Okay...the last post needs some clarification. I did not mean the notation '3' for the number '2+1' is unjustified. That is perfectly okay. We can have the symbol '4' for the number '3' or anything for that matter and the math would just be fine and consistent as it is now. So, denoting '3' for '2+1' is understandable and it is not the issue. The point was that how one can assume the existence of a set that satisfies the Peano's axioms? The set that satisfies the conditions of Peano's axioms could well be empty, isn't it? Thus natural numbers are not captured by those axioms rather the axioms hold for what we all know as natural numbers. That is we have to agree on such a basic set in order to move forward.
This points us to the fundamental question on the existence of numbers as basic as the natural numbers. Do they exist at all?
In my view, natural numbers are mathematical representation of the notion of duality that we all perceive, if not take it as a basic philosophical position. The mere fact that we see more than one thing is enough to justify the existence of numbers. In other words, numbers exist and they are real as much as you believe in reading this piece of writing.
This points us to the fundamental question on the existence of numbers as basic as the natural numbers. Do they exist at all?
In my view, natural numbers are mathematical representation of the notion of duality that we all perceive, if not take it as a basic philosophical position. The mere fact that we see more than one thing is enough to justify the existence of numbers. In other words, numbers exist and they are real as much as you believe in reading this piece of writing.
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