stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2007-06-29 04:38 pm
stoutfellow) wrote2007-06-29 04:38 pm
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Ramble, Part 27: All Is Flux
If, as I have said, Newton did not create the calculus out of whole cloth - if most of its elements already existed before he came on the scene - what, then, did he do to merit regard as its founder?
Those who had been nibbling at the corners of the subject included algebraists, like Viète and Fermat; geometers, like Barrow and Cavalieri; and a few stray physicists and astronomers, like Torricelli, Galileo, and Kepler. It was Newton who recognized that their work lay properly within an entirely new field: not algebra, the mathematics of quantity, nor geometry, the mathematics of shape, but a new calculus, a mathematics of change. More under the cut.
First, a brief linguistic digression. The word "calculus" derives ultimately from a Latin word meaning "pebble"; classical computational devices used small stones, and the word was thus transferred to mean any method of computation (I almost wrote "calculation", so ubiquitous is the word!). There are a number of branches of mathematics whose names contain the word, including the calculus of variations and the calculus of finite differences; however, the differential and integral calculus that begins with Newton has claimed the unadorned word as its own.
"Begins", I said; "begins", in the sense of a consciously recognized unity. Newton saw Cavalieri conceiving of area and volume in terms of infinitely thin cross-sections; Viète seeking to solve equations by tiny adjustments of an initial estimate; Fermat finding extrema by perturbations that were and were not equal to zero; and he saw the connections among these. The key lay in taking more seriously the word "variable" - a quantity which varies. Seizing the concept more robustly, Newton spoke of "fluents" - flowing quantities - and the rates, or "fluxions", at which they flowed. Areas could be computed by the accretion of indivisible cross-sections, as Cavalieri had seen, and quantities generated by the accumulation of infinitesimal "moments", linking the ideas of Viète and Fermat. Tangents sprang easily into view - the ad hoc techniques of de Roberval and Torricelli could be subsumed into a greater whole - and the theorem of Barrow linking areas and tangents was a natural consequence.
Newton struggled with these concepts for a long time, and his views as to their nature took more than one form. In the first edition of the Principia, he wrote not of fluents and fluxions, but of "genita" and "moments", and struggled to explain what the latter were:
He had not. That the calculus rested on shaky logical foundations was clear early on. Bishop George Berkeley - he of the falling tree in the deserted forest - in 1734 published The Analyst, a devastating critique of Newtonian calculus, concluding:
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Those who had been nibbling at the corners of the subject included algebraists, like Viète and Fermat; geometers, like Barrow and Cavalieri; and a few stray physicists and astronomers, like Torricelli, Galileo, and Kepler. It was Newton who recognized that their work lay properly within an entirely new field: not algebra, the mathematics of quantity, nor geometry, the mathematics of shape, but a new calculus, a mathematics of change. More under the cut.
First, a brief linguistic digression. The word "calculus" derives ultimately from a Latin word meaning "pebble"; classical computational devices used small stones, and the word was thus transferred to mean any method of computation (I almost wrote "calculation", so ubiquitous is the word!). There are a number of branches of mathematics whose names contain the word, including the calculus of variations and the calculus of finite differences; however, the differential and integral calculus that begins with Newton has claimed the unadorned word as its own.
"Begins", I said; "begins", in the sense of a consciously recognized unity. Newton saw Cavalieri conceiving of area and volume in terms of infinitely thin cross-sections; Viète seeking to solve equations by tiny adjustments of an initial estimate; Fermat finding extrema by perturbations that were and were not equal to zero; and he saw the connections among these. The key lay in taking more seriously the word "variable" - a quantity which varies. Seizing the concept more robustly, Newton spoke of "fluents" - flowing quantities - and the rates, or "fluxions", at which they flowed. Areas could be computed by the accretion of indivisible cross-sections, as Cavalieri had seen, and quantities generated by the accumulation of infinitesimal "moments", linking the ideas of Viète and Fermat. Tangents sprang easily into view - the ad hoc techniques of de Roberval and Torricelli could be subsumed into a greater whole - and the theorem of Barrow linking areas and tangents was a natural consequence.
Newton struggled with these concepts for a long time, and his views as to their nature took more than one form. In the first edition of the Principia, he wrote not of fluents and fluxions, but of "genita" and "moments", and struggled to explain what the latter were:
[Genita] I here consider as variable and indetermined, and increasing or decreasing, as it were, by a continual motion or flux; and I understand their momentary increments or decrements by the name of moments... But take care not to look upon finite particles as such. Finite particles are not moments, but the very quantities generated by the moments. We are to conceive of them as the just nascent principles of finite magnitudes. Nor do we in this Lemma regard the magnitude of the moments, but their first proportion, as nascent.If it is difficult to follow what Newton means here, it should be no surprise; he himself was dissatisfied with it, and in the second edition tried again.
These Lemmas are premised to avoid the tediousness of deducing involved demonstrations ad absurdum, according to the method of the ancient geometers. For demonstrations are shorter by the method of indivisibles, but because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the following Propositions to the first and last sums and ratios of nascent and evanescent quantities, that is, to the limits of those sums and ratios, and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is performed as by the method of indivisibles; and now those principles being demonstrated, we may use them with greater safety.This last is the key. Newton ardently wanted to use infinitesimals - indivisibles - but recognized the logical dangers involved in their use. He attempted, then, to show that it was possible to act as if indivisibles made sense, without succumbing to those dangers. The benefits infinitesimals promised were so great that he was willing to run the risk of illogic; and he believed that he had run that gantlet.
He had not. That the calculus rested on shaky logical foundations was clear early on. Bishop George Berkeley - he of the falling tree in the deserted forest - in 1734 published The Analyst, a devastating critique of Newtonian calculus, concluding:
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, nor yet nothing. May we not call them the ghosts of departed quantities?Several of Newton's successors - notably, Maclaurin, Euler, and Lagrange - attempted unsuccessfully to retrieve the situation; but their lack of success did not deter them and many others from applying and extending the calculus - it was simply too effective a tool to be neglected. They did not avoid the crisis; they merely postponed it for more than a century. We will return to this later.
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Thanks for the Ramble! Best wishes to Ben!
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Very interesting! Thanks!
Makes me remember my basics of calculus. Fun! It's quite sad when you reach your pinnacle of math knowledge at about 16-18. And realize you're now about twice that age.