Understanding Infinity
Mathmaticians and physicists tend to think of infinity as something really big. This is a very limited definition of infinity!
How a volume of water can become infinite or unbound
If you have a glass of water, that water is bound by gravity and the shape of the glass itself. While the individual molecules of that volume of water have freedom to move relative to each other within the bounds of the glass, they are limited by the glass. So, the molucules of water are infinite, or unbound, in one aspect, but limited in another.
Upturn the glass and the water falls due to gravity. Now it is no longer limited by the shape of the glass. The shapes it can assume are infinite. In other words, unlimited. When that volume of water lands on the floor it becomes bound in some ways, the surface of the floor limits or constriants it, but there are no limitations to how it spreads out. So, while it is limited in one aspect, it is infinite in another.
When triangles are no longer triangles
Looking at triangles, they are traditionally defined as having angles whose sum equals 180 degrees. In trigonometry, Tan (or tangent) is the ratio of the opposite side to the adjacent side.
From what I remember from my trigonometry classes, we can have a triangle with points, or vertices, labelled A, B, C. The sides opposite these points are labelled a, b, c. And so the side opposite to angle A is a. The side opposite to angle B is b and likewise for angle C and side c. Generally, we worked with triangles that had one 90 degree angle. The side opposite the 90 degree angle was the hypotenuse.
For angle B, the ratio of the opposite and adjacent side (so not the hypotenuse) is b/a.
If the angle of B approaches 90 degrees, this means the triangle is getting close to having more than 180 degrees. Tan is infinite for angles of 90 degrees because there are no triangles with that possibility.
Another way to put it is that the relationship of a triangle no longer holds true and as a result the ratio is no longer constrained by the definition of a triangle. Hence the tangent for that angle is any number possible. i.e. infinity.
An infinite rate of return?
Yet another example that inspired me to think more about what infinity means comes from Rich Dad's Robert Kiyosaki. He'd often talk of times where he used other peoples money to make money. He'd then say he had an infinite rate of return, implying that the rate of return was very big.
What made more sense to me was that he used one of an infinite number of ways of making money from money using other peoples money.
If you define rate of return as: the rate of return using your own money then when using someone elses money, the rate of return no longer applies.
Rather than an infinite rate of return, Kiyosaki plays with the infinite number of ways we can use other peoples money to make money.
Just quickly, here's the equation you use to calculate interest or rate of return: (final amount-initial amount)/initial amount. Since the initial amount was effectively zero (since it wasn't his money) the rate of interest or ratio of interest to the initial amount is infinite.
Put another way, there are an infinite number of answers to the equation of x/0 (where x is any number.)
Something else that is infinite in this regard is the amount of money he can make this way. It can range from zero to millions of dollars.
So why try to understand infinity in this way?
So why bother trying to understand infinity in this way. Okay, it doesn't mean really, really big, but so what!
It may help to understand that the opposite of infinity is limits and limitation. Limits are the things that we use to cut apart infinity.
Limits are generally things that we create with our mind. We define limits so that we can work within them. Generally, when we have limits, we give ourselves freedom to work within the limits we've defined.
A good example of this is games.
Games are just a set of limits that we can play within
Games generally involve a fixed set of limits. And these limits, or rules, are generally easy to learn. Once learned, we don't have to think about what the limits are. We can get on with playing inside of them.
One of the really nice thing about games is that often we can get into the flow state. Because we don't have to think about what we are doing, we can just get on with doing it.
We use limits to learn and problem solve
Another place where we use limits is when learning. We use (or we can use) limits to break complexs systems down into smaller sub systems. We can then learn the big system in terms of its smaller parts. And so understanding that we can define limits makes it easier for use to learn things. It also makes it easier for us to solve problems.
In terms of working from first principles, we can use limits to define ideas, relationships and systems. We can also use them to define signals.
Generally, if we aren't already familiar with the system in which the problem lies, we can learn the part of the system that seems to relate to the problem. We use limits to define the parts of the system that are of interest. And if we still can't figure out the problem, then one possible step is to redefine the limits.
Whether learning or problem solving, once we've learned, once we've solved the problem, we often no longer need the limits. We can get rid of them until we need them again.
Consciousness is limited
One of the first times I really began to think about the idea, the concept, of limits was when I heard the phrase consciousness is limited. I spent many years thinking about what that meant.
Like water in a glass, the amount of consciousness we have is limited. This could be because the machinery it runs in, our brain, is limited. Or for some other reason. The point is, we only have so much to spare.
Two very basic modes of consciousness are thinking and flowing.
I tend to think of thinking as occurring in imaginary space, or more accurately, in analytics space which is perhaps a subset of imaginary space. Meanwhile flowing happens in real space.
When we expand our consciousness fully into imaginary space, our thinking mind shuts of. The experience is like daydreaming. It can also happen while doing something rhythmic.
Our thinking mind also shuts off when we expand consciousness into real space. This is known as being in the flow, being present, or no mind.
In either case, it's like we've spilled the water (consciousness) out of the glass. However, instead of a glass, the limits that we've freed our consciousness from are those of time and space.
Our thinking mind creates limits
This might make more sense if we think of our thinking mind as the thing that creates limits. That's what it's for. It uses those limits to define things so that our brain can index and store models of things, including ourselves.
When we turn our thinking mind off, basically, when we shut the machinery off that it runs in and engage it fully in imaginary or real space, that's when we become infinite. Not infinitely big, just unlimited in the same way water spilled from a glass is limited.
Sure, it's limited by gravity pulling it down, but it's less limited that when it was in the glass.
The nice thing with the power of our thinking mind is that we can define the glass we pour our consciousness into. And we can redefine it at will.
Re-defining Freedom
Freedom isn't freedom from limits. Rather it is the freedom to create limits so that we can play within them. We can then change them when we're through.
There are an infinite number of ways to define limits. Freedom comes when we choose the limits we play within.
