> You can do math by hand with more precision than actually exists in the real world.
This claim can't mean anything because it disappears up its own backside. If you, here in the real world, can "do math by hand" that math is in the real world.
All you have in the real world is scribbles on paper.
I'm sorry, what did you mean by "how complicated the world we inhabit is" because I thought you were talking about physical interactions of matter and energy.
If you're including all math as "real world", then I think the claim that math teaches you something useful about the complexity of the real world is what actually disappears up its own backside.
Are you sure? Naively sure, you won't find sufficiently enormous circles and even if you could such a huge circle won't have the Pi ratio here, that's a property of Euclidean space and we don't live in a Euclidean space, ours is slightly off IIRC.
But Pi shows up in other places and I'm not at all sure you can show there are no such places which distinguish some arbitrary approximation from the actual ratio.
The most precise situations I can think of involve impossibly perfect measurements of volume. And even there, okay a cubic meter is 10^105 cubic planck units and the visible universe is 10^186. Finite math can easily throw a million digits at any problem. How do you reach a point where you need reals to describe actual things?
Pi, i and e show up with apparent perfect "precision" in all kinds of physics.
Waves are pervasive and described by relationships involving those numbers. They show up in other relationships. With important properties such as conservation of energy that any partial precision wouldn't be able to achieve.
Numbers are not just evident by single value measurement, but even more powerfully when they govern a system, where any problem with the definition would result in an easily recognizable failure of an entire theory.
I think "precision" is the wrong way to look at what can mean something or not.
I think the boundary between numbers that "make sense", relative those that don't is better found by looking at the progression of numbers.
From naturals, to integers, to rationals, to algebraic (both non-rational roots, and roots of negatives), all the way to limits and series. (Note that the infinite computation associated with expanding digits is not a definition problem. Even 1/3 requires infinite digits in decimal, but the relationship between 1 and 3 is clear.)
What is true about all these numbers is not precision, but that they emerge from a finite number of relationships.
They can be written exactly, defined perfectly, with finite numbers of symbols. (Meaning, abstracting away notation, with a finite number of relationships.)
And all those types of numbers do show up exactly (for all appearances), in waves, and other relationships. The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values.
So pi really exists. All kinds of physics would fail if it didn't. That doesn't mean we can make a perfect pi circle with plan length, since that would be an arbitrary test, and if the medium is discrete units, one chosen to a priori fail.
Contrast with: The uncomputable, undefinable numbers, which we can't define, can't measure, etc., and are introduced via shaky (relative to the general body of mathematics) means. They require infinite information to define exactly. Not just measure, but even to define. Which is a remarkable postulation, and is not needed to solve any problems they don't themselves introduce.
> With important properties such as conservation of energy that any partial precision wouldn't be able to achieve.
If your measurement of energy is 150 +/- 2, you only need a handful of digits to do calculations involving that value that preserve it just fine. Insisting that that billionth digit and more still match is no longer working with the real world.
> The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values.
You can make a prediction to infinite precision but it's not falsifiable. The infinite precision isn't real any more than aether theory is real.
Any powerful results that impact the real world don't need all that precision.
> if the medium is discrete units, one chosen to a priori fail
I choose discrete units because that's what the universe is, as far as we can measure.
If there's something more subtle than Planck, we can't measure it.
It's possible the universe does round at some point. We can't tell.
Can you describe any theoretical experiment that could tell the difference between perfect pi and thousand digit pi?
> Contrast with: The uncomputable, undefinable numbers,
I agree that there's a stark difference there. But I don't think computability is the specific point where it detaches from reality, it's just where the disconnect gets the most obvious.
> If your measurement of energy is 150 +/- 2, you only need a handful of digits to do calculations involving that value that preserve it just fine. Insisting that that billionth digit and more still match is no longer working with the real world.
You just repeated the misunderstanding.
Numbers like pi are not just magnitudes, but form critical relationships. And relationship tests offer (unimaginable) orders of magnitude more stringent testing.
"Weak" relationship test: The 3-body problem. There are stable modes, but even small discrepancies results in an unstable system falling apart. Accuracy rapidly compounds over observation or reconstructible time.
Strong example: If wave equations were not exact to pi, the discrepancy would be obvious in a nanosecond, much less thousands, millions or 14 billion years.
Pi isn't just a magnitude, it is a very special magnitude, where any offset completely destroys its properties. Properties that have held for billions of years of plank time intervals, themselves distributed over non-linear space time and all the other disturbances of the universe's complexities.
Try to come up with a non-pi number that does not radically alter quantum mechanics and chemistry. The maximum discrepancy you can come up with would be an unimaginable infinitesimal, shrinking faster and faster every Plank unit of time since the Big Bang. And also shrinking relative to the increasing volume, in Plank lengths, of observable space ever since the Big Bang.
There is no direct magnitude measurement that begins to compare with that.
It is impossible to create a circle made up of discrete lengths (Plank or not) in flat space, due to basic geometry. So using that as a test, when no theory predicts or depends on a "perfect" spacial circle, is a red herring. We already know it does not exist.
(If this does not make sense to you, point out the problem.)
> Numbers like pi are not just magnitudes, but form critical relationships.
Yes, relationships. But the only way to test relationships is to eventually get to measurements of magnitudes.
> And relationship tests offer (unimaginable) orders of magnitude more stringent testing.
How?
> "Weak" relationship test: The 3-body problem.
You can't prove the 3-body problem isn't rounding to the nearest planck unit. You can't measure it precisely enough.
> Strong example: If wave equations were not exact to pi, the discrepancy would be obvious in a nanosecond, much less thousands, millions or 14 billion years.
I don't think you're conceptualizing "a thousand digits" properly.
> where any offset completely destroys its properties
What's an experiment we could do that verifies pi doesn't have an offset of 1e-1000?
Also keep in mind that just the slightest bit of gravity or cosmic expansion has a much much bigger warping effect then 1e-1000 and yet physics keeps working the way we expect, and we can't tell the difference for small enough amounts of those things.
> The maximum discrepancy you can come up with would be an unimaginable infinitesimal, shrinking faster and faster every Plank unit of time since the Big Bang. And also shrinking relative to the increasing volume, in Plank lengths, of observable space ever since the Big Bang.
Why would it have to keep shrinking?
I think you're saying that for it to have no difference at all it would have to be that small. But my challenge is for an experiment that measures the difference. If some physical effect shifted over by 5 Planck units would you be able to tell? What if reality has just a tiny itty bit of jitter to it that ruins perfect numbers like Pi?
> It is impossible to create a circle made up of discrete lengths (Plank or not) in flat space, due to basic geometry. So using that as a test, when no theory predicts or depends on a "perfect" spacial circle, is a red herring. We already know it does not exist.
It's not a red herring when I'm suggesting that no test is even possible.
> What's an experiment we could do that verifies pi doesn't have an offset of 1e-1000?
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.
> 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
Pointing at entire fields is not helpful. Can you give me one specific measurement and an estimate of how far off it would be?
> really obvious impacts. As in: "we would not be here" impacts.
That sounds pretty nonsense to me. The range of possible values for life isn't that narrow. And relativity is already in there ruining any straightforward conservation of energy and mass by constantly shifting the weight of things as their state changes. But it still works just fine! And we don't know exactly how strong that effect is, which could hide all sorts of imprecision in the real world. It would not be obvious.
> Not everything can be tested with virtually unlimited precision, but basic mathematical constants in physics often can be.
I'm begging you, name a specific test. One that could tease out 1e-1000.
> 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.
And if the matter-antimatter imbalance was 1e-1000 it would be imperceptible. It would be less than one atom in the entire visible universe, by an unimaginable factor. It was somewhere around 1e-9, probably, sort of. Not that small at all.
> Pointing at entire fields is not helpful. Can you give me one specific measurement and an estimate of how far off it would be?
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.
> 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.
And if you had two separate copies of the universe, you could measure this compounding.
But we only have one universe. How do we know which one we're in?
> 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.
Or it just knocks off the frequency by an absurdly small amount.
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.
> 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.
It's possible something could lead to a discrepancy that you could measure if you knew exactly where to look. (I'd still want to see a non-vague theory for how that could happen, but I'll put that aside.) But even if you knew where to look, you'd just get a value, and that value wouldn't tell you which side of the discrepancy you're on. If the universe had very subtle shifts in how things align, or did very subtle rounding of certain numbers, anyone stuck inside the universe without external knowledge would not be able to tell whether physics matches the pure mathematical expectations or not.
I understand that there might be a threshold in some cases. But given how much of the universe is already effectively random when you get down into the itty bitty details, and that you can only make an experiment so big to tame the statistics, there could easily be untold rough edges that are invisible to us. Some particle went a different way than it "would have", but you can't measure "would have", it still looks like a normal interaction with a normal distribution of outcomes.
Even if you had access to a parallel universe and you knew either you or them had 'correct' physics and the other didn't, I don't see how you'd be able to figure out which is which.
This claim can't mean anything because it disappears up its own backside. If you, here in the real world, can "do math by hand" that math is in the real world.