“1) ‘2’ leads to confusion as to whether we are representing a real or a natural number. That is, whether we are counting discrete items or we are representing a value on a continuum.”
If I recall correctly, this “confusion” is what allowed modern, atomic chemistry. Chemical substances—measured as continuous quantities—seem to combine in simple natural-number ratios. This was the primary evidence for the existence of atoms.
What is the practical negative consequence of the confusion you’re trying to avoid?
You also say:
“2) If it is clear that we are representing numbers on a continuum, I could see the number of significant digits used as an indication of the amount of uncertainty in the value. For any real problem there is always uncertainty caused by A) the measuring instrument and B) the representation system itself such as the computable numbers which are limited by a finite amount of digits (although we get to choose the uncertainty here as we choose the number of digits). This is one of the reason the infinite limits don’t seem useful to me. They don’t correspond to reality. The implicit limits seems to lead to sloppiness in dealing with uncertainty in number representation.”
But wouldn’t good sig-fig practice round 1.999… up to something like 2.00 anyway?
Benoit Essiambre,
You say:
“1) ‘2’ leads to confusion as to whether we are representing a real or a natural number. That is, whether we are counting discrete items or we are representing a value on a continuum.”
If I recall correctly, this “confusion” is what allowed modern, atomic chemistry. Chemical substances—measured as continuous quantities—seem to combine in simple natural-number ratios. This was the primary evidence for the existence of atoms.
What is the practical negative consequence of the confusion you’re trying to avoid?
You also say:
“2) If it is clear that we are representing numbers on a continuum, I could see the number of significant digits used as an indication of the amount of uncertainty in the value. For any real problem there is always uncertainty caused by A) the measuring instrument and B) the representation system itself such as the computable numbers which are limited by a finite amount of digits (although we get to choose the uncertainty here as we choose the number of digits). This is one of the reason the infinite limits don’t seem useful to me. They don’t correspond to reality. The implicit limits seems to lead to sloppiness in dealing with uncertainty in number representation.”
But wouldn’t good sig-fig practice round 1.999… up to something like 2.00 anyway?