Musings on the Impossible, part 2
Infinity—what is it good for?
Let’s be clear from the start: in this Universe, infinity—an endless progression of things—does not exist. According to today’s scientific understanding, the observable Universe contains about 3.28 x 1080 particles. This is a stupendously large number, but not infinite. Nor is (almost) empty space stretching out forever, as the size of the observable Universe is a sphere with a radius of 13.8 billion light years. But what about the parts of the Universe that have expanded out of reach (due to light speed limitations)? Estimations vary, but the Bayesian average of all the models puts it at 7 trillion light years across. And even if the Universe expanded at light speed during inflation, its size would be 1023 light years. Another stupendous, almost unimaginable number, but not infinite.
So, according to the scientific consensus, the Universe is large, but not infinite. (Or is it, but in another way? More on that later.) Yet the concept of infinity plays a large role in mathematics (which effortlessly discerns different types of infinity) and infinity sometimes crops up in formulas that scientists use to determine an aspect of reality.
The earliest recorded idea of the concept of infinity dates back to the early Greek philosophers (some six hundred to three hundred years BC), beginning with Anaximander, while Aristotle, Euclid and others possibly had a ‘horror of the infinite’, or were skeptical about it. And sometimes the concept of infinity arises in unexpected places, such as Zeno’s paradox of Achilles and the tortoise.
It goes like this: Achilles is in a footrace with a tortoise, Achilles gives the tortoise a head start—say 10 metres—while the tortoise goes at half of Achilles’s speed. Then as Achilles runs 10 metres, the tortoise is 5 metres ahead; Achilles runs 5 metres, the tortoise is 2.5 metres ahead, etcetera ad infinitum. The paradox assumes that no matter how close Achilles comes, the tortoise always maintains an infinitesimal lead, so it is impossible for Achilles to overtake it.
This paradox only works if that series of approaches is indeed infinite, but in our reality it is not. In our reality the smallest possible distance is the Planck length, which is about 1.616 x 10-35 metres. So, at some point, Achilles approaches the tortoise at 2 Planck lengths, and as Achilles crosses those two Planck lengths, the tortoise moves one Planck length ahead. Now Achilles has approached the tortoise up to one Planck length. A shorter distance is not possible, according to our current understanding of the laws of physics.
Therefore, while the tortoise covers another Planck length (it cannot cover a shorter distance), Achilles covers 2 Planck lengths, and overtakes the tortoise. As such Achilles overtaking the tortoise (or any moving object overtaking a slower moving object) can be seen as a refutation of infinity.
Another example, π is one of the most well-known expressions of an ‘infinite’ number through the formula:
C = π d
Where C is the circumference of a circle and d is the diameter. Infamously, the mathematical constant π has an infinite number of decimals. So is it proof of the concept of infinity? I don’t think so, as there’s no such thing as a perfect circle—or a perfect sphere—in physical reality. The circumference of the circle, or the surface of the globe cannot be infinitely smooth due to the above-mentioned Planck length limitation. Therefore, the mathematical constant π does not need to have an infinite number of decimals to describe the circumference (or globe surface). In our reality, π has a limited—even if very long—number of decimals.
One might wonder if getting rid of the infinitely small—by way of the Planck limits of quantum mechanics—would also get rid of the infinitely large. While Carl Friedrich Gauss quipped that “Infinity is merely a way of speaking” and Max Tegmark explains infinity’s lure in “Infinity is a Beautiful Concept—And It’s Ruining Physics”, many physicist and mathematics are reluctant to let go of it.
To wit, a current interpretation of Richard Feynman’s path integral proposes that “. . . Our Reality May Be a Sum of All Possible Realities” (article in Quanta Magazine). This interpretation helps explain the results of the infamous double slit experiment.
However, Dutch theoretical physicist Renate Loll is not the first to consider the infinite shadow of quantum mechanics. British physicist David Deutsch argued that the double slit experiment is proof of the multiverse—Hugh Everett’s many-worlds interpretation of quantum mechanics, to be precise—in the opening chapter of his book The Fabric of Reality. Personally—keep in mind that I’m an engineer not a physicist—I think these interpretations of the results of the double slit experiment (and other quantum phenomena) are the polar opposite of Occam’s razor in that the explanation does not use the smallest possible set of elements, but rather—literally—an infinite amount of them. It’s far from elegant. Also, I suspect that the double slit experiment proves something else, entirely. But more on that in a future set of essays.
In conclusion, I view the concept of infinity as a useful tool, an imaginative, virtual entity that does not exist in physical reality. Like the concept of higher dimensions, an impossibility that can give us fresh insights about our current reality, but does not truly exist within it. A weapon of the imagination that has led to actual results, but remains imaginary nonetheless.
