“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 2, of the number 3 and so on; the law is evident, we say, and it has the same warranty as every physical law based on observations, whose number is very great but limited. But there is an essential difference. Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demonstration by recurrence, on the contrary, imposes itself necessarily, because it is only the affirmation of a property of the mind itself.”
“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
“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 inducti...”
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