This month, we talk recreational mathematics with Haim Gaifman, Professor of Philosophy at Columbia University. Click here to listen to our conversation.

Are numbers mind-independent entites, or are they just social constructs? A mountain is definitely real–you can climb it, take pictures of it, fall off it, show it to your friends, and so on. It wasn’t brought into existence by people agreeing that anything about it was true. But not everything is like that. The law not to cross the street when the light is red, for instance, only is the way it is because we all agree that that should be the law. If we were to change our minds about it, the law would *thereby* change. This raises an interesting question: which of these two things are numbers more like? Are they the way they are independently of what human beings decide, or are facts about numbers constituted by the conventions we decide to adopt?

In this episode, Haim Gaifman argues that mathematical reasoning just doesn’t make any sense unless we suppose that numbers are mind-independent entities. It’s just built into the nature of mathematics that before we prove some fact about numbers, either the conclusion of the proof is true or it isn’t. It’s not like having proved something *makes it true*.

Join us our guest argues for this view on mathematics by taking us through a delightful array of mathematical puzzles that all have the common feature of being difficult to solve, but easy to understand once solved.

*Matt Teichman*