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This month, we pull up our chairs and sit down once again with Robert May, Distinguished Professor of Philosophy and Linguistics at the University of California, Davis. Click here to listen to our conversation.

It seems sublime, unbelievable, groundbreaking – but maybe it actually doesn’t mean anything at all:

$$e^{i\pi} + 1 = 0$$

That’s Euler’s identity. Its proof is so deep that it’s beyond this podcast, or at least what this blogger knows about math. Yet without this podcast, we might worry that, once realized, Euler’s identity is as trivial as 0 = 0. For, after all, Euler proved that $$e^{i\pi} + 1$$ is the same as 0. He proved that $$e^{i\pi} + 1$$ refers to “0.” So when we write Euler’s identity, we can substitute the left-hand side of the equation to write that 0 = 0. Logical enough, but also dull. Disappointing.

And this worry extends beyond math. For instance, as the podcast mentions, Samuel Clemens is Mark Twain. So “Mark Twain” refers to “Samuel Clemens.” So have we really only said that Samuel Celemens is… Samuel Clemens? That’s trivial too. And so on for any identity, any expression of this equaling that.

At least, so said important 19th century philosophers, drawing on Leibniz (who, incidentally, Euler admired). They developed the problem of identity. But if you think that they must have missed something, so did their colleague Gottlob Frege. Frege worked on logicism, working math down to logic. And he did so by taking logic to reflect not only principles, but how we think.

In addressing the problem of identity, then, Frege argued not about reference. That is, he wouldn’t have argued about $$e^{i\pi} + 1$$ referring to 0. Rather, he argued about sense. So $$e^{i\pi} + 1$$ has a different sense than does 0 – in short, a different meaning given how we think. And he didn’t just argue this because we’re not great at math; he argued this because he saw the problem as epistemological. He saw the problem as addressing nothing less than how our thoughts reach knowledge.

Join us as Robert May explains how Frege worked out identity, and thus how we think and know.

Dominic Surya