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unskilledplay t1_j6bcf5t wrote

Math, like every logic system, must have axioms. These are statements that are true only because they are stated to be true, not because they can be shown to be true.

The imaginary unit is axiomatic. There is nothing special about axioms. I think people are just thrown off by the unrelated associations people have with the term "imaginary" and how/when it is taught in school.

In this case, imaginary numbers don't just lead to interesting structures and algebras, they are exceedingly useful in physics and might even be required.

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glootech t1_j6dc37u wrote

You're mistaking something having a definition by being axiomatic. Imaginary unit is a construct that's a consequence of first defining natural numbers using Peano's axioms and then (in layman's terms) further "creating" other, more complex structures based on your previous results.

I agree with your other statement - imaginary numbers are just an ordinary mathematical object and there's nothing special about them. I consider real numbers to be the really, really weird ones (transcendental numbers anyone?). Imaginary numbers are just a simple extension of that weirdness. And they are also very useful, so that's a big plus.

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unskilledplay t1_j6e1oft wrote

It’s fine to call it axiomatic. You can get into that Hilbert style formalism all you want. That’s all just backwards justification of math.

There is a reason nobody picked it up and continued that work when he abandoned it.

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