Infinity + Philosophy = Fun

I want to share some mind-blowing thoughts about seemingly mundane concepts of infinity and how they paved the foundation for math. I use the word mundane because of the strong intuitive feeling that the concept of infinity has no practical use. We live in a seemingly finite universe and issues concerning us don’t involve the use of infinity.

The big question is: “Does infinity even exist in the universe?”

The answer is yes; space and time in the physical universe is infinite. The answer may not make sense because the notion of ‘infinity’ is not made clear. The dictionary roughly defines infinity as, “A sequence or distance without bound.” Again, this definition may not satisfy you with my answer. Let us examine what it means for something to be boundless.

We have a region of space from point A to point B (assume the line is a continuous whole i.e. no discontinuities).

A---------------------------------------B


There are an infinite number of points between A and B. Be careful at what you’re intuition might be telling you, the spatial region is not made up of points, but can be infinitely divisible by them. Points are not parts of a spatial regions they are simply boundaries. To get a better idea of what I mean, imagine what would happen if I got rid of that line that is in-between point A and B. When that line disappears, so does point A and B! They are never actualized until you create a bound or limit, in this case is the color of the line against the background of the page.

Lets actualize some more bounds on our line

A---------B------------------------C------D

Between AB, BC, BD or any other lines there is always and infinite amount of points that is the line can be divided into. The key concept to draw from this is that a point or bound always separates one region of space from another and only have a potential existence unless actualized as discontinuities. Now imagine if the universe is finite in size and there is a boundary or limit at the end of it. But wait, spatial boundaries are always boundaries of one region of space to another (this thought is just to amuse you).

Time is similarly infinite to that of space. You can think of time as a line with two points. One point is 7:35PM and the other is 9:00PM.

7:35PM---------------------------------------9:00PM

Similar to our spatial line, this line is bounded by an instant of time. An instant of time is not a part of time of which time can be divided into. An instant is more like a point on a line; it is the boundary between two periods of time. The interesting reality of time is the fact that it feels as if it will go on forever. At any point in time there is a succeeding moment and a succeeding moment giving the illusion that time is always ‘becoming’. Aristotle called this the ‘potentially infinite’, versus the actual infinite. Space could also be considered potentially infinite only if space and time were considered dependent to one another. If space were potentially infinite it would mean you could be walking in a straight line forever and always covering new ground. Now if space is independent of time then it cannot be potentially infinite without being actually infinite. If you are walking in a straight line you will be covering new ground every time because there is ground to be covered.

If it still doesn’t make sense that space and time can be actually infinite, rather than potentially infinite, than consider what happens when infinity is applied in mathematics. In mathematical theory, there is no room for a potential infinity. I explained how the potentially infinite space does not work when space is inseparable of time. A mathematician doesn’t define a circle as the locus of a points moving in a plane equidistant from a fixed point by supposing the motions of a point moving along a plane in time. He doesn’t define the circle by imagining in time, he supposes it to exist as a completed whole. How can a statement about all natural numbers be true if there is no completed totality of numbers in reality? How can we prove that there are an infinite number of prime numbers without actually counting them? Well we can prove it, but we can’t do it by supposing a potential infinity. Potential infinities are never totalities, so it would be impossible to prove there is a totality of prime numbers if it weren’t actually infinite.

Whether the universe is actually infinite or not may not make a difference on the practical level, but it does make a difference on the theoretical level. Previously, I said that time feels as if it is potentially infinite. After concluding that potential infinities don’t make sense in the mathematical world, I have to cast doubt on my intuitive notion of time. This leads me to conclude that time is thus in a sense wholly actual even though it is not simultaneously present. This may switch your point of view of time being more of a spatial dimension rather than something else.