Philosophy of mathematics

The philosophy of mathematics is a kind of philosophy that tries to study what mathematics really is. Philosophers of math ask themselves questions like:

• Does math really make sense?
• Do numbers really exist, or are they just made up?
• Can math explain everything?
• Why do people do math?
• Why do some people think math is "beautiful"?
• Is mathematics really true, or just something that we think is true?

None of those questions has a clear answer, so mathematicians are free to choose which answers they like. For example:

Mathematicians who believe in Platonism think that numbers are real, just like rocks and trees. We can't see them, but we can represent them with symbols, and use them to study the world.

Mathematicians who believe in Fictionalism think that math is just made up. Numbers, shapes and other things in math are just ways that we use to make science possible, but they don't really exist.

Mathematicians who believe in Logicism believe that math is true because it is based on true sentences (called axioms). It does not matter if numbers and shapes exist, or not, just that they represent true things.

Mathematicians who believe in Finitism think that everything that makes sense has a beginning and an end. Things like infinity don't exist.

• Gödel's incompleteness theorems
• Mathematical logic
• Metaphilosophy
• Philosophy of science