It's now 10 years since my interest in mathematics became an interest in the philosophy of mathematics, but I'm not much closer to knowing why mathematicians standardly assume an axiom of infinity (which is described in the second comment below)—not the historical or sociological reasons, but what philosophical (or amateur-scientific) reasons those applying standard mathematics have, for thinking that an axiom of infinity is true.
......That they have found few problems with that axiom in a hundred years, for example, is an answer in a different ball-park—cf. how medieval astronomers found few problems with assuming (what seems clear) that the stars circle (and the planets epicircle) a fixed Earth. Ultimately the interesting question was does the Earth turn? And while our being able to imagine a star in each cubic lightyear of some infinite space is at least a metaphysical reason (e.g. see Benardete's "Infinity: An essay in metaphysics"), mathematicians have naturally been moving away from their previous reliance upon such geometrical intuitions (towards greater rigour).
......Not that standard mathematicians need any very good metaphysical reasons for finding most interesting those models of arithmetic that contain such an axiom (e.g. there are benefits to having a common language, and there was originally the epistemic possibility to explore) but if they don't have any such reasons then physicists and metaphysicians should perhaps not presume that some such axiom is true. And note that the question arises given that numbers are objective (that proof aims at, but is not to be identified with, truth), so that it is hardly an answer to have included Constructive mathematics within academia.
Great moments in obscure rock 'n' roll: Black Widow, "Sacrifice," 1970 - British hard rock/progressive band of the late 1960s and early 1970s, this is the title track of their second, and most successful album:
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