Infinity - Is a physical interpretation posible?

Infinity is an interesting subject indeed but its one that is lightly brushed over in most mathematical treatments.

Over the course of numerous discussions I have had with peers and friends, it has become apparent that by and large the "average" understanding of infinity is

1) It is a very large number

2) Its obtained as a result of an undefined operation (like division by zero)

These are wrong and do not even give a cursory view of what Infinity is. The subject of Infinity is beyond the scope of a single blog post but I will attempt to highlight a few of the special properties of Infinity which I think are fundamentally important. I will also briefly allude to more interesting characteristics but will have to leave that for some other day.

Alright, so lets see, what do we start off with? Lets start with two basic statements

1) Infinity is not a number, it is a concept

2) There are no infinities in physical reality

Lets illustrate these two aspects using a simple divergent series, also called the Grandi Series which is just a series of alternating negative and positive ones. Let us call this series S.

S = 1-1+1-1+1-1+1-1+1-1...............................∞

Now, if asked to express the sum to n terms of this series, it is quite apparent that there are two ways of looking at it

S = (1-1)+(1-1)+(1-1)+................................∞ = 0

or

S = 1+(-1+1)+(-1+1)+...................................∞ = 1

so essentially, depending on whether we choose to terminate the series at an even or an odd number we have either 0 or 1. Pretty simple isn't it?

Lets take another look at it now.

Let us take S and add it to itself so that we have

 

  S =  1-1+1-1+1-1+1-1+1-1...............................∞

+S =       1-1+1-1+1-1+1-1+1-1............................∞

----------------------------------------------------------------------------------

2S = 1+0+0+0+0....................................................0

 

Here, the second S had been slightly shifted to allow for ease of demonstration. All terms after the first, cancel each other out. Thus we have

2S = 1

S = 1/2

In essence, the sum of a infinite series comprising of integers is 1/2? Doesn't that sound surprising? It should. In fact it should be shocking, because this is true. Why do two methods yield such different results?

In the first case, we treated the series as infinite but forced it terminate at an even or an odd number of terms (which essentially is NOT infinity as that is not a number). The second proof made no such assumption and the series was continued ad infinitum.

Proof of this and other such sums are consistently used in M-Theory (String theory) and yield results.

The biggest conundrum is how do we reconcile this result with the physical world? Can we?

 

Let us assume that the 1 and -1 represent the states of a switch that turns on a light so that the light is either on or off and someone is rapidly flipping the switch.

If the switch is being flipped very very rapidly and you are asked if the light is on or off? What would you say?

Lets look at it another way. What would you say is the probability that the light is ON or OFF for that matter? 1/2 isn't it?

 

This does not imply that the 1/2 obtained as a sum of the series is somehow linked to the probability distribution of a light going on and off but nevertheless, this thought experiment is a bridge between a completely mathematical construct and physical reality.

 

Infinity is far more interesting with there being orders of infinity as well. So essentially there are some infinities which are "bigger" than other infinities. Interested folks can read up on Cantors theories. I will leave them for another day.

 

Adios....