By Tuvshe Bold

This is a brief paper on the philosophy of mathematics. The whole of philosophy of mathematics consists of two areas of studies:

      
The interpretation of mathematical concepts:
numbers, infinity, and certainty.1
The nature of mathematical propositions:
theorems and proofs.

      The first explores mathematical elements, the nature of numbers and other mathematical concepts. The second is concerned with what mathematics does, and how mathematical elements work with each other to produce mathematical propositions. Specifically, it is about mathematical statements, theorems and operations, including addition, division, subtraction. The importance of this distinction is quite crucial because it helps to organize the philosophy of mathematics and clarifies exactly what we want from it. Frank Ramsey first pointed out this distinction in his paper "The Foundations of Mathematics," in which he also offers a solution to Russell's Paradox.2 The above two issues are equally fundamental in the philosophy of mathematics, but this paper will concentrate on the first issue, mathematical concepts.

      Mathematics has a beauty and certainty that no other science can offer. However, as Russell put it once, we practice mathematics without knowing what we are practicing. As such, I find philosophical interpretations of the unique characteristics of mathematics extremely interesting. Until the late 19th century, there had not been any serious and successful attention given to this sacred subject. In fact, with the exception of Kant's work on philosophy of mathematics, most discussion of this topic was in platitudes. Maybe history had its reasons for not giving room to philosophy of mathematics, but I will leave that for the historians. However, thanks to early 20th Century philosophers, like Frege and Russell, people began to recognize philosophy's potential to reveal great secrets of mathematics. More and more people became involved, and three dominant views rapidly emerged: logicism, formalism, and intuitionism.

      All three schools tried to explain mathematics; none was fully successful. Formalism and intuitionism argued that mathematics is an invention of the mind, like backgammon or chess, an enterprise that is disconnected from the world. On the other hand, the logicists came close to proving that mathematics was a branch of logic. They concentrated on providing an account of mathematical propositions through logic, in the process of which logical definitions of mathematical concepts arose. Their philosophical interpretation of mathematical concepts is not quite satisfying, as we shall see. In the coming discourse, I wish to first examine the main premises of the four schools and their mistakes. Second, I will try to provide a theory that is simpler and holds more explanatory power. Once again, this paper concentrates on philosophy of concepts of mathematical elements.

I. Different schools on mathematical concepts