We need to back up a little here and talk about what a number is. This relates directly to our pursuit of what a number is and subsequently to what infinity is.
For starters, a number system has rules. These rules apply to what are regarded as object, although that is problematic for me. Just one example of a rule pertains to how two objects are added.
Next, the rules must apply to ALL objects.
One example of a number system is Natural Numbers. These are 1,2,3.....
The numbers here are possible answers to the question "How many?"
There are other number systems.
Next, we need to understand what existence means if we are to have any hope of understanding infinity. The "objects" mentioned earlier do not exist in the same way as other objects do. Mathematical objects are abstract. We need to keep this in mind if we ask whether a mathematical object exists.
These abstract objects can be seen as concepts, so when we ask whether a mathematical object like infinity exists, we are essentially asking, given this set of concepts, does the one we are asking about exist?
The issue goes like this. Suppose we ask whether there is a number between 1 and 2. The answer depends on what we mean by number. If we ask "does there exist a natural number between 1 and 2",the answer is "No". You cannot, for example, go to the market and buy more than one apple but fewer than two.
We start to branch off here into imaginary numbers when we discuss infinity, such that the question becomes "Can anything REAL be infinite?"
Think of it this way. If you believe in infinity, you believe in God. |