*In this sequel, Adam Schroeder argues that despite its pristine appearance in textbooks and the classroom, mathematics is a human endeavour, often stimulated by debate and controversy. He suggests that the critique of Isaac Newton’s work on differential calculus by the 18th century Irish philosopher, Bishop George Berkley, triggered the search for more formal methods in the foundations of differential calculus.*

Recently, you may have read or seen a few articles regarding various historical figures, such as Sir Isaac Newton, and the breakthroughs they made while in self-isolation during a pandemic. I want to focus on a contradiction in one of Newton’s discoveries while in isolation, in the field of Differential Calculus. A bit of historical context will help. In 1665 Newton was in his twenties studying at Trinity College, Cambridge, when the Great Plague of London struck – the final resurgence of the bubonic plague in England. Newton was forced to cease his studies at Cambridge and go into isolation at his home, Woolsthorpe. It was here that the literal and metaphoric ‘apple fell from the tree’. This isolation period of a year and a half was one of his most productive, in which he started to develop his theories of Newtonian Mechanics, Optics and Differential Calculus. In my previous article, “On the History of Proving False Theorems*”, *I argued that:

“[M]athematics is a human endeavour, sometimes stimulated by debate and controversy over the truth of theorems, despite its inhumanly pristine appearance in textbooks and the classroom.”^{[i]}

The development of Differential Calculus is a paradigm example of what I meant. It may be surprising to hear that Newton’s development of Calculus was not initially as ”pristine” as we may have thought. Rather, in 1734 the Philosopher George Berkeley in *The Analyst*^{[ii]} argued that Newton’s development of Calculus relied on a contradiction. So, in this paper I want to argue for two things: (1) The Philosopher Berkeley, a mathematical outsider, successfully argued that there was an error in the foundations of Differential Calculus as constructed by Newton; and (2) The identification of this error stimulated a fruitful debate amongst eighteenth-century British mathematicians over these foundations. This is, as I will suggest, what initially motivated the need for further developments in rigor for Calculus

The contradiction Berkeley identified centres around Newton’s notion of a *fluxion* of a *fluent* in his *Method of Fluxions*. A *fluent* was essentially a time varying function, e.g. *y* = *t*^{3} where the values of *t* are units of time, and a *fluxion *was the derivative of this function, e.g. *y’* = 3*t*^{2}. For simplicity, I will mostly drop Newton’s terminology and use conventional descriptions henceforth. To begin with, let me supply a rough approximation of how Newton found a derivative. Consider *f*(*t*) = *t*^{2}. Firstly, we need to find the ratio (or rate of change) of the formula between *t* and the increment *h*:

Then let *h* = ** o **where

**is an ‘infinitely small change in time’ (or infinitely small increment). That is,**

*o***is an ‘infinitesimal’, so importantly not the same as 0. Thus, we can do the following:**

*o*Berkeley, though an amateur Mathematician at best, was not satisfied with this reasoning. In *The Analyst*, he supplied a critique of two central applications of Newton’s *Method of Fluxions*, namely, the derivative of a product and the derivative of a power.^{[iii]} For brevity, I will only outline Newton’s reasoning, followed by Berkeley’s criticism of derivatives of powers, but as will be uncovered the criticism can be generalised to any derivative. Let us begin with Newton’s reasoning:

Consider *f*(*x*) = *x ^{n}* and demonstrate that

*f*′(

*x*) =

*nx*

^{n}^{-1}:

- Let
*x*increase by an increment of, such that it becomes*o**x*+;*o* - As
*x*is increased to*x*+,*o**x*is increased in due proportion until it becomes (^{n}*x*+)*o*;^{n} - Apply binomial expansion to (
*x*+)*o*to get:^{n} - The change of
*f*(*x*) =*x*between^{n}*x*and*x*+can be displayed as:*o* - So the rate of change is given by the ratio of two increments:
- Let both increments be divided by the common divisor
, which gives us:*o* - Since the
is a zero-increment it can be eliminated from the formula leaving us with:*o*^{[iv],[v]}*nx*^{(n-1)}

** Q.E.D**

Now Berkeley began by pointing out that it was initially assumed that “*x* hath a real increment, that ** o** is something.”

^{[vi]}This is what allowed Newton complete steps 1 – 6, since

**here is not 0 in the usual sense, rather it is something ‘infinitesimally small’. But then he continued to point out that Newton would implicitly do the following:**

*o*“I now beg leave to make a new supposition contrary to the first, *i.e.*, I will suppose that there is no increment of *x*, or that ** o** is nothing; which second supposition destroys my first, and is inconsistent with it, and therefore with everything that supposeth it.”

^{[vii]}

The crux above is that Newton supposed ** o** had some value, though infinitely small, in order to move through steps 1-6, but in order to arrive at the answer “

*nx*

^{n}^{-1}” he had to contradict his initial supposition about the value of

**and assume it had no value. But the increment**

*o***cannot be both an ‘infinitesimal’ and 0. Clearly, a contradiction – at least in a sense unacceptable to eighteenth-century mathematicians.**

*o*^{[viii]}

To generalise this argument to other derivatives, let us have a close look at Newton’s *Method of Fluxions*. First, the ratio of two increments of a pair of variables is taken, i.e.

As the increments grow smaller, the ratio comes closer to the desired limiting value, i.e. the rate of change at point *x*. Then, in order for the ratio to equal the limit, the two increments have to ‘vanish’. Herein lies the general worry that Berkeley identified. To find a ratio, the increments had to have some value, i.e. ** o** is something; however the increments also had to be of no value, i.e.

**is nothing, to get the ratio to achieve the desired limit.**

*o*^{[ix]}Berkeley’s point was simply that Newton could not have his cake and eat it. Due to this he made the following retort about Newton’s

**increments:**

*o*“They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?”^{[x]}

So the conclusion that Berkeley reaches in his critique of Newton’s *Method of Fluxions* is that either the limit, as a derivative, is not attained or it is attained through Ghosts of departed Quantities.^{[xi]} Neither is a satisfactory basis to rest Differential Calculus on.

However, as you read this you may be suspicious of Berkeley’s critique. Not because you think his reasoning is wrong, but because he possibly misinterpreted Newton’s *Method of Fluxions*. I do not deny that there is fuel for these suspicions. How could such a prestigious mathematician like Newton have missed such a seemingly simple error? Furthermore, in hindsight, the truth of Differential Calculus in indisputable. But it may interest you to mention the thoughts of an eminent mathematician here – Sir William Rowan Hamilton. In a letter sent to De Morgan he wrote:

“When your letter arrived this morning, I was deep in Berkeley’s ‘Defence of Freethinking in Mathematics’; … I think there is more than mere plausibility in the Bishop’s criticism … and that it is very difficult to understand the *logic* by which Newton proposes to prove [the derivative of a function.] His mode of getting rid of [** o**] appeared to me long ago (I must confess it) to involve so much

*artifice*, as to deserve to be called

*sophistical*; although I should not like to say so publicly.”

^{[xii]}

Other early supporters of Berkeley’s critique were Hermann Weisenborn and Colin MaClaurin. The number of discontent mathematicians, on the other hand, is far too large to list. So it goes without saying that Berkeley’s polemic gave rise to a controversy, one which raged on for nearly a century. During this time, key figures resolute on finding more rigorous foundations for the *Method of Fluxions* were Robin and MaClaurin and they made great leaps in formalising aspects of it. In spite of this, “All the eighteenth-century expositions of the foundations of the calculus – even the British – are defective”, though I will not argue for this point.^{[xiii]} Owing to these preceding developments, they paved the way for Cauchy’s definition of continuity in 1821: “The function *f*(*x*) is continuous between two given limits, if for each value of *x* that lies between these limits, the numerical value of the difference *f*(*x* + *h*) – *f*(*x*) diminishes with h in such a way as x to become less than every finite number.”^{[xiv]} This definition of continuity supplied the beginnings of a firm foundation that Differential Calculus required, though the reasons for this will be left for another time.

However, what is historically important here is Berkeley’s role in revealing the shaky foundations of Newton’s Differential Calculus. This engendered a controversy amongst British mathematicians either in agreement with Berkeley or in strong disagreement. The outcome of this was that Berkeley was proven correct in his critique and Newton’s use of an ‘infinitesimal zero-increment’ had to be eliminated and replaced with amendments.^{[xv]} So it is not unfathomable to think that a great mathematician, such as Newton, could have proposed a logical absurdity.^{[xvi]} Additionally, though Berkeley’s critique was entirely destructive, it helped influence the focus of attention in eighteenth-century mathematics in order to develop a more rigorous basis for Differential Calculus. Herein lies his contribution. Without such a contribution, the value of Newton’s Differential Calculus would have continued to be built on “Ghosts of departed Quantities”.^{[xvii]} No dispute would have arisen, thus no developments in rigor would be required. Therefore, unless someone else identified the contradiction, the indelible truth of Differential Calculus today would not be attained. More generally, sometimes deep errors need to be first committed and then critiqued before mathematical progress can begin to occur. Why should we expect mathematics to be flawless from the start?

### Adam Schroeder

MSocSc in Philosophy,

UNIVERSITY OF CAPE TOWN

[i] Schroeder, Adam. “On the History of Proving False ‘Theorems’.” Wisaarkhu. https://wisaarkhu.co.za/article/on-the-history-of-proving-false-theorems/ (accessed April, 2020).

[ii] Berkeley, George. *The Analyst: A Discourse Addressed to an Infidel Mathematician.* (London: Printed for J. Tonson in the Strand, 1734)

[iii] Ibid.

[iv] Ibid.

[v] Wisdom, J. O. “Berkeley’s Criticism of the Infinitesimal.” *The British Journal for the Philosophy of Science 4,* no. 13 (1953): 22-23.

[vi] Berkeley, George. *The Analyst: A Discourse Addressed to an Infidel Mathematician.*

[vii] Ibid.

[viii] It is important to note that there was no formal notion of contradiction at the time of Berkeley’s critique. Viz. “no formal notion of contradiction”, roughly, in the sense that it was not defined in a formal language. Nonetheless, mathematicians before this formalisation still used a notion of contradiction derived from Aristotle (cf. *Metaphysics* IV.3). A clear articulation of this idea comes from Leibniz: “a proposition cannot be true and false at the same time, and that therefore *A* is *A* and cannot be not *A*”. The contradiction Berkeley identified was under this definition and not acceptable by the standards of mathematicians, even before the first formal notion of contradiction appeared in Gottlob Frege’s *Begriffsschrift* (1879). So all that is required here for my argument is the historical norm held by mathematicians that “*A* is *A* and cannot be not *A*” which is exactly what Newton’s “** o**” violated. Leibniz, Gottfried Wilhelm.

*Philosophical Essays*. Translated and edited by Roger Ariew and Dan Garber. Indianapolis: Hackett, 1989, 321.

[ix] Wisdom, J. O.. “THE “ANALYST” CONTROVERSY: BERKELEY’S INFLUENCE ON THE DEVELOPMENT OF MATHEMATICS.” *Hermathena 29,* no. 54 (1939): 12.

[x] Berkeley, George. *The Analyst: A Discourse Addressed to an Infidel Mathematician.*

[xi] Wisdom, J. O. “THE “ANALYST” CONTROVERSY: BERKELEY’S INFLUENCE ON THE DEVELOPMENT OF MATHEMATICS,” 12.

[xii] Cajori, Florian. A History of the Conceptions of Limits and Fluxions Britain from Newton to Woodhouse. Quoting from R. P. Graves: Life of Sir William Rowan Hamilton, Vol. Ill, 1889 (Chicago, 1919): 91-92.

[xiii] Cajori, Florian. A History of the Conceptions of Limits and Fluxions Britain from Newton to Woodhouse (Chicago, 1919): 279.

[xiv] Cauchy, A. L. *Cours d’ Analyse*, Paris, 1821, Ch. II, § 2.

[xv] Wisdom, J. O. “Berkeley’s Criticism of the Infinitesimal,” 23.

[xvi] It should be noted that this contradiction was not intentional. Furthermore, historically it seems unavoidable given that the concept of continuity was unavailable to Newton. Without this concept he could not supply a rigorous notion of limit, since the concept of limit and continuity go hand in hand.

[xvii] Berkeley, George. *The Analyst: A Discourse Addressed to an Infidel Mathematician.*

You have not proven Berkeley’s criticism wrong. One can never make the mini increment zero…ever. And when it appears in the quotient eg 2x + dx the little dx cannot be removed by zeroing it. Yet it must be removed.

But how? Well it can be proven that that whenever we divide any numerator we ALWAYS get an average answer. Since the initial numerator is always a secant and not a tangent the quotient always contains the secant’s averaging error. The quotient is not yet a tangent that we seek in differential calculus. Since increasing or decreasing the size of the secant by increasing or decreasing the size of dx only changes the dx part of the quotient and never the 2x part, it can be argued that all the averaging error must reside entirely in the quotient’s dx. So we remove the dx not by making it zero but because it represents the secant’s averaging error leaving 2x as the unvarying tangent.

The indefinitely small increment h or dX or “o” can never be zero. Yet when it appears in the answer or differential coefficient (DC) it is made zero by “artifice or sophistry” per Hamilton. Thus the DC of X^2 is 2X + dx and the dX is conveniently zeroed when we were earlier assured that it CANNOT be zero. No amount of “twisting and turning” can make dX or h or “o” a zero.

Yet we HAVE to get rid of dX when it appears in the DC but without making it zero.

So how? There is a perfectly logical way called elimination of the “averaging error”.

All divisions give an AVERAGE quotient. Thus dividing 100 by 20 = 5. All the divided numbers are 5, an average. But when differentiating a curve like X^2 all the quotients are NOT the same but vary. Division only gives the average quotient which perforce has the averaging error. Sticking to the function X^2 the larger the secant we use meaning the larger the increment or size of dX the greater the averaging error and vice versa. So decreasing the increment or size of dX decreases the averaging error. But here is the crucial point: No matter how much we vary the increment dX the quotient, which is always a bi or polynomial has a nonvariant and variant terms. It can be argued that ALL the error is in the variant term and none in the nonvariant term. So if the DC is 2X +dx the 2X does not vary with variation of the increment dX. All the variation is in the DC’s term dX.

This term though never zero represents all the averaging error and should be removed keeping the non varying, error free term 2X. Thus the dX term in the DC is eliminated NOT because it is zero but because it has all the averaging error.