seeking a comment from David Bibby as to the history of math and some of its major contribuotrs such as Newton and Leibniz

[John Raymaker]

Hi all--seeking a clarification from David Bibby.    David you are the mathematician in residence on this site. Strangely, last night I had a dream about numbers as to the role of numerals and zeroes. Upon waking up, I googled and found, e. g. the following, I quote:

In the history of calculus, the calculus controversy (German: Prioritätsstreit, lit. 'priority dispute') was an argument between mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus. The question was a major intellectual controversy, beginning in 1699 and reaching its peak in 1712. Leibniz had published his work on calculus first, but Newton's supporters accused Leibniz of plagiarizing Newton's unpublished ideas. The modern consensus is that the two men independently developed their ideas. Their creation of calculus has been called arguably "the greatest advance in mathematics that had taken place since the time of Archimedes."   

Newton stated he had begun working on a form of calculus (which he called "The Method of Fluxions and Infinite Series") in 1666, at the age of 23, but the work was not published until 1737 as a minor annotation in the back of one of his works decades later (a relevant Newton manuscript of October 1666 is now published among his mathematical papers^[2]).^[3] Gottfried Leibniz began working on his variant of calculus in 1674, and in 1684 published his first paper employing it, "Nova Methodus pro Maximis et Minimis". L'Hôpital published a text on Leibniz's calculus in 1696 (in which he recognized that Newton's Principia of 1687 was "nearly all about this calculus"). Meanwhile, Newton, though he explained his (geometrical) form of calculus in Section I of Book I of the Principia of 1687,^[4] did not explain his eventual fluxional notation for the calculus^[5] in print until 1693 (in part) and 1704 (in full).

Today, the consensus is that Leibniz and Newton independently invented and described calculus in Europe in the 17th century, with their work noted to be more than just a "synthesis of previously distinct pieces of mathematical technique, but it was certainly this in part".^[6]

It was certainly Isaac Newton who first devised a new infinitesimal calculus and elaborated it into a widely extensible algorithm, whose potentialities he fully understood; of equal certainty, differential and integral calculus, the fount of great developments flowing continuously from 1684 to the present day, was created independently by Gottfried Leibniz.

— Hall 1980: 1

One author has identified the dispute as being about "profoundly different" methods:

Despite ... points of resemblance, the methods [of Newton and Leibniz] are profoundly different, so making the priority row a nonsense.

— Grattan-Guinness 1997: 247

On the other hand, other authors have emphasized the equivalences and mutual translatability of the methods: here, Niccolò Guicciardini (2003) appears to confirm L'Hôpital (1696) (already cited):

the Newtonian and Leibnizian schools shared a common mathematical method. They adopted two algorithms, the analytical method of fluxions, and the differential and integral calculus, which were translatable one into the other.

— Guicciardini 2003, on page 250^{[7]"-- end of quoting.}

^{It seems that the numbers 4 and 8 play important but  distinct roles in math.(is this true or false?) Historians have various views on this subject.  Would you briefly comment, noting whether Lonergan was  "concerned" about this history, John}

[David Bibby]

Dear John,

You honour me too highly by referring to me as a "mathematician in residence", but I find your comments intruiging. The priority dispute between Leibniz and Newton has an interesting connection to insight, because the dispute revolves around who discovered it first, or who had the earlier insight. It seems to me that they both had an insight but it was expressed differently, and what we now call calculus is a fusion of their two modes of thought.

The dispute may serve as a lesson on the distinction between common sense and science. From the latter perspective, what is relevant to the insight is its content, but from the former, what is relevant to the insight as related to us is its occurrence, that "aha!" moment when suddenly things fall into place. The occurrence is real, but only in a historical context when it forms part of the narrative of events. It is therefore perfectly acceptable for the same insight to find different expressions in different scientists, and when they start asking "who discovered it first", it is inevitable from the perspectivism of history that different narratives exist - it doesn't strictly have to be one or the other.

I was also intrigued that you asked about the importance of 4 and 8 - any reason for those numbers in particular? I'm sure there are many interesting applications; I would only observe that 4=2^2 and 8=2^3, with the numbers 2, 3 in the exponent, which relates to the discovery of a higher dimension. One could also associate them with quarternions and octonions. 

On the role of numerals and zeros, I will reflect on whether there is any connection with the Riemann Hypothesis, which is a statement about the zeros of a mathematical function, because zeros only emerge from insight, there is "nothing" there...

Thanks for the stimulating thoughts.

Best wishes,

David

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