Berkley’s Critique of 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 = t³ where the values of t are units of time, and a fluxion was the derivative of this function, e.g. y’ = 3t². 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) = t². 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 o 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:

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) = xⁿ and demonstrate that f ′(x) = nx^n-1:

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

“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 o 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.^[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. o is nothing, to get the ratio to achieve the desired limit.^[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 o increments:

“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.