Pi topic

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eMTe
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Pi topic

Post by eMTe »

It all started with this pm. And this pm is actually quoted post from another forum. I had to create separate topic to not pollute "random" topic more.

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"pi is certainly a finite real number. It's between 3 and 4. It's not even a particularly large number.

You are correct that it takes an infinite amount of information to specify it in decimal notation. That is, for every counting number n, there's a particular decimal digit in place n. (To the right of the decimal point. We can ignore the 3 for this.) So in decimal notation it takes infinitely much information to specify pi.

But that's just a limitation of the decimal notation. There are many finite ways to define pi. The Internet is full of them. I'm too new to post links, but if you google "formulas for pi" you will find lots of relatively short (in particular, finite) expressions that completely characterize pi.

My favorite is sum(1/n^2; n = 1, 2, 3, ...) = pi^2/6. That's really a cool fact, don't you think? Euler discovered it in 1735. That number is zeta(2), for fans of Riemann's zeta function.

From zeta(2) = pi^2/6 it follows that

pi = sqrt(6 * zeta(2))

I admit that we are still hiding an infinitary process in this notation. We're summing an infinite series of real numbers. But these days, modern math has figured out how to logically derive the operations on infinite series from first principles, using nothing more than the laws of logic and the axioms of set theory. We now have a comprehensive theory of how to represent infinitary processes using finite strings. Or computer programs if you like. We could easily program any of these formulas in a computer and generate as many digits of pi as we want, given constraints on time and electricity. That's a major human intellectual achievement. Not everyone's completely on board with it yet

We have finitized the infinite. That's the important philosophical point. And it's a relatively recent historical development.

Pi isn't "random." Its decimal digits are the result of a deterministic process. Even if the distribution of the digits turns out to be statistically random; those digits are still generated by a deterministic process. And it doesn't require infinite information to specify ... a short mathematical formula does the job perfectly well.

Generally speaking, it's helpful to put aside our high-school understanding of pi as being defined by a circle.

Rather, pi is a particular real number that arises naturally in the study of many mathematical phenomena. It's half the period of the cosine function, for example. And the cosine is not the high-school cosine that's defined by triangles. It's the modern cosine, defined by an infinite series; expressible in finitely many symbols; and (at least in principle) reducible to logic.

In other words, our modern understanding of pi is that it's an algorithm. It doesn't terminate, but it cranks out as many digits as you want. And algorithms are real and solid and they run the world. In that sense, we've tamed pi.

Hope something in here sheds some light."
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Zyx wrote:We have a way to accurately define pi and it is π. That's it, no infinite amount of information needed.

We haven't finitized the infinite, on the contrary, mathematics have found a way to work with infinity to the point that they can handle an infinite number of infinites.

However, this does not mean that pi is finite. Curiously, this doesn't matter. Really, it doesn't. The funniest thing ever is that for any calculation we need pi for, 40 digits is enough. And by enough I don't mean good enough, just literally enough for anything in the universe...
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Well, pi is finite as it was logically concluded in the first post. It's a number between 3 and 4, or even more precisely between 3.141 and 3.142. And if it isn't then we have a problem, because either there's something wrong with maths, with concept of infinity or with our understanding of Universe structure.

Logically, there must be a finite number of digits to describe what is pi, even if it is an extremely huge number of digits.

If the number of digits would be "infinite" than probably also all elementary particles we know at the moment would be built of infinite number of infinitely small particles. This is of course philosophically possible. Mathematically too. Physically, well, who knows. Anyway, it's against intuition that things can be infinite and infinitely small without defined and measurable borders.

Since I am predominantly interested in languages we use to communicate (maths is also a language) I think that "pi" is simply a paradox, similar to Zeno's. "Infinite" number of digits we possibly need to depict Pi means that language is not precise and Pi is just another abstract concept, simple observation of the world around us just like that the snow is cold and sand is hot.

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The problem is that "it doesn't matter whether pi is finite or infinite" doesn't fully resolve the issue. It is indeed taming, but not explaining. It's the same kind of expression like "I am happy being a non-believer". You may be happy, but your happiness doesn't prove that God doesn't exist. :wink:
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Re: Pi topic

Post by Zyx »

eMTe wrote:Well, pi is finite as it was logically concluded in the first post. It's a number between 3 and 4, or even more precisely between 3.141 and 3.142. And if it isn't then we have a problem, because either there's something wrong with maths, with concept of infinity or with our understanding of Universe structure.
How come? Pi is not the only real number which has a never-ending decimal representation. There are infinite amount of those, like 1/3. In fact, there are infinite amount of never-ending decimal numbers between 3.141 and 3.142.
If the number of digits would be "infinite" than probably also all elementary particles we know at the moment would be built of infinite number of infinitely small particles. This is of course philosophically possible. Mathematically too. Physically, well, who knows. Anyway, it's against intuition that things can be infinite and infinitely small without defined and measurable borders.
Intuition is often wrong. You are mistaking mathematics as the truth about the physical world, when it's just a tool that physics use to explain it. There are many things in mathematics that irrational, like a whole group of irrational numbers which have no relation to the real world like i (square root of -1). You will not find a circle in the physical world which has exactly pi as the ratio of its circumference to its diameter.
"Infinite" number of digits we possibly need to depict Pi means that language is not precise and Pi is just another abstract concept, simple observation of the world around us just like that the snow is cold and sand is hot.
Why would a finite decimal representation of pi be a requirement for precise language? π defines pi precisely, it is the ratio of a circle's circumference to its diameter. You're again bringing your snow is cold statement, which in my opinion can only be proven by circular reasoning whereas pi is an abstract concept, but is a true statement.
The problem is that "it doesn't matter whether pi is finite or infinite" doesn't fully resolve the issue. It is indeed taming, but not explaining. It's the same kind of expression like "I am happy being a non-believer". You may be happy, but your happiness doesn't prove that God doesn't exist. :wink:
There is no problem, pi, in its decimal representation (why you're so hung up on the decimal representation is beyond me) is never-ending. However, we know precisely how much pi is (but we can't have the full decimal representation).
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Re: Pi topic

Post by eMTe »

I am hung up on the decimal representation, because decimal representation is a language. We are probably interested in different issues and here lies the root of misunderstanding. 8)
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Re: Pi topic

Post by Drasir-Vel »

Yeah, there are computers calculating the numbers of pi. So we've just got to use the formula they're using to define it.
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Re: Pi topic

Post by eMTe »

Well, computers always "calculate". They use the brute force method to "calculate". They read, write, explain, understand, reread, rewrite, return, copy, paste, undo and do whatever we need them for and suddenly they make this heavily personalised weird ** sound to announce us the solution in a language we have learned them to use.

On a topic. It's always kind of amazing for me that people don't find the difference between graphical explanation and decimal explanation of the same mathematical constant fascinating.

"Pi is approximation" is true for its decimal representation, but its graphical representation is perfection, not the approximation.
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