Ghosts of Departed Errors:Early Criticisms of the Calculus of Newton and Leibniz

Eugene Boman, Assoc. Prof. of MathematicsPenn State, Harrisburg campus

ecb5@psu.edu

1 tan(x)sin(x)

First a Lemma:

x

sin (𝑥 )≤ 𝑥≤ tan (𝑥 )

1≤𝑥

sin (𝑥 )≤1/cos (𝑥)

1≥sin (𝑥 )𝑥

≥ cos (𝑥)

So, by the Squeeze Theorem:

Theorem: sin(x)=cos(x)Modern proof

Corollary:

x

Theorem: sin(x)=cos(x)Modern proof continued

𝐷𝑥sin ( x )= limh→0

sin (𝑥+h )− sin (𝑥 )h

=¿limh→0

sin (𝑥 ) cos (h )+cos (𝑥 )sin (h)−sin (𝑥)h

]

=1 =0

¿cos (𝑥 )

Theorem: sin(x)=cos(x)How would we have done this in the 18th century?

cos(x)

sin (𝑥)

𝑑(sin (𝑥 ))𝑑𝑥

1

𝑥

𝑥

So by proportional triangles:𝑑 (sin (𝑥 ))

𝑑𝑥=¿cos (𝑥)1

=cos (x )

dx

Wasn’t that nice?

For this:

Why would we give up this:

In the late 17th 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 formankind?

• 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 +𝑦 𝑑𝑥 +

0Berkeley 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

−∆ 𝑥2

∆ 𝑥2

−∆ 𝑦2

∆ 𝑦2

x

y

y

y

Subtracting gives the Product Rule:

y

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 toadmit. . . . .

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

𝑦 2=𝑥

T P

B

N

R

y

dy

z

dx

Newton:

Berkeley: is not a triangle so this is clearly wrong. The correct formula is:

Missing isthe first error.

𝑑𝑦= 𝑑𝑥2 𝑦

−( 𝑑𝑦2 𝑦 )2

Missing is the second error.

Grattan-Guiness shows in 1969 that this argument can be extended to any real analytic function.

But if then

so that

𝑃𝑇=𝑦𝑑𝑥𝑑𝑦+ 𝑧

Next, Calculus gives: (which we know is correct).

Berkeley then invokes Prop. 31 from On Conics by Apollonius to show that

Thus the errors cancel:

=0

¿𝑦𝑑𝑥

𝑑𝑥2 𝑦

−(𝑑𝑦2 𝑦 )2

+𝑧¿𝑦𝑑𝑥𝑑𝑥2 𝑦

¿𝑦 𝑑𝑥𝑑𝑦

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 18th 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 18th century approach (Newton’s) was no better

BC

Let B move toward C with a (possibly variable) velocity b

The ultimate ratio, a/b, of their respective velocities in the last instant beforethey 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, whichin 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.”