As I stated earlier in Part 1, the philosophy of mathematics is the branch of philosophy that relates to the foundation of mathematics, deals with the nature of mathematics, and ponders whether mathematics is discovered or invented. Part 2 focuses upon the nature of mathematics. The books Arithmetic by Paul Lockhart and Beautiful Math: The Surprisingly Simple Ideas Behind the Digital Revolution in How We Live, Work, and Communicate by Chris Bernhardt are entries into the nature and beauty of mathematics.
In his chapter on language, Lockhart states that the story of arithmetic is a history of representation. The distinction between a number in and of itself and the way in which it is represented must be maintained. He admits, “I can easily understand how a conflation of the number three and the symbol 3 can occur.” Then Lockhart asks, “If 3 is merely a symbol representing the number three, then what exactly is three itself, independent of its representation?” This question deals with the very nature of mathematics.
In what sense does the number three exist? In what sense does any number exist? These are ontological questions, from the Greek word ontos which means “being.” Whereas the foundation of mathematics relates to the logical basis for mathematics, the nature of mathematics deals with the ontological basis for mathematics.
There was no demarcation between mathematics and philosophy in Classical Greece. However, the Platonist and Aristotelian traditions continue to differ on the nature of mathematics. Platonism holds that numbers have a real existence independent of human thought. Aristotelianism holds that numbers are a human modeling of patterns they find in the world around them. Next in Part 3 we ponder whether mathematics is discovered or invented while pondering the viewpoints of Plato and Aristotle.
~ Boethius ~