A classic case is that of the inference by Leverrier and Adams for unexplained irregularities in the orbit of Uranus to the existence of the planet Neptune. [footnote 7 omitted HS] These irregularities were not, needless to say, taken as refuting Newtonian mechanics, but rather as motivating a search for the erroneous assumption, a search that culminated in the prediction of the existence and orbit of Uranus. More telling still, Leverrier also noticed deviations in the orbit of Mercury. But here no such explanation was found prior to the replacement of Newtonian with relativistic mechanics. Thus this latter case proved in the end to constitute, by our current lights, a genuine falsification of Newtonian mechanics. But no one took it as such until a new theory that actually predicted the phenomena in question had been developed as a serious rival to the Newtonian account.
John Dupré. 1993. The Disorder of Things. Cambridge MA: Harvard University Press. Pages 230-1.
That this aspect of scientism -- perhaps we should call it "mathematicism" -- is a sociologically significant contributor to scientific prestige seems hard to dispute. It is again perhaps best illustrated by the preeminent influence of economics, with its characteristic appeal to abstruse mathematical models of little empirical worth, among the social sciences. A number of factors probably contribute to this situation. Most relevant to my general theme, more or less inchoate commitments to reductionism surely play some part in the continuing role of physics, and indirectly its mathematical methods, as the paradigm to the methods of which other sciences should aspire. [footnote 1 A phenomenon often referred to as physics envy. [...]] More broadly still, believers in an ultimately orderly universe often suppose that mathematics provides the language necessary to capture such metaphysical order. Straightforwardly sociological factors surely also contribute to mathematicism. It seems very plausible, for example, that mathematical complexities the mastery of which requires substantial training, by setting up barriers to entry (to borrow a concept, this time, from economics), serve to increase the financial and other rewards that accrue to membership of scientific professions. Or even more simply, a mystifying veneer of mathematics will sometimes serve to conceal the banality of what is offered as scientific wisdom. At any rate, the omnipresent neo-Pythagoreanism of contemporary science is surely not adequately justified by its empirical successes. If it is motivated by any legitimately theoretical considerations, I suspect that these amount to some kind of commitment to a universe amenable to one systematic and orderly description; a universe in the existence of which, I have argued, we have every reason to disbelieve.
John Dupré. 1993. The Disorder of Things. Cambridge MA: Harvard University Press. Pages 223-4.