Infinity – The Infinitely Big

May 23rd, 2009 • Related • Filed Under

The infinite is a subject of great interest. Studied since man has been able to think, argued over for equally as long, yet ask anybody if they understand the infinite then most people would respond with a yes. It is full of contradiction, full of surprises and lies at the heart of most philosophical discussion. Mathematicians disagree as to what status the infinite should be afforded and scientists tend to sweep it under the carpet whenever it arises. Full of paradox and unable to be defined, the infinite is both understood and misunderstood in equal measure, often by the same people.

So what is the problem? This short series is not the best place to go in to great detail but is designed as an introduction to one of the most perplexing objects ever studied. Aristotle said that the infinite was central to the scheme of things. What he meant was that if we truly want to understand the world then an understanding of the infinite is crucial.

There are seven different types of infinity which are easily identifiable. Three of them are mathematical infinities, three metaphysical infinities and the final one is probably in the metaphysical camp. That is not to say that it is indisputable that they are mathematical or metaphysical objects but they are candidates for being so depending on who you are talking to. In this article I shall deal with just the first and probably most commonly known infinity and its metaphysical correlate.

This first infinity is can be expressed by the sequence:

1+2+3+4+5…..

where it is assumed the last term in the sequence is the object we seek to define; infinity. This is the infinite as expressed by the natural numbers so is called infinity N.

Note that it is impossible for me to define the infinite in clear unambiguous terms. The dots which appear after the five in the sequence mean ‘and so on and so on an infinite number of times.’ So in my definition of infinity I have had to use the thing I am defining to define it. (It is a good job you know what the infinite is or I could never explain it!)

The string of numbers represents a mathematical infinity. Now it is controversial as to whether this is a true mathematical object; we can call it N and let that mean ‘all numbers’ but it is by no means clear that we could ever gather together all numbers into one group. What we can say is that for whichever number we have, no matter how big, we can always find a number which is bigger. Because of this inability to collect together all objects of N into one group this type of infinity is often referred to as the potential infinity. It is only ever an infinity in becoming and never actually gets to the point of really being infinite.

The metaphysical equivalent of N is in objects whose size or distance or temporal duration have the same internal structure. So for example we may encounter this type of infinity in the real world if we were to have an infinite number of some object, or if we were to travel in a straight line away from earth – to infinity – or if we were to look back or forward in time to an infinite past or forwards to an infinite future. If any of these situations do occur in the real world then that would be a metaphysical example of infinity N.

The kind of concepts we will most often see being associated with this type of infinity are for example, unlimitedness – (As in unlimited hosting accounts), endlessness, immeasurable, eternal and boundlessness.

Despite this account of infinity probably being the most well known it is also probably the least likely to be represent anything which is truly ‘infinite.’ As a mathematical object it fails to exist in any coherent way and as a metaphysical or real world object it is probably impossible that it should exist to describe any aspect of reality.

Think what it would mean to have an infinite number of a thing; grains of sand for example, or any object in infinite quantity. Infinite distance seems unlikely and an infinite period of time either forward or backwards doesn’t quite stand up to logical scrutiny. There are also a number of paradoxes we run into if we treat this kind of infinity seriously.

For example: How many even numbers are there in comparison to natural numbers? The answer would seem to be less or we might even think a half but that is just not true. If we pair off the numbers against the even numbers we discover that we have the same amount of both.

0..1..2..3..4..5….6….n
|    |   |   |    |    |     |       |
0..2..4..6..8.10.12….2n

It seems as though there are as many even numbers as there are ordinary numbers and we know it is true because there are an infinite number of each yet…. it just doesn’t seem quite right..

We can devise more examples of infinite trickery. For example, think of a hotel with an infinite number of bedrooms. The hotel is full and new guests arrive, no problem. Just move the occupants in room 1 into room 2, the occupants in room 2 into room 3 and so on and put the late arrivals in the first room. In fact you would never have a problem finding a room in this hotel, even if you arrived with an infinite number of friends on a bus with infinite number of seats. In this case you just need to point out to the hotel manager all he has to do is move the guests in room 1 into room 2, the guests in room 2 into room 4, the guests in room 3 into room 6; in effect the guest in room n to room 2n and there’s enough space created for you and your infinite number of friends to all stay the night at this magical hotel. (Pity the chambermaid, but not too much. She would work hard, but only for one shift, just think of all the tips she would receive!)

There is also the example of the two men who are destined to spend all of eternity, one in Heaven and the other in Hell. The keepers of the eternal realm however have a policy of allowing the resident of hell to spend Christmas day in heaven whilst the resident of heaven has to spend that day in hell just as a reminder of how fortunate he is. On the surface it would seem that one man spends a lot more time in heaven whilst the other spends most of his time in hell. But on closer inspection, because this is an eternal arrangement they both spend an equal amount of time in each.

In temporal terms there are a few ideas to think about which makes eternal past and futures seem incoherent. Imagine someone walked up to you and said; ‘2,…4,…1,…3 finished!’ and then explained that they had just recited the decimal part of pi backwards! This draws attention to the difficulty we have in contemplating an infinite regress of time. The forward part we can take on board a little easier. For someone to begin reciting the decimal expansion of pi and continuing ‘for all time’ is easier to conceptualize. We can just assume that no matter how far he has got, he will always have more to do. But for someone to have already gone through an infinite sequence to have arrived in the ‘here and now,’ strains our imagination more than we can cope with. Just when did our eternal reciter begin his task and at which point in the sequence!

This is the infinite of the very large and the associated paradoxes are the paradoxes of the infinitely big. The infinite of this kind does not stand up to scrutiny. It makes no real sense from a mathematical perspective and if it was taken seriously logically, it has nothing to offer us in our investigations into reality. It seems likely that this kind of infinity could not exist in the universe as an object of math or as a description of things, events or time.

That is the infinite in its most common conception and it should be clear that it is probably something of a falsehood both mathematically and metaphysically. I shall deal with the other types of infinity on this blog later in the next few days / week.

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