César Frederico dos Santos

Realismo, em lógica, refere-se à tese que afirma que as teorias lógicas devem refletir uma realidade autônoma, independente da mente, da linguagem, de práticas de inferência e de quaisquer outras convenções humanas. Tradicionalmente, a realidade lógica tem sido pensada como um reino platônico de entidades abstratas, mas versões mais recentes propõem que a própria realidade natural contém, de alguma forma, os elementos que tornam as leis lógicas verdadeiras. Este artigo apresenta uma breve introdução crítica ao realismo lógico e aponta como a concepção realista está presente no debate atual em filosofia da lógica, especialmente na corrente denominada anti-excepcionalismo sobre a lógica. A principal dificuldade do realismo diz respeito à indefinição da natureza da realidade lógica e, por conseguinte, do conjunto de dados que as teorias lógicas deveriam levar em conta. Como alternativa ao realismo lógico, apresentamos o convencionalismo lógico que parte do pressuposto de que as leis lógicas não respondem a qualquer realidade, sendo antes convenções estabelecidas pelos lógicos. O convencionalismo, contudo, também enfrenta seus próprios problemas. O artigo conclui com a constatação das dificuldades que cercam a metafísica da lógica.

Realismo, em lógica, refere-se à tese que afirma que as teorias lógicas devem refletir uma realidade autônoma, independente da mente, da linguagem, de práticas de inferência e de quaisquer outras convenções humanas. Tradicionalmente, a realidade lógica tem sido pensada como um reino platônico de entidades abstratas, mas versões mais recentes propõem que a própria realidade natural contém, de alguma forma, os elementos que tornam as leis lógicas verdadeiras. Este artigo apresenta uma breve introdução crítica ao realismo lógico e aponta como a concepção realista está presente no debate atual em filosofia da lógica, especialmente na corrente denominada anti-excepcionalismo sobre a lógica. A principal dificuldade do realismo diz respeito à indefinição da natureza da realidade lógica e, por conseguinte, do conjunto de dados que as teorias lógicas deveriam levar em conta. Como alternativa ao realismo lógico, apresentamos o convencionalismo lógico que parte do pressuposto de que as leis lógicas não respondem a qualquer realidade, sendo antes convenções estabelecidas pelos lógicos. O convencionalismo, contudo, também enfrenta seus próprios problemas. O artigo conclui com a constatação das dificuldades que cercam a metafísica da lógica.

The field of numerical cognition provides a fairly clear picture of the processes through which we learn basic arithmetical facts. This scientific picture, however, is rarely taken as providing a response to a much-debated philosophical question, namely, the question of how we obtain number knowledge, since numbers are usually thought to be abstract entities located outside of space and time. In this paper, I take the scientific evidence on how we learn arithmetic as providing a response to the philosophical question of how we obtain number knowledge. I reject the view that numbers are abstract entities located outside of space and time and, alternatively, derive from the scientific evidence a novel account of the nature of numbers. In this account, numbers are reifications of the counting procedure and arithmetic statements are seen as describing the functioning of counting and calculation techniques.

In the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into account that there are non-numerical methods of assessing numerosity, which opens up the possibility that cognitive agents lacking numerical abilities may still be able to represent numerosity. In this paper, I distinguish between numerical and non-numerical methods of assessing numerosity and show that the most common models of the internal mechanisms of the so-called number sense rely on non-numerical methods, despite the claims of their proponents to the contrary. I conclude that, even if it is established that agents attend to numerosity, rather than continuous properties of stimuli correlated with it, an answer to the question of the existe...

In the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into account that there are non-numerical methods of assessing numerosity, which opens up the possibility that cognitive agents lacking numerical abilities may still be able to represent numerosity. In this paper, I distinguish between numerical and non-numerical methods of assessing numerosity and show that the most common models of the internal mechanisms of the so-called number sense rely on non-numerical methods, despite the claims of their proponents to the contrary. I conclude that, even if it is established that agents attend to numerosity, rather than continuous properties of stimuli correlated with it, an answer to the question of the existence of the number sense is still pending the investigation of a further issue, namely, whether the mechanisms the brain uses to assess numerosity qualify as numerical or nonnumerical.

Dados da psicologia são relevantes na abordagem de alguns problemas em filosofia da lógica mesmo que não se assuma, de antemão, uma posição psico-logista. Para ilustrar um método de abordar problemas na filosofia da lógica que faz largo uso de dados da psicologia, neste artigo considero a questão sobre a metafísica da lógica—qual o objeto de estudo da lógica?—à luz de resultados da psicologia do pensamento e da psicologia do desenvolvimento. Estes resultados nos permitem concluir que sistemas lógicos não são meramente descritivos de aspectos gerais da linguagem natural, do mundo ou da racionalidade humana. Os dados são consistentes com a hipótese de que sis-temas lógicos dedutivos sejam conjuntos de regras de caráter primordialmente não-descritivo, criadas mediante reflexão ativa de filósofos, matemáticos e lógicos, visando garantir inferências dedutivas em certos contextos.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Filosofia e Ciências Humanas, Programa de Pós-Graduação em Filosofia, Florianópolis, 2012Em um certo sentido, a matemática usual pode ser reduzida à teoria dos conjuntos, isto é, os objetos matemáticos usuais podem ser definidos como conjuntos, e os teoremas que governam esses objetos podem ser provados em ZFC. Todavia, existem enunciados importantes - a hipótese do contínuo é o mais célebre deles - que são independentes de ZFC. A existência de enunciados independentes levanta a questão da necessidade e conveniência de estender ZFC pela adição de novos axiomas. Nesse contexto, é interessante dispor de critérios claros que permitam avaliar candidatos a axioma para uma extensão da teoria dos conjuntos que decidiria enunciados independentes. Maddy tem se dedicado intensamente à filosofia da teoria dos conjuntos, com destaque para a investigação de critérios para seleção de axiomas. Nesta dissertação, nosso objet...

Do numbers exist? Most of the answers to this question presented in the literature of the last decades have relied on a priori methods of investigation, where scientific data and theories about the human experience of numbers are irrelevant. These a priori approaches, however, have been inconclusive. In this dissertation, I adopt an empirically informed approach in which scientific descriptions of the human experience of numbers—as provided by cognitive sciences, linguistics, developmental psychology, and mathematics education—provide valuable information on the existence and status of what we call “numbers.” These scientific descriptions allow for the conclusion that numbers, conceived of as independent, non-spatiotemporal objects, do not exist. What exist are certain human-made techniques which engender in us the idea that a special class of objects called numbers exists. I show that, just as counting procedures and other arithmetical operations are cognitive tools that allow us to go beyond the limits of our genetically endowed cognitive skills, the very idea that numbers exist as independent objects is a cognitive tool that facilitates calculation—in other words, a useful reification. The ontological hypothesis suggested by the scientific description of the human experience of numbers is that operations such as counting and calculating procedures are the objective subject matter underlying arithmetic, rather than a putative class of non-spatiotemporal objects. Thus, the claim is that applied and pure arithmetical statements are true of the counting procedure and other arithmetical operations, rather than true of non-spatiotemporal numbers. In contrast to other attempted answers to the question of the ontological status of numbers, the hypothesis defended in this dissertation is accountable towards empirical data, and can thus be improved or refuted on an empirical basis

Anti-exceptionalists about logic claim that logical methodology is not different from scientific methodology when it comes to theory choice. Two anti-exceptionalist accounts of theory choice in logic are abductivism (defended by Priest and Williamson) and predictivism (recently proposed by Martin and Hjortland). These accounts have in common reliance on pre-theoretical logical intuitions for the assessment of candidate logical theories. In this paper, I investigate whether intuitions can provide what abductivism and predictivism want from them and conclude that they do not. As an alternative to these approaches, I propose a Carnapian view on logical theorizing according to which logical theories do not simply account for pre-theoretical intuitions, but rather improve on them. In this account, logical theories are ameliorative, rather than representational.

In 1939, the influential psychophysicist S. S. Stevens proposed definitional distinctions between the terms 'number,' 'numerosity,' and 'numerousness.' Although the definitions he proposed were adopted by several psychophysicists and experimental psychologists in the 1940s and 1950s, they were almost forgotten in the subsequent decades, making room for what has been described as a "terminological chaos" in the field of numerical cognition. In this paper, I review Stevens's distinctions to help bring order to this alleged chaos and to shed light on two closely related questions: whether it is adequate to speak of a number sense and how philosophers can make sense of the claim by cognitive scientists that numbers are perceptual entities. Moreover, I o↵er further support to Stevens's distinction between numerosity and numerousness by showing that they are relational properties that emerge through di↵erent relationships between agents and their environments. The final conclusion is that, by adopting Stevens's distinctions, numbers do not need to be seen as perceptual entities, since the so-called number sense is better described as a sense of numerousness.

Resumo: A filosofia segunda de Penelope Maddy parece ir de encontro aos avanços em filosofia da ciência que sucederam a obra de Thomas Kuhn ao defender que não cabe à filosofia criticar a atividade científica a partir de um ponto de vista filosófico. Neste artigo pretendemos mostrar que tal oposição aparente não se sustenta, e que há uma forte semelhança entre esses dois programas filosóficos notadamente no que se refere à concepção de racionalidade científica. Palavras-chave: filosofia segunda, Penelope Maddy, racionalidade científica. Abstract: Penelope Maddy's second philosophy seems to be contrary to the developments carried out by the new philosophy of science after Thomas Kuhn's works. She defends that philosophy can't criticize science from a first philosophical point of view. In this paper we aim to show that there isn't such disagreement, quite the contrary, there is a striking similarity between the two philosophical programs, mainly concerning the concepti...

In the literature on enculturation—the thesis according to which higher cognitive capacities result from transformations in the brain driven by culture—numerical cognition is often cited as an example. A consequence of the enculturation account for numerical cognition is that individuals cannot acquire numerical competence if a symbolic system for numbers is not available in their cultural environment. This poses a problem for the explanation of the historical origins of numerical concepts and symbols. When a numeral system had not been created yet, people did not have the opportunity to acquire number concepts. But, if people did not have number concepts, how could they ever create a symbolic systemfor numbers? Here I propose an account of the invention of symbolic systems for numbers by anumeric people in the remote past that is compatible with the enculturation thesis. I suggest that symbols for numbers and number concepts may have emerged at the same time through the re-semantif...

In Thomasson’s “easy” approach to ontology, recalcitrant ontological problems are purportedly solved through trivial and straightforward inferences from putatively uncontroversial premises. Easy ontology aims at putting aside the metaphysical quarrels that, according to Thomasson, have led philosophers to think that existence questions were hard to answer. In this paper, I argue that, even if we refrain from engaging in metaphysics and limit our investigations to conceptual and empirical matters, as the “easy” approach recommends, we cannot expect to answer disputed existence questions by trivial and straightforward inferences. The problem is that putative easy-arguments leave room for many contentious issues for which there is no trivial and straightforward answer. To illustrate this point, I discuss some aspects of the debates on the existence of human races and numbers.

No Discurso sobre as Ciências e as Artes, seu primeiro discurso, Rousseau defende a polêmica tese de que o progresso das ciências e das artes, contrariamente ao que pretendia o Iluminismo, estava contribuindo mais para a degeneração dos costumes e da sociedade do que para seu aperfeiçoamento. O Primeiro Discurso foi escrito em 1749, há quase 300 anos. Nesse período, a ciência e a nossa compreensão sobre ela mudaram profundamente. Mais importante, nesse período surgiu da ciência algo imprevisto para Rousseau no Primeiro Discurso: a tecnologia moderna. A proposta deste trabalho é, pois, revisitar as críticas de Rousseau à ciência com um olhar contemporâneo, buscando avaliar o quanto daquelas críticas ainda faz sentido nos panoramas científico e social atuais. Para tanto, classificamos esquematicamente as críticas de Rousseau em dois grupos: as que acusam a inutilidade das ciências e as que acusam como nociva a sofisticação que as ciências produzem na sociedade. Defenderemos que o surg...

Anti-exceptionalists about logic claim that logical methodology is not different from scientific methodology when it comes to theory choice. Two anti-exceptionalist accounts of theory choice in logic are abductivism (defended by Priest and Williamson) and predictivism (recently proposed by Martin and Hjortland). These accounts have in common reliance on pre-theoretical logical intuitions for the assessment of candidate logical theories. In this paper, I investigate whether intuitions can provide what abductivism and predictivism want from them and conclude that they do not. As an alternative to these approaches, I propose a Carnapian view on logical theorizing according to which logical theories do not simply account for pre-theoretical intuitions, but rather improve on them. In this account, logical theories are ameliorative, rather than representational.

In the literature on enculturation--the thesis according to which higher cognitive capacities result from transformations in the brain driven by culture--numerical cognition is often cited as an example. A consequence of the enculturation account for numerical cognition is that individuals cannot acquire numerical competence if a symbolic system for numbers is not available in their cultural environment. This poses a problem for the explanation of the historical origins of numerical concepts and symbols. When a numeral system had not been created yet, people did not have the opportunity to acquire number concepts. But, if people did not have number concepts, how could they ever create a symbolic system for numbers? Here I propose an account of the invention of symbolic systems for numbers by anumeric people in the remote past that is compatible with the enculturation thesis. I suggest that symbols for numbers and number concepts may have emerged at the same time through the re-semantification of words whose meanings were originally non-numerical.

In describing numerosity as “a kind of ersatz number,” Clarke and Beck fail to consider a familiar and compelling definition of numerosity, which conceptualizes numerosity as the cognitive counterpart of the mathematical concept of cardinality; numerosity is the magnitude, whereas number is a scale through which numerosity/cardinality is measured. We argue that these distinctions should be considered.

In 1939, the influential psychophysicist S. S. Stevens proposed definitional distinctions between the terms 'number,' 'numerosity,' and 'numerousness.' Although the definitions he proposed were adopted by several psychophysicists and experimental psychologists in the 1940s and 1950s, they were almost forgotten in the subsequent decades, making room for what has been described as a "terminological chaos" in the field of numerical cognition. In this paper, I review Stevens's distinctions to help bring order to this alleged chaos and to shed light on two closely related questions: whether it is adequate to speak of a number sense and how philosophers can make sense of the claim by cognitive scientists that numbers are perceptual entities. Moreover, I o↵er further support to Stevens's distinction between numerosity and numerousness by showing that they are relational properties that emerge through di↵erent relationships between agents and their environments. The final conclusion is that, by adopting Stevens's distinctions, numbers do not need to be seen as perceptual entities, since the so-called number sense is better described as a sense of numerousness.

Do numbers exist? Most of the answers to this question presented in the literature of the last decades have relied on a priori methods of investigation, where scientific data and theories about the human experience of numbers are irrelevant. These a priori approaches, however, have been inconclusive. In this dissertation, I adopt an empirically informed approach in which scientific descriptions of the human experience of numbers—as provided by cognitive sciences, linguistics, developmental psychology, and mathematics education—provide valuable information on the existence and status of what we call “numbers.” These scientific descriptions allow for the conclusion that numbers, conceived of as independent, non-spatiotemporal objects, do not exist. What exist are certain human-made techniques which engender in us the idea that a special class of objects called numbers exists. I show that, just as counting procedures and other arithmetical operations are cognitive tools that allow us to go beyond the limits of our genetically endowed cognitive skills, the very idea that numbers exist as independent objects is a cognitive tool that facilitates calculation—in other words, a useful reification. The ontological hypothesis suggested by the scientific description of the human experience of numbers is that operations such as counting and calculating procedures are the objective subject matter underlying arithmetic, rather than a putative class of non-spatiotemporal objects. Thus, the claim is that applied and pure arithmetical statements are true of the counting procedure and other arithmetical operations, rather than true of non-spatiotemporal numbers. In contrast to other attempted answers to the question of the ontological status of numbers, the hypothesis defended in this dissertation is accountable towards empirical data, and can thus be improved or refuted on an empirical basis.

Dados da psicologia são relevantes na abordagem de alguns problemas em filosofia da lógica mesmo que não se assuma, de antemão, uma posição psicologista. Para ilustrar um método de abordar problemas na filosofia da lógica que faz largo uso de dados da psicologia, neste artigo considero a questão sobre a metafísica da lógica---qual o objeto de estudo da lógica?---à luz de resultados da psicologia do pensamento e da psicologia do desenvolvimento. Estes resultados nos permitem concluir que sistemas lógicos não são meramente descritivos de aspectos gerais da linguagem natural, do mundo ou da racionalidade humana. Os dados são consistentes com a hipótese de que sistemas lógicos dedutivos sejam conjuntos de regras de caráter primordialmente não-descritivo, criadas mediante reflexão ativa de filósofos, matemáticos e lógicos, visando garantir inferências dedutivas em certos contextos.

In Thomasson's "easy" approach to ontology, recalcitrant ontological problems are purportedly solved through trivial and straightforward inferences from putatively uncontroversial premises. Easy ontology aims at putting aside the metaphysical quarrels that, according to Thomasson, have led philosophers to think that existence questions were hard to answer. In this paper, I argue that, even if we refrain from engaging in metaphysics and limit our investigations to conceptual and empirical matters, as the "easy" approach recommends, we cannot expect to answer disputed existence questions by trivial and straightforward inferences. The problem is that putative easy-arguments leave room for many contentious issues for which there is no trivial and straightforward answer. To illustrate this point, I discuss some aspects of the debates on the existence of human races and numbers.

Resumo E lugar comum, no meio filosofico, a afirmacao de que a teoria dos conjuntos constitui os fundamentos da matematica. No entanto, a nocao de fundamento, historicamente cercada de uma extensa rede de significados e anseios filosoficos, pode dar margem a interpretacoes inadequadas do papel desempenhado por essa teoria na matematica. Nosso objetivo, nesse artigo, e afastar algumas dessas confusoes a luz de resultados matematicos da propria teoria, e afirmar um sentido matematico em que a teoria dos conjuntos pode ser dita propriamente integrante dos fundamentos da matematica. Palavras-chave Fundamentos da matematica, teoria dos conjuntos. Abstract It's common in philosophical circles to claim that set theory is in the foundations of mathematics. However, the idea of foundations has historically been related to various philosophical meanings and expectations, and this can give room to erroneous interpretations of the role of set theory in mathematics. In this article, our purp...

In Maddy's philosophy, mathematics is autonomous, i.e., it is not subordinated to either science or philosophy. Mathematics establishes and pursues its own goals and must be judged on its own terms. This leads Maddy to admit, in Naturalism in Mathematics (1997) and also in Second Philosophy (2007), that, even if mathematicians choose to pursue a goal that could seem improper from the philosophical or scientific point of view, there is nothing to be done except accepting the new state of affairs. In Defending the Axioms (2011), nevertheless, Maddy changes her position regarding this issue. She claims to have found the basis from which to assess the adequacy of mathematical goals. From this basis, if mathematicians choose to pursue what seems to be an improper goal, the philosopher could claim that they are going astray. In this paper, I will review Maddy's positions in these books; and, especially regarding Defending the Axioms, I will sustain that the institution of a permanent parameter for the judgment of mathematical goals goes against the alleged autonomy of mathematics and other important traits of her philosophy.

Do ponto de vista matemático, pode-se dizer que há uma resposta satisfatória, ainda que incompleta, ao problema ontológico: existe em matemática tudo aquilo que pode ser provado na teoria dos conjuntos. Porém, é certo que a resposta conjuntista, ainda que satisfatória matematicamente, é insatisfatória filosoficamente. Essa constatação nos leva a defender neste ensaio que o problema ontológico em matemática tem duas dimensões irredutíveis entre si: a dimensão matemática e a dimensão filosófica. Sugerimos que uma resposta matematicamente satisfatória pode ser filosoficamente insatisfatória, e vice-versa. Concentramo-nos, então, em identificar que características são esperadas de uma resposta filosófica minimamente satisfatória. Para tal, esboçamos uma versão de naturalismo, de inspiração quiniana, mas divergente deste. A partir dessa versão de naturalismo, especificamos quatro requisitos: (i) evidência empírica; (ii) enquadramento das entidades matemáticas no cenário da ontologia das ciências naturais; (iii) explicação do sucesso dos métodos matemáticos dedutivos usuais e (iv) explicação das relações entre métodos científicos e métodos matemáticos.

A filosofia segunda de Penelope Maddy parece ir de encontro aos avanços em filosofia da ciência que sucederam a obra de Thomas Kuhn ao defender que não cabe à filosofia criticar a atividade científica a partir de um ponto de vista filosófico. Neste artigo pretendemos mostrar que tal oposição aparente não se sustenta, e que há uma forte semelhança entre esses dois programas filosóficos notadamente no que se refere à concepção de racionalidade científica.

Quais das críticas que Rousseau faz à ciência em seu "Discurso sobre as Ciências e as Artes" ainda se aplicam à ciência de hoje? Sugerimos que o surgimento da tecnologia moderna desautoriza algumas das críticas de Rousseau, ao mesmo tempo que fortalece outras.

Quine e Kuhn têm visões diferentes sobre a ciência e deram origem a linhas de pesquisa em filosofia da ciência comumente percebidas como divergentes. Mas, apesar das diferenças, é possível identificar no trabalho de ambos semelhanças metodológicas. Neste ensaio, sustentamos que certos aspectos metodológicos da filosofia histórica da ciência de Kuhn harmonizam-se com o espírito da epistemologia naturalizada de Quine. O naturalismo de Maddy, defendemos ao final, pode ser visto como combinando aspectos das duas abordagens.

É lugar comum, no meio filosófico, a afirmação de que a teoria dos conjuntos constitui os fundamentos da matemática. No entanto, a noção de fundamento, historicamente cercada de uma extensa rede de significados e anseios filosóficos, pode dar margem a interpretações inadequadas do papel desempenhado por essa teoria na matemática. Nosso objetivo, nesse artigo, é afastar algumas dessas confusões à luz de resultados matemáticos da própria teoria, e afirmar um sentido matemático em que a teoria dos conjuntos pode ser dita  propriamente integrante dos fundamentos da matemática.

Do numbers exist? Most of the answers to this question presented in the literature of the last decades have relied on a priori methods of investigation, where scientific data and theories about the human experience of numbers are irrelevant. These a priori approaches, however, have been inconclusive. In this dissertation, I adopt an empirically informed approach in which scientific descriptions of the human experience of numbers—as provided by cognitive sciences, linguistics, developmental psychology, and mathematics education—provide valuable information on the existence and status of what we call “numbers.” These scientific descriptions allow for the conclusion that numbers, conceived of as independent, non-spatiotemporal objects, do not exist. What exist are certain human-made techniques which engender in us the idea that a special class of objects called numbers exists. I show that, just as counting procedures and other arithmetical operations are cognitive tools that allow us to go beyond the limits of our genetically endowed cognitive skills, the very idea that numbers exist as independent objects is a cognitive tool that facilitates calculation—in other words, a useful reification. The ontological hypothesis suggested by the scientific description of the human experience of numbers is that operations such as counting and calculating procedures are the objective subject matter underlying arithmetic, rather than a putative class of non-spatiotemporal objects. Thus, the claim is that applied and pure arithmetical statements are true of the counting procedure and other arithmetical operations, rather than true of non-spatiotemporal numbers. In contrast to other attempted answers to the question of the ontological status of numbers, the hypothesis defended in this dissertation is accountable towards empirical data, and can thus be improved or refuted on an empirical basis.

Karenleigh A. Overmann's book, The Materiality of Numbers: Emergence and Elaboration from Prehistory to Present , undertakes what can be described as a "Copernican Revolution" in the way we understand the relationship between numbers and the material devices we use to record and manipulate them. Its main thesis is that, rather than an a priori concept of number stored in the brain being the cause of the external representations of numbers created by humans since prehistory, it was the external material devices for counting and calculation, created in prehistory, that caused the

Logical theories are usually seen as true or false in relation to the phenomenon they aim to describe, namely, validity. An alternative view suggests that the laws governing validity are "legislated-true," making logical theories conventional. In this chapter, I propose an intermediate standpoint in which logical systems exhibit a blend of descriptive and conventional aspects, balanced to fulfill their primary purpose, which is not seen as theoretical, but practical: assisting us in generating and identifying valid inferences. From this perspective, logics are cognitive tools, human creations designed primarily to facilitate the performance of cognitive operations. This viewpoint aligns logic more closely with technology than science and supports a pluralistic approach: the diversity of alternative technical solutions for issues related to the creation and recognition of valid inferences allows for the coexistence of multiple logics.