Filosofia De Las Matematicas

Philosophy of Mathematics
First published Tue Sep 25, 2007
If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy ofscience. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge usinginductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problemsof a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics.
• 1. Philosophy of Mathematics, Logic, and the Foundations of Mathematics
• 2. Four Schools
o 2.1 Logicism
o 2.2 Intuitionism
o 2.3 Formalism
o 2.4 Predicativism
• 3. Platonism
o 3.1 Gödel's Platonism
o 3.2 Naturalism andIndispensability
o 3.3 Deflating Platonism
o 3.4 Benacerraf's Epistemological Problem
o 3.5 Plenitudinous Platonism
• 4. Structuralism and Nominalism
o 4.1 What Numbers Could Not Be
o 4.2 Ante Rem Structuralism
o 4.3 Mathematics Without Abstract Entities
o 4.4 In Rebus Structuralism
o 4.5 Fictionalism
• 5. Special Topics
o 5.1 Philosophy of Set Theory
o 5.2 Categoricity
o 5.3Computation and Proof
• Bibliography
• Other Internet Resources
• Related Entries
________________________________________
1. Philosophy of Mathematics, Logic, and the Foundations of Mathematics
On the one hand, philosophy of mathematics is concerned with problems that are closely related to central problems of metaphysics and epistemology. At first blush, mathematics appears to study abstractentities. This makes one wonder what the nature of mathematical entities consists in and how we can have knowledge of mathematical entities. If these problems are regarded as intractable, then one might try to see if mathematical objects can somehow belong to the concrete world after all.
On the other hand, it has turned out that to some extent it is possible to bring mathematical methods to bear onphilosophical questions concerning mathematics. The setting in which this has been done is that of mathematical logic when it is broadly conceived as comprising proof theory, model theory, set theory, and computability theory as subfields. Thus the twentieth century has witnessed the mathematical investigation of the consequences of what are at bottom philosophical theories concerning the nature ofmathematics.
When professional mathematicians are concerned with the foundations of their subject, they are said to be engaged in foundational research. When professional philosophers investigate philosophical questions concerning mathematics, they are said to contribute to the philosophy of mathematics. Of course the distinction between the philosophy of mathematics and the foundations ofmathematics is vague, and the more interaction there is between philosophers and mathematical logicians working on questions pertaining to the nature of mathematics, the better.
2. Four Schools
The general philosophical and scientific outlook in the nineteenth century tended toward the empirical. Platonistic aspects of rationalistic theories of mathematics were rapidly losing support. Especially...