Hi Smith. I read your post. I am not a mathematician, so cannot deeply comment on this subject as such.
However, from the purely logical point of view, in deductive logic, ‘p implies q’ and its contraposite ‘not-q implies not-p’ are exactly equivalent, both meaning no more or less than “the conjunction ‘p and not-q’ is impossible” (or, identically, “the conjunction ’not-q and p’ is impossible’).
For example, in syllogistic theory, we use direct reduction to the first figure to validate fourth figure moods, and indirect reduction (reductio ad absurdum) to the first figure to validate second figure moods. See here: http://thelogician.net/FUTURE-LOGIC/Syllogisms-Validations-10.htm. Neither method is better or worse than the other in terms of logical certainty. They are just conveniences.
In inductive logic, things are more complicated as already pointed out in my previous reply to you. Notably, any claim to having an exhaustive list of hypotheses implying a certain phenomenon is, of course, itself a mere hypothesis. Indeed, any claim that this or that is a credible hypothesis for that phenomenon is itself, ultimately, a mere hypothesis. So, every conclusion is in the realm of the merely probable, even if some possibilities are more probable than others.
As regards the mathematical case presented, the apparent contradiction seems easy to resolve (if I understand it correctly). Both p and p+1 could well be primes. The fact that the symbol p is used in both terms does not affect this possibility. That is to say, ‘p+1’ could well be a case of ‘p’ – symbols do not signify fixed values but all values of a kind. So, the two series are really one and the same. They differ only superficially in symbols used.
As regards the claim that, in the said mathematical context, a proof viewed as more ‘direct’ might be more informative or productive than one viewed as more ‘indirect’ – well, ok, that might well happen. But the logical validity of both avenues can still be equal, even so, if the logic involved is sufficiently strict.
I hope that helps.
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