Infinite Regress: How To Really Understand It? - A whiteboard illustration showing a series of Russian dolls on top, where a smaller doll recursively appears out of each bigger doll (left to right). At the same time, there is an image of a square-based fractal below. In between these two images, the phrase "Infinite Regress" is presented in a box. Both of these concepts seem to be linked by infinite regress.

The notion of infinite regress has long been a source of arguments and controversies in the fields of philosophy, logic, and mathematics. While mathematicians have been able to work out issues with infinite regress over the years, logicians and philosophers still seem to be divided on this topic.

Furthermore, when we look at mainstream interpretations of infinite regress, we see great creativity, awe, and fascination come out of them (you can find examples later in the essay). But we also see deep and divisive existential questions pop up.

This is where folks from the worlds of theology, artificial intelligence, etc., visit the topic as well. In short, this is a highly fascinating topic with a wealth of contributions from many fields.

Consequently, it has plenty of real-life applications as well. Why don’t we start by taking a look at some of the mainstream takes on infinite regress?

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Mainstream Interpretations of Infinite Regress

Here is an interesting two-sentence story that gives you an intuitive feel for the notion of infinite regress:

“Might I offer you some tea, Mr. Butler?”, asked the butler, whose name was also Butler.

“Why don’t we ask Mr. Butler?”, answered Mr. Butler rhetorically.

Of course, this story does not necessarily lead to an infinite regress in a strict sense. But it has the potential to do so. You can also sense a certain recursive logic at play here. In fact, numerous plays and artists have taken advantage of recursive logic and infinite regress in the past.

E. Nesbit wrote a short story titled, “The Town in the Library in the Town in the Library”. In Lewis Carroll’s “Alice Through the Looking Glass”, as Alice dreams about the Red King, the Red King dreams about Alice.

In Henry Hasse’s story “He Who Shrank”, a scientific experiment from a higher cosmic dimension than ours starts shrinking a man recursively. Each time he shrinks, he lands in a new sub-universe. After shrinking several iterations, he arrives in Cleveland for just long enough to recite his story.

He wonders how long this will go on and hopes that all of these dimensions are interconnected by some sort of a cosmic loop. This would enable him to eventually get back home. Before long, he shrinks and vanishes again.

Some of the most prominent artistic works of infinite regress that I know of are from M.C. Escher. Just google the phrase “M.C. Escher infinite regress” and see the fascinating images for yourself. I would have posted a few of these images here in this essay had they been in the public domain. But trust me, the image search is worth it!

That’s enough mainstream take on this topic for now. Why don’t we move on to more nuanced interpretations of infinite regress?

Intellectual Interpretations of Infinite Regress

When I was a child, I asked my parents the following innocent question:

“Who made our world with blue skies and vast beaches?”

My parents answered: “God did.” I thought for a moment and then replied with yet another innocent question:

“Who made God?”

Even though this was an innocent child’s question, it lies at the core of the intellectual discussion on infinite regress. Often, we define a stopping case to avoid infinite regress. In this case, my parents needed “God” (without them realizing it) probably to avoid an endless loop of questions.

In fact, this is the same logic that modern mathematics uses for a nuanced computer science/programming concept known as recursion. I have covered the technical details of this concept in my essay on how to really understand recursion. If you are interested, check it out.

Moving back through history, Greek philosopher Agrippa famously argued that nothing, even in mathematics, can ever be proved. This is because, according to him, every proof must be proved valid, and this would mean that its proof must be proved, and so on ad infinitum.

German theologian Rudolf Otto used the term mysterium tremendum (it sounds like a Harry Potter spell, I know) to describe the ultimate limitations of life and things we might never understand.

Bertrand Russell had the following famous mathematical take on infinite regress:

“The man who says ‘I am telling a lie of order n’ is telling a lie, but of order n+1.”

In my opinion, the most inspiring use case for infinite regress came from Kurt Gödel in his famous incompleteness theorems. His work eventually became an indispensable pillar of modern logic and mathematics.

After this rather engaging and serious discussion, I’d like to explore some fun applications of infinite regress next.

How to Have Fun with Infinite Regress?

Why don’t we play a fun little game? Let us start with a square, chop its sides off at the middle-third of their lengths respectively and extend the sides outward as follows:

Infinite Regress: How To Really Understand It? — A square on the left transforms into a cross-stitch curve on the right. The middle third of each side is extended forward to create three more sides, each one-third the length of the original sides.
Cross-stitch curve after iteration 1 — Geometry illustrated by the author

The curve we get as a result of this activity is known as the cross-stitch. Now, the beauty of the cross-stitch is that we can recursively erect newer sides by extending the middle third of each side (also the newer ones previously generated). Here is what the geometry looks like if we do it for one more iteration:

Infinite Regress: How To Really Understand It? — The same logic from before is applied once more to iteration 1 of the cross-stitch curve. The second iteration of the cross-stitch curve appears like diamond with 8 square holes, 2 touching each of the diamond’s sides.
Cross-stitch curve after iteration 2— Geometry illustrated by the author

And we could keep going. Below, you can see the next two iterations from left to right:

Infinite Regress: How To Really Understand It? — The third iteration of the cross-stitch curve is presented on the left, while the fourth iteration is presented on the right. As the iteration count increases, the curve looks more like a fractal art resembling some sort of a Mandala created by a human artist.
Cross-stitch curve after iteration 3 and 4— Geometry illustrated by the author

Now, let us say that we started with a square that occupied an area of 8 square units. After applying the recursive cross-stitch algorithm through infinite iterations:

1. What would be the area enclosed by the cross-stitch curve?

2. What would be the perimeter of the cross-stitch curve?

Spoiler Alert:

Beyond this section, I will be revealing the answers to the above questions. So, continue reading ONLY when you are ready to learn/discuss the answers.

The Answers to the Cross-Stitch Problem

At the limit of infinity, the cross-stitch curve is bound by double the original area. In this case, we would arrive at 16 square units.

On the other hand, the perimeter is unbound at infinity, which means that the perimeter diverges to an infinite length.

An interesting property of this curve is that as the iteration count goes up, it appears as if the sides run diagonally. But in actuality, the entire geometry is only made of vertical lines and horizontal lines.

Another interesting property of this curve is that it belongs to a larger family of curves known as the Snowflake Curves. Infinite regress and recursion lie at the heart of such curves.

Final Comments

Westarted our journey with mainstream interpretations of infinite regress such as in short stories and Broadway plays. Then, we slowly moved into more intellectual takes on the topic in the form of philosophy, theology, logic, and mathematics.

Finally, we had some fun applying infinite regress through recursive logic to generate a geometrical curve that has artistic visual appeal. I actually generated those figures using a short program that I wrote.

I am working on a web version of this code so that readers may personally play with the model if they wish to. I will update this essay with more information as soon as I have completed this.

For now, I hope you had fun reading this essay and could take a thing or two with you.

Reference and credit: Martin Gardner.

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