Depends on what "it" is. The formula doesn't change, but the output of the formula does, depending on how many times you iterate.
1.41 is different than 1.414 is different than 1.4142, etc.
The formula stays the same. The output changes.
Post by twetch#3375
Ok, so? is sqrt(2) a quantity or is it the output of a formula? If it is a quantity then there is no finite formula for it. If it is the output of a formula then it is something that converges on the quantity but never reaches it.
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Arithmetic is broken. Modular arithmetic is ok though.
"The square root of two" is a way to describe a formula. The formula is a construction that generates outputs. The outputs differ depending on how many iterations you run.
There is no "converging" on a quantity. The outputs just gain decimals.
Perspective A: "The sqrt(2) is a quantity whose value we don't know, can never know, and can only approximate"
Perspective B: "The sqrt(2) is a formula that generates outputs of increasing decimal length, starting with 1.41 and never repeating"
If I draw the diagonal of a square, do you think that this line has no distance, or that the distance along the line is not a quantity, or something like that? Incommesurate ratios were studied by Euclid.
Euclid could show that his GCD algorithm would not always halt for any two lines that he could draw. But we still have a concrete representation of the irrational quantity in the diagonal of a square. https://en.wikipedia.org/wiki/Euclidean_algorithm
en.wikipedia.org Euclidean algorithm - Wikipedia each step of the computation results in a number with a finite representation that is closer to the quantity than the previous one. There is no positive distance that we don't get closer to eventually.
There is no reason to posit "getting closer to a quantity."
Imagine we're talking about English sentences. I can give you a formula to construct a sentence. After each iteration, it contains more words. That doesn't imply it's "approaching" anything.
The greeks knew that pi phi ... are incommensurable magnitudes and not numbers. There is always another digit and it goes on indefinitely so you are not really getting close to anything.
Sentences aren't associated with quantities, so that's an invalid analogy.
if you exist on a circle but as you walk across it, you observe an infinite “non countable” line, this doesn’t mean there is no circle. it merely means there is bias
you would have to exist outside the circle to call it a circle (in this scenario)
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