To clarify the comparison I made, consider the set of integer solutions to linear equations (or to be more precise, the Diophantine set of a linear polynomial).

To clarify the comparison I made, consider the set of integer solutions to linear equations (or to be more precise, the Diophantine set of a linear polynomial). For example, for the equation x+y=k, the set of integer solutions k is merely the set of all integers. For the equation 10x+6y=k, you get the set of all even integers. For the equation 10x+6y+7z=k, again you get the set of all integers. So the sets of integers that you can obtain as the respective solutions to linear equations are all very simply structured. (If I'm not mistaken, they are always simply the multiples of the gcd of the coefficients of the variables, possibly translated by some constant offset e.g. if one is considering 6x+9y+15z+7=k, the set of obtainable k (by plugging in various values for (x, y, z)) is “all integers that are 1 mod 3” — so the set is always simply an arithmetic progression.) Similarly for quadratic equations — you still get sets that are easily parametrized. Something like this is what motivated Hilbert to ask the question in the first place. Nothing so far would have suggested that if you go up to higher-degree polynomials, you should eventually be able to obtain any computable (i.e. recursively enumerable) set as the set of integer solutions to some polynomial equation. Yet that's the case. The contrast was never with real numbers, so mentioning real numbers is neither here nor there. (Though I understand how one can get that impression, if one thinks that what is being highlighted is “Diophantine solutions” in the sense of “integer solutions” as opposed to “real solutions” or whatever.) The surprising fact being highlighted by the number-theorist in the quote (and one of the highlights of 20th-century mathematics) is that the structure “the set of integer solutions to a polynomial equation” is rich enough to encode all possible computation — i.e. polynomials are not as “simple” as one may think. (Please take a look at those links if you haven't; it's really beautiful and super surprising that a program can be encoded as the set of integer roots of an integer polynomial.)