Mathematics is one of the oddest phenomena you will ever encounter. Consider the following two peculiarities:
First, mathematical claims seem to be beyond even the possibility of falsehood. Provided we take the symbols in their usual sense, there could be no evidence that two plus two does not equal four. If you put two biscuits in a tin, and another two biscuits in a tin, and then find there are five biscuits in the tin, you will consider any option other than that you have proved that, in this instance, two plus two equalled five. Why is this? Why is mathematics so certainly true? What actually are mathematical claims about?
That last question leads me to the second peculiarity. Why is mathematics so unreasonably good at describing the world? A mathematician can sit down, fiddle about with lots of squiggles on a page, and predict that a planet will be at a certain place at a certain time. That is just weird; why can we do this with mathematics when we can’t do it with gazing into crystal balls?
When I say ‘the door is open’ I am talking about the door. When I say ‘two plus two equals five’, what am I talking about? What is this realm that is beyond all evidence, and which somehow picks out reality? There have been several attempts to answer this question. One way into the debate is to consider whether you think we discover mathematics, or whether you believe we invent it.
The first option takes finding truths in mathematics to be akin to explorers finding new bits of the world. The Victoria Falls were there before Livingstone discovered them, as bits of mathematics are there before we discover them. The second option takes the truths of mathematics to be a bit like the rules of etiquette. Once we have invented them, we are landed with them. It's crucial, though, that we invented them; they did not exist out there for us to discover.
For very simple reasons it looks as if the first option is the most satisfactory explanation. The truths of mathematics seem to be independent of us: two plus two would equal four even if nobody existed to know it. When we learned about it, we discovered something. The same cannot be said for the rules of etiquette: we did not discover an eternal truth when we formulated the rule that Archbishops should be addressed as ‘your Grace’.
However, that only deepens the mystery. The claim is that there is a realm of objects, that is outside time and space, that is necessary and unchangeable, and which exists completely independently of us. Even if there were such a realm (and one might already start wondering whether one can really believe that there is), what would it have to do with the way we do maths? Livingstone found the Victoria Falls by going there. It might have been difficult, but at least we can understand what he did. How are we to get to, and learn about, a necessary and unchanging realm, outside time and space? How would we go about getting there?
The other option is that the claims mean only what it would take to prove them: there is nothing more to the claims of mathematics than what is given by the way we use them in our mathematical practice. However, in the same way that there are no rules of etiquette that are "true" and not known by anyone, it would follow that there are no truths of mathematics that are true and not proven by anyone. This follows, because, according to this viewpoint, maths comes into existence only with it being proved. This seems counterintuitive because, surely, there is mathematics that we don’t yet know?
These are only some of the problems that go under the general heading of ‘the philosophy of mathematics’. There are plenty of others. Is maths the same as logic? What exactly are numbers? These are questions that have puzzled people for as long as there have been civilizations, and have inspired some of the cleverest and subtlest threads of thought ever produced. It is also, oddly, a topic that has not yet produced much agreement as to which answers are the best ones.