It is perplexing to me that Newton and Leibniz get so much credit for “inventing” Calculus, when at most, they merely added a few finishing touches to a masterpiece which had been centuries in the making. I suppose it probably boils down to the obvious… it’s just easier to credit one or two people with something than to actually distribute the credit more justly. Solon was “the” man who gave Ancient Athens its laws. Einstein was “the” man who produced relativistic thought (here, I’m thinking of the terribly slighted Lorentz). And Newton and/or Leibniz “invented” Calculus.
Whereas, in truth, after reading Boyer’s book, The History Of The Calculus, I see that the different parts cobbled together to make Calculus were provided by a long series of contributors. To name merely the most obvious:
Pythagoras, Zeno, Plato, Aristotle, Eudoxus, Archimedes, Calculator, Oresme, Stevin, Valerio, Kepler, Galileo, Cavalieri, Torricelli, Barrow, and Fermat.
In fact, Fermat has his fans, who believe that HE deserves credit for “inventing” calculus if anyone does– and Boyer implies that Barrow would also be a contender for that honor (it was Barrow who first clearly recognized the connection between tangents and integrals). Boyer contends that the work of Newton and Leibniz… “differed from the corresponding methods of their predecessors, Barrow and Fermat, more in attitude and generality than in substance and detail.”
It is true enough that Newton and Leibniz offered simplifying algorithms for doing the heavy lifting of Calculus, but Boyer believes that by their time, “infinitesimal considerations were so widely employed and had developed to such a point that, given a suitable notation, a unifying analytic algorithm was almost bound to follow.” To oversimplify: Leibniz blessed us with a rational notational system and some sweet algorithms for finding differentials (he also gave Calculus its name), and Boyer says that Newton also offered us ways of “facilitating the operations” of the math. Boyer makes sure to point out that neither man did much by way of “clarifying the conceptions” of the group of methods which would become known as the “Calculus.” He goes so far as to state that neither Newton nor Leibniz are responsible “for the ideas and definitions underlying the subject at the present time, for these basic notions were to be rigorously elaborated only after two centuries of further effort in this direction.”
The main thing it seems to me that Newton and Leibniz did was to UNIFY many mathematical threads into one strong rope. It appears that no one before their time, including Barrow, had really driven home the point that the derivative and the integral were part of one over-arching continuum of operations.
For Boyer, Newton and Leibniz were just the right men living at the right time to catch the zeitgeist and ride it to glory. “The time was indeed ripe in the second half of the seventeenth century,” says Boyer, “for someone to organize the views, methods, and discoveries involved in the infinitesimal analysis into a new subject characterized by a distinctive method of procedure.”
For most of Europe’s recorded history, men have been capable of finding, fairly exactly, the areas contained by curves. Dynamism was added to the study of curves by the great Galileo. Descartes and others allowed us to describe curves by equations. Euler then decided to make the equations, themselves, the focus of Calculus. Before Euler, the shape was the thing… the equation describing it was merely convenient. Soon after Euler, it would become actually irrelevant if Calculus functions were graphable– or even imagineable– at all.
Another reason that Newton and Leibniz suck-up so much credit when it comes to Calculus is due to their dispute over which of them deserved the most credit for inventing Calculus. Their dispute, I contend, has side-tracked objective analysis of the history of Calculus.
Once we are lured into the question as to whether it was Newton or Leibniz who invented Calculus, we automatically concede that… 1) there was an “inventor”, and 2) we narrow the list of possible inventors to two. When we ask, “Newton or Leibniz?” we are NOT asking the question, “Who were all the people who contributed to the development of Calculus?”
The ability to replace the real and fundamental question by a less-in-depth question was a lesson learned by politicians –and other disputants– a long time ago.