1. I was just thinking about how computers calculate Pi so exactly. Unless I'm misunderstanding something, would the nature of a circle mean that either the length of the circumference or the length of the diameter itself has an infinite number or significant figures?

The idea that information can be calculated to such incredibly small numbers makes me think about how there is no such "Planck scale" when it comes to mathematical calculation. Or is there? Would we ever reach a limit where we could only ever (for example) calculate Pi to a maximum of xy decimal places, even if the number of decimal places themselves is infinitely long?

2.

3. Planck scale ia about physical quantities. π is a mathematical constant, defined in terms of mathematical objects, circle and diameter.

4. Sorry, I guess I didn't really explain my question all that clearly.

Is there a smallest possible unit of information that can determined before anything smaller/finer is inherently unknowable simply because it's too small to discern any more individual difference?

In the same way that Planck Units refer to the physical Universe, are there any similar restrictions on the nature of information that is inherently knowable?

Could we calculate Pi to a Googol of decimal places, but the Googol+1st decimal place is too small to tell what number it really is without a margin of error?

Or is the idea of a "smallest possible unit of information" something that just doesn't apply to the mathematical realm?

5. Originally Posted by Daecon
Sorry, I guess I didn't really explain my question all that clearly.

Is there a smallest possible unit of information that can determined before anything smaller/finer is inherently unknowable simply because it's too small to discern any more individual difference?

In the same way that Planck Units refer to the physical Universe, are there any similar restrictions on the nature of information that is inherently knowable?

Could we calculate Pi to a Googol of decimal places, but the Googol+1st decimal place is too small to tell what number it really is without a margin of error?

Or is the idea of a "smallest possible unit of information" something that just doesn't apply to the mathematical realm?
Mathman actually answered your quite-insightful question, but subtly (evidently too much so). The mathematical realm, being independent of physical reality (although remarkably useful to describe physical reality), operates under a quite different (and less restrictive) set of rules than "reality." There is no fundamental restriction on the precision with which one might compute pi. We can keep going as long as we wish. There's even an algorithm that yields the nth (binary) digit of pi (in hex) -- with no restriction on the value of n -- without having to compute the (n-1) preceding digits.

So, the "granularity" of mathematics is not the granularity of physics.

6. Ah, I see. Thanks for that. Score one for Maths.

7. Originally Posted by Daecon
Ah, I see. Thanks for that. Score one for Maths.

A fun exercise for a rainy day is to express various quantities in Planck units. For example, the size of the universe is some 61 orders of magnitude of Planck units. We've already computed pi to more than a trillion decimal digits, so there's a concrete example of a case where the resolution in calculating a mathematical quantity is much, much, much finer than that of nature (if you treat the Planck length as the fundamental grain size). If you repeat this exercise for other quantities, you find similarly small numbers of orders of magnitude, relative to the numerical dynamic range of maths.

8. I admit I'm still struggling to wrap my head around the "how" of such a small number being able to be calculated with such incredible accuracy. Obviously it's impossible to actually measure the value on so small a scale, and wouldn't either the circumference or the diameter itself also have a value with an infinite amount of decimal places...? I'm just not a mathematical person.

That's not actually a real question, just a lamentation on my lack of comprehension.

9. Originally Posted by Daecon
I admit I'm still struggling to wrap my head around the "how" of such a small number being able to be calculated with such incredible accuracy. Obviously it's impossible to actually measure the value on so small a scale, and wouldn't either the circumference or the diameter itself also have a value with an infinite amount of decimal places...? I'm just not a mathematical person.

That's not actually a real question, just a lamentation on my lack of comprehension.
It's not a lack of comprehension at all! Clearly, you are thinking deeply about what that precision implies, and are properly astonished. It is practically absurd to compute pi to so many digits. Just 10 digits means that we can compute the circumference of the earth to a fraction of a millimeter. To compute a trillion decimal digits (as we have) is to take absurdity to new extremes of absurdity, if one were to insist connecting that precision to any practical measurement problem. We do it entirely for other reasons.

10. Originally Posted by tk421
Originally Posted by Daecon
I admit I'm still struggling to wrap my head around the "how" of such a small number being able to be calculated with such incredible accuracy. Obviously it's impossible to actually measure the value on so small a scale, and wouldn't either the circumference or the diameter itself also have a value with an infinite amount of decimal places...? I'm just not a mathematical person.

That's not actually a real question, just a lamentation on my lack of comprehension.
It's not a lack of comprehension at all! Clearly, you are thinking deeply about what that precision implies, and are properly astonished. It is practically absurd to compute pi to so many digits. Just 10 digits means that we can compute the circumference of the earth to a fraction of a millimeter. To compute a trillion decimal digits (as we have) is to take absurdity to new extremes of absurdity, if one were to insist connecting that precision to any practical measurement problem. We do it entirely for other reasons.
And there is also the complication of relativity. If we construct a 4-dimensional hypercube of Planck unit sides, then in a different frame of reference, we have to deal with length contraction and time dilation. This is because 4-dimensional hypercubes are not Lorentz-invariant objects. Thus, one can't make spacetime granular at the Planck scale by means of a 4-dimensional grid because such a grid would form a preferred frame of reference in violation of special relativity. Such a grid is not even rotationally invariant, so the problem still exists even in 2- or 3-dimensional space (consider the hypotenuse of a right-triangle with Planck length sides).

11. BTW, since and is irrational (meaning it can't be expressed as a fraction), then either c or d must be irrational. (Rearrange to get and note that the ratio of two rational numbers is rational.) Irrational numbers have infinite non-repeating decimal expansions.

12. I confess I have no idea how relativity is relevant here. Try this........

Forget geometry (allowable because the circle is a manifold and hence locally homeomorphic to the real line ), and let's consider an arbitrary field endowed with a total ordering.

Suppose are non-empty subsets of such that

1. is the set of all lower bounds for

2. is the set of all upper bounds of

This called a "Dedekind cut"

Suppose is a least upper bound for ; obviously . But then is the smallest element in so it is a lower bound for all and hence .

Under this circumstance I will say that the "cut" goes through .

Since is arbitrary this implies that if is a least upper bound for and is a greatest lower bound for then and therefore there are no "gaps", of Planck size or otherwise

13. I'm sorry, I didn't explain clearly. Relativity isn't relevant to my question.

I'm not actually talking about Physical Plank-scale measurements, but rather the mathematical idea of a "smallest possible unit of measurement ever" akin to the kind of scales that Planck lends his name to, but applied to the concept of information as opposed to the concept of physical reality.

The idea of, on an epistemological level (I so hope I'm using the right word), that we can't measure or compute/calculate something smaller than some "smallest" value of information because mathematical measurement is no longer capable of determining the result with sufficient accuracy.

For example, calculating Pi to a googol of decimal places leads us to eventually being unable to determine whether that googol and first number after the decimal point has a margin of error because the value is too small to calculate accurately, as opposed to physically measuring it using scales that apply to physical reality. However I accept the explanation that the calculations of mathematics are not bound by the limitations of the physical Universe's quantum scale.

Although I am amazed as to how we can calculate the value so accurately to begin with when the values of either the circumference or diameter having what must be an infinity of decimal figures itself.

14. Originally Posted by Daecon
I'm sorry, I didn't explain clearly. Relativity isn't relevant to my question.

I'm not actually talking about Physical Plank-scale measurements, but rather the mathematical idea of a "smallest possible unit of measurement ever" akin to the kind of scales that Planck lends his name to, but applied to the concept of information as opposed to the concept of physical reality.

The idea of, on an epistemological level (I so hope I'm using the right word), that we can't measure or compute/calculate something smaller than some "smallest" value of information because mathematical measurement is no longer capable of determining the result with sufficient accuracy.

For example, calculating Pi to a googol of decimal places leads us to eventually being unable to determine whether that googol and first number after the decimal point has a margin of error because the value is too small to calculate accurately, as opposed to physically measuring it using scales that apply to physical reality. However I accept the explanation that the calculations of mathematics are not bound by the limitations of the physical Universe's quantum scale.

Although I am amazed as to how we can calculate the value so accurately to begin with when the values of either the circumference or diameter having what must be an infinity of decimal figures itself.
"smallest possible unit of measurement ever" There is no such thing in mathematics.

15. As I've learned, thanks to this thread.

Infinitesimals are truly infinite.

16. quoting KJW

"the problem still exists even in 2- or 3-dimensional space (consider the hypotenuse of a right-triangle with Planck length sides)"

>That hypotenuse would have a lenght in decimals of 1.414213562 x 1 Planck unit

Yet in Planck units this distance cannot be expressed (It is one unit or 2 units, noting in between available)

Does that imply that a Planck unit must be subdividable into even smaller units ?
(Let's call them sub-Planck-units)
Because that hypotenuse distance is still a real distance, no ?
This would suggest a fractal nature of space, systems ever dividable into smaller units, no ?

Because : I could take 2 such 'sub-Planck-units' and put those distances on a right-triangle again,
and then a new hypotenuse would appear, in need of again smaller subdivisions and so on.

(No intention to start on spacetime, no place for that here, just mathematical implications that are interesting)

17. Originally Posted by Noa Drake

Does that imply that a Planck unit must be subdividable into even smaller units ?
(Let's call them sub-Planck-units)
Because that hypotenuse distance is still a real distance, no ?
This would suggest a fractal nature of space, systems ever dividable into smaller units, no ?

Because : I could take 2 such 'sub-Planck-units' and put those distances on a right-triangle again,
and then a new hypotenuse would appear, in need of again smaller subdivisions and so on.
You should take another look at post #2 by Mathman.

18. Originally Posted by Daecon
I'm sorry, I didn't explain clearly. Relativity isn't relevant to my question.

I'm not actually talking about Physical Plank-scale measurements, but rather the mathematical idea of a "smallest possible unit of measurement ever" akin to the kind of scales that Planck lends his name to, but applied to the concept of information as opposed to the concept of physical reality.
I was quite aware that you were talking about the purely mathematical realm, but I took your question to be based on an underlying assumption about the nature of a discrete physical reality. I wanted to show that even for physical reality, the notion of Planck-sized discreteness was untenable at least in the simple case of a grid, and that any notion of Planck-sized discreteness was not going to be conceptually straightforward.

19. Originally Posted by Noa Drake
quoting KJW

"the problem still exists even in 2- or 3-dimensional space (consider the hypotenuse of a right-triangle with Planck length sides)"

>That hypotenuse would have a lenght in decimals of 1.414213562 x 1 Planck unit

Yet in Planck units this distance cannot be expressed (It is one unit or 2 units, noting in between available)

Does that imply that a Planck unit must be subdividable into even smaller units ?
(Let's call them sub-Planck-units)
Because that hypotenuse distance is still a real distance, no ?
This would suggest a fractal nature of space, systems ever dividable into smaller units, no ?

Because : I could take 2 such 'sub-Planck-units' and put those distances on a right-triangle again,
and then a new hypotenuse would appear, in need of again smaller subdivisions and so on.
I don't know what I was saying means about physical reality at the smallest scales. All I can say is whatever it means is not going to conceptually straightforward.

20. Originally Posted by KJW
Originally Posted by Daecon
I'm sorry, I didn't explain clearly. Relativity isn't relevant to my question.

I'm not actually talking about Physical Plank-scale measurements, but rather the mathematical idea of a "smallest possible unit of measurement ever" akin to the kind of scales that Planck lends his name to, but applied to the concept of information as opposed to the concept of physical reality.
I was quite aware that you were talking about the purely mathematical realm, but I took your question to be based on an underlying assumption about the nature of a discrete physical reality. I wanted to show that even for physical reality, the notion of Planck-sized discreteness was untenable at least in the simple case of a grid, and that any notion of Planck-sized discreteness was not going to be conceptually straightforward.
Thank you for the clarification.

21. Originally Posted by tk421
Just 10 digits means that we can compute the circumference of the earth to a fraction of a millimeter.
And only about 45 decimal places allows us to calculate the circumference of the observable universe to an accuracy of the size of a proton.

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