I had always been told that Newton had invented calculus out of thin air in order to help him develop his theories of mechanics, yet after reading about the development of calculus in the seventeenth century in “A Concise History of Mathematics” by Dirk J. Struik that appears not to be so.
From my reading, it appears that a lot of the concepts in calculus were already up in the air, so to speak at the time, that Newton’s methods weren’t particularly rigorous and that his notation and choice of language were so cumbersome (“fluxions”, etc.) that they held up British mathematics for a while whereas the continent raced ahead using Leibniz’s notations (IIRC).
Is the role of Newton in the development of calculus overplayed by patriotic British historians / mathematicians?
In Fermat’s Enigma, the author, Simon Singh says that Newton himself credited Pierre Fermat’s work as helping considerably in Isaac’s development of Calculus.
It’s a good book IMHO, and you can get a used copy for less than $5 at amazon. I’m going to read it again, if I can get my copy back from my son-in-law.
And if you like that one, you might also enjoy Singh’s, The Code Book. Check 'em out.
The credit for inventing/discovering calculus still gets assigned to Newton and Leibniz (independently), but neither of them created it “out of thin air”; a number of earlier mathematicians (including Archimedes! as well as Fermat, Pascal, and Descartes) anticipated many of the ideas that became calculus. And then other mathematicians came later to expand on calculus, find new uses for it, and eventually set it on the firm theoretical footing (the formulation you’ll find in a modern Calc textbook, thanks to Cauchy, Weierstrass, and Riemann).
I seem to recall reading somewhere that at least part of the reason Newton and Leibniz get the credit for the invention of calculus is that they were the first to make the connection between differential and integral calculus in what we know as the Fundamental Theorem of Calculus.
Basically, the ancient Greeks didn’t like to any kind of arithmetic/algebra, feeling that it was inferior to geometric reasoning. If Archimedes hadn’t had that blind spot, he probably would’ve come up with the notion of a limit, and from there, who knows?
+MDI will probably wring my neck for this, but let me ask: If we were able to transport a few college mathematics texts back to Archimedes–covering, say, differential and integral calculus, differential equations, linear algebra, etc., how quickly would things likely click? (I’m not sure if the language barrier would really present a barrier as much as the paradigm shift.)
In short, would things click almost instantly with such a mathematical genius or would it more likely have taken him awhile to reorient to an entirely different paradigm?
Even Newton would take a while to go through a modern calculus textbook. There was a lot of revision of the calculus in 1870 or so, and that’s about when the notion of a limit was introduced. He’d need time to grasp that–perhaps not much, but time nonetheless.
The notion that Newton’s influence over his British successors put them at a disadvantage over their Continental colleagues in the years after his death is certainly something that’s often been repeated by writers on the history of mathematics. However, this has been questioned by specialists in recent years, notably by Niccolo Guicciardini in his The Development of Newtonian Calculus in Britain, 1700-1800 (Cambridge, 1989). The argument is that, rather than this being a period of stagnation as suggested in the conventional picture, a detailed look at it reveals lots of fruitful work being done. Guicciardini does accept that there was a crucial divergence between the British and French traditions, with the latter’s contributions going on to dominate 19th century developments, but dates this rather late in the 18th century and absolves Newton of the blame.
But wasn’t Archimedes a great mechanic? I seem to recall reading that this snobbery towards algebra and arithmetic was a later trend in Greece, well after Archimedes had died. Perhaps I’m mistaken.
I’m pretty sure that the Greek anti-algebra attitude goes back before Archimedes. If you want a definite answer, read E.T. Bell’s Men of Mathematics, a true classic in the field.
> I had always been told that Newton had invented calculus out of thin air . . .
Nobody ever invented anything out of thin air. The whole notion of the history of science as a series of a few great men producing important developments with no context is wrong. There are always precursors to every important invention.
Actually, both Newton’s and Leibniz’ motivations for the calculus were largely geometric rather than as tools for physics, as is usually supposed. That curves describe the motions of particles was well-established, but the techniques for deriving (hah) those curves were open.
This work can hardly claim to be the definite answer. It is not an accurate historical work and Bell let’s his prejudices guide his pen. The book has since been discredited.
Tony Rothman’s dismantling of Bell’s mythical version of Galois is the best known critique of Men of Mathematics.
Bell’s undoubtedly an entertaining (and influential) writer, but, no, he doesn’t have a particularly high reputation for factual accuracy.
Excelent link, bonzer, thanks. I was having trouble finding a citation. The depiction of Galois is one of the most important parts of Bell’s book.
I’m a critic of Bell because, while entertaining, he presents Men of Mathematics as an historical book and then offers a lot of myth and distortion as fact. I have read such opinions from other authors but couldn’t tell you where from memory.
I recommend this great book for a more scholarly work on the history of mathematics.
