"Dedekind proves mathematical induction, while Peano regards it as an axiom. This gives Dedekind an apparent superiority, which must be examined. ...not because of any logical superiority, it seems simpler to begin with mathematical induction. And it should be observed that, in Peano's method, it is only when theorems are to be proved concerning any number that mathematical induction is required. The elementary Arithmetic of our childhood, which discusses only particular numbers, is wholly independent of mathematical induction; though to prove that this is so for every particular number would itself require mathematical induction. In Dedekind's method, on the other hand, propositions concerning particular numbers, like general propositions, demand the consideration of chains. Thus there is, in Peano's method, a distinct advantage of simplicity, and a clearer separation between the particular and the general propositions of Arithmetic. But from a purely logical point of view, the two methods seem equally sound; and it is to be remembered that, with the logical theory of cardinals, both Peano's and Dedekind's axioms become demonstrable."
Quote Details
Added by wikiquote-import-bot
Unverified quote
0 likes
Original Language: English
Available Languages (1)
Sources
Bertrand Russell,The Principles of Mathematics (1903) Vol. 1 p.248
https://en.wikiquote.org/wiki/Mathematical_induction
Revision History
No revisions have been submitted for this quote.
Categories
Mathematical induction
17 quotes on TrueQuotesView all quotes by Mathematical induction →
Related Quotes
"Few contemporaries were as profoundly read in the history of mathematics as was De Morgan. No subject was too insigni…"
"One who extended the theory of equations somewhat further than Vieta was Albert Girard... Like Vieta this ingenious a…"
"A more modern attempt to explain the fruitfulness of mathematical reasoning is that of Poincaré, who finds it all due…"
"It is absolutely certain that if a proposition is established by mathematical induction, it will never be disproved, …"
"The propositions of arithmetic, the... operations, for instance, which play such a fundamental rôle even in the most …"
"It is significant that we owe the first explicit formulation of the principle of recurrence to the genius of Blaise P…"
"Despite the age-long tyranny exercised by the Aristotelian logic... Of all argument forms, there is one which, viewed…"
"This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is…"
"We can not... escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contr…"
"But, one will say, if raw experience can not legitimatize reasoning by recurrence, is it so of experiment aided by in…"