"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
Academics from SwitzerlandPhysicists from SwitzerlandMathematicians from SwitzerlandAstronomers from SwitzerlandLogicians
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/Leonhard_Euler
Revision History
No revisions have been submitted for this quote.
Categories
Leonhard Euler
Leonhard Euler (15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for
34 quotes on TrueQuotesView all quotes by Leonhard Euler →
Related Quotes
"Quanquam nobis in intima naturae mysteria penetrare, indeque veras caussas Phaenomenorum agnoscere neutiquam est conc…"
"Madam, I have come from a country where people are hanged if they talk."
"Now I will have less distraction."
"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have re…"
"All the greatest mathematicians have long since recognized that the method presented in this book is not only extreme…"
"To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence ther…"
"La construction d'une machine propre à exprimer tous les sons de nos paroles , avec toutes les articulations , seroit…"
"It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical…"
"Till now the mathematicians tried in vain to discover some order in the sequence of the prime numbers and we have eve…"
"He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air."