How real are real numbers? (2004)

But you get there by a path of increasing fuzziness, and no clear boundaries, from numbers that we can both write down, and work with.

Then there is a jump to numbers that cannot be written down. Not even in principle.

Some people feel that that jump matters. Others don't. I feel it matters. But I accept that most mathematicians, don't.

Quantum mechanics, the wave equation. Electron shells, photochemistry, general chemistry, just about everything if we are talking about pi, or e, or i.

1 part off in trillions ^ trillions would impact the fusion of stars, the rates of chemical reactions, require adjustments to basic laws, violate conservation of energy as we know it, ... really obvious impacts. As in: "we would not be here" impacts.

Cosmology has run experiments for us that ran billions of years.

Contrast: Not everything can be tested with virtually unlimited precision, but basic mathematical constants in physics often can be. The gravitational constant is not testable like that. We don't have a mathematic derivation that we can leverage to test for violations like we do with pi, e, i, and other basic mathematical relationships that show up in physics.

But often, even a tiny difference becomes obvious. We exist because the production of matter and anti-matter at the beginning of the universe was off by a tiny amount. Despite the small discrepancy, that there was a discrepancy is very clear. Another "we would not be here" test.

You are dismissing my point, then asking me to make it.

You can think of tests/measurements of values as falling into different classes. The strongest tests of all are for the critical values of systems. Because any discrepancy would result in entirely different system behaviors.

Single highly accurate measurements are much lower on the rung. Complementing those are many tests with known statistical inaccuracy. Etc.

Single tests? Every experiment involving quantum mechanics tests pi's role in those equations to a much lesser extent. Similarly, any test involving gravity tests the gravitational constant. But we have much stronger tests for pi in quantum mechanics that we do for g in gravitation.

There isn't just one kind of measurement/test, there are many. And we want the strongest test we can make in any given situation.

But of course, we can always perform weaker tests.

Validating pi in quantum mechanics can be done with extreme robustness, because the entire theory depends on that value critically. Even the tiniest discrepancy would result in different physics compounding over all Plank space and time units, over billions of years and universe expansion, and we wouldn't be here.

Of course, we can't rule out any discrepancy. But in this case, we can rule out discrepancies down to unimaginable infinitesimals. I doubt anyone even knows how to characterize how much of a discrepancy from pi would still be consistent with what we know. That tiny.

The criticality is what gives us this far stronger test. Non-critical values cannot be tested this way. Pi can. Particular tiny ranges of stable constants in a stable 3-body system can (to a lesser extent, given the smaller system and higher bounds on criticality).

Another way to view the systemic criticality of pi, is to recognize that pi is not just a representation for a particular magnitude, but a representation of conserved cyclic behavior. Any deviation from pi breaks cyclic behavior. Thus, the implications of pi in a theory, and our ability to test pi, are profoundly greater than for most other constants. Because the difference between cyclic vs. non-cyclic behaviors, is profound. Not just slightly different behavior, but entirely different behavior.

> And if you had two separate copies of the universe, you could measure this compounding.

> But even if it did ruin cyclic behavior, how long are your cycles? The universe is only 1e61 planck times old. A discrepancy of 1e-100 would have no effect yet, let alone 1e-1000.

When this is true:

When there are no threshold conditions, then discrepancies accumulate just as you have described. And for any given t, a small enough discrepancy can be chosen so that it does not impact measurements of a given accuracy.

When it is not true:

For any system with thresholding conditions, any discrepancy can have profound immediate effects. A collision can result in a particle heading off in an entirely different direction after a collision, cascading into an entirely different system state.

How critical constraints change structure:

When pi, i and e are used structurally, i.e. "pi" represents traversal of a cycle, "i" a quarter turn traversal, "e" a positive feedback traversal, each of them represent something invariant: for "pi" some sum of two squared units is conserved (the squared radius on two units), for "i" a position may be conserved while an orientation rotates, for "e" some feedback value maintains an invariant relation between its position, its rate of change, and its accumulation. These relations define the system itself, not just some proportions.

So for those cases, where constants define structure, any change to those constants changes the structure. Suddenly, there are differences where they did not exist before, non-unity proportions where they did not exist before. The system has new state values, new interactions. The system itself has changed, not just proportions. The system likely has more states.

Structural change is threshold change:

Changing the system's structure is the most significant threshold-type change one might imagine. The entire system is different starting at time = zero.

Can a structural constant be changed in a way that leaves it only proportionally changed? So that discrepancies simply accumulate, until they are measurable? Sometimes, yes.

But will that hold in general? No. In general, the difference between interactions that exist, vs. interactions that do not exist, states that exist vs. states that do not, includes systems that can behave qualitatively differently from the very first time step.

Does that make sense? (I rewrote this several times!)

Sometimes numbers define structure. They define what interacts with what. And what does not interact. Not just proportions.

Changing structure is a threshold-type change: 0 to something, equal to unequal. A state that didn't exist, to one that exists. No interaction, to interaction. That might result in a system that simply accumulates discrepancy. But it may also result in entirely different behavior from the first step onward.