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It is to replace, in proofs, standard mathematical reasoning which, in essence, relies on hand-waving arguments (it should be obvious to everyone that B follows from A) with formal logic.
ECONOMIST: Just what does it mean to prove something?
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His mathematical reasoning now in high gear, no doubt nudged along by very motivated reasoning elsewhere in his brain, Gould soon convinced himself he would be a long-term survivor, a conclusion that would prove to be accurate.
FORBES: Do Oncologists Lie to Their Patients About Their Prognoses?
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People who were asked to reflect on their many past successes or a specific failure scored roughly 10% better on tests of mathematical ability, as well as verbal, spatial, and abstract reasoning, than those who reflected on either many past failures or a single specific success.
FORBES: Yesterday Influences Your Performance Today in Surprising Ways