I'm currently reading Thinking about Mathematics by Stewart Shapiro. In chapter 1, it says:
"For an intuitionist, (statement 1) The content of a proposition stating that not all natural numbers have certain property P is that it is refutable that one can construct a number x and show that P does not hold of x. (statement 2) The content of proposition that there is a number that lacks P is that one can construct a number x and show that P does not hold of x.
Intuitionist agree that the latter proposition entails the former, but they balk at the converse because it is possible to show that a property cannot hold universally without constructing a number for which it fails."
I don't understand what he is trying to say here in the last paragraph. So intuitionists agree that statement 2 is the consequence of statement 1 but dont agree that statement 1 is a consequence of statement 2?
Can someone please provide clarity on Shapiro's last paragraph, please!