Dear Colleagues,

Mathematics poses specific philosophical problems including ontological and epistemological ones. What is the nature of mathematical objects? What are the fundamental laws that govern them? How do we acquire mathematical knowledge about them? These are three questions that the Philosophy of Mathematics tries to answer. While philosophers and mathematicians have been interested in answering similar questions in the course of human history, it is important to note that mathematics is not necessarily concerned with the space-time dimension. For this reason, mathematical philosophy occupies a different place in the philosophy of science than the philosophies of the natural sciences, whose object of study involves space-time. There are various philosophical conceptions of mathematics including Platonism, intuitionism (constructivism), logicism, formalism, empiricism, predicativism, conventionalism, nominalism, fictionism, naturalism, and structuralism.

Ancient–medieval logic and a large part of modern logic have developed philosophically and are conceived today as a feature of discursive reason. Previously, logic was considered an introduction to philosophy and as its instrument. Within that philosophical horizon, there was never a sharp separation between form and content, although logic had been aptly described as formal logic. It was a comprehensive logic of content, where notions were universalized in intention. The evolution of modern science and, especially, the development of mathematical thought gives rise to logic as an exact discipline.

Mathematical logic is a branch of mathematics that emerged in the 20th century that The Mathematics Subject Classification divides into the following areas: the philosophy of mathematics, general logic (which includes fields such as modal logic and fuzzy logic), model theory, computability theory, set theory, the theory of demonstration and constructive algebraic mathematics, and non-standard models. Mathematical logic is today considered an important embodiment of our cultural world and is used increasingly in computer applications and automatic mechanisms. It is created by mathematicians using arithmetic, algebraic, analytical, topological, and axiomatic methods. Thus, with respect to the content, a greater type of abstraction and greater formal autonomy are achieved: it is no longer purely formal; it is formalized.

Prof. Dr. Josep Lluis Usó-Doménech
Guest Editor

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