos, psicológicos e cosmológicos durante o mundo antigo, e ainda mais durante o período medieval e a hegemônica Igreja Católica. No mundo moderno e embasado pelo “Luz da razão”, a metafísica começa a ser refutada pelos pensadores da época em questão, dentre eles o célebre cético Hume, e o emblemático Kant, além de Descartes e seus cartesianistas. Kant e seu criticismo incorporaram à Metafísica uma nova interpretação, esta que fez a mesma entrar em declínio e sair da centralidade do mundo filosófico. Kant, graças às indagações e refutações de Hume, pôde acordar do célebre “sono dogmático”. O que fez Kant se importa com a Metafísica. Kant trouxe à luz o fim da Metafísica clássica e deu legado ao início da metafísica contemporânea alcunhada de Ontologia. No decorrer da história da filosofia, a Metafísica teve intensa ascenção e vertiginosos declives, sendo seu maior declive o crítico Kant. Por fim, muitos filósofos se apropriaram de ser heterodoxos e enfrentaram de frente o mundo metafísica e a base do catolicismo sobre a realidade. Posta realidade que hoje, com o legado de Kant, da fenomenologia de Husserl, etc., originaram a nova escola filosófica chamada de Ontologia. -/- REFERÊNCIAS BIBLIOGRÁFICAS ABBAGNANO, N. Dicionário de filosofia. 5ª. ed. Trad. Ivone Castillo Benedetti. São Paulo: Martins Fontes, 2007. ARANHA, M. L. de A.; MARTINS, M. H. P. Filosofando: introdução à filosofia. 5ª. ed. São Paulo: Moderna, 2013. ARISTÓTELES. Metafísica. Trad. L. Vallandro. Porto Alegre: Globo, 1969. BELO, R. dos S. Filosofia. Vol. Único. 2ª. ed. São Paulo: FTD, 2016. (Coleção Filosofia). BENOIT, L. O. Augusto Comte: fundador da física social. 1ª. ed. São Paulo: Moderna, 2002. CHAUÍ, M. Iniciação à Filosofia. 1ª. ed. São Paulo: Ática, 2010. COMTE, A. Curso de filosofia positiva. In: FERNANDES, F.; FILHO, E. de M. Comte: sociologia. São Paulo: Ática, 1978. (Coleção Grandes Cientistas Sociais). COTRIM, G.; FERNANDES, M. Fundamentos de filosofia. 2ª. ed. São Paulo: Saraiva, 2013. FERREIRO, H. Siete ensayos sobre la muerte de la Metafísica: Una introducción al idealismo absoluto a partir de la ontología. 1ª. ed. Porto Alegre: Editora Fi, 2016. FIGUEIREDO, V. de. (Org.); et. al. Filosofia: Temas e Percursos. Vol. Único. 2ª. ed. São Paulo: Berlendis & Vertecchia, 2016. FILHO, J. S. Filosofia e filosofias: existência e sentidos. 1ª. ed. Belo Horizonte: Autêntica, 2016. HUME, D. Investigações sobre o entendimento humano e sobre os principios da moral. São Paulo: Unesp, 2004. ________. Tratado da natureza humana: uma tentativa de introduzir o método experimental de raciocínio nos assuntos morais. São Paulo: Unesp, 2001. JAPIASSÚ, H.; MARCONDES, D. Dicionário básico de filosofia. 3ª. ed. Rio de Janeiro: Zahar, 2001. KANT, I. Crítica da razão prática. Lisboa: Edições 70, 1989. [Textos filosóficos]. ______. Crítica da razão pura. São Paulo: Abril Cultural, 1979. (Col. Os Pensadores). MARIAS, J. História da filosofia. 2ª. ed. Trad. Claudia Berliner. São Paulo: Martins Fontes, 2015. MELANI, R. Diálogo: primeiros estudos em filosofia. Vol. Único. 2ª. ed. São Paulo: Moderna, 2016. OTERO, J. G. Compendio de filosofía: para uso de los jóvenes estudiosos. 1ª. ed. Bogotá: Imprensa de San Bernardo, 1919. SILVA, F. L. e. Descartes, a metafísica da modernidade. 1ª. ed. São Paulo: Moderna, 2006. WITTGENSTEIN, L. Tratado lógico-filosófico. 2ª. ed. Lisboa: Fundação Calouste Gulbenkian, 1995. -/- Emanuel Isaque Cordeiro da Silva – estudante, pesquisador, professor, agropecuarista, altruísta e defensor dos direitos humanos e dos animais. -/- Alana Thaís Mayza da Silva – estudante, potterhead, pesquisadora, projetista, musicista, filantropa, defensora dos direitos humanos e dos animais, LGBT. (shrink)

Analytical philosophy is ruled by the alliance of logic, linguistics and mathematics since its beginnings in the syllogistic calculus of terms and premises in Aristotle's Analytica protera, in the theories of medieval logic that dealt with what are Proprietatis Terminorum (significatio, suppositio, appellatio), in the theological apologetics of argumentation with the combinatorics of symbols by Raymundus Llullus in the work Ars Magna, Generalis et Ultima (1305-08), in what is presented as Theologia Combinata (cf. Tomus II.p.251) in Ars Magna Sciendi (...) sive Combinatoria by Athanasius Kircher (1669), in analytical combinatorics based in Leibniz's dissertation Ars Combinatoria (1666), where language is understood as a lingua characteristica and as a calculus ratiocinator, in the combinatorics of classes that have the meanings 1 or 0 given in the idea of Boolean functions (1854), in Frege's work Begriffschrift, eine der arithmeticeschen nachgebildete Fomelsprache des reinen Denkens (Concept letter, the language of formulas of pure thought made according to the model in arithmetical language), in Cantor's study of set theory / Grundlagen einer allgemeinen Mannigfaltigkeitstheorie (1883) and up to algorithms of fuzzy logic as a "means of approximate thinking that is in the human prior" (Zadeh, 1990) and fuzzy linguistics in computer semantics (Burghard B. Rieger, 1993). These are all the philosophical ideas of mathematicians and logicians that founded what we call logical pragmatism here as a demand that logic be absolutely used in every way in which it will dominate the grammar of language until logic itself becomes the philosophical grammar of the mind!!! (shrink)

This paper centers on the implicit metaphysics beyond the Theory of Relativity and the Principle of Indeterminacy – two revolutionary theories that have changed 20th Century Physics – using the perspective of Husserlian Transcedental Phenomenology. Albert Einstein (1879-1955) and Werner Heisenberg (1901-1976) abolished the theoretical framework of Classical (Galilean- Newtonian) physics that has been complemented, strengthened by Cartesian metaphysics. Rene Descartes (1596- 1850) introduced a separation between subject and object (as two different and self- enclosed substances) while Galileo and Newton (...) did the “mathematization” of the world. Newtonian physics, however, had an inexplicable postulate of absolute space and absolute time – a kind of geometrical framework, independent of all matter, for the explication of locality and acceleration. Thus, Cartesian modern metaphysics and Galilean- Newtonian physics go hand in hand, resulting to socio- ethical problems, materialism and environmental destruction. Einstein got rid of the Newtonian absolutes and was able to provide a new foundation for our notions of space and time: the four (4) dimensional space- time; simultaneity and the constancy of velocity of light, and the relativity of all systems of reference. Heisenberg, following the theory of quanta of Max Planck, told us of our inability to know sub- atomic phenomena and thus, blurring the line between the Cartesian separation of object and subject, hence, initiating the crisis of the foundations of Classical Physics. But the real crisis, according to Edmund Husserl (1859-1930) is that Modern (Classical) Science had “idealized” the world, severing nature from what he calls the Lebenswelt (life- world), the world that is simply there even before it has been reduced to mere mathematical- logical equations. Husserl thus, aims to establish a new science that returns to the “pre- scientific” and “non- mathematized” world of rich and complex phenomena: phenomena as they “appear to human consciousness”. To overcome the Cartesian equation of subject vs. object (man versus environment), Husserl brackets the external reality of Newtonian Science (epoché = to put in brackets, to suspend judgment) and emphasizes (1) the meaning of “world” different from the “world” of Classical Physics, (2) the intentionality of consciousness (L. in + tendere = to tend towards, to be essentially related to or connected to) which means that even before any scientific- logical description of the external reality, there is always a relation already between consciousness and an external reality. The world is the equiprimordial existence of consciousness and of external reality. My paper aims to look at this new science of the pre- idealized phenomena started by Husserl (a science of phenomena as they appear to conscious, human, lived experience, hence he calls it phenomenology), centering on the life- world and the intentionality of consciousness, as providing a new way of looking at ourselves and the world, in short, as providing a new metaphysics (as an antidote to Cartesian metaphysics) that grounds the revolutionary findings of Einstein and Heisenberg. The environmental destruction, technocracy, socio- ethical problems in the modern world are all rooted in this Galilean- Newtonian- Cartesian interpretation of the relationship between humans and the world after the crumbling of European Medieval Ages. Friedrich Nietzsche (1844-1900) comments that the modern world is going toward a nihilism (L. nihil = nothingness) at the turn of the century. Now, after two World Wars and the dropping of Atomic bomb, the capitalism and imperialism on the one hand, and on the other hand the poverty, hunger of the non- industrialized countries alongside destruction of nature (i.e., global warming), Nietzsche might be correct: unless humanity changes the way it looks at humanity and the kosmos. The works of Einstein, Heisenberg and Husserl seem to be pointing the way for us humans to escape nihilism by a “great existential transformation.” What these thinkers of post- modernity (after Cartesian/ Newtonian/ Galilean modernity) point to are: a) a new therapeutic way of looking at ourselves and our world (metaphysics) and b) a new and corrective notion of “rationality” (different from the objectivist, mathematico- logical way of thinking). This paper is divided into four parts: 1) A summary of Classical Physics and a short history of Quantum Theory 2) Einstein’s Special and General Relativity and Heisenberg’s Indeterminacy Principle 3) Husserl’s discussion of the Crisis of Europe, the life- world and intentionality of consciousness 4) A Metaphysics of Relativity and Indeterminacy and a Corrective notion of Rationality in Husserl’s Phenomenology . (shrink)

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by appealing to what the (...) world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (shrink)

K denotes both the knowledge predicate satisfied by every currently known theorem and the finite set of all currently known theorems. The set K is time-dependent, publicly available, and contains theorems both from formal and constructive mathematics. Any theorem of any mathematician from past or present forever belongs to K. Mathematical statements with known constructive proofs exist in K separately and form the set K_c⊆K. We assume that mathematical sets are atemporal entities. They exist formally in ZFC theory although their (...) properties can be time-dependent (when they depend on K) or informal. The true statement "K is non-empty" is outside K forever because K is not a formal set. Paul Cohen proved in 1963 that the equality 2^{\aleph_0}=\aleph_1 is independent of ZFC. Before 1963, the statement "There is a known constructively defined integer n⩾1 such that 2^{\aleph_0}=\aleph_n" was outside K. Since 1963, this statement is outside K forever. The true statement "There exists a set X⊆{1,...,49} such that card(X)=6 and X never occurred as the winning six numbers in the Polish Lotto lottery" refers to the current non-mathematical knowledge and is outside K forever. In Martin-Löf's terminology, every currently known theorem is actually true whereas every theorem (known or unknown) is potentially true. Actual truth is knowledge dependent and tensed. Potential truth is knowledge independent and tenseless. Algorithms always terminate. We explain the distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known. By using this distinction, we obtain non-trivially true statements on decidable sets X⊆N that belong to constructive and informal mathematics and refer to the current mathematical knowledge on X. For every such statement, its truth concerning the current mathematical knowledge is implied by a true statement of the form: (the conjunction of i conditions of the form α∈K)∧(the conjunction of j conditions of the form β∉K)∧(the conjunction of k conditions of the form γ∈K_c)∧(the conjunction of l conditions of the form δ∉K_c), where i,j,k,l∈N and α,β,γ,δ are mathematical statements. In constructive mathematics and the traditional Brouwerian intuitionism, the truth of a mathematical statement means that we know a constructive proof. Therefore, the truth of a mathematical statement depends on time, where the statement is formally stated in the classical mathematics without the predicate K. In this article, mathematical statements on decidable sets X⊆N refer to time because they refer to the current mathematical knowledge on X. They cannot be formally stated in the classical mathematics without the predicate K and their logical values may change in time. For a set X⊆N whose infiniteness is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(X)=ω" is outside K. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. No set X⊆N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove the assumption of the above statement. We prove that the set X={n∈N: the interval [-1,n] contains more than 29.5+(11!/(3n+1))∙sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. We present a table that shows satisfiable conjunctions of the form #(Condition1)∧(Condition 2)∧#(Condition 3)∧(Condition 4)∧#(Condition 5), where # denotes the negation ¬ or the absence of any symbol. We prove that there exists a naturally defined set C⊆N which satisfies the following conditions (6)-(11). (6) A known and simple algorithm for every k∈N decides whether or not k∈C. (7) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(C)=ω" is outside K. (8) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(N\C)=ω" is outside K. (9) It is conjectured, though so far unproven, that C is infinite. (10) For every known algorithm A with no input, the statement "A returns an integer n satisfying card(C)<ω ⇒ C⊆(-∞,n]" is outside K. (11) For every known algorithm A with no input, the statement "A returns an integer m satisfying card(N\C)<ω ⇒ N\C⊆(-∞,m]" is outside K. The set C={k∈N: 2^{2^k}+1 is composite} is naturally defined and satisfies conditions (6)-(11). (shrink)

The reconstruction of Leibniz’s metaphysics that Deleuze undertakes in The Fold provides a systematic account of the structure of Leibniz’s metaphysics in terms of its mathematical foundations. However, in doing so, Deleuze draws not only upon the mathematics developed by Leibniz—including the law of continuity as reflected in the calculus of infinite series and the infinitesimal calculus—but also upon developments in mathematics made by a number of Leibniz’s contemporaries—including Newton’s method of fluxions. He also draws upon a number of subsequent (...) developments in mathematics, the rudiments of which can be more or less located in Leibniz’s own work—including the theory of functions and singularities, the Weierstrassian theory of analytic continuity, and Poincaré’s theory of automorphic functions. Deleuze then retrospectively maps these developments back onto the structure of Leibniz’s metaphysics. While the Weierstrassian theory of analytic continuity serves to clarify Leibniz’s work, Poincaré’s theory of automorphic functions offers a solution to overcome and extend the limits that Deleuze identifies in Leibniz’s metaphysics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. This essay is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

In these days, there is an increasing technological development in intelligent tutoring systems. This field has become interesting to many researchers. In this paper, we present an intelligent tutoring system for teaching mathematics that help students understand the basics of math and that helps a lot of students of all ages to understand the topic because it's important for students of adding and subtracting. Through which the student will be able to study the course and solve related problems. An evaluation (...) of the intelligent tutoring systems was carried out and the results were encouraging. (shrink)

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are vulnerable (...) to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

I introduce an argument for Platonism based on intra-mathematical explanation: the explanation of one mathematical fact by another. The argument is important for two reasons. First, if the argument succeeds then it provides a basis for Platonism that does not proceed via standard indispensability considerations. Second, if the argument fails it can only do so for one of three reasons: either because there are no intra-mathematical explanations, or because not all explanations are backed by dependence relations, or because some form (...) of noneism-the view according to which non-existent entities possess properties and stand in relations-is true. The argument thus forces a choice between nominalism without noneism, intra-mathematical explanation and a backing conception of explanation. You can have any two, but not all three. (shrink)

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to (...) express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what (...) does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

The encounter of the medieval Muslims with Greek philosophy undeniably shaped the course of their philosophical and theological thought. This encounter led to the complex and contentious issue of ‘philosophy versus theology’. Medieval Muslim thinkers needed to develop a response to the issue of philosophy versus theology. The present article will first highlight the response of the Islamic theologians to their encounter with Greek philosophy in the form of three major trends in medieval Islamic theology: (1) strong (...) opposition to the application of reason and rationalist approach to Islamic doctrines, and strict adherence to the actual text of the Qur’an and the Hadith, (2) the adoption of Greek philosophy, and the application of reason and rationalist approach to explain and defend Islamic religion and (3) acknowledging the significance of reason in exploring the matters related to the natural world but, at the same time, stressing the subordination of reason to revelation. This article will discuss Atharism, Muʿtazilism and Ashʿarism as the representatives of the first, second and third trends, respectively. The response of the medieval Islamic theologians to the issue of philosophy versus theology serves as a context in which medieval Muslim philosophers carried out their philosophy–theology debate. The article will proceed to show that some medieval Muslim philosophers, such as Abu Bakr Al-Razi, subordinated religion or revelation to philosophy or reason. Other medieval Muslim philosophers, such as Al-Ghazali, subordinated philosophy to theology. The third group of medieval Islamic philosophers represented by Alfarabi argued for the reconciliation and harmonious co-existence of philosophy and religion. Contribution: This article highlights the response of medieval Islamic theologians and philosophers to the issue of philosophy versus theology that was caused by their encounter with Greek philosophy. (shrink)

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most (...) one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to key (...) intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (shrink)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, we go (...) through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness. (shrink)

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) mathematical truths are not (...) truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which are essential (...) to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic-neutral reasoning. These in turn are classified under seven headings: retroduction, citation, intuition, meta-argument, closure, generalization, and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed. (shrink)

Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object, and how (...) we can have knowledge about an unchanging object. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, I argue (...) that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

Published in 1903, this book was the first comprehensive treatise on the logical foundations of mathematics written in English. It sets forth, as far as possible without mathematical and logical symbolism, the grounds in favour of the view that mathematics and logic are identical. It proposes simply that what is commonly called mathematics are merely later deductions from logical premises. It provided the thesis for which _Principia Mathematica_ provided the detailed proof, and introduced the work of Frege to a wider (...) audience. In addition to the new introduction by John Slater, this edition contains Russell's introduction to the 1937 edition in which he defends his position against his formalist and intuitionist critics. (shrink)

Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘ mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine-Putnam (...) indispensability argument and Baker’s enhanced indispensability argument. (shrink)

My aim in this paper is to advance our understanding of medieval approaches to consciousness by focusing on a particular but, as it seems to me, representative medieval debate. The debate in question is between William Ockham and Walter Chatton over the existence of what these two thinkers refer to as “reflexive intellective intuitive cognition”. Although framed in the technical terminology of late-medieval cognitive psychology, the basic question at issue between them is this: Does the mind (or (...) “intellect”) cognize its own states via higher-order (or “reflexive”) representational states? Their debate is representative both because it highlights the central dialectical issues and alternatives at play in medieval discussions of consciousness generally and because it showcases the two main types of approach to consciousness one finds in the later medieval period, namely, those that explain consciousness in terms of intentionality (typically, higher-order intentionality), and those that understand consciousness as a non-intentional, sui-generis mode of awareness. (shrink)

In 1989, Cambridge University Press announced the publication of a new, three-volume book series: The Cambridge Translations of Medieval Philosophical Texts. The first volume – edited by Norman Kretzmann and Eleonore Stump, and dedicated to logic and the philosophy of language – contained 15 medieval texts, of which 15 were composed by Christian authors. The second volume in the series, this time focusing on ethics and political philosophy, appeared in 2000. Seventeen of the 17 texts included in this (...) collection – edited by Arthur S McGrade, John Kilcullen and Matthew Kempshall – were authored by Christian writers. Late-medieval Jewish or Islamic texts on ethics or politics? Not in our school. (shrink)

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according (...) to some speculative research programs. (shrink)

This contribution aims to explore the historical predecessors of the Five Percenter model of self-realization, as popularized by Hip Hop artists such as Supreme Team, Rakim Allah, Brand Nubian, Wu-Tang Clan, or Sunz of Man. As compared to frequent considerations of the phenomenon as a creative mythological background for a socio-political struggle, Five Percenter teachings shall be discussed as contemporary interpretations of historical models of self-realization in various philosophical, religious, and esoteric systems. By putting the coded system of the tenfold (...) Supreme Mathematics as one of its core teachings in context with the Pythagorean Tetractys, an arrangement of ten points in four lines, the commonalities between the sequence and concepts attributed to the respective numbers will be demonstrated. (shrink)

With regard to the new directions in the Humanities, here I am going to consider and examine the approach of al-Farabi as a medieval thinker in introducing a new outlook to “language” in difference with the other views. Thereby, I will explore his challenges in the frame of “philosophical humanism” as a term given by Arkoun (1970) and Kraemer (1984) to the humanism of the Islamic philosophers and their circles, mainly in the tenth and eleventh centuries. Al-Farabi’s conception of (...) philosophical humanism, in which philosophy is thick and religion is thin, makes its agony with the other versions of humanism and also orthodox Islam. It means that his introduction of a humanistic understanding of language should be placed in such a multi-level contested environment. According to al-Farabi, language as a universal category has relation with reason that logic should function as its proper instrument. As a result, there is no specific privileged predetermined language, but the position of any language is shaped by its relation with human reason and formal logic that is something human-made. And such a conception means language in human terms. Key Terms: Al-Farabi; Language; Philosophical Humanism; Arabic: Greek . (shrink)

This article presents a challenge that those philosophers who deny the causal interpretation of explanations provided by population genetics might have to address. Indeed, some philosophers, known as statisticalists, claim that the concept of natural selection is statistical in character and cannot be construed in causal terms. On the contrary, other philosophers, known as causalists, argue against the statistical view and support the causal interpretation of natural selection. The problem I am concerned with here arises for the statisticalists because the (...) debate on the nature of natural selection intersects the debate on whether mathematical explanations of empirical facts are genuine scientific explanations. I argue that if the explanations provided by population genetics are regarded by the statisticalists as non-causal explanations of that kind, then statisticalism risks being incompatible with a naturalist stance. The statisticalist faces a dilemma: either she maintains statisticalism but has to renounce naturalism; or she maintains naturalism but has to content herself with an account of the explanations provided by population genetics that she deems unsatisfactory. This challenge is relevant to the statisticalists because many of them see themselves as naturalists. (shrink)

Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician (...) Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify a conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed. (shrink)

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and the (...) epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics’ unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true—no one doubts that 2 + 3 = 5—but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about. (shrink)

In this handbook, I put into practice my philosophical views on children's mathematics. The handbook contains brief instructions and examples of mathematical activities. In the INSTRUCTIONS section, instructions are given on how, and in part why that way, to help preschool children in their mathematical development. In the ACTIVITIES section, there are examples of activities through which the child develops her mathematical abilities.

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...) about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

The study determined the influence of innate mathematical characteristics on the number sense competencies of junior high school students in a Philippine public school. The descriptive-correlational research design was used to accomplish the study involving a nonrandom sample of sixty 7th-grade students attending synchronous math sessions. Data obtained from the math-specific Learning Style and Self-Efficacy questionnaires and the modified Number Sense Test (NST) were analyzed and interpreted using descriptive statistics, Pearson’s Chi-Square, and Simple Linear Regression analysis. The research instruments and (...) statistics were all validated and tested for reliability. The analysis revealed that the students are visual learners, had no or slight self-efficacy, and their number sense competency level is poor. They encountered difficulty in all the components and domains of the NST. Moreover, the students’ mathematical self-efficacy is significantly related and may influence their number sense competency level. Building upon the learners’ self-efficacy to further their understanding and skills in number sense is necessary. (shrink)

During the 13th century, several logicians in the Latin medieval tradition showed a special interest in the nature of impossibility, and in the different kinds or ‘degrees’ of impossibility that could be distinguished. This discussion resulted in an analysis of the modal concept with a fineness of grain unprecedented in earlier modal accounts. Of the several divisions of the term ‘impossible’ that were offered, one became particularly relevant in connection with the debate on ars obligatoria and positio impossibilis: the (...) distinction between ‘intelligible’ and ‘unintelligible’ impossibilities. In this article, I consider some 13th-century tracts on obligations that provide an account of the relation between impossibility and intelligibility and discuss the inferential principles that are permissible when we reason from an impossible – but intelligible – premise. I also explore the way in which the 13th-century reflection on this topic survives, in a revised form, in some early 14th-century accounts of positio, namely, those of William of Ockham, Roger Swineshead and Thomas Bradwardine. (shrink)

Francis Ysidro Edgeworth’s unduly neglected monograph New and Old Methods of Ethics (1877) advances a highly sophisticated and mathematized account of social well-being in the utilitarian tradition of his 19th-century contemporaries. This article illustrates how his usage of the ‘calculus of variations’ was combined with findings from empirical psychology and economic theory to construct a consequentialist axiological framework. A conclusion is drawn that Edgeworth is a methodological predecessor to several important methods, ideas, and issues that continue to be discussed in (...) contemporary social well-being studies. (shrink)

I discuss ways in which the linguistic form of mathimatics helps us think mathematically.