What Really Happens When You Invent Infinite Infinities?

‘Infinite Infinities’ is a crazy-sounding concept. Most people have issues wrapping their heads around the concept of infinity. Why are we then talking about an infinite number of infinities? Let me clarify. In my earlier article, Does Division By Zero Really Lead To Infinity?, I argued that we cannot treat zero just like any other number, especially when it comes to division.

If we treat zero just like any other number, firstly, 1/0 leads to infinity, and then, 2/0 also leads to infinity, and so on. This would imply that 1 = 2. Similarly, we could prove that any number is equal to every other number. All of this is, of course, absurd.

My point of travelling down this path was to show that 1/0 cannot be infinity. In the original article, I argued that division by zero leads to an ‘undefined’ result. But then, the following question arises: what if 1/0 leads to an infinity that is different from the one that arises from 2/0? This would lead to an infinite number of infinities.

Well, even considering the validity of that question, division by zero leads to other contradictions which makes it difficult to justify any conclusion other than an ‘undefined’ result. However, the concept of infinite infinities does not require division by zero. Infinite Infinities actually prevail! In order to understand what makes this possible, we need to start with the story that started it all.

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The Story of the Infinity and Anti-Infinity

Yes, you read that right! What could be anti-infinity? The story begins with the Greek Mathematician, Archimedes. In his famous letter to Eratosthenes, The Method of Mechanical Theorems, Archimedes defined a number x as infinite, if it satisfies the following condition-set: |x|>1, |x|>1+1, |x|>1+1+1,… (where |x| represents the absolute value of x).

He then went on to define anti-infinity as a number that is non-zero and if it satisfies the following condition-set: |x|<(1/1), |x|<(1/(1+1)), |x|<(1/(1+1+1)),…(where |x| again represents the absolute value of x).

This concept of anti-infinity is necessary for the existence of infinity. It is analogous to shadows that come to existence because of light (when blocked by objects). Later on, the concept of anti-infinity came to be formally known as Infinitesimals. Throughout history, the concept of infinitesimals has played a strong role in calculating areas, among others. Before we move onto infinite infinities, it is useful if we look at the brief history of infinitesimals.

Infinitesimals Through History

Our story jumps forward in time by many centuries, and moves to Johannes Kepler, a key scientific figure from the 17^th century. Kepler was a German astronomer, mathematician, philosopher, and music writer, among others. In short, he was a polymath.

He took a specific interest in planetary motion and wanted to calculate how planets move about in the sky. He eventually figured out that planetary orbits are elliptical in nature, which led him in pursuit of calculating elliptical areas. Back then, there was no formula available for the area of an ellipse, so he had to work around it. He tried to do this starting from the area of a circle and happened to further develop the practical application of infinitesimals. For ease of understanding, I’ll explain the concept using a simplified example.

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Calculating Areas Using Infinitesimals

Consider a situation where you have a device that is capable of cutting tiles into rectangles and/or triangles of varying sizes. Using this device, and some raw tiles, you need to fill in the following floor space with tiles.

One way to do this would be to split the floor space as shown below, where the blue line indicates a split.

However, do you notice the region circled in purple? That part of the floor space is a bit curved. And your tile-cutting device is only capable of producing rectangles and/or triangles. So how shall we proceed? How about introducing yet another section like this?

It appears that we have now arrived at 3 sections: two triangles and one rectangle. That’s no problem, right? Hold your horses! There is one small detail that we have missed here. Let me clarify:

Take a note of the small region that is left unfilled by the triangular tile. This is pretty much the problem that Kepler was facing. He was trying to calculate the area of non-regular geometries (an ellipse in his case) using areas of regular geometries (like circles, triangles, rectangles, etc.). In the end, he proposed that if one keeps dividing the geometry finer and finer, the approximation error would turn out to be negligible for practical purposes. He was practically doing something like this (except that he did it for an entire circle):

If you have come this far, you might be wondering what all this has to do with infinite infinities. Let’s get to that next.

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The Precursor to Infinite Infinities

Kepler went a step further to show that if the number of divisions increases, the curve would be divided into smaller and smaller infinitesimal segments. And if the infinitesimal segment is small enough, the curve would become a straight line, and this would essentially eliminate the error. It is important to note that mathematically speaking, there is a world of difference between practically negligible error and eliminating the approximation error. Later on, Gottfried Wilhelm Leibniz and Isaac Newton used a similar definition to invent calculus (each, individually). Even though this notion of infinitesimal was practically useful, many mathematicians did not buy it. They felt that the concept of infinitesimals was not rigorous and consistent with the mathematical rules of the time.

They faced the same problem (among others) that we saw at the beginning of the article. If we define an infinitesimal as ϵ, we end up with the following situation:

This led to huge controversial discussions, and the mathematicians ended up banning the concept of infinitesimals. It seemed like the concept of infinitesimals was practically useful, but not mathematically acceptable.

Infinite Infinities

Fast forward to today, not only has the concept of infinitesimals come back with a vengeance but has led to an entire branch of mathematics known as Nonstandard Analysis. Following the footsteps of Leibniz, 20^th century mathematician Abraham Robinson led the charge in perfecting the mathematical definition of infinitesimals to be consistent and rigorous.

The result was that a new number system called the hyperreal number system was born. In this system, an infinite number of infinitesimals, and consequently, an infinite number of infinities coexist with real numbers. One of the notable features of the hyperreal number system is that it allows for mathematical operations to be performed on infinities and infinitesimals just like on real numbers.

Having said this, the story connecting Leibniz, Newton, and Robinson is way too good to compress into this article. I will cover that, and the 20^th century mathematical treatment of infinitesimals in a follow-up article. For now, I’ll conclude by saying that the concept of infinite infinities and infinitesimals do really exist and have proved their right to exist in the mathematical world.

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Further reading that might interest you: What Exactly Is Zero Raised To The Power Zero? and How Many Decimal Digits Of Pi Do We Really Need?