The Calculus Wars

One definition of calculus is "an advanced branch of mathematics that deals mostly with rates of change and with finding lengths, areas, and volumes".  This entire branch of mathematics is at its core, about how things change.  It is somewhat ironic, that the two men who battled in Jason Socrates Bardi’s "The Calculus Wars", Isaac Newton and Gottfried Leibniz, failed to recognize the incremental and gradual change that allowed them to "invent" this branch of mathematics.  The idea of invention suggests a brand new idea, which may radically change how we live or how we understand the world.  By fighting and arguing over who invented calculus, Newton, Leibniz, and their contemporaries overlooked the significant contributions of dozens of other men made in the years leading up to their own work.  In their argument about calculus, the study of change, they failed to address how this change to mathematics really happened.  

“The Calculus Wars” tracks the lives and careers of the two men credited with inventing calculus, Isaac Newton and Gottfried Leibniz.  Bardi makes the case that there were two core mistakes or misunderstandings that allowed the calculus conflicts to escalate into the calculus wars.  If Newton had published his work on calculus when he wrote them, in 1666, the conflict could have been avoided entirely.  He had, in his personal notes, ample evidence of his work on “fluxions and fluents”, which make up his claim to the creation of calculus.  However, Leibniz was the first to publish his ideas of calculus in the year 1684, allowing him to claim ownership of the field.  This publication did, however, come after he had limited access to Newton's journals, which would later lead to claims of plagiarism.  The conflict over which man created calculus is more complicated than just these two events, but the timeliness of Newtown’s publishing and Leibniz’s alleged access to those unpublished ideas were the catalyst for a fairly ugly academic battle.  The conflict would warp to include nationalism, academic name-calling, and claims on both sides of stolen work.  Those initial claims of plagiarism would prove to be the spark that would lead to the rising accusations that would divide these great minds for the rest of their lives. 

These men are each undeniable geniuses, both in mathematics and elsewhere in the academic world.  Newton is a familiar name for any student who has taken physics, where he established his still famous laws of motion and gravitation.  Leibniz's talents were wide ranging and profound.  He established foundational ideas in Earth science and geology, and was a renowned scholar on politics, philosophy, and even Chinese culture and history.  The abilities of these men, in such varied fields, prove that neither discovery was an isolated achievement.  It seems unlikely that either man was intellectually dishonest about their work in calculus.  The most likely order of events saw Newton as the earliest "inventor" but not publishing, with Leibniz publishing first, marginally (if at all), influenced by Newton’s journal.  Leibniz went on to create, simplify and improve the mechanics and symbolism of calculus.  He introduced enough new elements to calculus, including the familiar elongated 'S' many recognize (and dread) from integration, to prove his own merits as a mathematician.  

Prior to the work of Newton and Leibniz, there were many mathematicians working on topics very similar to calculus.  As Bardi notes, Johannes Kepler, building on the work of earlier Greek geometry experts, used infinitely small shapes to determine the volume of larger shapes, closely approximating a method of integration.  Bonaventura Cavalieri “considered a line to be an infinity of points: an area an infinity of lines; and a solid an infinity of solids” (p. 8).  This approach reflects the dimensional relationships that are at the heart of differential calculus, specifically the power rule.  Newton and Leibniz owed a debt to René Descartes, who created analytical geometry, which “allowed the analysis of geometrical shapes through mathematical equations”.  Pierre Fermat and Blaise Pascal both created methods for finding maxima and minima, and did extensive work with tangents to curves, all of which are taught in thousands of calculus classrooms across the world.  Gilles Personne de Roberval worked with geometric shapes and volumes, as did Evangelista Torricelli.  John Wallis, Johann Hudde, Christian Huygens, Isaac Barrow, and René François de Sluse and no doubt numerous others made contributions to what we understand to be calculus, all within about 100 years of Newton and Leibniz.  

Looking at the list of mathematicians who contributed to what we know as calculus, the idea of “inventing” calculus becomes even cloudier.  Both Newton and Leibniz claimed individual credit, but both owed a debt of gratitude to a long list of scholars.  Beyond the contemporaries mentioned earlier, there is a long history, a continuum of mathematical understanding that has developed, sometimes slowly, and sometimes rapidly.  It is no accident that each man claims to have created calculus in the same time period.  Kepler, Descartes, Fermat, and many others had set the stage for calculus to emerge in the 17^th century.   It is also important to note that the conflict between Newton and Leibniz only escalated in 1713, 30 years after Leibniz published his work and almost 50 years after Newton had done his.  Both men had established impressive careers and reputations across Europe, and the conflict seemed to be more about protecting those reputations than anything else.  Both men relied upon proxies to argue on their behalf, and were fairly careful to not directly attack the other man.  This professional politeness and respect seems to undercut the idea that either man really thought the other had done anything malicious.  

The “Calculus Wars”, like most wars, was destructive and unnecessary.  Two men, Isaac Newton and Gottfried Leibniz stood out among a long list of brilliant mathematical minds in the 17^th and 18^th century.  They contributed imaginative and important work to the field of mathematics that should have been enough to establish their place in history.  Instead, an ugly feud ensued over credit for work that both most likely created independently.  If each man had the benefit of hindsight, they may see that they were part of an innovative time for mathematics, and that their own contributions were significant parts of a much larger whole.