Can physics deal with the existence of Pi?

[richard9678]

Summary:

Can physics deal with a question on the existence of Pi

Hi. I'm not sure if physics/cosmology can deal with my question. I suspect not, but I'll ask it anyway. The answer could be "No" and that would be "end of".

Is there any situation, where Pi = 3.142...does not exist as a fact? Thanks. Rich

 

[PeroK]

I think there is a confusion of ideas here. $\pi$ is a number.

 

[richard9678]

Yes, it a number. We've discovered it. It's discoverable a long time. But was there ever a time it was undiscovered because of some physics reason? Does it require space for it to "exist"?

 

[Ibix]

Why would physics prevent the study of geometry? You can calculate the value of $\pi$ yourself if you know enough calculus to derive the Taylor series for $\tan^{-1}$.

 

[BWV]

Circles are physically impossible as well, but we still have them

 

[PeroK]

Yes, it a number. We've discovered it. It's discoverable a long time. But was there ever a time it was undiscovered because of some physics reason? Does it require space for it to "exist"?

Pi exists, as a number, as a consequence of the axioms of number theory. It's very useful and has some physical interpretations, but mathematics itself doesn't depend on physics.

 

[richard9678]

Just because no-one is there to discover it, does not mean it's not real or that it does not exist. With that thought in mind, let's say we are 100,000 years after the big bang, is there anything in physics knowledge that says Pi cannot have existed. I think the basic premise would be, if we have space Pi must exist. If the answer is "no" Pi cannot have not existed, we go farther back in time until we say "yes". If that's possible.

 

[etotheipi]

Huh?

The only space you need is $\mathbb{R}$.

 

[PeroK]

Just because no-one is there to discover it, does not mean it's not real or that it does not exist. With that thought in mind, let's say we are 100,000 years after the big bang, is there anything in physics knowledge that says Pi cannot have existed. I think the basic premise would be, if we have space Pi must exist. If the answer is "no" Pi cannot have not existed, we go farther back in time until we say "yes". If that's possible.

Are you thinking of $\pi$ as the ratio of the circumference to the diameter of a "real" circle?

 

[Andrew Mason]

Pi is defined as the ratio of the circumference of a circle to its diameter in a Euclidean plane. The diameter of a circle is defined as twice the radius, the radius being the shortest distance between the centre of the circle and a point on the circle as measured in the Euclidean plane defined by the circle. Since all Euclidean planes are indistinguishable, this ratio does not change. So Pi does not change.

However, the earth surface is not a Euclidean plane and geodesic paths in real space-time (the shortest space-time metric between two points) do not follow a Euclidean plane. So the ratio of a circle to its diameter as measured in curved space-time or in curved space will, generally, be different than Pi and will vary depending the curvature. But Pi will not change.

AM

 

[fresh_42]

Just because no-one is there to discover it, does not mean it's not real or that it does not exist. With that thought in mind, let's say we are 100,000 years after the big bang, is there anything in physics knowledge that says Pi cannot have existed.

The existence of numbers has nothing to do with a physical existence. Even $1$ does not physically exist-.

I think the basic premise would be, if we have space Pi must exist. If the answer is "no" Pi cannot have not existed, we go farther back in time until we say "yes". If that's possible.

This makes no sense.

Are you thinking of $\pi$ as the ratio of the circumference to the diameter of a "real" circle?

There is no physical circle, it simply does not exist. It's always a model (path of motion), and if realized (circles in the sand), not round anymore under an electron microscope.

 

[fresh_42]

I suspect this is what @PeroK was hinting to the OP

I know. I wasn't really addressing @PeroK here. I just had to take the words to somehow emphasize the different meaning of existence for the OP.

 

[vanhees71]

$\pi$ has nothing to do with physics. It's simply defined by the definition of the cosine function via its power series, [EDIT: typo corrected in view of #15]
$$\cos z=\sum_{k=0}^{\infty} \frac{1}{(2 k)!} (-1)^k z^{2k},$$
such that it's the smallest positive real number, for which $\cos \pi=-1$, which implies btw. that $\cos(\pi/2)=0$. So you can define $\pi/2$ as the smallest positive real zero of cos.

 

[S.G. Janssens]

Yes, the issue was settled in post #2.

I remember about an experiment I had to do in school at some point. To my horror, it involved the "experimental determination of $\pi$". In hindsight, this may have contributed to my decision to switch to mathematics at the end.

(On the other hand: Later on, when I studied physics first, one of the teachers that showed most sympathy for my stubborness and pedantry was an experimental condensed matter prof. that I still think about with a lot of sympathy.)

 

[etotheipi]

$\pi$ has nothing to do with physics. It's simply defined by the definition of the cosine function via its power series,
$$\cos z=\sum_{k=0}^{\infty} \frac{1}{(2 k)!} (-z)^{k},$$

I think there's a small typo, that$$\cos z=\sum_{k=0}^{\infty} \frac{(-1)^k}{(2 k)!} z^{2k},$$

 

[A.T.]

I remember about an experiment I had to do in school at some point. To my horror, it involved the "experimental determination of $\pi$".

Like this?

 

[Delta2]

Just because no-one is there to discover it, does not mean it's not real or that it does not exist. With that thought in mind, let's say we are 100,000 years after the big bang, is there anything in physics knowledge that says Pi cannot have existed. I think the basic premise would be, if we have space Pi must exist. If the answer is "no" Pi cannot have not existed, we go farther back in time until we say "yes". If that's possible.

Numbers exist only within the human mind or the human brain if you want. To our best knowledge they correspond to electrochemical or electromagnetic signals inside our brains. When we measure a piece of rod or a piece of a string and we find it to be $\pi$ (there are many different ways to construct geometrical a line segment that equals $\pi$) it doesn't mean that it exists in the physical reality. In the physical reality exist only the molecules of the rod or the string which we used.

 

[S.G. Janssens]

Like this?

I wish it had been that tasty, then it would perhaps have been forgivable.

 

[jbriggs444]

Numbers exist only within the human mind or the human brain if you want.

This point of view is attractive, but it leads one away from useful mathematics.

Suppose that we decide that all numbers have physical existence as concepts -- biochemical patterns existing in a brain somewhere. Then the Peano axioms are false. Not every integer has a successor. Or a predecessor. Not every integer which exists today existed yesterday. Nor may some of them exist tomorrow. That's a pretty wishy washy background within which to do mathematical work.

Edit: here is an example of an integer that did not exist yesterday, may not exist tomorrow [depending on disk erasure details] and which has neither successor nor predecessor at present.

fly:3:~$ openssl genrsa 2048 > temp.key
Generating RSA private key, 2048 bit long modulus
...........+++
...........+++
e is 65537 (0x10001)
fly:4:~$ rm temp.key

Normally, one ignores the question of physical existence of numbers, decides that they exist in some Platonic realm or other and gets on with the business of solving the problem at hand.

 

[Delta2]

This point of view is attractive, but it leads one away from useful mathematics.

I could never imagine this as a consequence of what i wrote

Suppose that we decide that all numbers have physical existence as concepts -- biochemical patterns existing in a brain somewhere. Then the Peano axioms are false. Not every integer has a successor. Or a predecessor. Not every integer which exists today existed yesterday. Nor may some of them exist tomorrow. That's a pretty wishy washy background within which to do mathematical work.

Not sure here, i ll have to think this when i have slept better (unfortunately i am suffering from central sleep apnea and its totally random when i manage to sleep well) you might be right

Normally, one ignores the question of physical existence of numbers, decides that they exist in some Platonic realm or other and gets on with the business of solving the problem at hand.

I fully agree with the above.

 

[A.T.]

Numbers exist only within the human mind or the human brain if you want.

This point of view is attractive, but it leads one away from useful mathematics.

It's also very human-centric. Some other species on our planet (and potentially many on other planets) have developed the idea of numbers.

Not every integer which exists today existed yesterday.

Well, if it's not here:
https://en.wikipedia.org/wiki/List_of_numbers
then it doesn't exist.

 

[vanhees71]

Yes, the issue was settled in post #2.

I remember about an experiment I had to do in school at some point. To my horror, it involved the "experimental determination of $\pi$". In hindsight, this may have contributed to my decision to switch to mathematics at the end.

(On the other hand: Later on, when I studied physics first, one of the teachers that showed most sympathy for my stubborness and pedantry was an experimental condensed matter prof. that I still think about with a lot of sympathy.)

Which kind of experiment was this? What's interesting from a mathematical point of view is this experiment where you throw a needle on a floor with parallel strips painted on it and then getting $\pi$ from probality theory. The only problem is that this is very slowly converging ;-)).

https://en.wikipedia.org/wiki/Buffon's_needle_problem

 

[A.T.]

Which kind of experiment was this? What's interesting from a mathematical point of view is this experiment where you throw a needle on a floor with parallel strips painted on it and then getting $\pi$ from probality theory. The only problem is that this is very slowly converging ;-)).

https://en.wikipedia.org/wiki/Buffon's_needle_problem

Here is another one:

 

[A.T.]

And here with one optics, but about intensity, not ray geometry:

 

[Gordianus]

When I was 10/11 (memory fails), I "measured" Pi with a piece of string, several pipes and a ruler. A mathematician may scream, but I still remember it as a wondeful "experiment".