B Can physics deal with the existence of Pi?

[mpresic3]

There is a common joke, but I am sure it is apocryphal, (i.e. fiction), regarding a physicist giving a lecture to a group of naval officers regarding how ships move in a channel. For example, it might have been how to disperse the forces in a quick and efficient manner. The physicist put a statistical transport equation on the blackboard (in those days). The physicist assumed the process was a uncorrelated gaussian process and put on the board a density function for the normal distribution which involves a factor of pi for normalization.
The admiral asked what the funny symbol in the equation was.
Physicist: That is the greek symbol for pi.
Admiral: What is pi
Physicist: Pi is the the ratio of the circumference of a circle to the diameter.
Admiral: What can the ratio of the circumference of a circle to the diameter have anything to do with ships leaving a channel.

Admirals of my acquaintance are much smarter and informed than the one in the joke, but the point is pi shows up in many diverse contexts. It is likely that if it were somehow removed from geometry, it would have appeared elsewhere.

BTW there is another version of the joke where the physicist puts a stochastic differential equation for the transport of ships through the channel and admiral asks:

Admiral: what is the term on the right hand side of the transport equation?
Physicist: That is the collision term
Admiral: (indignantly) Our ships don't collide.

 

[Andrew Mason]

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

Given we live in non-Euclidean spacetime, real Euclidean circles are hard to come by.

$\pi$ is defined purely mathematically; it doesn't rely on the physical universe.

Any mathematical definition of $\pi$ must depend on a set of axioms. So $\pi$ can be defined mathematically starting with certain axioms of Euclidean space. Those axioms may have been developed as a true statement about our local physical space but whether they are "true" or not in any sense is irrelevant. All we need is the starting point (axioms) for the application of logic (mathematics). For example: $\pi$ can be expressed as the limit of an infinite sum of certain defined terms, or it can be expressed as the limit of the a sum that is related to the perimeter of a Euclidean regular n-sided polygon where $n \rightarrow \infty$. It is all just mathematics (logic) applied to a set of axioms.

What is particularly interesting about $\pi$ is the number of ways that it can be expressed mathematically. eg: $\pi =\frac{ \ln{(-1)}}{i}$ or $e^{\pi i} = -1$ . But, again, that is just one of many ways to define it mathematically.

AM

 

[berkeman]

$e^{\pi i} = -1$

Wait, are you saying that @etotheipi is negative? I find him to be quite positive.

 

[r731]

Can physics deal with a question on the existence of Pi

If Pi is measurable as a fact, perfect circles must exist in nature.

 

[etotheipi]

If Pi is measurable as a fact, perfect circles must exist in nature.

This thread is a perfect circle, 35 posts in and you've taken us right back to where we started.

 

[Arjan82]

Personally I'm a big fan of A Perfect Circle, since they exist, Pi must exist as well.

 

[hilbert2]

The question whether numbers are "discovered" or "invented" is a philosophical question, like "does a perfect circle somehow exist as an idea that is independent of humans thinking about it".

 

[sophiecentaur]

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".

At that age, it would have been pointless to hit you with the maths involved in calculating Pi so it was a 'timely' experiment. It could have been one of the first time that you got to learn the meaning of the term "accurate enough".

Science is always finding itself at the interface between when we measure / observe and what Maths tells us. Good Maths and Good experimentation will often make that interface very thin. Otoh, there are times (Chaos Theory, for instance) where the boundary is very wide in places.

It can get very uncomfortable to try to reconcile maths with what we find in Science. There seem to be some basic ideas in Maths - like Symmetry - where the Maths seems magically to predict what we find in Physics. Sometimes. I'm glad to have been an Engineer because Engineers don't lose sleep over things like that. Using Maths as a black box is an experience at a reassuring level. It's when you start to worry about "what things really are" that the pain can come on.

 

[A.T.]

Personally I'm a big fan of A Perfect Circle, since they exist, Pi must exist as well.

I just measured Pi from their logo, as 4.

 

[Andrew Mason]

Wait, are you saying that @etotheipi is negative? I find him to be quite positive.

You may have found the essential test to distinguish physics and mathematics: in the real world @etotheipi is a positive individual but mathematically he is a negative one. Or, as your very smart President has observed with respect to physical (COVID) tests, a negative one is really a positive one.

AM

 

[etotheipi]

I mean, hey, at least I'm real!

 

[sophiecentaur]

Here’s a bit of whimsy. Pi is not a rational number but it is still a ratio. You could wonder which of the two numbers is rational and which is transcendental.
No sensible replies please.

 

[jbriggs444]

Here’s a bit of whimsy. Pi is not a rational number but it is still a ratio. You could wonder which of the two numbers is rational and which is transcendental.
No sensible replies please.

Just move to Indiana where the squared circle nearly had circumference 32 and diameter 10.

 

[sophiecentaur]

Just move to Indiana where the squared circle nearly had circumference 32 and diameter 10.

There’s irrational and there’s irrational. And IT’S VOTING DAY in some places in the World.

 

[richard9678]

So, Pi is a number. Numbers are real, but not physical. Pi can be calculated mathematically, without there need for space. Yet, Pi can be calculated by physical means, even though a circle does not exist in space/physics. That's what I'm understanding. So, when we "see" in space a circle, that is our mind saying that. A physical circle an interpretation of the mind.

 

[anorlunda]

https://www.sciencefriday.com/articles/estimate-pi-by-dropping-sticks/

You can even discover $\pi$ in the game of pick-up-sticks.

 

[jbriggs444]

So, Pi is a number. Numbers are real, but not physical. Pi can be calculated mathematically, without there need for space. Yet, Pi can be calculated by physical means, even though a circle does not exist in space/physics. That's what I'm understanding. So, when we "see" in space a circle, that is our mind saying that. A physical circle an interpretation of the mind.

Regardless of what pi "is", its decimal digits still begin with 3.14159... The rest is philosophy.

The same sort of thing might be said of circles. We may not find anything which "is" a perfect circle, but we can physically measure how much various physical objects deviate from being one. That is good enough for practical purposes.

 

[sophiecentaur]

The relationship of Pi to Science is no different, in principle, from 2, Root 2 , minus 2 or any other number. Thing is that Maths gives accurate predictions about what happens in Science. That is sufficient justification to be using it. It's so much more useful than "Nature abhors a vacuum" ever was.

 

[cmb]

I think perfect circles are pointless.

 

[Andrew Mason]

I think perfect circles are pointless.

If I get your point, I think you are speaking about a pointed point and not a set of points that are equidistant from a central point (in all directions on a Euclidean plane).

AM

 

[A.T.]

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

Interesting video on this:

 

[vanhees71]

Perfect points are already pointless ;-).

 

[danielhaish]

with pie being irrational there isn't any problem to describe the world because like the circles you draw the circles in nature are not perfect .
https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ .
But more amazing fact is that the square root of two is also irrational and so if you take two equally lines put then in angle of 90 degrees , and then connect their edges with a line the line should be exactly the square root of two( Pitagoras sentence)

 

[A.T.]

But more amazing fact is the square root of two is also irrational and so if you take two equally lines put then in angle of 90 and then connect their edges with a line the line should be exactly the square root of two( Protagoras sentence)

It's Pythagoras, and the legend is, that he hated irrational numbers so much, that he had one of his students drowned, for proving they exist:

https://en.wikipedia.org/wiki/Hippasus#Irrational_numbers

 

[Andrew Mason]

with pie being irrational there isn't any problem to describe the world because like the circles you draw the circles in nature are not perfect .

Perhaps e is not an irrational number:

= pie = 2pi
So:
e = 2pi/pi = 2

AM

 

[A.T.]

pie = 2pi

This obvious contraction is one more reason to use tau.

 

[haushofer]

Like this?

Such a waste of pi.

 

[DanielMB]

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 $\tan^{-1}$.

Yes, but if you do that you are using geometric considerations, "Pi" like "e" are derived from geometry and geometry is affected by the stress-energy tensor in GRT, except if you define you are living in a manifold and locally you see "flatland" and in flatland (yes) Pi = 3.14.., and all our formulae where Pi appears are laws from flatland

 

[DanielMB]

Given we live in non-Euclidean spacetime, real Euclidean circles are hard to come by.

$\pi$ is defined purely mathematically; it doesn't rely on the physical universe.

It is not, are geometric considerations expressed in the language of maths, it is about the physical universe, all our formulae where Pi appears mean "Pi is a constant 3,14... in Euclidean space"

 

[jbriggs444]

It is not, are geometric considerations expressed in the language of maths, it is about the physical universe, all our formulae where Pi appears mean "Pi is a constant 3,14... in Euclidean space"

A statement that $e^{i\pi}=-1$ is not a statement about the physical universe.

Edit: A mathematician is likely to be supremely dis-interested in the question of what definition of pi is canonical and which definitions are merely provably equivalent.