Calculus (IV): How To Replace Limits With Infinitesimals?

While mainstream calculus uses the concept of limits, an alternative version differs to replace limits with infinitesimals. And it is just as effective. Some mathematicians even report that this alternative version solves certain problems faster as compared to the mainstream version.

In my essay on the notion of limits, I had mentioned that I will be covering the infinitesimal calculus in a separate essay. So, here we are. This also happens to be the fourth entry in my calculus series.

I will begin by giving you a brief historical journey of infinitesimal calculus. Following this, I will proceed to cover the intuitive difference between calculus using infinitesimals and limits.

Finally, I will touch upon the advantages and disadvantages of using this alternative version practically. Without any further ado, let us begin.

This essay is supported by Generatebg

What are Infinitesimals?

It is easy for any modern calculus practitioner to think that calculus has always used the notion of limits. But this was certainly not the case. In fact, the inventors of calculus never used limits; they used infinitesimals.

Consider the concept of fractions. Most fractions we deal with are finite fractions of finite numbers. However, there exist infinitely small fractions such that the number 1, for example, may be divided into infinitely many parts.

These infinitesimals are such infinitely small fractions that are infinitely closer to zero but are NOT zero. Swiss mathematician Johann Bernoulli had the following intuitive explanation about infinitesimals:

“…if a quantity is increased or decreased by an infinitesimal, then that quantity is neither increased nor decreased.”

— Johann Bernoulli.

A Brief History of Infinitesimal Calculus

Newton, one of the inventors of calculus (Leibniz being the other one) called derivatives “fluxions”. The eighteenth-century British philosopher and Bishop George Berkeley had the following to say about fluxions:

“And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?”

— George Berkeley.

And it was not just Berkeley. For two-hundred odd years, the majority of the mathematical community rejected the notion of infinitesimals/fluxions because it sounded “sketchy” and not mathematically well-defined.

As most established mathematicians and people of science opposed the usage of infinitesimals, there remained a minority of reputed mathematicians/scientists who supported it. Edmond Halley, the astronomer after whom the famous comet is named, was Newton’s close colleague and a strong supporter.

The American mathematician Charles Peirce strongly supported the notion of infinitesimals as well. He had the following to say about it.

“The doctrine of infinitesimals is far simpler than the doctrine of limits.”

“…As a mathematician, I prefer the method of infinitesimals to that of limits, as far easier and less infested with snares.”

— Charles Peirce.

In 1960, Abraham Robinson of Yale University shocked the mathematical world by developing a rigorous system of analysis that included infinitesimals and infinite numbers (collectively known as hyperreal numbers) consistently with existing mathematical axioms. This meant that infinitesimals were vindicated once and for all!

This developed into a field of its own and is now known as “nonstandard analysis”. You can read more about it in this essay.

In mathematics, analysis is a term that refers to calculus and advanced calculus. If you indeed choose to use infinitesimals instead of limits, nonstandard analysis would be my recommendation to you.

---

How to Replace Limits with Infinitesimals?

Consider the following function:

y = f(x) = sin(x)/x

Let us say that we are trying to answer the following question:

What happens when x approaches zero?

If we simply plug in zero, it leads to 0/0 which is undefined. If you are interested in learning the mathematical complications behind this, check out my essay on this topic.

There are several ways in which we could handle this. Usually, Taylor series is a pretty flexible option. However, for the sake of ease, I will be using a simpler approach. Let us begin by plotting sin(x) in the interval of x = -1 to x = +1:

Notice how the plots of y = x and y = f(x) = sin(x) align as the value of x gets closer and closer to zero. It turns out that this wiggly alternating trigonometric function can be approximated close to zero using a simple straight line:

So, we can actually ‘approximate’ our original function very close around zero as follows:

y = f(x) = sin(x)/x ~ x/x = 1

What Does the Limit Do?

What we have done here is convert a complex model into a simpler model that just provides sufficiently accurate results for our needs. In calculus, limits achieve this by dynamically recreating a simpler model at a resolution that is just beyond our tolerances for error.

My goal here is not to actually show you how to solve this problem using calculus, but just give you the intuition for now. We will have enough time to look at the calculus side of things in later essays.

Imagine that you are measuring the length of a curve using a ruler. It is sufficient if you measure it using a minimum scale of 1 centimetre. This is because the highest resolution anyone can measure in your world is 1 centimetre.

But for this task, the limit equips you with a special ruler that is capable of measuring with a scale of 1 millimetre. This will not be the mathematically perfect measure by any means.

But still, no one in your world of 1 centimetre scales will ever be able to know or detect any deviations. Having said this, the prime focus of this essay is not on limits, but on infinitesimals. How would they fare?

What Do Infinitesimals Do?

You see, both limits and infinitesimals attack the same problem: the issue of error tolerances around zero. “Why is this of significance?”, you ask?

Well, imagine that we measure your weight now and one second away from now. If we use your normal weighting scale, the difference is likely to be zero.

However, if we use a hyper-precise scale capable of measuring differences of up to atomic masses, it is very unlikely to be zero. This is because you continuously perspire, cells leave your body, molecules attach and detach to your skin, etc.

So, what is the actual difference between your weights between two seconds? Zero or not zero? Well, it depends on the scale you use and your requirement.

Zero is practically relative, and calculus takes advantage of this relativity to compute complex problems in simpler terms. We just saw how limits dynamically operate outside of your error tolerance range at a higher resolution.

Infinitesimals operate in a completely different dimension altogether. They create a model of the problem and solve it in the hyperreal number plane.

No human measurement application can ever require a scale with better precision than that offered by the hyperreal dimension (by definition, pretty much).

Once the problem is solved in the hyperreal dimension, the solution is transported to your problem’s finite dimension and the world turns as usual.

---

How to Choose Between Limits and Infinitesimals?

Before we answer that question, let me quickly summarise what we have so far:

1. Limits solve complex problems dynamically at a higher resolution just out of the reach of our error tolerances.

2. Infinitesimals transport the problem to the hyperreal dimension, solve it there, and then transport the solution back to the problem’s original dimension.

So, the choice between the two is quite simple: it depends on your preference. Do you prefer talking about values that converge on a limit or about quantities that are infinitely small yet not zero?

Practically speaking, because of the rough history behind infinitesimals, they come with baggage. Robinson vindicated infinitesimals in the 1960s. Because of this reason, the majority of the reputed texts on calculus do not even feature infinitesimals.

This means that the options for reference on infinitesimal calculus are very limited. While we are on the topic of references, I have read that “Nonstandard Analysis” by Martin Davis and Reuben Hersh is a good reference on the topic.

When I first learned calculus in school, I did not even know that infinitesimal calculus existed. Now, I am glad to know that it exists. But because of practical reasons, I still stick with limits.

We do not live in the twentieth century any more. So, if you really wish to give infinitesimals a try, no one will try and stop you. I often hear from supporters that infinitesimals are more intuitive than limits.

Having said this, because of the aforementioned practical reasons, I will be sticking with limits in my future essays on calculus.

---

Reference and credit: Martin Gardner.

If you’d like to get notified when interesting content gets published here, consider subscribing.

Further reading that might interest you:

• What Happens When Projective Geometry Meets Information Theory?
• How To Perfectly Predict Impossible Events?
• How To Actually Solve The Königsberg Bridge Problem?

If you would like to support me as an author, consider contributing on Patreon.