Ghosts of Departed Errors: Early Criticisms of the Calculus of Newton and Leibniz Eugene Boman, Assoc. Prof. of Mathematics Penn State, Harrisburg campus

1 tan(x) sin(x) x x

=1 =0

cos(x) So by proportional triangles: dx Wasn’t that nice?

For this: Why would we give up this:

In the late 17 th century calculus is invented and... It’s Christmas morning and we’ve got this great new toy. Construct tangents Describe the motion of the moon and the motion of the planets Explain the tides Add up infinite series Solve all manner of physical problems which had been wholly intractable before Let’s Play!! We can use this new toy to: Calculus fuels the burgeoning Scientific Revolution. It is seen as the quintessential product of pure reason. What else might Reason do for mankind?

Politics Finance or... Religion? Suppose we apply our ability to reason to non-physical problems like: The Deistic movement (also known in England as Freethinking) is the “use of Understanding, in endeavoring to find out the Meaning of any Proposition whatsoever, in considering the Evidence for or against it, and in judging of it according to the seeming Force or Weakness of the Evidence.” The Age of Reason

The Deist Criticisms of Christianity Mysteries are accepted without examination: e.g. Virgin birth, the existence of a soul Deference is given to established authorities solely by virtue of that authority: e.g. St. Augustine, or Thomas Aquinas Illogical reasoning

Although his intent is to defend his church he is widely seen as criticizing Newton personally. George Berkeley, the Anglican Bishop of Cloyne, Ireland and a philosopher of some renown responds to the Deist critique in The Analyst in 1734 and sets off a firestorm. Quite a lot, actually. But we’ll only be looking at his comments on two specific problems: The Product Rule and the Power Rule So what exactly did he criticize?

The Product Rule: dA = x dy + y dx x dx y dy A = xy x dy y dx (dx)(dy ) dA = x dy + 0 Berkeley derides this by quoting Newton himself: “In mathematical matters the smallest of errors are not to be scorned.” Then he says, in his most famous line: “... And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?” Illogical reasoning Mysteries

A=xy The Product Rule in the Principia, or the silliest thing Newton ever said x y Subtracting gives the Product Rule: A=xy “Such reasoning as this for Demonstration, nothing but the obscurity of the Subject could have induced” Newton “to put upon his Followers. Berkeley And nothing but an implicit deference to... Authority could move them to admit..... If a Man Shall have satisfied himself of the usefulness of certain Rules; which he afterwards shall propose to His Disciples... His Disciples may,... be inclined to confound the usefulness of a Rule with the certainty of Truth. Augustus DeMorgan called this the ‘magic ½’ argument. Mysteries Deference to authority If a Man Shall have satisfied himself of the usefulness of certain Rules; which he afterwards shall propose to His Disciples... His Disciples may, to save themselves the trouble of thinking, be inclined to confound the usefulness of a Rule with the certainty of Truth. Deference to authority

Compensating Errors T P B N R y dy z dx Grattan-Guiness shows in 1969 that this argument can be extended to any real analytic function. =0

Berkeley’ conclusions: With what appearance of Reason shall any man presume to say, that Mysteries may not be Objects of Faith, at the same time that he himself admits such obscure Mysteries to be the Object of Science? He who can digest a second or third Fluxion (derivative), a second or third Difference, need not, methinks, be squeamish about any point in Divinity. All these points, I say, are supposed and believed by certain rigorous exactors of evidence in religion, men who pretend to believe no further than they can see. You who are at a loss to conduct yourselves, cannot with any decency set up guides to other Men.

The Structure of the Controversy: Who said what? George Berkeley; The Analyst, Published 1734 James Jurin; Geometry No Friend to Infidelity, A Defence of Sir Isaac Newton and the British Mathematicians J. Walton; A Vindication of Sir Isaac Newton’s Principles of Fluxions Against the Objections Contained in the Analyst George Berkeley; A Defense of Free-Thinking in Mathematics James Jurin; The Minute Mathematician or, The Free-Thinker no Just-Thinker J. Walton; The Catechism of the Author of the Minute Philosopher Fully Answer’d George Berkeley; Reasons for not Replying to Mr. Walton’s Full Answer Colin Maclaurin; Treatise on Fluxions; 1742 In fact, every book about calculus for the rest of the 18 th century paid its respects to The Analyst Benjamin Robins; A Discourse Concerning the Nature and Certainty of Sir Isaac Newton’s Methods of Fluxions and of Prime and Ultimate Ratios; 1736 It was not until Weierstrauss wrote down the modern definition of a limit, about 200 years later, that Berkeley’s objections were finally (mostly) answered. But that is another story.

Questions?

The other 18 th century approach (Newton’s) was no better B C Let B move toward C with a (possibly variable) velocity b The ultimate ratio, a/b, of their respective velocities in the last instant before they reach C is the fluxion (derivative) of A with respect to B at C Or as he put it in the Principia: Quantities, and ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal. A Let A move toward C with a (possibly variable) velocity a C

Deism was perceived as a direct attack on Christianity “The fable of Christianity... was now so exploded in England that any man of fashion or condition would have been almost as much ashamed to own himself a Christian as formerly he would have been to profess himself none.” – Lord Hervey’s Memoirs of the Court of George II “It is come, I know not how, to be taken for granted by many persons that Christianity is not so much as a subject for inquiry; but that now at length, discovered to be fictitious,” and “nothing remained but to set it up as a principal subject of mirth and ridicule.” – Analogy of Religion, Butler And many of the leading scientists and mathematicians of the day (eg. Halley) proclaimed themselves Deists. (Halley was in fact an avowed atheist as well.) “In no area was the application of reason more needed, claimed the freethinkers, than in theology, where tradition, superstition, and vested interest had prevailed.”