“A more modern attempt to explain the fruitfulness of mathematical reasoning is that of Poincaré, who finds it all due to the principle of mathematical induction. This principle of mathematical induction is undoubtedly of wide application, though there are many regions even in arithmetic where it is difficult to see its application, e.g., the science of prime numbers, a science dealing entirely with non-recurring individuals. But the important thing to observe is that this principle of mathematical induction is entirely different from the induction that prevails in the physical sciences.”
“Few contemporaries were as profoundly read in the history of mathematics as was De Morgan. No subject was too insignificant to receive his attention. ...In [his] article "Induction (Mathematics)," fir...”
Mathematical induction
“One who extended the theory of equations somewhat further than Vieta was Albert Girard... Like Vieta this ingenious author applied algebra to geometry, and was the first who understood the use of nega...”
Mathematical induction
“It is absolutely certain that if a proposition is established by mathematical induction, it will never be disproved, i.e., if a general proposition is true of n + 1 whenever it is true of n,...”
Mathematical induction
“We can not... escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction. ...Neither can this rule come to us from experience... This rule, inacces...”
Mathematical induction
“But, one will say, if raw experience can not legitimatize reasoning by recurrence, is it so of experiment aided by induction? We see successively that a theorem is true of the number 1, of the number ...”
Mathematical induction