In my previous post on the use of mathematics to describe nature I closed with the following remark about Newton’s Principia:
Thirdly the ultimate mathematical book of nature written in the 17th century, Newton’s Mathematical Principles of Natural Philosophy, was cast in the form it has, propositions deduced logically from axioms, in order to give it the status of an Aristotelian scientia.
This provoked the following comment from Brian (didn’t Monty Python film his autobiography?):
Now, be gentle with me here (I’m but a Whiggish twat), but wasn’t that because Newton didn’t want to expose himself to withering attack on two fronts? If he’d unleashed his fluxions (differential calculus or thereabouts) as well as his cosmology then he’d be in a right pickle. From what I understand, he didn’t like being pickled by critics.
Now we have here two problems. Firstly Brian is confusing form and content and secondly he is repeating one of the most widespread myths concerning Newton and the composition of the Principia.
In my original remark I said that the Principia is cast in the form of an Aristotelian scientia that is axiomatic and deductive but naturally the content is principally mathematically and that is not Aristotelian. Brian is of course referring to the content, the mathematics.
In his comment Brian is repeating a myth that can be read all over the place and was unfortunately put into the world by Newton himself. As is well known Newton is the co-inventor (discoverer) of the calculus he is also the author of what is considered to be one of the most important books on mathematical physics ever written. Strangely this book is not written in the language of the then brand new, and as already stated from the author himself invented, analytical mathematics but at least superficially in the style of the classical Greek synthetic mathematics; strange to say the least. The standard explanation is that proffered by Brian the Principia was created using fluxions (Newton’s version of the calculus) but then translated into classical geometry so as not to offend, confuse, irritate or whatever the readers. This version of the story was, as already mentioned, created by Newton, who wrote in 1715:
By the help of the new Analysis Mr. Newton found out most of the Prepositions in his Principia Philosophiae: but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the System of the Heavens might be founded upon good Geometry. And this makes it now difficult for unskilful men to see the Analysis by which these propositions were found out.
Now D. T. Whiteside who edited all eight large format volumes of Newton’s Collected Mathematical Papers could find no evidence that Newton ever wrote or developed any part of the Principia in any other way than the published form, a conclusion that is confirmed by other Newton experts such as I. B. Cohen and Richard S. Westfall, from which it follows that Newton lied! So there are two questions that need to be answered, why did Newton lie and why did he develop and write the Principia in synthetic geometry and not the modern analysis?
The answer to the first question lies in the infamous calculus priority dispute with Leibniz. The Newton quote above, in which he refers to himself in the third person, is his review of the Commercium Espistolicum published in the Royal Society’s Philosophical Transactions. The dispute, which had slowly come to the boil during the 1690s and the first decade of the 18th century had involved a series of steadily ruder attacks by the supporters of the two principals. Finally Leibniz provoked by a series of particularly vicious attacks by John Keil had appealed to the Royal Society, of which he was a foreign member, to clear his name. This was not a clever move as Newton was the President of the Royal Society so the committee that he set up to investigate the mutual accusations of plagiarism was more than biased; in fact Newton wrote their supposed neutral report of the affair, the Commercium Espistolicum from 1712, himself naturally finding Leibniz guilty and himself innocent! This was not enough and he returned to subject in 1715 with his review quoted above. In this review and in later writings he claimed to have developed the Principia using analysis in order to counter critics from Leibniz’ side who said if Newton had invented the calculus how come he didn’t use it to develop the Principia? Their answer being that he had not invented the calculus and his mastery of Leibniz’ invention was so weak that he could use it for the Principia! To counter this slur Newton lied!
This still leaves the question open as to why he didn’t use it. The answer to this question is hinted at in the passage quoted above and to understand why we have to examine Newton’s mathematical development. Neither Newton nor Leibniz invented the calculus or analytical mathematics as is usually claimed. The analytical methods evolved slowly from the middle of the 15th century onwards with many mathematicians making contributions to the development of symbolic algebra, algebraic geometry and finally calculus. As a comparatively young mathematician in the 1660s, Newton developed the work of others such as Descartes, van Schooten, Fermat, John Wallis, Isaac Barrow and James Gregory extending and improving the analysis until he reached the fundamental theory of the calculus, to be fair before Leibniz. Somewhat later with more maturity Newton came to question the philosophical underpinning of the new analytical methods and turned back to what he saw as the philosophically preferable synthetic approach of the Greeks, as he says in the quote above. He didn’t reject the new analytical methods, to which he had contributed so much, but proceeded to develop a synthetic calculus, strong traces of which can be found in the proofs in the Principia. Put simply Newton had serious doubts about the reliability of the new analytical mathematics and that is why he didn’t use it for his magnum opus.
Anyone interested in learning more can read about Newton’s lie in A Guide to Newton’s Principia by I. Bernard Cohen, section 5.8 and Richard S. Westfall Never at Rest: A Biography of Isaac Newton p. 424 ff. For a detailed analysis of Newton’s rejection of analytical methods in mathematic then I heartily recommend, Niccolò Guicciardini, Reading the Principia, CUP, 1999, but with the warning that it’s not an easy read!