The philosophy of mathematics considers what is behind the math that we do. What is mathematics? Is it some cosmic truth we discover, or is it created by humans? Do mathematical objects such as numbers and functions really exist, or are they just symbols we have invented? Two of the great debates in the history of mathematical philosophy center around ontology and epistemology. Where did mathematics come from? How do we know that it is true?
Where did mathematics come from? Is it discovered or created? Ontological questions are concerned with the nature and status of mathematical objects. Some people believe that the numbers, functions, and other mathematical objects that we talk about actually exist, either in the world or “out there” somewhere. There are things about them that we need to learn. Statements like 2 + 2 = 4 are true whether we know about them or not, and they are waiting to be discovered. This is called independent truth. It can be likened to the perennial question, “If a tree falls in the forest with no one around, does it make a sound?” Here, “If there are no humans to study mathematics, does mathematical truth still exist?” Those who believe in independent truth say yes. Others believe that mathematical symbols have been made up to make our calculations possible, and that mathematical truth is a human invention. These debates can get very heated, sometimes taking on an almost religious tone. To some, denying the existence of independent truth is dangerously close to denying the role of God as creator of all.
How do we know that mathematics is true? How do we know that what we think is true really is? Epistemological questions address issues of mathematical justification and knowledge. In the mathematical community, the primary means of justification is proof. A proof begins with a set of axioms, and generates a sequence of statements, through inference, that ends in the given proposition. The debate that arises surrounds the nature of the knowledge that results. The traditional philosophy of mathematics has been one which asserts that mathematics offers absolute certainty through proof. This certainty is called absolute truth. But how can we be sure that what we have come to know is true? Some philosophers say we are sure because we proved it using deduction and reasoning. Others say we are sure because of the way mathematics models scientific phenomena. There is also a group of philosophers who believe that we can’t be sure that what we know is actually true, but that it doesn’t matter.
In this report, I will summarize several positions in the philosophy of mathematics, both historical and contemporary. The diversity of opinions provides an intriguing look at the world of mathematics. These philosophers offer a sense that there is more going on behind “ x +5 = 8” than meets the eye.
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Wilding-Martin, Erin, "The Philosophy of Mathematics" (2008). Sabbaticals. 2.