3 ms·

If you're going to be pedantic, do it right. The average of 0 and 2π is π, so an average can certainly be π. Yes, the average of a finite number of rational nu

If you're going to be pedantic, do it right. The average of 0 and 2π is π, so an average can certainly be π. Yes, the average of a finite number of rational numbers cannot be π since π is irrational. But why would the sinuosity of any river be rational? The sinuosity of a circle is exactly π, for example. The true sinuosity of any given river is almost certainly irrational as well, since irrational numbers vastly outnumber rational ones in a very relevant sense: if you pick a real number uniformly at random between 0 and 1, there is literally zero chance that you will pick a rational number. Similarly, there is zero chance that the sinuosity of any river will be π or that the average of any finite number of rivers will be π. However, what they mean when they say this is this more precise fact: if there were an infinite number of rivers formed like those on Earth, the average sinuosity of that infinite collection of rivers would be exactly π. There. That's how to be pedantic.

matt_kantor 11y ago
Good call, I'm not sure why I hadn't considered that. I blame the whiskey!
- StefanKarpinski 11y ago
  No worries. You gave me an excellent excuse to exercise my pedantry in a way that I usually try to avoid so as not to be a social pariah :-)
plonh 11y ago
If you are going to be pedantic, please explain how anything in the Heisenberg Uncertainty-bounded universe can have a transcendental size.
pash 11y ago
> If you're going to be pedantic, do it right. Let's do it. > The true sinuosity of any given river is almost certainly irrational as well, since irrational numbers vastly outnumber rational ones in a very relevant sense: if you pick a real number uniformly at random between 0 and 1, there is literally zero chance that you will pick a rational number. Apparently you have access to measuring devices that can spit out irrational numbers. I'm impressed. No other scientist has ever seen such a thing. Unfortunately, because the computable numbers are countable, the set of irrational numbers that you will almost surely see in your setup will almost surely be uncomputable. In other words, not only will you almost surely select a number that cannot be the result of a measurement of finite precision, but you will almost surely select one that has no finite description at all. So you will almost surely never get a measurement of the length of a river. The average of an empty set is undefined, and oofabz's objection above pertains: rivers almost surely lack lengths. (Less pedantically: matt_kantor's pedantry was essentially correct, even if in an unintended way. Irrational numbers do not exist in the world of physical measurements. Abstracting from this reality, as StefanKarpinski did, can lead to ridulous models, because the real numbers are wholly artificial. Make probabilistic assertions about physical realities modeled with real numbers at your own peril.)
- black_knight 11y ago
  > > if you pick a real number uniformly at random between 0 and 1, there is literally zero chance that you will pick a rational number. > Unfortunately, because the computable numbers are countable, the set of irrational numbers that you will almost surely see in your setup will almost surely be uncomputable. That only applies if you assume some continous distribution. The world may for all we know be discrete and finitary, and so it may be that any real number which pops up is computable. And surely there is nothing preventing a measuring instrument to produce irrational numbers. It is just that it will still be an approximation, and thus there are rational numbers which are just as close to the real value.
- dogecoinbase 11y ago
  This is a trivially absurd argument for two reasons: First, it would be extremely straightforward to make a measuring device that spits out irrational numbers. Take the output, truncate at half the accuracy, and append an irrational to the output. Second, outputting an irrational number as a measurement does not imply that it's able to output any member of the complement of the rationals in the reals. You also conflate the computable numbers with the describable numbers -- but I'll give you the benefit of the doubt and assume you believe strong Church-Turing and aren't just committing an elementary error. There are ways in which set theoretic concerns apply to the real world, but they are few and far between, and this is not among them. You're essentially in line with people who attempt to use Goedel's proofs to make grandiose pronouncement about human thought. It. Does. Not. Apply.
  - pash 11y ago
    > ... outputting an irrational number as a measurement does not* imply that it's able to output any member of the complement of the rationals in the reals.* It's my fault, I'm sure, but you have missed the point entirely. The argument to which I responded, and which I extended absurdly, does imply that every real number is a valid output of a chance setup, and further assumes that each real number, including "any member of the complement of the rationals in the reals", is an equally likely outcome of that setup. That is anyway the conventionally understood definition of "pick a real number uniformly at random between 0 and 1". I would attempt to clarify the rest of what I wrote, but your condescension dissuades me.
eru 11y ago
> The true sinuosity of any given river is almost certainly irrational as well, since irrational numbers vastly outnumber rational ones in a very relevant sense [...] Though, at these levels of precision any physical quantity stops being a number, and becomes more like a distribution that varies over time.

matt_kantor 11y ago

Good call, I'm not sure why I hadn't considered that. I blame the whiskey!

StefanKarpinski 11y ago
No worries. You gave me an excellent excuse to exercise my pedantry in a way that I usually try to avoid so as not to be a social pariah :-)

plonh 11y ago

If you are going to be pedantic, please explain how anything in the Heisenberg Uncertainty-bounded universe can have a transcendental size.

pash 11y ago

> If you're going to be pedantic, do it right. Let's do it. > The true sinuosity of any given river is almost certainly irrational as well, since irrational numbers vastly outnumber rational ones in a very relevant sense: if you pick a real number uniformly at random between 0 and 1, there is literally zero chance that you will pick a rational number. Apparently you have access to measuring devices that can spit out irrational numbers. I'm impressed. No other scientist has ever seen such a thing. Unfortunately, because the computable numbers are countable, the set of irrational numbers that you will almost surely see in your setup will almost surely be uncomputable. In other words, not only will you almost surely select a number that cannot be the result of a measurement of finite precision, but you will almost surely select one that has no finite description at all. So you will almost surely never get a measurement of the length of a river. The average of an empty set is undefined, and oofabz's objection above pertains: rivers almost surely lack lengths. (Less pedantically: matt_kantor's pedantry was essentially correct, even if in an unintended way. Irrational numbers do not exist in the world of physical measurements. Abstracting from this reality, as StefanKarpinski did, can lead to ridulous models, because the real numbers are wholly artificial. Make probabilistic assertions about physical realities modeled with real numbers at your own peril.)

black_knight 11y ago
> > if you pick a real number uniformly at random between 0 and 1, there is literally zero chance that you will pick a rational number. > Unfortunately, because the computable numbers are countable, the set of irrational numbers that you will almost surely see in your setup will almost surely be uncomputable. That only applies if you assume some continous distribution. The world may for all we know be discrete and finitary, and so it may be that any real number which pops up is computable. And surely there is nothing preventing a measuring instrument to produce irrational numbers. It is just that it will still be an approximation, and thus there are rational numbers which are just as close to the real value.
dogecoinbase 11y ago
This is a trivially absurd argument for two reasons: First, it would be extremely straightforward to make a measuring device that spits out irrational numbers. Take the output, truncate at half the accuracy, and append an irrational to the output. Second, outputting an irrational number as a measurement does not imply that it's able to output any member of the complement of the rationals in the reals. You also conflate the computable numbers with the describable numbers -- but I'll give you the benefit of the doubt and assume you believe strong Church-Turing and aren't just committing an elementary error. There are ways in which set theoretic concerns apply to the real world, but they are few and far between, and this is not among them. You're essentially in line with people who attempt to use Goedel's proofs to make grandiose pronouncement about human thought. It. Does. Not. Apply.
- pash 11y ago
  > ... outputting an irrational number as a measurement does not* imply that it's able to output any member of the complement of the rationals in the reals.* It's my fault, I'm sure, but you have missed the point entirely. The argument to which I responded, and which I extended absurdly, does imply that every real number is a valid output of a chance setup, and further assumes that each real number, including "any member of the complement of the rationals in the reals", is an equally likely outcome of that setup. That is anyway the conventionally understood definition of "pick a real number uniformly at random between 0 and 1". I would attempt to clarify the rest of what I wrote, but your condescension dissuades me.

eru 11y ago

> The true sinuosity of any given river is almost certainly irrational as well, since irrational numbers vastly outnumber rational ones in a very relevant sense [...] Though, at these levels of precision any physical quantity stops being a number, and becomes more like a distribution that varies over time.