The Problem of Infinity

In this paper I will be attempting to analyze the concept of infinity. My goal is not to undermine common intuitions about infinity, but to offer my own insight on this seemingly unknowable concept. I am aware that man’s history with infinity is all but brief and there have been many schools of science, mathematics, and thought dedicated to finding some truths about it. The purpose of this paper is not to challenge any of these ideas but to analyze the way infinity may manifest itself in our universe. I would like to make clear that the nature of infinity is something that we don’t and as of right now can’t know much about. I am also aware that many of my arguments consist of the fact that a lot of what we know about infinity cannot be proven, and for this reason you may try to turn my argument around on me by saying that nothing I am saying can be proven either. To which I must again clarify I am not proposing anything as truth or fact, just a very grounded alternative possibility that I have come to particularly like. Throughout this paper I will (as briefly as I can) lay out what we generally accept about infinity, ask questions, and propose alternatives. I will often refer to infinity as we believe in to be in the realm of science and mathematics. It is important to know that this is not necessarily what I am trying to analyze. The infinity I am speaking of is a state of existence. My core argument is this; much of our common understanding of infinity is linear in its characteristics. I am arguing for an alternative way in which infinity manifests itself and this is as a loop.

Aristotle first tackled the problem of infinity in Physics. He made the distinction between potential and actual infinity. Potential infinity is a group of numbers or things that continue on forever and ever without end whereas actual infinity is a set of infinite numbers or things within a confined space with a beginning and end. To Aristotle, actual infinity was a paradox that just could not exist. Potential infinity is the common understanding that we hold even today.

George Cantor later revisited the idea of infinity in his Set Theory. Here he proved that there are many different types of infinity, that infinity can exist relative to other infinities, that is, you can always find a larger infinite set of numbers than ever before. This tower of climbing infinities all led up to an “absolute infinity.” This absolute infinity is unreachable.

Both of these theories see infinity as a linear concept. Keep in mind I am not disputing their accuracy. I do not wish to be interpreted as claiming that they are incorrect in any way, I am only saying that they and many other theories in mathematics and physics do not consider infinity as being a loop. And to consider it one way is not to reject it the other way, as Cantor proved that infinity can vary in to size it is also possible that it could vary in manifestation. That is, in some, many, or all instances it may exist as a loop.

This problem with infinity first came to for me light when reading Aquinas’s argument for the existence of God. He explains five reasons why God must exist one of these being the idea of “effective causation.” (Perry, Bratman, Fisher 43) This is the idea that in order for something in our universe to change, something must have initiated that change. Something cannot spontaneously change on its own. Therefore, effective causation cannot go back to infinity because there would have been nothing to initiate the first change and that there must have been something (which he claims is god) outside of this law that would have done so. This seems to be an intrinsic assumption for our picture of infinity, that it must begin somewhere. For numbers to go to infinity we must start somewhere (i.e. one) or for our universe to expand infinitely it must have begun the expansion with the Big Bang. This is sort of a linear depiction of infinity in the sense that for a line to go on infinitely in either direction we must begin at a point. The problem of infinite regression- a sequence of reasoning that never comes to an end- is a result of a linear understanding of infinity. This idea of infinity as it exists in our universe is what we have based much of our knowledge of the universe on. In accepting the possibility of infinity existing as a loop allows for a way out of the problem of infinite regression.

It is important to understand exactly what I mean by infinite loop. A perfect visualization is the Ouroboros, a snake eating its own tail. It is like tracing a circle’s circumference on and on into eternity.

You may respond, well yes, I can agree that you can trace a circles circumference infinitely many times, but you must still have at some point drawn the circle for the first time or begun at a specific point. But by responding this way, you fail to recognize that infinity, in its very nature, does not require a beginning if understood nonlinearly. We seek a starting point for us to be able to understand the way infinity works but that does not mean it is necessary to its existence. It is a result of our punctuality. How do we know that in counting to infinity we may someday not find ourselves back at 1 again when the numbers get large enough? The truth is we don’t, although we base much of our understanding of infinity and the universe on this assumption.

Now I understand where our concept of infinity being linear and only linear in the universe would come from. Numbers as far as we understand begin at a number and move linearly in either direction, either increasing infinitely or decreasing infinitely. Time for man is of course linear and moves in only one direction. But being as we are finite, I would argue that our concepts alone are not enough to truly understand the nature of the infinite.

An important cosmological possibility to consider for a moment is a counterpart to the Big Bang, it is the idea of a Big Crunch. This is the assumption that our universe may someday stop expanding and contract back into another infinity like that which begun the big bang. If this is true, it could come to prove that the universe truly exists in a state of infinity and this infinity exists as a loop. It begins with an infinite density that expands into a smaller infinity which contracts back into an infinite density and back again.

It is important to point out that up until now I have often been speaking of infinity as it pertains to science and mathematics. Again, the infinity that I would like to examine is not just a number but a state of existence. Historically, science and mathematics have formed the basis of much of our understanding of the universe. Therefore, I have attempted to formulate an argument that allows for the possibility of infinity existing as a loop based on our knowledge of math and science. But to try and understand the nature of infinity in terms of math and science is similar to trying to understand God in these terms. Both are states of being that manifest themselves outside of the realm that us finite beings can understand, therefore this approach just won’t work.

I now suggest that you disregard numbers, mathematics, and science. I am proposing that you think of infinity only as a state of existence. For space to exist in a state of infinity or for time to exist as a state of infinity is where I would argue that infinity could exist as a loop. And you may wish to entertain this possibility or you may find it absurd but if you find yourself in the category with the latter I must pose a final question. Why? To accept the possibility of infinity being a loop is not to reject all common understanding of infinity, as there may even be more than one way it manifests itself. To accept this loop helps us out of the problem of infinite regression and allows us to explore many more possibilities of the nature of our universe and how we have come to exist in it.

Works Cited

Bratman, Michael, and John Martin Fischer. “God and Evil.” Introduction to Philosophy, by John Perry, 7th ed., Oxford University Press, 1993, pp. 42–44.