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- TL;DR Summary
- Discuss the use of different number systems (rationals vs reals) in science

A couple of weeks ago we had an interesting thread where a tangent developed discussing whether real-valued measurements were possible. I would like to generalize that discussion a bit in this one and discuss all scientific purposes, not just measurements.

1) What is a measurement anyway? Is it the physical interaction, or is it the number we assign to the interaction. For example, in a galvanometer, is the measurement the deflection of the needle or is it the number that we assign to the deflection? This is probably a matter of opinion, so it probably is important just to state one's opinion.

2) One thing that I found in my research on the topic is that there is only one axiom that the reals satisfy and the rationals do not. If you have a bounded set of reals then the bound is a real, but there are bounded sets of rationals whose bound is irrational. For example, the set of all rationals such that ##\left(\frac{p}{q}\right)^2<2##. This set is bounded from above, but the bound is ##\sqrt{2}## which is irrational. So, for measurements it is unclear how this applies. There are no infinite sets of measurements, and for any finite set of numbers the bound is an element of that set. So the one axiom that distinguishes reals and rationals doesn't apply to measurements.

3) Because precision is finite, measurements don't have limits in the epsilon-delta sense. Even means of measurements don't have such limits. So speaking of large sets of measurements in the limit as the number of measurements goes to infinity still doesn't give convergence, bounding, or limits even in principle.

4) Suppose that we have two scientific models, one based on real numbers and one based on rational numbers. Suppose further that whenever the real model predicts a value, the rational model predicts the rational value that is closest to that real value. There is no experimental measurement which can provide evidence for one of these models and against the other. Any measurement is compatible with an infinite number of reals and an infinite number of rationals.

5) There are theoretical reasons to prefer the reals over the rationals. For example, you cannot do integration of rational functions, or rather the resulting integrals can be irrational. If we prefer reals over rationals for theoretical reasons, and if our models are compatible with both, can we not declare our measurements to be reals?

6) What about hyperreals? We might make theoretical arguments favoring hyperreals. The same compatibility and preference arguments would apply. Can we declare our measurements to be hyperreals? What about surreals? Is there a line, and if so where do we draw it and why?

1) What is a measurement anyway? Is it the physical interaction, or is it the number we assign to the interaction. For example, in a galvanometer, is the measurement the deflection of the needle or is it the number that we assign to the deflection? This is probably a matter of opinion, so it probably is important just to state one's opinion.

2) One thing that I found in my research on the topic is that there is only one axiom that the reals satisfy and the rationals do not. If you have a bounded set of reals then the bound is a real, but there are bounded sets of rationals whose bound is irrational. For example, the set of all rationals such that ##\left(\frac{p}{q}\right)^2<2##. This set is bounded from above, but the bound is ##\sqrt{2}## which is irrational. So, for measurements it is unclear how this applies. There are no infinite sets of measurements, and for any finite set of numbers the bound is an element of that set. So the one axiom that distinguishes reals and rationals doesn't apply to measurements.

3) Because precision is finite, measurements don't have limits in the epsilon-delta sense. Even means of measurements don't have such limits. So speaking of large sets of measurements in the limit as the number of measurements goes to infinity still doesn't give convergence, bounding, or limits even in principle.

4) Suppose that we have two scientific models, one based on real numbers and one based on rational numbers. Suppose further that whenever the real model predicts a value, the rational model predicts the rational value that is closest to that real value. There is no experimental measurement which can provide evidence for one of these models and against the other. Any measurement is compatible with an infinite number of reals and an infinite number of rationals.

5) There are theoretical reasons to prefer the reals over the rationals. For example, you cannot do integration of rational functions, or rather the resulting integrals can be irrational. If we prefer reals over rationals for theoretical reasons, and if our models are compatible with both, can we not declare our measurements to be reals?

6) What about hyperreals? We might make theoretical arguments favoring hyperreals. The same compatibility and preference arguments would apply. Can we declare our measurements to be hyperreals? What about surreals? Is there a line, and if so where do we draw it and why?