Is Pi Infinite and What Does It Mean for Science?

[quantumdude]

I'm just glad Donde isn't here.

 

[Integral]

Originally posted by Tom
I'm just glad Donde isn't here.

Funny you should menttion that.. I was thinking of him also..

 

[Jug]

Is pi finite?

Originally posted by Tom
I'm just glad Donde isn't here.

What is the intrinsic value of any number, transcendental, irrational, etc., beyond what it can prove by empirical evaluation? For example, can the degree of arc be better defined by the purely abstract irrational pi than it can by the rational value of 355/113?

 

[selfAdjoint]

Originally posted by Jug
What is the intrinsic value of any number, transcendental, irrational, etc., beyond what it can prove by empirical evaluation? For example, can the degree of arc be better defined by the purely abstract irrational pi than it can by the rational value of 355/113?

Yes it can. And when pi comes up in formulas that physicists use, you don't wnat to complicate the logic by assuming pi is some approximation like 22/7 or 355/113, or 3.1416. Physics assumes space is a continuum and that the values it discusses can take on all real numbers. The exception is action, which can only take on integer multiples of h.

What is true is that physicists can never tell whether some particular value they work with is irrational, unless math tells them it is. So pi and e are known to be irrational (and transcendental), but h and alpha are not sure.

 

[Jug]

Have to disagree, SelfAdjoint. What I hear you saying is that we shouldn't complicate logic with the truth.

But as regards the subject proper, pi is of course infinite. In that regard then, and not being a maths person myself, I can only relate to the basics from which our mathematics derive - the perfect ratios given by Pythagoras: 0:1:1:2:3:4 - and where it is assumed that the cipher was intended by Pythagoras to represent the finite condition; which is to suggest that the cipher in place of denoting a lack of magnitude might in fact denote total magnitude.

My contention then is that the true ratio of pi cannot be arbitrarily determined but must in fact conform exactingly to some full set of ratios describing the finite condition. Just some thoughts on the thing...

 

[matt grime]

If that is what you believe, Jug, then you are commiting several common mistakes. The first is presuming that there is something special about decimal expansions (or any other base). pi is as easily descibed, and as easily calculted as 1/3. That is to say, given a line with integer units marked on it pi and 1/3 are just as easily located. in fact, I'd say pi is more easilyl located; look at the proof of existence of the first transcendental number.

pi, 1/3 sqrt(2) are just symbols reprsenting some cuachy sequence of rational numbers, they are all as useful as each other, pi perhaps more so. How is 22/7 more accurate than pi?

pi is not infinite. Its decimal expansion is not eventually periodic, but that isn't important. I can more easily construct sqrt(2) than 1/3 given a ruler and compass.

The rest of your post is mostly uninterpretably to my mind.

 

[Hurkyl]

My contention then is that the true ratio of pi cannot be arbitrarily determined but must in fact conform exactingly to some full set of ratios describing the finite condition. Just some thoughts on the thing...

If you mean that π is the limit of a sequence of fractions that approximate it, it may interest you to know that one model of the real numbers is just that; a real number is a sequence of 'converging' fractions.

 

[Jug]

Pi is infinite

MG, we appear to be equally incoherent to one another. By example, you ask:

1) "How is 22/7 more accurate than pi?" It is not, regardless of what value you give to pi.

2) "Pi is not infinite." Then what is its finite value?

 

[Hurkyl]

"Pi is not infinite." Then what is its finite value?

Pi.

pi is of course infinite.

Then prove it's bigger than 4.

 

[Jug]

Originally posted by Hurkyl
Pi.




Then prove it's bigger than 4.

Pi cannot possibly be bigger than 4. Prove that 4 is finite.

 

[matt grime]

Originally posted by Jug
MG, we appear to be equally incoherent to one another. By example, you ask:

1) "How is 22/7 more accurate than pi?" It is not, regardless of what value you give to pi.

2) "Pi is not infinite." Then what is its finite value?

I don't think 'equally' is the word you want.

First as pi is a number between 3 and 4, it is not infinte, is it? It is finite. If you think it is infinite, then you do not know what infinite means.

You were the person that suggested that some fraction was 'better' than pi.

I can tell you what pi is exactly: it is the ratio between a circle's diameter and its circumference. Just as sqrt(2) is the number (positive) which squares to two, just as 22/7 is the number which when multiplied by 7 gives the answer 22.

Look at Tim Gowers's discussions on what is solved when one solves an something.

Your replies to Hurkyl's post indicate that you ought to learn about things that you are speaking on.

 

[Hurkyl]

The definition of finite for ordinal numbers is:
n is finite if and only if n is a natural number.
4 is a natural number, thus as an ordinal number, it is finite.


The definition of finite for the extended real numbers is:
n is finite if and only if n is a real number.
4 is a real number, thus as an extended real number, it is finite.


The definition of finite for hyperreal numbers is:
n is finite if and only if there is a natural number m with -m < n < m.
5 is a natural number, and -5 < 4 < 5.
Thus, as a hyperreal number, 4 is finite.


The definition of finite for sets is:
S is finite if and only if it is bijective to a natural number¹.
4 is a natural number, and is bijective to itself.
Thus, as a set, 4 is finite.


Um... I can't think of any other (standard) mathematical terms labelled 'finite' or 'infinite' that apply to something labelled '4'. Is that good enough for you?


Hurkyl

1: In set theory, there is a "standard" model for the natural numbers in terms of sets. Since it is fairly common to label these sets by the symbol used to label the natural number it models, I figured I might as well include this definition too.

 

[Jug]

What y'all say sounds reasonable, standard and as curricula for an abstract, logarithmic progression of numbers for the defining of pi; yet, IMO, showing no relationship to any finite conclusion, particulrly as pertaining to a numbers set of such finality. How can you say that the pi ratio is finite when it merely represents one particular ratio of such finality?

I have to hold to the Pythagorean concept of mathematics as the science of describing exacting relationships. The concept, at least by my interpretation, allows for but one finite condition, all participles of which emanate from a finite fundamental and in return to which can only find final resolve. Furthermore, I would suggest that all participles emanating from the fundamental must demonstrate a repetitive numbers set of infinite progression. In short, whatever its value the pi ratio can only be infinite.

 

[HallsofIvy]

"Furthermore, I would suggest that all participles emanating from the fundamental must demonstrate a repetitive numbers set of infinite progression."

Isn't it remarkable how one can put together a sentence that sounds like it actually means something!

There are several problems I have with that sentence. First is the fact that "fundamental" is an adjective, not a noun so I cannot find a subject in it. On the other hand "numbers" is a noun rather than an adjective so I have no idea what "a repetitive numbers set" is. I guess it would be too much to point out that a "set", by definition, cannot be "repetitive".

Finally, I can't see how a "conclusion" can be either "finite" or infinite. What I would like is for Jug to tell us explicitely what definitions of "finite" and "infinite" he is using. They are clearly not the standard ones.

 

[Hurkyl]

I know the first digit in the decimal representation of pi is 3. Therefore, I know -4 < pi < 4. Thus, pi is finite.


If you would like to supply an alternative definition of "infinite" (either by saying what "infinite" is, or properties "infinite" has), we can discuss whether pi is finite or infinite with respect to that definition.

However, even if pi is infinite according to your definition, that does not change the fact that pi is finite according to the definitions I quoted.


(And for the record, your post made very little sense)

 

[Jug]

Originally posted by HallsofIvy
"Furthermore, I would suggest that all participles emanating from the fundamental must demonstrate a repetitive numbers set of infinite progression."

Isn't it remarkable how one can put together a sentence that sounds like it actually means something!

There are several problems I have with that sentence. First is the fact that "fundamental" is an adjective, not a noun so I cannot find a subject in it. On the other hand "numbers" is a noun rather than an adjective so I have no idea what "a repetitive numbers set" is. I guess it would be too much to point out that a "set", by definition, cannot be "repetitive".

Finally, I can't see how a "conclusion" can be either "finite" or infinite. What I would like is for Jug to tell us explicitely what definitions of "finite" and "infinite" he is using. They are clearly not the standard ones.

I was hoping we might avoid the pedantry. By "fundamental" is of course meant fundamental wavelength. Do we really need to engage in a discussion over ordinary physics terms?

As to your claim that a "set" cannot be repetitive, what is 1.185185185...ad infinitum, if not a repetitive set?

 

[Hurkyl]

I was hoping we might avoid the pedantry. By "fundamental" is of course meant fundamental wavelength. Do we really need to engage in a discussion over ordinary physics terms?

Oh, of course, it was so obvious.

As to your claim that a "set" cannot be repetitive, what is 1.185185185...ad infinitum, if not a repetitive set?

A decimal expansion? A sequence of digits, perhaps?

 

[matt grime]

the solution of 1000(x-1)-185=x-1?

I'm still also at a loss to understand how a ratio is infinite, in whatever sense you use.As yoy adopt the pythagorean attitude of exactitude, whatever that might be, can you get more precise than the statement pi is the ratio of the circumference to the diameter of a circle?

 

[Jug]

Is pi finite?

Originally posted by matt grime
the solution of 1000(x-1)-185=x-1?

I'm still also at a loss to understand how a ratio is infinite, in whatever sense you use.


As yoy adopt the pythagorean attitude of exactitude, whatever that might be, can you get more precise than the statement pi is the ratio of the circumference to the diameter of a circle?

If any number is the ratio to some fundamental that is itself finite, how can that number be said to also be finite? Y'all may be perfectly right in your assertion but, with all due respect, I'm not going to take your word for it. The academia is ever being corrected.

 

[matt grime]

With least due respect, are you trying to be a crank or a troll?

You perhaps want to change some of the prepositions in that last post. You at least want to rewrite it so it makes sense, anyway.

1 is finite, I think, though it's rather unclear what you think finite means. It is also a ratio, of 1/1, and I think it's fundamental, in some sense...

 

[Jug]

If any number is a ratio to some fundamental that is itself finite, how can that number be said to be finite? Y'all might be right in your assertions but, with all due respect, I'm not going to take your word for it. Saying that pi is finite is like saying that the sky is finitely blue.

 

[quantumdude]

Originally posted by Donde (let's stop kidding ourselves, shall we?
Saying that pi is finite is like saying that the sky is finitely blue.

What are you talking about? Earlier in this very thread, you acknowledged that p is less than 4. Of course it is finite, by your own admission!

p is simply the ratio of the circumference of a circle to its own diameter. Consider any given circle. Since the circumference is not infinite and the diameter is not zero, p is finite.

This question was answered in the first two posts. Why on Earth is this silly debate still going on?

 

[Zero]

Crackpot rule #1: when you make up your own language, no one can ever prove you wrong!

 

[Hurkyl]

If any number is a ratio to some fundamental that is itself finite, how can that number be said to be finite?

By proving that number satisfies the definition of finite.

Y'all might be right in your assertions but, with all due respect, I'm not going to take your word for it.

We're not asking you to take our word for it... there's a reason we use something called a "proof".

Saying that pi is finite is like saying that the sky is finitely blue.

Is it? In what way?

 

[lethe]

Originally posted by selfAdjoint
Physics assumes space is a continuum and that the values it discusses can take on all real numbers. The exception is action, which can only take on integer multiples of h.

i know that in Bohr-Sommerfeld quantization, this was the approach, but does that have any resemblance to modern quantum theory? i have not heard said about modern quantum theory that the action must be discrete. i would be surprised to find this is true.

 

[Zero]

The word "infinite" has a very specific mathematical meaning. No number that falls between 3 and 4 can ever satisfy that meaning. What else is there to talk about?

I mean really, even from a common sense viewpoint, if you asked any person with an IQ over 90 if "infinite" was bigger than 4, what do you think the right answer would be?

 

[Michael D. Sewell]

Is it your contention that a circle with a radius of 1 meter contains within it an infinite amount of area? Can a silo that is 4 Meters in diameter and 10 meters high hold an infinite amount of grain? How much coffee can I fit in my cup? Is the volume of my cup equal to the volume of the silo? How large is a baseball?

If this discussion started pi hours ago and ends pi hours from now, does that mean that we are half-way done? Or will it go on forever...?

 

[Jug]

Originally posted by Tom
What are you talking about? Earlier in this very thread, you acknowledged that p is less than 4. Of course it is finite, by your own admission!

p is simply the ratio of the circumference of a circle to its own diameter. Consider any given circle. Since the circumference is not infinite and the diameter is not zero, p is finite.

This question was answered in the first two posts. Why on Earth is this silly debate still going on?

Tom, 1) Please do not put words in my mouth. I have never admitted to any such (ridiculous) thing as pi being finite.

2) If pi were the "finite" value for describing ratio of circumference to diameter, are you then saying that there is no other value that is capable of describing that same relationship? For if there were, then the pi ratio could not possibly be finite, could it?

 

[Hurkyl]

1) Please do not put words in my mouth. I have never admitted to any such (ridiculous) thing as pi being finite.

You have admitted that pi satisfies a condition which is (one) definition of finite.

If pi were the "finite" value for describing ratio of circumference to diameter, are you then saying that there is no other value that is capable of describing that same relationship?

There is no other value that describes the relationship, but that has nothing to do with the finiteness of pi.

By Finite we mean that both ends of some interval are reachable.

No, Organic, that is what you mean, not what we mean.



Anyways, Jug: If you want to talk math, then do so. If you want to continue talking about your own personal ideas and theories, I suggest moving over to an appropriate forum, such as Theory Development or one of the Philosophy forums.

 

[quantumdude]

Jug,

Learn to read, will you please? I said that you admitted to pi being less than 4. That automatically means that it is finite.

Here is where you said it:

Originally posted by Jug
Pi cannot possibly be bigger than 4. Prove that 4 is finite.