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monographic section i

1. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Mary Leng

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2. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Concha Martínez Vidal

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Putnam rejects having argued in the terms of the argument known in the literature as “the Quine-Putnam indispensability argument”. He considers that mathematics contribution to physics does not have to be interpreted in platonist terms but in his favorite modal variety (Putnam 1975; Putnam 2012). The purpose of this paper is to consider Putnam’s acknowledged argument and philosophical position against contemporary so called in the literature ‘fictionalist’ views about applied mathematics. The conclusion will be that the account of the applicability of mathematics that stems from Putnam‘s acknowledged argument can be assimilated to so-called ‘fictionalist’ views about applied mathematics.
3. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
José Miguel Sagüillo

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This paper focuses on Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. Putnam’s 1971 book Philosophy of Logic came one year later than Quine’s homonymous volume. In the first section, I compare these two Philosophies of Logic which exemplify realist-nominalist viewpoints in a most conspicuous way. The next section examines Putnam’s views on modality, moving from the modal qualification of his intuitive conception to his official generalized non-modal second-order set-theoretic concept of logical truth. In the third section, I emphasize how Putnam´s “mathematics as modal logic” departs from Quine’s “reluctant Platonism”. I also suggest a complementary view of Platonism and modalism showing them perhaps interchangeable but underlying different stages of research processes that make up a rich and dynamic mathematical practice. The final, more speculative section, argues for the pervasive platonistic conception enhancing the aims of inquiry in the practice of the working mathematician.
4. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Otávio Bueno

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Crucial to Hilary Putnam’s realism in the philosophy of mathematics is to maintain the objectivity of mathematics without the commitment to the existence of mathematical objects. Putnam’s indispensability argument was devised as part of this conception. In this paper, I reconstruct and reassess Putnam’s argument for the indispensability of mathematics, and distinguish it from the more familiar, Quinean version of the argument. Although I argue that Putnam’s approach ultimately fails, I develop an alternative way of implementing his form of realism about mathematics that, by using different resources than those Putnam invokes, avoids the difficulties faced by his view.
5. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Sorin Bangu

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When considering mathematical realism, some scientific realists reject it, and express sympathy for the opposite view, mathematical nominalism; moreover, many justify this option by invoking the causal inertness of mathematical objects. The main aim of this note is to show that the scientific realists’ endorsement of this causal mathematical nominalism is in tension with another position some (many?) of them also accept, the doctrine of methodological naturalism. By highlighting this conflict, I intend to tip the balance in favor of a rival of mathematical nominalism, the mathematical realist position supported by the ‘Indispensability Argument’ —but I do this indirectly, by showing that the road toward it is not blocked by considerations from causation.
6. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Susan Vineberg

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This paper discusses Baker’s Enhanced Indispensability Argument (EIA) for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such use still does not support realism about mathematical objects.
7. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Matteo Plebani

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Two conceptions of the nature of mathematical objects are contrasted: the conception of mathematical objects as preconceived objects (Yablo 2010), and heavy duty platonism (Knowles 2015). It is argued that some theses defended by friends of the indispensability argument are in harmony with heavy duty platonism and in tension with the conception of mathematical objects as preconceived objects.

monographic section ii

8. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
María de Paz, José Ferreirós

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9. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Rafael Núñez

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La matemática forma un cuerpo único de conocimiento. Entre otras cosas, es abstracta, exacta, eficaz, simbolizable y proporciona sorprendentes aplicaciones al mundo real. En el campo de la filosofía de la matemática el estudio de la práctica matemática ha devenido gradualmente una importante área de investigación. ¿Qué aspectos de la mente y el cuerpo humano hacen posible la particular práctica matemática? En este artículo, reviso brevemente algunas dimensiones cognitivas que juegan un papel crucial en la creación y consolidación de la matemática.
10. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Markus Pantsar

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In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez (2000), I propose one particular conceptual metaphor, the Process → Object Metaphor (POM), as a key element in understanding the development of mathematical thinking.
11. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Roy Wagner

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This paper considers two models of embodied mathematical cognition (a modular model and a dynamic model), and analyses the image of mathematics that they support.
12. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
José Ferreirós, Manuel J. García-Pérez

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Se discutirán críticamente algunas tesis recientes sobre cognición geométrica, específicamente la tesis de la universalidad planteada por Dehaene et al., y la idea de una “geometría natural” empleada por Spelke. Argumentaremos la necesidad de distinguir entre cognición visuo-espacial y conocimiento geométrico básico, y más aún, afirmaremos que este último no se puede identificar con la geometría euclidiana. El propósito principal del artículo es proponer una caracterización de la geometría básica, para lo cual se requiere una combinación de experimentos en cognición visuo-espacial con estudios en arqueología cognitiva e historia comparativa. Ofreceremos ejemplos de estos campos, con especial énfasis en la comparación de ideas y procedimientos geométricos de la antigua China y Grecia.
13. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Valeria Giardino

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In the first part of the article, a semiotic reading of the embodied approach to mathematics will be presented. From this perspective, the role of the sensorimotor in mathematics will be considered, by looking at some work in experimental psychology on the segmentation of formulas and at an analysis of the practice of topology as involving manipulative imagination. According to the proposed view, representations in mathematics are cognitive tools whose functioning depends on pre-existing cognitive abilities and specific training. In the second part of the paper, the view of cognitive tools as props in games of “make-believe” will be discussed; to better specify this claim, the notion of affordance will be explored in its possible extension from concrete objects to representations.
14. Theoria: An International Journal for Theory, History and Foundations of Science: Volume > 33 > Issue: 2
Sorin Costreie

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Frege writes in Numbers and Arithmetic about kindergarten-numbers and “an a priori mode of cognition” that they may have “a geometrical source.” This resembles recent findings on arithmetical cognition. In my paper, I explore this resemblance between Gottlob Frege’s later position concerning the geometrical source of arithmetical knowledge, and some current positions in the literature dedicated to arithmetical cognition, especially that of Stanislas Dehaene. In my analysis, I shall try to mainly see to what extent (Frege’s) logicism is compatible with (Dehaene’s) intuitionism.