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“That this blind and aging man forged ahead with such gusto is a remarkable lesson, a tale for the ages. Euler's courage, determination, and utter unwillingness to be beaten serves, in the truest sense of the word, as an inspiration for mathematician and non-mathematician alike. The long history of mathematics provides no finer example of the triumph of the human spirit.”
― Euler: The Master of Us All
― Euler: The Master of Us All
“Never was his remarkable memory more useful than when he could see mathematics only in his mind's eye.”
― Euler: The Master of Us All
― Euler: The Master of Us All
“In his eulogy, the Marquis de Condorcet observed that whosoever pursues mathematics in the future will be "guided and sustained by the genius of Euler" and asserted , with much justification, that "all mathematicians...are his disciples.”
― Euler: The Master of Us All
― Euler: The Master of Us All
“Already uneasy over the foundations of their subject, mathematicians got a solid dose of ridicule from a clergyman, Bishop George Berkeley (1685-1753). Bishop Berkeley, in his caustic essay 'The Analyst, or a Discourse addressed to an Infidel Mathematician,' derided those mathematicians who were ever ready to criticize theology as being based upon unsubstantiated faith, yet who embraced the calculus in spite of its foundational weaknesses. Berkeley could not resist letting them have it:
'All these points [of mathematics], I say, are supposed and believed by certain rigorous exactors of evidence in religion, men who pretend to believe no further than they can see... But he who can digest a second or third fluxion, a second or third differential, need not, methinks, be squeamish about any point in divinity.'
As if that were not devastating enough, Berkeley added the wonderfully barbed comment:
'And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, not yet nothing. May we not call them the ghosts of departed quantities...?'
Sadly, the foundations of the calculus had come to this - to 'ghosts of departed quantities.' One imagines hundreds of mathematicians squirming restlessly under this sarcastic phrase.
Gradually the mathematical community had to address this vexing problem. Throughout much of the eighteenth century, they had simply been having too much success - and too much fun - in exploiting the calculus to stop and examine its underlying principles. But growing internal concerns, along with Berkeley's external sniping, left them little choice. The matter had to be resolved.
Thus we find a string of gifted mathematicians working on the foundational questions. The process of refining the idea of 'limit' was an excruciating one, for the concept is inherently quite deep, requiring a precision of thought and an appreciation of the nature of the real number system that is by no means easy to come by. Gradually, though, mathematicians chipped away at this idea. By 1821, the Frenchman Augustin-Louis Cauchy (1789-1857) had proposed this definition:
'When the values successively attributed to a particular variable approach indefinitely a fixed value, so as to end by differing from it by as little as one wishes, this latter is called the limit of all the others.”
― Journey through Genius: The Great Theorems of Mathematics
'All these points [of mathematics], I say, are supposed and believed by certain rigorous exactors of evidence in religion, men who pretend to believe no further than they can see... But he who can digest a second or third fluxion, a second or third differential, need not, methinks, be squeamish about any point in divinity.'
As if that were not devastating enough, Berkeley added the wonderfully barbed comment:
'And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, not yet nothing. May we not call them the ghosts of departed quantities...?'
Sadly, the foundations of the calculus had come to this - to 'ghosts of departed quantities.' One imagines hundreds of mathematicians squirming restlessly under this sarcastic phrase.
Gradually the mathematical community had to address this vexing problem. Throughout much of the eighteenth century, they had simply been having too much success - and too much fun - in exploiting the calculus to stop and examine its underlying principles. But growing internal concerns, along with Berkeley's external sniping, left them little choice. The matter had to be resolved.
Thus we find a string of gifted mathematicians working on the foundational questions. The process of refining the idea of 'limit' was an excruciating one, for the concept is inherently quite deep, requiring a precision of thought and an appreciation of the nature of the real number system that is by no means easy to come by. Gradually, though, mathematicians chipped away at this idea. By 1821, the Frenchman Augustin-Louis Cauchy (1789-1857) had proposed this definition:
'When the values successively attributed to a particular variable approach indefinitely a fixed value, so as to end by differing from it by as little as one wishes, this latter is called the limit of all the others.”
― Journey through Genius: The Great Theorems of Mathematics
“To this day the existence of odd perfect numbers remains unsolved.”
― Euler: The Master of Us All
― Euler: The Master of Us All
“I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.”
― The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities
― The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities
