An overview of the major views on foundations of mathematics:
AbstractMathematics is said to be language of all sciences and is commonly believed to be an objective discipline which works on some very clear laws and in which there are no chances of confusion or ambiguous results. However there are some mathematicians and philosophers who have raised concerns regarding some of the fundamental assumptions of mathematics which makes it vital for everyone engaged in these subjects, to understand its foundational issues and be aware of its inherent limitations. Philosophy and Mathematics has a long history of mutual engagement, ever since the time of early Greek thinkers. The nature of interaction between Mathematics and Philosophy has also taken many shapes and forms. However the chief philosophical challenges to Mathematics are basically from Metaphysics and Epistemology. In the matter of Metaphysics, the questions raised belong to existence or being of numbers, sets, points, lines, functions and other such mathematical entities which are freely used as if they have an objective identity. These questions belong to the set of problems covered under the foundations of mathematics which deals with the issue of ontological status of mathematical entities. In the matters of Epistemology, the questions raised are even more fundamental in nature and deal with the meaningfulness of mathematical statements as well as nature of mathematical truth. Since all mathematics is abstract, it can then be argued that all mathematics is a purely mental exercise. However, there are mathematicians and philosophers who do not accept these explanations and they have presented logical and structural arguments to establish the proof and certainty of mathematical facts. This paper will cover some of the issues mentioned above and attempt to provide an overview of the historical and contemporary debates dealing with the same. In the process, the endeavor will be to familiarize with the most common concerns of Philosophy of Mathematics and get a glimpse into some of the challenges that exist in resolving the same.
2. Hart, W. D., editor (1997), The Philosophy of Mathematics. Oxford: Oxford University Press.
3. Russell, Bertrand (1993/1919), Introduction to Mathematical Philosophy. Minneola, NY, Dover Publications.
4. Benacerraf, Paul, and Hilary Putnam, editors (1983), Philosophy of Mathematics: selected readings, second edition. Cambridge: Cambridge University Press.
5. Schirm, Matthias, editor (2003), The Philosophy of Mathematics Today. Oxford: Oxford University Press.
6. Shapiro, Stewart, editor (2005), The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford: Oxford University Press.
7. Shapiro, Stewart (1997), Philosophy of Mathematics: structure and ontology. New York: Oxford University Press.
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