Nanfan Yi

In the previous three articles (links at the bottom of the page), we’ve talked about answers to the question “What is infinity?”, and hopefully you now have some new ideas about this infamous number. So, what comes next? Have you ever thought about anything beyond infinity? How did humans first come up with this concept? Why do we care about infinity when it is very abstract and may not even exist? Why do scientists, mathematicians, writers, philosophers and everyone else discuss it? All are very good questions, and here I will hopefully provide some peripheral insights.

On the one hand, in the mathematical reality, infinity is more or less a “well-defined” object to be used. As we have seen, infinity can be captured naturally via geometrical objects: from point to line, line to plane, and plane to space. Georg Cantor defined infinity via set theory (how-many situations): there are infinitely many natural numbers and real numbers, but the size of the set of real number represents a larger infinity. Its usual definition can be captured via the sum of infinite series (how-large situations): sometimes we have the sum being a finite value, other times, infinity itself. In addition, we can dive into the world of calculus: Leibniz and Newton studied continuous curves, and using the idea of infinitesimally small to define the derivative and integral of such curves. Infinity does exist in the mathematical reality, in the sense that many mathematical objects have properties associated with infinity and studying them requires rigorous definitions and treatments of the concept.

On the other hand, in the physical reality, infinity is somewhat abstract, but does lie within how we understand the world around us. If we have a scale for measuring the size of matter in this world, going infinitely close to 0 led to the discoveries of particles. We asked: “Hey, is everything made up from something?”. And we started to decompose this world into smaller and smaller units: molecules – proteins, carbohydrates, fatty acids, carbon dioxide; elements; atoms; nuclei; protons and neutrons; electrons; quarks/neutrinos… Anything smaller? Perhaps one day your fellow physics friend from university will tell you that they proved, or failed to prove, that there is an infinitesimally small building block of the universe. Going in the other direction, where we have big stuff going on, we asked: “Is the universe infinite?” (Even we assumed the universe is infinite in the first article of the series.) For now, there is no way of verifying the infinitude of the universe, simply because we are not able to observe anything beyond the so-called observable universe. Read a more solid argument here, and a weird one here.

Though we have no doubt put in lots of effort to make sense of it, infinity remains mystical to many human minds. This, perhaps, is the key reason why the idea of infinity is mind-triggering. Quoting from Blaise Pascal:

“When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there . . . now instead of then”.

 Article 1: Infinity at First Glance (Perhaps a Stare)

Article 2: Different Sizes of Infinities?!

Article 3: Infinity: Go Home or Go Wild!