Abstract
This article has two objectives. The first one is a review of some of the most important questions in the contemporary philosophy of mathematics, including: What is the nature of mathematical objects? How do we acquire knowledge about these objects? Should mathematical statements be interpreted differently than ordinary ones? And finally, how can we explain the applicability of mathematics in science? The topic that guides these reflections is the debate between mathematical realism and anti-realism. The second objective of this article is to discuss the arguments that use the applicability of mathematics in science to justify mathematical realism, and show that none of them reaches its objective. To this end, three aspects of the problem of the applicability of mathematics are distinguished: the (mere) utility of mathematics in science; the unexpected utility of some mathematical theories; and the apparent indispensability of mathematics in our best scientific theories, and in particular, in our best scientific explanations. Then I argue that none of these aspects constitutes a reason to adopt mathematical realism.
Highlights
The first one is a review of some of the most important questions in the contemporary philosophy of mathematics, including: What is the nature of mathematical objects? How do we acquire knowledge about these objects? Should mathematical statements be interpreted differently than ordinary ones? And how can we explain the applicability of mathematics in science? The topic that guides these reflections is the debate between mathematical realism and anti-realism
The second objective of this article is to discuss the arguments that use the applicability of mathematics in science to justify mathematical realism, and show that none of them reaches its objective
I argue that none of these aspects constitutes a reason to adopt mathematical realism
Summary
En la filosofía de las matemáticas existe una distinción entre el valor de verdad de las afirmaciones matemáticas y el estatuto ontológico de los objetos sobre los cuales dichas afirmaciones cuantifican. Realismo en ontología es la postura que sostiene que las afirmaciones matemáticas cuantifican sobre objetos reales, tales como números, conjuntos, etcétera. Por ejemplo, que, si uno cree que dichos objetos matemáticos existen, las afirmaciones acerca de ellos tendrán un valor de verdad objetivo. Uno puede ser antirrealista respecto de los objetos matemáticos y aun así otorgar un valor de verdad objetivo a las afirmaciones matemáticas. Si se entiende a las matemáticas como un mero conjunto de afirmaciones sobre entidades ficcionales, entonces dichas afirmaciones matemáticas pueden tener un valor objetivo sin que esto implique que los objetos. El punto principal de este artículo es que, incluso si uno asumiera que estas posturas son correctas, uno no tendría razones para ser realista matemático basado en la aplicabilidad de las matemáticas en ciencia. El naturalismo, el realismo científico, etcétera, para refutar la idea de que la aplicabilidad de las matemáticas en ciencia justifica el realismo matemático
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