Discussions about constructive mathematics are usually derailed by philosophical opinions and meta-mathematics. But how does it actually feel to do constructive mathematics? A famous mathematician wrote that "taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists." Was he right? In this talk we shall visit the astounding worlds of constructive mathematics. As tourists from the classical world we will at first be shocked by the bizarre and unfamiliar phenomena, but hopefully also left curious about the new possibilities and dimensions of mathematics that the classical dogma declares heretical or even non-existent.