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> 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 n

by 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.
LukeShu 11y ago
> measuring devices that can spit out irrational numbers. ... No other scientist has ever seen such a thing. There are devices that indirectly measure something, and relate it to the desired measurement involving pi. Stupid example: Take a measuring wheel that counts encoder clicks. Say the encoder has 300 clicks/revolution. The device is quite likely calibrated to output clicks(2pi*radius/300).
- pash 11y ago
  This sort of thing exists and is apparently what other commenters also have in mind, but I would argue that in your example the measuring device's output is a natural number n in the range [0, 300] and that its interpretation as a fraction of a circle's circumference is just that. Clearly the existence of describable irrational numbers implies that we can describe a measurement using an irrational number. That does not make a finite measurement essentially irrational in any meaningful sense.

by pash 11y ago

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.

LukeShu 11y ago

> measuring devices that can spit out irrational numbers. ... No other scientist has ever seen such a thing. There are devices that indirectly measure something, and relate it to the desired measurement involving pi. Stupid example: Take a measuring wheel that counts encoder clicks. Say the encoder has 300 clicks/revolution. The device is quite likely calibrated to output clicks(2pi*radius/300).

pash 11y ago
This sort of thing exists and is apparently what other commenters also have in mind, but I would argue that in your example the measuring device's output is a natural number n in the range [0, 300] and that its interpretation as a fraction of a circle's circumference is just that. Clearly the existence of describable irrational numbers implies that we can describe a measurement using an irrational number. That does not make a finite measurement essentially irrational in any meaningful sense.