B Can physics deal with the existence of Pi?

[richard9678]

TL;DR 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".

 

[etotheipi]

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".

Wow, when I was 10.91 months old, I definitely could not do that!

 

[cmb]

Depends what branch of physics! ... the OP question reminds me of this

https://xkcd.com/2205/

 

[richard9678]

Hi. When I posted the question, my thinking was this: It seemed to me that Pi existed, because space existed. Pi is related to a circle, it's diameter & it's circumference. Space being thought of as a subject of physics. I did not consider that Pi might exist or derive from mathematics. The issue for me, or one issue, was can Pi exist if there is no space. My personal view is yes, it's a latent thing, (latent because it's possible) but that's perhaps more a philosophical thing, than scientific.

 

[richard9678]

Are you thinking of π as the ratio of the circumference to the diameter of a "real" circle?

Yes, I am.

 

[PeroK]

Yes, I am.

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.

 

[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