I'm currently reading "Mathematics Without Numbers" by Hellman, G., and I'm on pages 26-27. It seems like Hellman is discussing opposition to viewing mathematical proofs solely through the interpretative structure of if-then. To avoid misunderstanding, I'll show you the passage first.
A categorical assumption to the effect that “$ω$-sequences are possible” is indispensable and of fundamental importance. Without it, we would have a species of “if-thenism”, i.e. a modal if-thenism, and this would be open to quite decisive objections, analogous to those which can be brought against a naïve, non-modal if-then interpretation. Consider the latter. Suppose it represents sentences A of arithmetic by means of a material conditional, say, of the form, $∧PA2⊃A$, or some refinement thereof. Suppose also that, in fact, there happen to be no actual $ω$-sequences, i.e. that the antecedent of these conditionals is false. (This could be “by accident” as it were. For the sake of argument, do not insist upon Cantor's universe of sets as “necessary existents” (please!). Consider, instead, the stance of the “if-thenist” who seeks to avoid platonism.) Then, automatically, the translate of every sentence A of the original language is counted as true, and the scheme must be rejected as wildly inaccurate.
However, isn't it true that in a material conditional, even if the antecedent is false and the entire conditional is true, the truth value of the consequent isn't necessarily true? Why does negating the antecedent and affirming the conditional automatically argue that the consequent, an arithmetic statement A in this case, is true? I'm having trouble understanding what I'm missing here. The only unused clue seems to be that ω-sequences are not possible, which Hellman refers to in terms of numbers here, but this argument doesn't seem to involve that concept directly. What am I missing?
