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Kindly comments on Proofs, Implications, and Models - Less Wrong
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"Take the square root of both sides" is always a truth-preserving operation, but subsequently simplifying sqrt(x^2) to x is not.
So when you take the square root of both sides in, e.g., x^2=25, you really get sqrt(x^2)=5. Simplifying that to x=5 is where you lose generality.
sqrt(x^2)=5 or -5, along with an infinite number of complex values.
No. Even in the complex plane there are only two possible square roots. Moreover, if one wants to sqrt to be a function one needs a convention, hence we definite sqrt to be the non-negative square root when it exists.
You could define sqrt as a multi-valued function, in which case, when you apply it to x^2 = 25, you will get +/-x = +/-5, but you don't have to. We can take the positive square root (which is what people usually mean) and get sqrt(x^2)=sqrt(25)=5, and the operation is truth-preserving. sqrt(x^2) then simplifies to |x|.
Complex numbers are a bit trickier, but you don't get "an infinite number of complex values". Even over the complex numbers, the only square roots of 25 are 5 and -5. In general, there are two possible square roots, and the popular one to take is the one with positive real part. So everything I said above is still true, except sqrt(x^2) doesn't simplify nicely -- it's no longer equal to |x|. So when you take square roots of complex numbers it's probably better to go the multi-valued function approach with the +/-.
Squaring is not truth-preserving (Although I think raising to any power not an even number is, at least for real numbers). Why would even roots be truth-preserving?
What? For any function f, if x=y, then f(x) = f(y). Squaring is a function. Do you mean something else by truth-preserving?
Squaring can introduce truth into a falsehood. For example, if we write -5 = 5, that's false, but we square both sides and get 25=25, and that's true. Furthermore, squaring doesn't preserve the truth of an inequality: -5 < 3, but 25 > 9.
Ah- if you don't define the (principle) square root to be the inverse of squaring, the apparent contradiction goes away.
I concluded that you wanted to be able to preserve falsehood as well. Squaring preserves falsehood in the domain of the nonnegative reals, exactly like multiplication and division by positive values does.