The fifth century BC was the most exuberant time in the history of Athens, when some of the most well known Greek achievements in art, politics and philosophy were developed. We argue that this was an equally seminal period for the mathematical sciences, through examples from the works of Anaxagoras, Hippocrates, Hipias, Architas, Hippasus and Zeno. The adjective `heroic' refers to the contrast between the depth of the problems pursued by these mathematicians and the techniques available to them at the time.