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On Teaching Calculus

Almost universally, when Calculus is taught to modern students, we preface the entire subject by introducing those students to a concept known as a “limit.” The reason for this, historically, was to ensure that mathematics was taught in a rigorous and well-defined manner. When Leibniz (and, independently, Newton) first developed methods for performing calculus, the concept of a limit was nowhere to be found. However, the tool which these men did utilize in their work was something which they had not rigorously defined, at the time. Newton called it a “fluxion” and Leibniz called it a “differential,” but the concept was the same: a number which was not zero, but which was so small that adding it to any Real number did not yield a different Real number.

Many other mathematicians and philosophers of the time rightfully balked at the notion. It seemed entirely ludicrous. Bishop George Berkeley famously scoffed at Newton, asking if his fluxions were “the ghosts of departed quantities.” However, it was quite plain that the mathematics which Leibniz and Newton presented worked. When the results which could be found from the methods of Calculus were able to be confirmed using other methods, they were found to be accurate and true. Indeed, the Calculus was such a powerful tool that even most mathematicians and philosophers who recognized its flaws continued to utilize it in their work. Many began searching for some way to make the Calculus just as rigorous as the rest of mathematics. These efforts culminated in the work of Karl Weierstrass, who found a way to base Calculus upon a different tool. Instead of the Newtonian “fluxion” or the Leibnizian “differential,” Weierstrass gave mathematics a well-defined notion of the limit.

It is Weierstrass’ method of limits which is still taught, even to this day, in nearly every Calculus textbook in the world; but perhaps it is time to abandon this notion and return to the concept which Newton and Leibniz pioneered.

In the 1960’s, a mathematician named Abraham Robinson developed a rigorously well-defined number system called the Hyperreal numbers. This number system included numbers which are larger than any given Real number– known as “infinite” or “unlimited” numbers– as well as their reciprocals, which are greater than zero but nonetheless smaller than any real number– known as “infinitesimals.” Robinson explicitly noted that his development of the Hyperreals came out of a desire which he had for better understanding Leibniz’s thought processes. Indeed, the infinitesimals of the Hyperreal numbers look very much like the “fluxions” and “differentials” of that early Calculus. In 1986, H. Jerome Keisler wrote a textbook for the subject, Elementary Calculus: An Infinitesimal Approach, in which he provides a method for teaching Calculus without the need for limits, while still maintaining the rigor desired in mathematics.

Unfortunately, Dr. Keisler’s work has not yet gotten much of a foothold in the educational system. The method of limits has been taught for so long that it would be exceedingly difficult to displace it. However, there are some very distinct pedagogical advantages in Keisler’s approach which may make the whole ordeal well worth the effort.

Let’s look at a simple example. One early Calculus problem with which every student is presented is to find the derivative of the function y=f(x)=x^2. For those who don’t remember, the derivative of a function, f'(x), tells us how much the value of that function changes with respect to a change in the value of x. So, let’s say that the value of y increases by some amount which we will call \Delta y when the value of x is increased by some amount \Delta x. Algebraically, we would write this as y+\Delta y=(x+\Delta x)^2, for the equation we are discussing. We can then take this new equation, and solve it for the value of \frac{\Delta y}{\Delta x} as follows:

  1. y+\Delta y=(x+\Delta x)^2
  2. \Delta y=(x+\Delta x)^2-y
  3. \Delta y=(x+\Delta x)^2-x^2
  4. \Delta y=x^2+2x(\Delta x)+\Delta x^2-x^2
  5. \frac{\Delta y}{\Delta x}=\frac{2x(\Delta x)+\Delta x^2}{\Delta x}
  6. \frac{\Delta y}{\Delta x}=2x+\Delta x
  7. f'(x)=2x

It is these final three steps which the mathematicians of Newton and Leibniz’s day found to be offensive. According to the Calculus, the derivative was the function which results from \frac{\Delta y}{\Delta x}. If we are to say that \frac{\Delta y}{\Delta x}=\frac{2x(\Delta x)+\Delta x^2}{\Delta x} actually has a value, then it must be true that \Delta x \neq 0, because division by zero is undefined. However, if 2x+\Delta x=2x, then it must be true that \Delta x=0, because that is the only additive identity. Thus, we are left with a contradiction if we claim that f'(x)=\frac {\Delta y}{\Delta x}.

Later mathematicians, culminating in Weierstrass, resolved this issue by redefining the derivative to be a limit as the change in x approaches zero. Specifically, they said that f'(x)=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}. Of course, this raises a new question: what, precisely, is a limit? Well, if f(x) is defined on an open interval about x_0, except possibly at x_0 itself, then \lim_{x\rightarrow x_0} f(x)=L if, for every number \epsilon >0, there exists a corresponding number \delta >0 such that for all x it is true that 0<|x-x_0|<\delta implies that |f(x)-L|<\epsilon. Needless to say, this is a fairly complex idea, which is why a large amount of time needs to be spent on teaching students how to properly find and evaluate limits.

Keisler’s resolution to the derivative problem we presented is somewhat simpler, and quite a bit more intuitive. In his Elementary Calculus, the \Delta x in the equations above is defined to be a non-zero infinitesimal. The derivative is then defined to be f'(x)=st(\frac{\Delta y}{\Delta x}) where st() means “the standard part of…” The standard part of a finite Hyperreal number, a, is the Real number which is infinitely close to a; and two numbers are infinitely close if they only differ by an infinitesimal value. Looking again at Step 6 from our work above, we had the expression 2x+\Delta x. Since we know that \Delta x is infinitesimal, we know that 2x+\Delta x is infinitely close to 2x. Thus, for any Real number, x, we can see that f'(x)=st(2x+\Delta x)=2x.

 From a pedagogical standpoint, it would seem that Keisler’s method is superior. Hyperreal variables can be manipulated algebraically in exactly the same way students are already familiar with manipulating Real variables. The standard part function is quite a bit easier and more intuitive to learn than the limit function. The method is inordinately closer to the original ideas which created Calculus, in the first place, and it is just as rigorous a treatment as is the method by limits. Keisler and others have reported that they’ve seen students take to the material more easily, in this manner. Perhaps the time has come to leave off of the use of limits, and to return to the method of infinitesimals for teaching Calculus.

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