One approach that I like uses two methods and looks for the convergence. One formula errors slightly above Pi for the result, and one formula errors slightly below Pi for the result. If BOTH formulas produce the same value then you can be pretty confident that you at least have Pi to that particular decimal place. If they are different then one or both of the formulas needs to adjust the formula by adding or subtracting a polynomial of the equation corresponding to the decimal location that produced the error.
The caveat? ... after adjusting the formula, I'm not sure you can pick up where you left off and continue. I think you must start all over again.
@$WMc%, you are not alone in that feeling. The ancient Greeks despised the concept of zero and it's twin infinity. There's a whole philosophy of mathematics called finitism http://en.wikipedia.org/wiki/Finitism which seeks to avoid the concept of infinity.
You seem to be skilfully holding two opposite points of view at the same time:)
On the one hand we have:
The ratio of a circle's circumference to its diameter is universal and invariant. So I would say that the value has always been there for us to discover, not to invent.
Which says to me that "there has always been pi".
...pi has nothing to do with the physical measurements of real circles of any kind, but only with the ideal measurements of those Platonic circles in flat, isotropic space that mathematicians talk about.
Which suggests pi does not exist "out there" but only in the minds of mathematicians.
Therefore, when there were no mathematicians there was no pi i.e. "there has always not been pi"
Kronecker,
God created the natural numbers, all else is the work of man.
I have to disagree. Man created the natural numbers as well. They are a high level concept that relies on awareness that one thing can be distinguished from another, that things can be placed in sets, the idea of succession and probably a few other things I have not thought of.
Again, with out sentient beings to be aware of all this there are no natural numbers.
You seem to be skilfully holding two opposite points of view at the same time:)
It's called superposition. Thoughts are quantum mechanical entities, after all.
So, to resolve this apparent dichotomy, we have to ask, "Does a Platonic ideal exist absent a mind to conceive it?" I would say "yes", if it can be shown that the concept is inevitable, IOW arrived at independently in two completely disjoint parts of the universe. Since imperfect circles exist everywhere (e.g. as projections of celestial orbs of various sorts), it would not be a stretch to assume that any advanced extraterrestrial would, at some point, contemplate the ideal, perfect circle (albeit not named after Plato), along with its properties, including diameter and circumference.
Lu is from China. My Chinese friend also has this power. I was teaching him English and he brought in 24 pages of Obama speeches that he had hand written. I was very impressed. But not as impressed as I was about to become. He said it was for English practice and I should speak some words on any of 24 pages. I grabbed something like page 12 and read a few words. He immediately began speaking the rest of the speech! He had memorized all 24 pages! Chinese schools are heavy into advanced memorization and I simply don't know how they do it.
I was joking, Humanoido. The recitation of PI to 67,890 places took 24 hours, 4 minutes. The process of memorizing PI to 67890 places took much longer.
I have to disagree. Man created the natural numbers as well. They are a high level concept that relies on awareness that one thing can be distinguished from another, that things can be placed in sets, the idea of succession and probably a few other things I have not thought of.
Again, with out sentient beings to be aware of all this there are no natural numbers.
Many mathematicians and philosophers would disagree with you, they feel that mathematical entities exist independently of human minds.
I was joking, Humanoido. The recitation of PI to 67,890 places took 24 hours, 4 minutes. The process of memorizing PI to 67890 places took much longer.
I know. It took him a year to do the memorization and an hour to recite it. Pi contests are quite popular where students memorize digits of pi. They usually top off around 200 digits. There are many web sites devoted to Pi contests.
There is a world record web site (year by year) regarding the memorization of Pi here. Even calculating Pi took a long time historically speaking. A German mathematician Ludolph van Ceulen used geometry to calculate 35 digits and it took most of his life. (1540-1610) He was so proud of his accomplishment that he had the digits engraved on his tombstone.
Humanoido,
One approach that I like uses two methods and looks for the convergence. One formula errors slightly above Pi for the result, and one formula errors slightly below Pi for the result. If BOTH formulas produce the same value then you can be pretty confident that you at least have Pi to that particular decimal place. If they are different then one or both of the formulas needs to adjust the formula by adding or subtracting a polynomial of the equation corresponding to the decimal location that produced the error.
Great idea. A technique like this could be automated including the polynomial corrections. However, what is the validity of each method? How do we know? I have not researched the programs used to calculate Pi to world records of digits in the 5 trillion or above range. I really wonder how they handled this issue of knowing the reliability of their programs.
Many mathematicians ... feel that mathematical entities exist independently of human minds.
Yes but Kronecker is stating that natural numbers were always there, created by God and hence independent of human minds. All other higher mathematical entities he says are dependent on the existence of man.
My main point is that Kronecker sets the level at which God's work stops and mans work begins rather too high, at the level of integers. That is to say that integers are already complicated mathematical concepts that are built on much simpler foundations. Perhaps we learn numbers and counting at such a young age and live with them so long that we forget how complicated they are.
As my young said when we were going through a childrens counting book together, "No dad, that's not three ducks, that's just another duck". Well, I could not fault him on the truth of that.
Followers of George Spencer-Brown's "Laws of Form" might tend to agree. Spencer-Brown starts his mathematical explorations from something a lot simpler than the integers.
67890 digits / 3600 seconds = 18.86 digits/second. He must be one of these fast talkers you hear about.
He used a man'ed biological serial interface with a vocal tract. It made only one mistake in the bit transmission. Interesting how words were used throughout, instead of Longs after 65,536.
>Jazzed wrote:
>Now, what is the probability of finding similar encodings in sqrt(2) or even "War and Peace"
There is a class of numbers called the "normal numbers" that have this property, that in a given base, all digits of that base are equally probable, furthermore, all n-tuples of digits are also equally probable in the long run.
The reason they are called "normal" is that "almost all" real numbers are of that type. Other classes of numbers are empty in comparison: integers, rational numbers, or the set of all decimal numbers missing the digit 9, they are all sets of "measure zero" in relation to the "normals".
However, from Wolfram, linked above: <As stated by Kac (1959), "As is often the case, it is much easier to prove that an overwhelming majority of objects possess a certain property than to exhibit even one such object....It is quite difficult to exhibit a 'normal' number!" (Stoneham 1970).>
It has apparently never been proven that pi or even sqrt(2) in particular are normal (and that, I guess, affects the chances that they do or do not encompass "War and Peace".)
Leon, ah, I see that you did. So, I feel in good company to mention it again and to amplify. The proof takes less than 3 pages in Hardy & Wright, "Introduction to the Theory of Numbers". On topic, the flip side is that pi has never been shown to be normal despite much evidence that it is, given the multitude of computed digits that have poured forth in the last few years.
I see that you also brought in Kronecker's famous quote, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk". Kronecker might not have found the elementary proof very convincing, precisely because of the leap from the "almost all numbers" back to constructing or exhibiting (hardly any) specific examples. Kronecker had a fundamental problem with the notion of the continuum. Maybe he was thinking of God's universe as an infinite but enumerable lattice of points in space and time with only quantum leaps between them. Could be.
Thanks for the post.
Hi guys, Im a newbie. Nice to join this forum.
Hi Tracy, welcome to the Parallax Forum. I'm honored to have your first post in my thread. We look forward to hearing more from you, about your Parallax projects and various ideas in the upcoming days.
Comments
One approach that I like uses two methods and looks for the convergence. One formula errors slightly above Pi for the result, and one formula errors slightly below Pi for the result. If BOTH formulas produce the same value then you can be pretty confident that you at least have Pi to that particular decimal place. If they are different then one or both of the formulas needs to adjust the formula by adding or subtracting a polynomial of the equation corresponding to the decimal location that produced the error.
The caveat? ... after adjusting the formula, I'm not sure you can pick up where you left off and continue. I think you must start all over again.
You seem to be skilfully holding two opposite points of view at the same time:)
On the one hand we have:
Which says to me that "there has always been pi".
Which suggests pi does not exist "out there" but only in the minds of mathematicians.
Therefore, when there were no mathematicians there was no pi i.e. "there has always not been pi"
Kronecker,
I have to disagree. Man created the natural numbers as well. They are a high level concept that relies on awareness that one thing can be distinguished from another, that things can be placed in sets, the idea of succession and probably a few other things I have not thought of.
Again, with out sentient beings to be aware of all this there are no natural numbers.
So, to resolve this apparent dichotomy, we have to ask, "Does a Platonic ideal exist absent a mind to conceive it?" I would say "yes", if it can be shown that the concept is inevitable, IOW arrived at independently in two completely disjoint parts of the universe. Since imperfect circles exist everywhere (e.g. as projections of celestial orbs of various sorts), it would not be a stretch to assume that any advanced extraterrestrial would, at some point, contemplate the ideal, perfect circle (albeit not named after Plato), along with its properties, including diameter and circumference.
-Phil
I was joking, Humanoido. The recitation of PI to 67,890 places took 24 hours, 4 minutes. The process of memorizing PI to 67890 places took much longer.
Many mathematicians and philosophers would disagree with you, they feel that mathematical entities exist independently of human minds.
There is a world record web site (year by year) regarding the memorization of Pi here. Even calculating Pi took a long time historically speaking. A German mathematician Ludolph van Ceulen used geometry to calculate 35 digits and it took most of his life. (1540-1610) He was so proud of his accomplishment that he had the digits engraved on his tombstone.
Bummer! It could be in a perpetual loop!
Yes but Kronecker is stating that natural numbers were always there, created by God and hence independent of human minds. All other higher mathematical entities he says are dependent on the existence of man.
My main point is that Kronecker sets the level at which God's work stops and mans work begins rather too high, at the level of integers. That is to say that integers are already complicated mathematical concepts that are built on much simpler foundations. Perhaps we learn numbers and counting at such a young age and live with them so long that we forget how complicated they are.
As my young said when we were going through a childrens counting book together, "No dad, that's not three ducks, that's just another duck". Well, I could not fault him on the truth of that.
Followers of George Spencer-Brown's "Laws of Form" might tend to agree. Spencer-Brown starts his mathematical explorations from something a lot simpler than the integers.
http://www.lawsofform.org/lof.html
67890 digits / 3600 seconds = 18.86 digits/second
He must be one of these fast talkers you hear about.
>Now, what is the probability of finding similar encodings in sqrt(2) or even "War and Peace"
There is a class of numbers called the "normal numbers" that have this property, that in a given base, all digits of that base are equally probable, furthermore, all n-tuples of digits are also equally probable in the long run.
The reason they are called "normal" is that "almost all" real numbers are of that type. Other classes of numbers are empty in comparison: integers, rational numbers, or the set of all decimal numbers missing the digit 9, they are all sets of "measure zero" in relation to the "normals".
However, from Wolfram, linked above: <As stated by Kac (1959), "As is often the case, it is much easier to prove that an overwhelming majority of objects possess a certain property than to exhibit even one such object....It is quite difficult to exhibit a 'normal' number!" (Stoneham 1970).>
It has apparently never been proven that pi or even sqrt(2) in particular are normal (and that, I guess, affects the chances that they do or do not encompass "War and Peace".)
I see that you also brought in Kronecker's famous quote, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk". Kronecker might not have found the elementary proof very convincing, precisely because of the leap from the "almost all numbers" back to constructing or exhibiting (hardly any) specific examples. Kronecker had a fundamental problem with the notion of the continuum. Maybe he was thinking of God's universe as an infinite but enumerable lattice of points in space and time with only quantum leaps between them. Could be.