Michael Spivak in "Calculus" asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by "exist"? Michael Spivak in "Calculus" asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by "exist"? How are you to prove that any number "exists"? Why can't we define $\sqrt2$ as a number that fits under some arbitrary definition of existence, while asserting that its most concise expression is with a functional root?

I'm sorry if these questions seem a bit sophomoric; in some ways it resembles an 8 year old repeatedly asking "why". But given that his prose is very concise and technical, his usage of "exist" was out of the ordinary.

(I used two tags representing the book's field of study; and one representing the actual relevant tag.)

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Oh, I'm sorry. I misquoted. My question still stands, though; how has he defined existence such that $\sqrt2$ might possible not be within it.

Direct quote: "We have not proved that any such number exists..." in reference to $\sqrt2$.
The point Spivak is making is that the properties of numbers that have been studied till that point in the book are not enough to prove that there is a number whose square is 2. This follows from the proof that no rational number will do for this task. Since the rational numbers satisfy all properties till that point, it is clear that some other property is needed (and that the rationals cannot have this property). That property is completeness and is a what characterizes real numbers. Using completeness, one can and does prove that there is a positive number whose square is 2; we call it $\sqrt2$.
I believe that the existence of √2 was proven in 300 B.C. give or take. 

Just draw a right triangle, with the sides being exactly 1 unit in length, then the hypotenuse of the triangle - using the Pythagorean theorem - will be √2 in length. 

I fully concede that I probably missed that whole point of this conversation, but that's how you prove the existence of √2 anyway :)