I've recently read that some mathematician's believe that there are "no necessary truths" in mathematics. Is this true? And if it is, what implications would it have on deductive logic, it being the case that deductive logical forms depend on mathematical arguments to some degree. Would in this case, mathematical truths be "contingently-necessary"?

Your question is tantalizing. I wish it had included a citation to mathematicians who say what you report them as saying. On the face of it, their claim looks implausible. Are there no necessary truths at all? If there are necessary truths, how could the mathematical truth that 1 = 1 not be among them?

One way to hold that mathematicians seek only contingent truths might be as follows. If some philosophers are correct that propositions are to be identified with sets of possible worlds, then there's only one necessarily true proposition, because there's only one set whose members are all the possible worlds there are. That single necessarily true proposition (call it "T") will be expressed by indefinitely many different sentences, including the sentences "1 = 1" and "No red things are colorless," and it will be contingent just which sentences express T. On this view, mathematicians don't try to discover various necessary truths, since there's just one necessary truth, T. Instead, they try to discover which of various mathematical sentences express T, and because any sentence is a contingent item of language, any mathematical sentence that expresses T does so only contingently. Might this be what the mathematicians you mentioned had in mind?

To answer your last two questions: (1) Mathematical arguments depend on deductive logic, not conversely, so developments in math aren't going to threaten deductive logic. (2) If no mathematical truths are necessary in the first place, then none of them are contingently necessary: any truth that possesses a property contingently possesses that property. But some philosophers say that some truths that are necessary are only contingently necessary, i.e., some truths are necessary in the actual world but not necessary in all possible worlds. These philosophers deny the characteristic axiom of the S4 system of modal logic that says "Whatever is necessarily true is necessarily necessarily true."

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