What is Infinity?

In mathematics, infinity (?) is an abstract idea representing a state of limitless extent, where a quantity or process can grow endlessly large or small without reaching a specific limit. Although it may seem abstract, Hermann Weyl, a German mathematician, emphasized its significance by stating:

> Mathematics is the science of the infinite.

I can illustrate the concept of infinity in several ways, but my favorite is through a number line. Consider the illustration below:

Imagine a number line. If we zoom in between the values of 3 and 4, we discover an even smaller segment within that distance. We can divide the space between 3 and 4 into ten equal parts, each increasing by 0.1. Further zooming in reveals that this process can continue indefinitely. That, dear reader, is infinity—an endpoint that remains forever out of reach for humanity. To be more precise, this example demonstrates uncountable infinity.

For those unfamiliar with the concept, there are various types of infinity. This is the first thing I want you to keep in mind.

Countable Infinity

Imagine you possess a magical bag that endlessly generates toys. You take out toys one by one: teddy bears, toy cars, action figures. No matter how long you keep pulling out toys, you can always count them sequentially—1, 2, 3, and so forth. This scenario illustrates countable infinity, where counting is possible because the toys come in an orderly fashion.

Uncountable Infinity

Now, picture a magical coloring box filled with a crayon that can produce any color imaginable. You start creating colors like red, blue, and green, but you can also mix colors, such as a hue that's half-red and half-blue! As you experiment with colors, you realize there are so many that you can’t assign each one a number. This exemplifies uncountable infinity—an overwhelming abundance of colors that resist numerical categorization.

While my examples may not be entirely precise, the aim is to keep things straightforward and digestible.

Understanding Arithmetic with Infinity

It’s crucial to grasp that basic arithmetic operations like addition, subtraction, multiplication, and division break down when applied to infinity. For instance, adding infinity to infinity does not yield 2 infinity; it simply remains infinity. The same principle applies to subtraction and multiplication.

The Physical Limitations of Infinity

Lastly, remember that nothing physical can truly be infinite; no tangible quantity can be infinite. This brings us back to our initial question: if nothing physical can embody infinity, how can the universe be infinite? It appears paradoxical. Despite the finite nature of the elements comprising the universe, many eminent scientists assert its infinite nature.

Don’t fret over this conundrum; I’ve spent countless nights pondering it and watching numerous videos on infinity.

To grasp this concept, let’s consider creating infinity ourselves.

Creating Infinity from the Finite!

We will explore one of the most beautiful offerings of mathematics: fractals.

Take a moment to watch this video by Mathigon:

Fractals are unique shapes that retain their appearance no matter how closely you examine them or how far you zoom out. Imagine a picture resembling a puzzle piece, where inspecting one section reveals the entire image. This notion contradicts the third point I mentioned earlier: finite resources cannot create something infinite. Yet, as the video zooms in on the fractal image, it seems to continue infinitely.

The computer generating this fractal has finite memory, processing power, and other physical constraints, but it nonetheless creates an infinite visual marvel.

So, how does this occur? Through rendering and recursion. The video reuses the same resources repeatedly, displaying only the portion currently observed, thus crafting an illusion of infinity. We cannot ascertain whether the previously viewed parts still exist; we trust that they do because the program ensures they are rendered.

If this seems complex, the next example will clarify things.

The Painter’s Paradox

Imagine being a painter assigned to color the inner walls of a room with infinite length before an inspection. What do you do?

Take a moment to think about it.

Many might guess the answer, but allow me to elaborate on the scenario:

1. The inspectors will only move from one end to the other.
2. The paint is magical and can be removed by saying “off” for reuse.
3. You can work at the speed of light.

While this may be amusing to physics enthusiasts, bear with me.

The painter can cover a finite distance with paint. As the inspectors move forward and reach the midpoint, the painter removes paint from the area they’ve traversed to cover the next section. This process continues infinitely, allowing the painter to use finite resources to paint an infinite room.

The intriguing part? The inspectors perceive a painted room, oblivious to the fact that the next area they will encounter is unpainted because, by the time they reach it, it is painted.

The stipulation that the inspectors can only move forward is essential. Their linear movement mimics a concept we’re all familiar with: time. Time flows in one direction, and our only option is to advance.

The Climax

Here’s the revelation: Time creates the illusion of infinity from finite resources, making the universe appear infinite by continuously recycling its limited elements. While this idea may initially seem counterintuitive, it gains plausibility when we acknowledge that the patterns and components of the universe are finite.

Although our perception may suggest infinity, it could very well be an illusion. While I lack mathematical proof for this assertion, various observations lend credence to the idea. Numerous patterns recur in nature, such as the Golden Ratio, waves, and fractals. Recently, the James Webb Telescope captured an image resembling a question mark in space!

How is it that something potentially existing for billions of years in the universe resembles the symbol we use for a question mark? The answer lies in human intelligence, creativity, and the interconnectedness of everything, achieved through reuse and repetition.

Our difficulty in grasping the concept of infinity stems from our linear existence. We cannot return to the past to verify if things remain as we left them, and when we venture into the future, we encounter the same reality we live in. Even if we could travel back in time, who’s to say the universe wouldn’t simply recreate the same reality using finite resources?

We are limited beings, and our capabilities are finite. To comprehend infinity, it may be best to view it as an illusion. The infinity we perceive could just as easily be an illusion. True infinity exists, and while it should be common sense, we are likely never meant to fully grasp it. As with everything, our minds also have limitations.

Philosopher Colin McGinn proposed that all minds experience a kind of “cognitive closure” regarding specific issues. Just as animals cannot comprehend concepts like prime numbers, humans too have boundaries in their understanding.