Mathematics, Foundations Of
The study of the foundations of mathematics comprises investigations, though probably not all possible investigations, that consist of general reflection on mathematics. The subject naturally proceeds by singling out certain concepts and principles as "fundamental" and concentrating attention on them, but of course the identification of fundamental concepts and principles is itself based on foundational research or may be revised in the light of it.
In this entry considerable emphasis will be placed on philosophical questions about mathematics, which undoubtedly belong to foundations. However, many, perhaps most, foundational investigations are mainly mathematical. In the last hundred years an important role has been played by mathematical logic. We shall not give a detailed exposition of mathematical logic, but we hope that our discussion will give an idea of the relation between the logical problems and results and the philosophical problems and an idea of some of the results of recent work in logic.
Two of the main qualities for which mathematics has always attracted the attention of philosophers are the great degree of systematization and the rigorous development of mathematical theories. The problem of systematization seems to be the initial problem in the foundations of mathematics, both because it has been a powerful force in the history of mathematics itself and because it sets the form of further investigations by picking out the fundamental concepts and principles.
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