Pi

HallsofIvy said:
pi can be clearly "expressed" by "the ratio of the circumference of any circle divided by the radius of that circle."
Click to expand...
Yes, but those circles must be perfect, in order to express pi (not approximately, but as pi). I read Jeff's comments to mean that we cannot generate pi by anything we could measure in the universe. Perfect circles exist only in our mind (no physical correlate). We can't do numerical calculations with pi (we need an approimation). Processes that generate pi through limit or infinite expressions give exact representations, but they can't generate pi in decimal form.

Earlier, did you imply that the exact value of pi is an approximation?

:)
 
HallsofIvy said:
I don't know what you mean by "cannot be expressed in the life of the universe". I would say that pi can be clearly "expressed" by "the ratio of the circumference of any circle divided by the radius of that circle." If you mean that it is an infinitely continuing decimal, that has nothing to with the existence of the number itself or whatever you mean by "physical correlate". The number \(\displaystyle \sqrt{2}\) has an infinitely continuing decimal and is the length of the diagonal of a square with side length 1. If you agree that "1" has a "physical correlate" then you must agree that "\(\displaystyle \sqrt{2}\)" has a "physical correlate" and that is no different from \(\displaystyle \pi\).
Click to expand...
What I mean by a physical correlate is that it can be physically observed. Any measurement you make of the circumference and ratio of a circle will be rational numbers, and their ratio will necessarily be a rational number. Numbers like pi exist only in the realm of Platonic ideas, and none are observable in the physical world.
 
JeffM said:
What I mean by a physical correlate is that it can be physically observed. Any measurement you make of the circumference and ratio of a circle will be rational numbers, and their ratio will necessarily be a rational number. Numbers like pi exist only in the realm of Platonic ideas, and none are observable in the physical world.
Click to expand...

I agree π is a figment of our "nightmares" - I think 0.99999..[repeat] is of the same class
 
Subhotosh Khan said:
I agree π is a figment of our "nightmares" - I think 0.99999..[repeat] is of the same class
Click to expand...
Actually, I have fewer mental qualms with 0.9999…. This probably explains something about my psyche because I am not sure I there is ANY logical reason for my different feelings. (Hey, I am just an historian who ended up in banking.)

The number one is definitely observable: I have one wife, not none or more than one. (Actually if I did have more than one, I suspect my wife would ensure that I only “enjoyed” (???) that state of affairs for a brief interval.)

Within the realm of Platonic ideas, I have no difficulty with

[math]\dfrac{1}{3} = 3 * \sum_{j=1}^{\infty}10^{-j} \implies 3 * \dfrac{1}{3} = 9 * \sum_{j=1}^{\infty} \dfrac{1}{10^j} \implies 1 = \sum_{j=1}^{\infty} \dfrac{9}{10^j}.[/math]
But of course I do not believe that sums with infinite terms are computable or observable. So it is quite probably irrational to distinguish between 0.999… and pi. (And yes, I am familiar with Archimedes’s application of the sandwich theorem with respect to pi.)
 
So, basically, no one here wants to deal with MATHEMATICS which deals mainly with things that are not "computable" or "observable"!
 
No. I think some people are saying that certain aspects of mathematics are NECESSARILY Platonic and do not deal with approximations at all and that other aspects are not necessarily Platonic and so must deal with approximations in certain cases. It was you who defined the “best approximation” as an exact value. Others perceive an approximation as being, by definition, inexact.
 
The exact value of \(\displaystyle \pi\) is \(\displaystyle \pi\). Just because we can't write it using the Hindu-Arabic numeral system (or the Roman, or the Egyptian, or ...) doesn't mean it doesn't exist!
 
Harry_the_cat said:
The exact value of \(\displaystyle \pi\) is \(\displaystyle \pi\). Just because we can't write it using [numerals] doesn't mean it doesn't exist!
Click to expand...
I've always suspected you're a purebred.


Folks will need to excuse me, as I've lost track of the point of this thread.

?
 
You must log in or register to reply here.
Top