 Thus consideration of the partition function has not brought a proof of GC any closer. However it does allow us to give an interesting twist to the argument of the previous section. The graph suggests that the hardest test cases for GC are likely to occur among the smallest numbers; hence the inductive sample for GC is biased, but it is biased against the chances of GC. Mathematicians’ confidence in the truth of GC is not based purely on enumerative induction. The values taken by the partition function indicate that the sample of positive instances of GC is indeed biased, and biased samples do not—as a general rule—lend much support to an hypothesis. But in this particular case the nature of the bias makes the evidence stronger, not weaker. So it is possible to argue that enumerative induction is unjustified while simultaneously agreeing that mathematicians are rational to believe GC on the basis of the available evidence. (Note that there is a delicate balance to maintain here because evidence for the behavior of the partition function is itself non-deductive. However the impression that G(n) is likely to be bounded below by some increasing analytic function is not based on enumerative induction per se, so the justification—while non-deductive—is not circular.)