"Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive."
Quote Details
Added by wikiquote-import-bot
Unverified quote
0 likes
Original Language: English
Available Languages (1)
Sources
George Pólya, Induction and Analogy in Mathematics (1954) Vol. 1 Of Mathematics and Plausible Reasoning
https://en.wikiquote.org/wiki/History_of_algebra
Revision History
No revisions have been submitted for this quote.
Categories
History of algebra
112 quotes on TrueQuotesView all quotes by History of algebra →
Related Quotes
"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the…"
"The precision of statement and the facility of application which the rules of the calculus early afforded were in a m…"
"The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pa…"
"We think only through the medium of words.—Languages are true analytical methods.—Algebra, which is adapted to its pu…"
"As regards algebra, the early Arabs failed to adopt either the Diophantine or the Hindu notations. An examination of …"
"Admitting the Hindu and Alexandrian authors [such as Diophantus], to be nearly equally ancient, it must be conceded i…"
"al-Khwārizmī “not having taken algebra from the Greeks,. . . must have either invented it himself, or taken it from t…"
"My specific... object has been to contain, within the prescribed limits, the whole of the student's course, from the …"
"The following Treatise... has been endeavoured to make the theory of limits, or ultimate ratios... the sole foundatio…"
"I have decided first to consider the majority of the authors who up to now have written about [algebra], so that I ca…"