Orientador: Marcelo de Carvalho Borba / Banca: Maria Aparecida Viggiani Bicudo / Banca: Nilson José Machado / Banca: Siobhan Victoria Healy / Banca: Marcus Vinicius Maltempi / Resumo: Os processos perceptivos que fundamentam a experiência humana podem nos parecer absolutamente naturais. Devido a isso, costumamos não tematizá-los. Este trabalho é um esforço de evidenciação da percepção na educação matemática, mais especificamente na geometria dinâmica e no ensino de funções com auxílio de representações dinâmicas. Percepção é entendida em uma concepção fenomenológica. Sustenta-se que: (a) é da natureza humana certa capacidade de perceber comportamentos de dependências entre eventos do mundo físico, isto é, existe um sentido de percepção de dependência; (b) as representações dinâmicas de funções, como os Dynagraphs (conhecidos na literatura) e os funcionetes (propostos no trabalho), são depreendidas pelo sentido de percepção de dependência; (c) o emprego de representações dinâmicas no auxílio ao ensino de funções abre novos sentidos para funções matemáticas, conceitos, propriedades e teoremas correlatos, justificando o interesse em sua aplicação; além disso, os sentidos abertos são perceptivos e, por isso, diretos, imediatos e evidentes (conforme a fundamentação fenomenológica); (d) existe um sentido de percepção de restrições ou impedimentos; (e) na resolução interativa (geometria dinâmica) de sistemas de restrições geométricas, o sentido de percepção de restrições apresenta, ao trazer perceptivamente as restrições para primeiro plano, as construções geométricas como uma combinação de restrições. No desenvolvimento dessas ideias: apresentam-se os funcionetes planos e sua aplicação na construção de uma abordagem pedagógica para o conceito (da álgebra linear) transformação linear, que é um tipo de função; abordam-se os tópicos: autovetores de um operador linear, propriedade de linearidade e núcleo de uma transformação linear, inclusive o teorema do núcleo e da imagem... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The perceptive processes that provide the basis for human experience can seem absolutely natural. Therefore we do not have the habit of focusing on them as the object of study. The present theoretical study aimed to make perception evident in the context of mathematics education, specifically dynamic geometry and teaching of functions using dynamic representations. Perception is understood as a phenomenological conception. It is maintained that: (a) it is of human nature to be able to perceive dependent behaviors among events in the physical world, i.e. a sense of perception of dependence; (b) dynamic representations of functions, such as Dynagraphs (known in the literature) and "funcionetes" (proposed here), are ascertained through the sense of perception of dependence; (c) the use of dynamic representations to aid in the teaching of functions opens up new senses for mathematical functions, concepts, properties and correlated theorems, justifying interest in its application; in addition, these newly-opened senses are perceptive in nature, and therefore direct, immediate and evident (according to foundations of phenomenology); (d) there exists a sense of perception of constraints or impediments; (e) in the interactive resolution (dynamic geometry) of constraint systems for geometric domain, the sense of perception of constraints presents geometric constructions as a combination of constraints as it perceptively brings the constraints to the foreground. The concept of "funcionetes planos" is presented and their use proposed as part of an approach for teaching the concept (from linear algebra) of linear transformation, which is a type of function. Topics addressed include: eigenvectors of a linear operator, the property of linearity and nucleus of a linear transformation, including the theorem of nucleus and of image, presented in a perceptive sense... (Complete abstract click electronic access below) / Doutor

No description available.

Made available in DSpace on 2014-06-11T19:31:43Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-11-03Bitstream added on 2014-06-13T20:02:50Z : No. of bitstreams: 1 figueiredo_oa_dr_rcla.pdf: 599343 bytes, checksum: 2107a2036f27ecde0dcfdbff1f4a2bf3 (MD5) / Os processos perceptivos que fundamentam a experiência humana podem nos parecer absolutamente naturais. Devido a isso, costumamos não tematizá-los. Este trabalho é um esforço de evidenciação da percepção na educação matemática, mais especificamente na geometria dinâmica e no ensino de funções com auxílio de representações dinâmicas. Percepção é entendida em uma concepção fenomenológica. Sustenta-se que: (a) é da natureza humana certa capacidade de perceber comportamentos de dependências entre eventos do mundo físico, isto é, existe um sentido de percepção de dependência; (b) as representações dinâmicas de funções, como os Dynagraphs (conhecidos na literatura) e os funcionetes (propostos no trabalho), são depreendidas pelo sentido de percepção de dependência; (c) o emprego de representações dinâmicas no auxílio ao ensino de funções abre novos sentidos para funções matemáticas, conceitos, propriedades e teoremas correlatos, justificando o interesse em sua aplicação; além disso, os sentidos abertos são perceptivos e, por isso, diretos, imediatos e evidentes (conforme a fundamentação fenomenológica); (d) existe um sentido de percepção de restrições ou impedimentos; (e) na resolução interativa (geometria dinâmica) de sistemas de restrições geométricas, o sentido de percepção de restrições apresenta, ao trazer perceptivamente as restrições para primeiro plano, as construções geométricas como uma combinação de restrições. No desenvolvimento dessas ideias: apresentam-se os funcionetes planos e sua aplicação na construção de uma abordagem pedagógica para o conceito (da álgebra linear) transformação linear, que é um tipo de função; abordam-se os tópicos: autovetores de um operador linear, propriedade de linearidade e núcleo de uma transformação linear, inclusive o teorema do núcleo e da imagem... / The perceptive processes that provide the basis for human experience can seem absolutely natural. Therefore we do not have the habit of focusing on them as the object of study. The present theoretical study aimed to make perception evident in the context of mathematics education, specifically dynamic geometry and teaching of functions using dynamic representations. Perception is understood as a phenomenological conception. It is maintained that: (a) it is of human nature to be able to perceive dependent behaviors among events in the physical world, i.e. a sense of perception of dependence; (b) dynamic representations of functions, such as Dynagraphs (known in the literature) and “funcionetes” (proposed here), are ascertained through the sense of perception of dependence; (c) the use of dynamic representations to aid in the teaching of functions opens up new senses for mathematical functions, concepts, properties and correlated theorems, justifying interest in its application; in addition, these newly-opened senses are perceptive in nature, and therefore direct, immediate and evident (according to foundations of phenomenology); (d) there exists a sense of perception of constraints or impediments; (e) in the interactive resolution (dynamic geometry) of constraint systems for geometric domain, the sense of perception of constraints presents geometric constructions as a combination of constraints as it perceptively brings the constraints to the foreground. The concept of “funcionetes planos” is presented and their use proposed as part of an approach for teaching the concept (from linear algebra) of linear transformation, which is a type of function. Topics addressed include: eigenvectors of a linear operator, the property of linearity and nucleus of a linear transformation, including the theorem of nucleus and of image, presented in a perceptive sense... (Complete abstract click electronic access below)

Alors que l’applicabilité des mathématiques est devenue un sujet d’intérêt pour le débat philosophique récent, celui-ci n’a pas encore clairement mis l’accent sur les questions épistémiques posées par l’intervention des mathématiques dans les sciences et dans la vie quotidienne. Ces questions peuvent être formulées comme suit : comment pouvons-nous connaître la vérité d’une proposition scientifique, ou plus généralement d’une proposition sur certaines caractéristiques du monde naturel, lorsque cette proposition comprend des éléments mathématiques? Quelle sorte de justification avons-nous pour les parties mathématiques de notre connaissance empirique? Cette thèse de doctorat a un double objectif : d’une part, offrir une systématisation critique du débat philosophique en cours sur l’applicabilité. D’autre part, clarifier le problème épistémique posé par l’applicabilité des mathématiques et le séparer des problèmes métaphysiques corrélés. La première partie du travail est consacrée à la formulation des questions propres à une enquête épistémologique sur les mathématiques appliquées, ainsi qu’à la présentation de l’analyse classique de l’applicabilité offerte par Steiner. Dans la partie II, la présentation du débat récent sur l’applicabilité est organisée autour d’une distinction entre les approches qui considèrent les mathématiques pures et appliquées sur le même niveau épistémique, et celles qui distinguent le niveau de mathématiques pures du niveau des applications. Les positions étudiées sont, respectivement, les points de vue fregéen et néo-fregéen, et ce que l’on considère comme des points de vue ‘structurels’, à savoir le structuralisme mathématique (à la fois ante rem et éliminatif), la position de Field et la théorie de la mesure. La partie III introduit le débat sur l’indispensabilité des mathématiques dans les sciences, pour montrer comment les différentes formulations des arguments d’indispensabilité et les critiques qui leur sont adressées renouvellent l’attention sur les questions philosophiques liées à l’applicabilité, et clarifient la séparation entre les questions épistémiques sur les mathématiques pures (par exemple, le problème de l’accès) et les questions épistémiques sur les applications (par exemple, la justification des parties mathématiques de notre connaissance scientifique). La position de Christopher Pincock, qui théorise un traitement épistémique distinct pour les mathématiques pures et appliquées, est spécifiquement analysée. Enfin, la dernière partie en conclut ce que peuvent être les caractéristiques d’une théorie épistémologique adéquate pour les mathématiques pures et pour les mathématiques appliquées, et présentent plusieurs problématiques connexes et cruciales pour de futures recherches. / While the applicability of mathematics has become a topic of great interest in recent philosophical debate, the debate has not yet clearly focused on the fundamental epistemic questions that arise from the use of mathematics in science and in daily life. These questions can be basicallystated as follows: how can we affirm to know the truth of a scientific statement, or more generally that of any statement about a feature of the natural world, when that statement includes some elements of mathematics? What kind of justification do we have for the mathematical portions of our empirical knowledge? My PhD dissertation has a twofold purpose: on the one hand, it offers a critical systematization of the on-going philosophical debate on applicability. On the other hand, the epistemic problem posed by the applicability of mathematics is clarified and separated from correlated metaphysical issues. The first part of the work is devoted to the definition of the specific epistemic questions and the presentation of the classic analysis of applicability problem(s) offered by Steiner. In Part II, the recent debate on applicability is organized around a distinction between those approaches that take pure and applied mathematics to be on the same epistemic level, and those that keep the level of pure mathematics separate from the level of application. The positions investigated are, respectively, Fregean and Neo-Fregean views for the one-stage side, and what I refer to as ‘structural’ views for the two-stageside, namely, mathematical structuralism (both ante rem and eliminative), Field’s account, and measurement theory. Part III takes into account the related debate on the indispensability of mathematics to science, showing how the different formulations of indispensability arguments and the criticisms led to renewed attention to the philosophical questions about applicability in the early 2000s, along with a clarification of the separation between epistemic questions about pure mathematics (e.g. the access problem) and epistemic questions about applications (e.g. the justification of the mathematical portions of scientific knowledge). The account offered by Christopher Pincock, which provides a separate epistemic treatment for pure and appliedmathematics, is specifically analyzed. Finally, in the last part of the work, we draw particular conclusions about what would be, following our analysis, the features of a suitable epistemological treatment of both pure and applied mathematics, while several connected issues are identified as crucial for further inquiry. / L’applicabilità della matematica è diventata un argomento di grande interesse per il dibattito filosofico recente, ma il dibattito non si è ancora focalizzato sulle fondamentali questioni epistemologiche poste dall’uso della matematica nella scienza e nella vita quotidiana. Queste domande possono essere formulate come segue: come possiamo dire di conoscere la verità di un asserto scientifico, o più in generale di qualsiasi asserto su alcune caratteristiche del mondo naturale, quando tale asserto include elementi matematici? Che tipo di giustificazione possiamo avere per le porzioni matematiche della nostra conoscenza empirica? La presente tesi di dottorato ha un duplice scopo: da un lato, si offre una presentazione sistematica deldibattito filosofico in corso sull’applicabilità. Dall’altro lato, il problema epistemico posto dall’applicabilità della matematica è chiarito e separato dai correlati problemi metafisici. La prima parte del lavoro è dedicata alla definizione delle specifiche domande epistemiche sulla matematica applicata; si presenta inoltre l’analisi classica dei problemi legati all’applicabilità offerta da Steiner. Nella seconda parte, la presentazione del dibattito recente sull’applicabilità è organizzata attorno ad una distinzione tra le posizioni che considerano matematica pura e applicata sullo stesso livello epistemico e quelle che mantengono il livello della matematica pura separato dal livello applicativo. Le posizioni indagate sono, rispettivamente, la posizionefregeana e neo-fregeana da un lato, e le posizioni che definiremo ‘strutturali’ dall’altro, ovvero lo strutturalismo matematico (sia ante rem che eliminativo), la posizione di Field e la teoria della misura. La terza parte del lavoro affronta il dibattito sull’indispensabilità della matematica nella scienza, mostrando come le diverse formulazioni degli argomenti di indispensabilità e le critiche ad esse rivolte contribuiscano a rinnovare l’interesse per le domande filosofiche sull’applicabilità, oltre che a chiarire la separazione tra domande epistemiche sulla matematica pura (ad esempio Il problema dell’accesso) e domande epistemiche sulle applicazioni (ad esempio la giustificazione delle porzioni matematiche della nostra conoscenza scientifica). La proposta teorica di Christopher Pincock, che tratta separatamente l’epistemologia di matematica pura e applicata, è analizzata in modo specifico. Nell’ultima parte del lavoro, si traggono alcuni conclusioni su quali potrebbero essere, in seguito allo studio svolto, le caratteristiche di un trattamento adeguato dell’epistemologia della matematica pura e applicata. Infine, alcuni ulteriori problemi connessi sono individuati come cruciali per indagini future.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-27T18:28:27Z No. of bitstreams: 1 TeseALN.pdf: 2172393 bytes, checksum: 0233779408d5d19f4ecc2c9d4de83bf0 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-04T17:28:56Z (GMT) No. of bitstreams: 1 TeseALN.pdf: 2172393 bytes, checksum: 0233779408d5d19f4ecc2c9d4de83bf0 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-04T17:29:02Z (GMT) No. of bitstreams: 1 TeseALN.pdf: 2172393 bytes, checksum: 0233779408d5d19f4ecc2c9d4de83bf0 (MD5) / Made available in DSpace on 2016-10-04T17:29:09Z (GMT). No. of bitstreams: 1 TeseALN.pdf: 2172393 bytes, checksum: 0233779408d5d19f4ecc2c9d4de83bf0 (MD5) Previous issue date: 2015-09-29 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / This thesis provides a reading and interpretation of Wittgenstein’s writings on mathematics at the beginning of his “middle period” (more precisely, at the “mathematical chapters” of Philosophische Bemerkungen), placing these writings in the context of two crises. The first, internal to his thought, consists of inconsistencies regarding what the Tractatus prescribed as the result of the application of logic and the effective logical analysis of certain domains of reality, which characterized, in Wittgenstein’s view, a crisis in the foundations of logic. On the other hand, controversies about the foundations of mathematics were intensified throughout the 1920s, and debates between three schools who attempted to impose their way not only of conceiving mathematics, but also of doing it, became increasingly frequent. This crisis, also called Grundlagenkrise der Mathematik, is an important historical and conceptual background for these early writings immediately after Wittgenstein’s return to philosophy in 1929. If in the Tractatus Wittgenstein had positioned himself only with regard to Frege’s and Russell’s logicism, in these writtings he tries in his own way to contrast his thought with the prevailing trends of his time: the intuicionism of Brouwer and Weyl, Hilbert’s formalism and, finally, Ramsey’s renewed logicism. This thesis develops, in its concluding Chapter, a reflection on Wittgenstein’s posture with respect to these three classical schools and with respect to the problems faced by them. / A tese fornece uma leitura e interpretação dos escritos de Wittgenstein sobre a matemática no início do seu “período intermediário” (mais precisamente, nos “capítulos matemáticos” das Philosophische Bemerkungen), situando estes escritos no contexto de duas crises. A primeira, interna ao pensamento do autor, diz respeito a inconsistências referentes `aquilo que o Tractatus prescrevera como resultado da aplicação da lógica e a análise lógica efetiva de certos domínios do real, o que configurava, aos olhos de Wittgenstein, uma crise nos fundamentos da lógica. Por outro lado, controvérsias acerca dos fundamentos da matemática se acirraram ao longo da década de 1920, e debates entre três escolas que buscavam impor o seu modo não apenas de conceber a matemática, mas também de fazê-la tornavam-se cada vez mais frequentes. Essa crise, que recebera o codinome de Grundlagenkrise der Mathematik, constitui um importante pano de fundo histórico-conceitual para estes primeiros escritos de Wittgenstein após seu retorno `a filosofia em 1929. Se, no Tractatus, Wittgenstein se posiciona apenas em relação ao logicismo de Frege e Russell, nestes escritos ele procura, a seu modo, contrapor seu pensamento em relação às tendências dominantes de sua época: o intuicionismo de Brouwer e Weyl, o formalismo de Hilbert e, por fim, o logicismo renovado de Ramsey. A tese desenvolve, em seu Capítulo conclusivo, uma reflexão sobre a postura de Wittgenstein ante estas três escolas clássicas e ante os problemas por elas enfrentados

In this thesis, I aim to motivate a particular philosophy of mathematics characterised by the following three claims. First, mathematical sentences are generally speaking false because mathematical objects do not exist. Second, people typically use mathematical sentences to communicate content the truth of which does not require mathematical objects to exist. Finally, in using mathematical language in this way, speakers are not doing anything out of the ordinary: they are performing straightforward assertions. In Part I, I argue that the role played by mathematics in our scientific explanations is a purely expressive one, merely allowing us to say more than we otherwise would be able to about, or yielding a greater understanding of, the physical world. Mathematical objects to not need to exist for mathematical language to play this role. This proposal puts a normative constraint on our use of mathematical language: we ought to use mathematically presented theories to express belief only in the consequences they have for non-mathematical things. In Part II, I will argue that what the normative proposal recommends is in fact what people generally do in both pure and applied mathematical contexts. I motivate this claim by showing that it is predicted by our best general means of analysing natural language. I provide a semantic theory of applied arithmetical sentences and show that they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which shows that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show the hermeneutic fictionalist position that emerges is preferable to any alternative which interprets mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects.

Orientador: Maria Aparecida Viggiani Bicudo / Resumo: A continuidade é um tema que sempre trouxe desafios aos filósofos e matemáticos, desde a Grécia antiga com os paradoxos sobre o movimento, que persistem até os dias atuais, quando nos encontramos discutindo a continuidade da consciência e do tempo. Com o advento da tecnologia digital uma perspectiva se abre nessa discussão, pois modelos matemáticos contínuos estão sendo aplicados a problemas numéricos computacionais, que são caracteristicamente discretos. Essa possível discretização do contínuo mostra-se de modo claro, levando-nos a investigar a presença do contínuo ao se produzir Matemática junto ao computador. Investigaremos esse assunto realizando um movimento característico do pensar filosófico, tomando o tema como uma lectio, entendida como momento de discussão sobre textos numa dimensão argumentativa filosófica sobre nossa interrogação de pesquisa, ou seja, nossa quaestio. Nossa compreensão dos textos é exposta de maneira articulada e dividida em três seções, que versam sobre os estudos realizados no âmbito das Ciências Matemática, Filosofia e Computação. Finalizaremos trazendo uma meta-compreensão dos estudos realizados, tomando como centro articulador da reflexão a interrogação formulada. Nosso objetivo com esse exercício filosófico é compreender o fenômeno “contínuo-discreto” na região de inquérito das Ciências Ocidentais e sua presença na computação / Abstract: Continuity has been a challenging topic to philosophers and mathematicians, since the ancient Greece, with paradoxes of movement, until present days when continuity of consciousness and of time are discussed. With the advent of digital technology another perspective has brought into the discussion, because continuous mathematical models are being applied in numerical computational problems, which are characteristically discrete. This possibility of continuos’ discretization is drawing our attention. Therefore, this research aims to understand the presentification of continuous when we are producing Mathematics with computers. We will investigate this subject via a philosophical approach. This thesis is constituted as a lectio, understood as a moment of discussion about texts in a philosophical argumentative dimension about our research question, that is, our quaestio. Our understanding of the texts is articulated and divided into three sections, which deal with the studies carried out in Mathematics, Philosophy and Computer Science. Our goal with this philosophical exercise is to explore the "continuous-discrete" phenomenon under Western Sciences influence and its presence in computation. / Doutor

As computers become a more prevalent commodity in mathematical research and mathematical proof, the question of whether or not a computer assisted proof can be considered a mathematical proof has become an ongoing topic of discussion in the mathematics community. The use of the computer in mathematical research leads to several implications about mathematics in the present day including the notion that mathematical proof can be based on empirical evidence, and that some mathematical conclusions can be achieved a posteriori instead of a priori, as most mathematicians have done before. While some mathematicians are open to the idea of a computer-assisted proof, others are skeptical and would feel more comfortable if presented with a more traditional proof, as it is more surveyable. A surveyable proof enables mathematicians to see the validity of a proof, which is paramount for mathematical growth, and offer critique. In my thesis, I will present the role that the mathematical proof plays within the mathematical community, and thereby conclude that because of the dynamics of the mathematical community and the constant activity of proving, the risks that are associated with a mistake that stems from a computer-assisted proof can be caught by the scrupulous activity of peer review in the mathematics community. Eventually, as the following generations of mathematicians become more trained in using computers and in computer programming, they will be able to better use computers in producing evidence, and in turn, other mathematicians will be able to both understand and trust the resultant proof. Therefore, it remains that whether or not a proof was achieved by a priori or a posteriori, the validity of a proof will be determined by the correct logic behind it, as well as its ability to convince the members of the mathematical community—not on whether the result was reached a priori with a traditional proof, or a posteriori with a computer-assisted proof.

In broadest terms, this thesis is concerned to answer the question of whether the view that arithmetic is analytic can be maintained consistently. Lest there be much suspense, I will conclude that it can. Those who disagree claim that accounts which defend the analyticity of arithmetic are either unable to give a satisfactory account of the foundations of mathematics due to the incompleteness theorems, or, if steps are taken to mitigate incompleteness, then the view loses the ability to account for the applicability of mathematics in the sciences. I will show that this criticism is not successful against every view whereby arithmetic is analytic by showing that the brand of "conventionalism" about mathematics that Rudolf Carnap advocated in the 1930s, especially in Logical Syntax of Language, does not suffer from these difficulties. There, Carnap develops an account of logic and mathematics that ensures the analyticity of both. It is based on his famous "Principle of Tolerance", and so the major focus of this thesis will to defend this principle from certain criticisms that have arisen in the 80 years since the book was published. I claim that these criticisms all share certain misunderstandings of the principle, and, because my diagnosis of the critiques is that they misunderstand Carnap, the defense I will give is of a primarily historical and exegetical nature. Again speaking broadly, the defense will be split into two parts: one primarily historical and the other argumentative. The historical section concerns the development of Carnap's views on logic and mathematics, from their beginnings in Frege's lectures up through the publication of Logical Syntax. Though this material is well-trod ground, it is necessary background for the second part. In part two we shift gears, and leave aside the historical development of Carnap's views to examine a certain family of critiques of it. We focus on the version due to Kurt Gödel, but also explore four others found in the literature. In the final chapter, I develop a reading of Carnap's Principle - the `wide' reading. It is one whereby there are no antecedent constraints on the construction of linguistic frameworks. I argue that this reading of the principle resolves the purported problems. Though this thesis is not a vindication of Carnap's view of logic and mathematics tout court, it does show that the view has more plausibility than is commonly thought.

Orientador: Marcos Vieira Teixeira / Resumo: Realizamos um histórico da introdução das “matemáticas mistas”, e identificamos que o filósofo Francis Bacon (1561-1626) é creditado por meio das publicações “Proficience and Advancement of Learnings” em 1605 e “De Dignitate et Augmentis Scientiarum” em 1623. Entretanto, para responder questões pertinentes da Filosofia Natural, essa classificação matemática gerou um conflito entre as ciências matemáticas e a metafísica. Desse modo, as ciências físico-matemáticas surgem como uma tentativa de utilizar a matemática para abordar tópicos relacionados as causas naturais do mundo real. No ano de 1751, Jean le Rond D'Alembert (1717-1783), realizou uma nova classificação dos conhecimentos humanos, divulgada na obra Discours Préliminaire, que foi o texto de abertura da primeira edição da Encyclopédie, editada em conjunto com Denis Diderot (1713-1784). No que se refere a matemática, foi mantida como uma ramificação da metafísica, mas essa nova classificação a dividiu em: “matemática pura”, “matemática mista” e “ciências físico-matemáticas”. Porém, no decorrer do século XVIII, estimulado principalmente pelas críticas de Kant (1724-1804) em relação ao conhecimento puro, se deu início a uma discussão quanto a metafísica, e consequentemente refletiu nas ciências que estavam subordinadas a esse ramo do conhecimento. Desse modo, o século XIX culminou no desuso da expressão “matemáticas mistas”, contudo, veremos que essas ciências forneceram elementos para fomentar o aparecimento das “matemáti... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: Our aim is to realize a history of the introduction of the "mixed mathematics", and we find that the philosopher Francis Bacon (1561-1626) is credited through the publications "Proficiency and Advancement of Learnings" in 1605 and "De Dignitate et Augmentis Scientiarum" in 1623. However, to answer pertinent questions of Natural Philosophy, this mathematical classification generated a conflict between the mathematical sciences and metaphysics. In this way, the physico-mathematics sciences appeared as an attempt to use mathematics to address topics related to the natural causes of the real world. In 1751, Jean le Rond D'Alembert (1717-1783), made a new classification of human knowledge, published in the Discours Préliminaire, which was the opening text of the first edition of the Encyclopedie, edited together with Denis Diderot 1713-1784). As far as mathematics is concerned, was maintained as a branch of metaphysics, but this new classification divided it into "pure mathematics", "mixed mathematics" and "physico-mathematics sciences". But in the course of the eighteenth century, stimulated mainly by Kant (1724-1804) and his critiques of pure knowledge, a discussion of metaphysics began, and consequently reflected in the sciences which were subordinate to this branch of knowledge. The 19th century culminated in the disuse of the expression "mixed mathematics," however, these sciences provided elements to foster the emergence of "applied mathematics" as well as contributed to the... (Complete abstract click electronic access below) / Doutor