 Marc Lange (2009a) has argued that a whole class of arguments fail to be explanatory, namely arguments by mathematical induction. Letting A(x) be a property, mathematical induction licenses the conclusion that for all n, A(n), from the premises A(1) and for all n, (if A(n) then A(n′)). Lange’s argument rests on contrasting ordinary mathematical induction with a form of upward and downward induction from a fixed number k (k≠1; say, 5), which is deductively equivalent to ordinary induction. He then claims that if proofs by ordinary mathematical induction are explanatory so are proofs by upward and downward induction from a fixed number k≠1. But if so, one ends up, so the argument concludes, with P(1) part of the explanation for P(k) and P(k) part of the explanation for P(1) and this contradicts the non-circularity of mathematical explanations (which is one of the premises about mathematical explanations defended by Lange). By reductio, the original proof by ordinary mathematical induction is not explanatory. Baker 2009b rejects the claim that the proof by ordinary mathematical induction and the proof by upward and downward induction from k are equally explanatory. He points out that the latter type of induction is more ‘disjunctive’ than the former and contains elements that are irrelevant to the explanation.  Cariani 2011 (see Other Internet Resources) also focuses on the explanatory asymmetry between the two types of induction but with considerations different from those of Baker. He insists on ‘simplicity’ as the explanatory tie-breaker between proofs of the same theorem by appeal to the two different types of induction and concludes with considerations leading one to suspect that there is no global answer to the question of whether proofs by mathematical induction are explanatory.