Does Pi terminate or never?

[shadron]

The value of pi is what it is, and not even God can change it. It would be extremely unlikely to find a string of a million 1s or a rasterized circle or any other such simple sequence or pattern in the first trillion or so digits, in any base. If we found such a thing I'm not sure what we would make of it, but I wouldn't call it a message from God. I can only say that God would probably be as astonished as us. (And yes, I know it should probably be "as astonished as we", but that just sounds silly.)

However, if we found a more complex coded message, such as the phrase "I am the Lord your God", spelled out in ASCII characters, then I think that would be very good evidence of God. Not that God could have put that particular string there either, but He certainly could have influenced the historical development of the English language and the choice of ASCII character representations so that the string, when we eventually found it, would have the meaning that it does.

Not so. The point of the rasterized circle in base 11 is that this could be found by any civilization, any intelligence in the universe from a simple series expansion and a computer (or their own brain given time and paper enough). No need to know ascii, no need to know english, no need to be air breathers or carbon based life. Just know what a circle is, enough math to have developed a series expansion for pi, and time. Your message in terms of information is much smaller than hers was; it was something like 51x51 base-11 digits long.

 

[MetalPig]

Nothing to add, just a 'thank you' to those who picked up the 'compression' idea and analyzed it beyond the point where I would have given up

 

[Ziggurat]

Here's something I've always wondered about Pi: Why do we use the circumference divided by the diameter and not the circumference divided by the radius? I began to wonder about that when I noticed how many times I encountered the expression "2π", such as the fact that there are 2π radians in a circle.

With 2pi radians in a circle, the power series expansion for sin(x) and cos(x) become nice and tidy, the derivatives of each just jump back and forth between them without introducing any prefactors, and they connect nicely to e^x. If you make it so there are pi radians in a circle, these relationships get cluttered.

 

[ddt]

Here's something I've always wondered about Pi: Why do we use the circumference divided by the diameter and not the circumference divided by the radius? I began to wonder about that when I noticed how many times I encountered the expression "2π", such as the fact that there are 2π radians in a circle.

Honestly it's just a convention I think.

Plus using the diameter we get (e^(pi*i))+1=0, which is pretty nifty.

With 2pi radians in a circle, the power series expansion for sin(x) and cos(x) become nice and tidy, the derivatives of each just jump back and forth between them without introducing any prefactors, and they connect nicely to e^x. If you make it so there are pi radians in a circle, these relationships get cluttered.

However, approximations of pi are known already from the ancient Egyptians and the ancient Greeks, among many others. They didn't yet do power series or imaginary numbers. Now, it can be of course they really made approximations of 2*pi - I don't know, I didn't read the ancient sources.

But there's another equation - about the area of a circle - which I think is very old:
A = pi * r^2
and that one would get a fraction of 1/4 in it when you'd use 2*pi as the constant.

 

[ddt]

The question becomes, can an ultimate creator fashion a universe where the laws of mathematics operate differently? Philosophically speaking I think so. I could probably describe a few myself. They wouldn't be very interesting universes, but that's not necessary for the point. On the other hand, a universe where pi works out mathematically the same except for a carefully crafted variation in the number sequence would function nicely.

She might fashion a universe in which the usual axioms were different. In some universes, for example, Lobachevskian geometry might be taught in junior high schools instead of Euclidean geometry, and pi might be regarded as an obscure transcendental number that comes up in the theoretical world of Euclidean geometry but not in the universe she has fashioned.

As late as Gauss - start 19th Century and contemporary of Lobachevsky - people regarded indeed (Euclidean) geometry as a description of reality. We now know better (think GR), and regard mathematics as "just" mind games, and it's up to the physicists to pick the right mathematical model.

Yes, you could think of a world where Lobachevskian geometry were the first type of geometry invented. That makes you think how that would influence the development of, e.g., calculus, or complex numbers. After all, pi is present all over the place: Euler's formula, Fourier series, etc.

In envisaging another world with different mathematics, you can go one step further. What about a different logic underlying what constitutes mathematical proof? There is such one, intuitionistic logic, which lacks the Law of excluded middle. That implies that you can't make a proof that "there is a number with property P" without, in fact, giving an algorithm to calculate that number. It also implies that reasoning about negatives becomes much more tedious. Take my above proof of sqrt(5) being irrational as an example. If you define "irrational" as "not rational", that proof is allowed; however, proving a number rational by contradiction, starting out with the assumption that the number is irrational, would not, as you'd at most get that the number is "not not rational" and you couldn't get away the double negative. Nevertheless, intuitionists have been able to recreate various parts of mathematics in an intuitionistic logical framework.

 

[3point14]

<SNIP>
No, it's easy to do.,SNIP>

You missed out the "for me" after that. Easy for you. For me, not so much, but I'm quite flattered you thought I might be able to understand all that squiggly stuff.

 

[W.D.Clinger]

Nevertheless, intuitionists have been able to recreate various parts of mathematics in an intuitionistic logical framework.

Indeed, all of the mathematics needed to formulate the laws of science and engineering appears to be available within a constructive framework.

Which is kind of fortunate, because we'd like to calculate with those laws.

 

[Lucky]

What I like about Sagan's idea is that it's logically possible, actually - just extremely unlikely if there's no god. That is, suppose it is the case (as everyone seems to believe) that pi is normal. That means that every finite sequence occurs in it infinitely many times.

So whatever the message is, it's in there somewhere, and all the creator has to do is make sure it occurs near enough to the beginning that we have a chance of noticing it by computing enough digits of pi. Since any sequence can be a significant message in some code, it's just a matter of creating life that will see some relatively early sequence as significant.

Off the top of my head, one example would be a string of a million 1's in the base 10 decimal expansion of pi. The frequency with which that would occur is something like 10^-1,000,000 if pi is normal, so if we find it it would be pretty good proof of the existence of god. But if we had 6 fingers, or 2, or 12, and used a number system with a different base, we might not notice it.

But that's not at all what Sagan suggested in Contact (you seem not to have read the book). For a start, the 'message' is found in the base 11 expansion of pi - which removes any possibility that Sagan had your idea in mind.

He refers to the pattern in pi as the 'artist's signature', and it's quite clear that he really is implying an intelligence that created pi (and presumably mathematics and logic as a whole), along with the physical universe. Even worse, he implies that pi is dependent on the properties of space in our universe, and that we discover its value by measuring it!

Do you seriously want to argue that there's any other way to interpret this line (from the book's conclusion):

"The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover a miracle — another circle, drawn kilometers downstream of the decimal point. In the fabric of space and in the nature of matter, as in a great work of art, there is, written small, the artist’s signature. Standing over humans, gods, and demons, subsuming Caretakers and Tunnel builders, there is an intelligence that antedates the universe."

I and the couple of others here who are trying to explain that pi is not a property of the physical universe seem to be getting nowhere.

 

[gnome]

I accept that it's not a property of the physical universe. What I'm exploring (and not convinced one way or the other) is of the universality of the concept of pi -- starting as a thought experiment in universes where it is not physically relevant.

 

[Robin]

However, approximations of pi are known already from the ancient Egyptians and the ancient Greeks, among many others. They didn't yet do power series or imaginary numbers. Now, it can be of course they really made approximations of 2*pi - I don't know, I didn't read the ancient sources.

But there's another equation - about the area of a circle - which I think is very old:
A = pi * r^2
and that one would get a fraction of 1/4 in it when you'd use 2*pi as the constant.

It could just be that the ratio of the circumference to the diameter was the most practically useful ratio in ancient times.

For example early approximations of pi might have been used by builders or artisans to estimate the amount of material needed to build a round structure or object of a certain width.

 

[ddt]

Indeed, all of the mathematics needed to formulate the laws of science and engineering appears to be available within a constructive framework.

Which is kind of fortunate, because we'd like to calculate with those laws.

Not all. For one example: Quantum Mechanics relies on functional analysis, in particular Hilbert spaces, and for that you need the Axiom of Choice, which is universally rejected by intuitionists/constructivists because it is non-constructive. In fact, it would re-introduce the law of the excluded middle through the backdoor.

 

[sol invictus]

But that's not at all what Sagan suggested in Contact (you seem not to have read the book).

I read it many years ago. What I recall is a hidden pattern in pi, not necessarily what it was. But I disagree that what I'm suggesting is significantly different.

For a start, the 'message' is found in the base 11 expansion of pi - which removes any possibility that Sagan had your idea in mind.

Umm, no. That was a simple example. Sagan's is more sophisticated.

He refers to the pattern in pi as the 'artist's signature', and it's quite clear that he really is implying an intelligence that created pi (and presumably mathematics and logic as a whole), along with the physical universe. Even worse, he implies that pi is dependent on the properties of space in our universe, and that we discover its value by measuring it!

No, I don't think so. Not necessarily at least.

Do you seriously want to argue that there's any other way to interpret this line (from the book's conclusion):

"The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover a miracle — another circle, drawn kilometers downstream of the decimal point. In the fabric of space and in the nature of matter, as in a great work of art, there is, written small, the artist’s signature. Standing over humans, gods, and demons, subsuming Caretakers and Tunnel builders, there is an intelligence that antedates the universe."

First off, I don't really have a problem with "measure" there. "Calculate" is more accurate, but he's being poetic. And surely that's made clear in the novel?

More importantly, I disagree that this suggestion is so different from the one I made, or necessarily at odds with any form of logic. Why do we focus on pi at all, and not its third root, or e^pi, or Riemann zeta(23)? Presumably because we live in a universe that's approximately flat, and we perceive circles as special, and we think it a certain sort of way, etc. I can easily imagine a universe in which the inhabitants would never bother to calculate pi and instead focus on some other number, or perhaps on some entirely different type of endeavour.

So perhaps, being omniscient, our putative creator noticed that the quantity we call pi had this special feature in its decimal (or base 11 or whatever) expansion, and created us so that we would eventually notice and appreciate the significance.

 

[W.D.Clinger]

Not all. For one example: Quantum Mechanics relies on functional analysis, in particular Hilbert spaces, and for that you need the Axiom of Choice, which is universally rejected by intuitionists/constructivists because it is non-constructive. In fact, it would re-introduce the law of the excluded middle through the backdoor.

No, you're confusing the nature of Hilbert spaces with the way they're usually taught.

There are lots of constructive Hilbert spaces that don't need the axiom of choice. To support your claim, you'd have to identify a scientific law that requires a Hilbert space that

1. isn't constructive, and
2. doesn't have a constructive analogue that would do just as well.

That would be hard to do. Errett Bishop, in his 1967 book on Foundations of Constructive Analysis, presented a constructive development of Hilbert spaces and commutative Banach algebras. In his review of that book, Gabe Stolzenberg wrote:

He is not joking when he suggests that classical mathematics, as presently practiced, will probably cease to exist as an independent discipline once the implications and advantages of the constructivist program are realized. After more than two years of grappling with this mathematics, comparing it with the classical system, and looking back into the historical origins of each, I fully agree with this prediction.

Constructive Analysis, by Errett Bishop and Douglas Bridges, is an updated version published in 1985. I have read only the original version, but the updated version might be easier to find in university libraries.

 

[sol invictus]

That would be hard to do. Errett Bishop, in his 1967 book on Foundations of Constructive Analysis, presented a constructive development of Hilbert spaces and commutative Banach algebras. In his review of that book, Gabe Stolzenberg wrote:

He is not joking when he suggests that classical mathematics, as presently practiced, will probably cease to exist as an independent discipline once the implications and advantages of the constructivist program are realized. After more than two years of grappling with this mathematics, comparing it with the classical system, and looking back into the historical origins of each, I fully agree with this prediction.

So why hasn't it happened?

 

[W.D.Clinger]

So why hasn't it happened?

Patience, grasshopper.


Conservatism and laziness, mostly.

Mathematicians were taught to use non-constructive methods, and it is often easier to prove something non-constructively.

Constructive proofs yield stronger theorems, because the theorem doesn't need non-constructive assumptions, but mathematical tradition has implicitly made non-constructive assumptions. That means the statement of the constructive theorem doesn't look any different from the traditional statement of the weaker non-constructive theorem, which means there has been little incentive to find constructive proofs.

In some areas, notably computer science and logic, mathematicians are beginning to understand that constructive proofs yield stronger and more useful theorems, but this is going to be a long slow process, much as the shift to machine-assisted or machine-verified proofs will be long and slow.

Part of the problem is that many departments of mathematics have regarded logic as the province of the philosophy department and computer science as the province of the computer science department. At MIT, for example, Marvin Minsky had to move from the math department to the department of electrical engineering despite his impeccable mathematical pedigree. That particular math department eventually realized its mistake and made amends; for example, Michael Sipser has been a recent chair of the department.

 

[drkitten]

Not quite--she was looking for a sequence that had a length that was the product of two primes...

Er, 9 (3x3) is the product of two primes.....

 

[stilicho]

Proof that sqrt(5) is not rational is pretty easy, and goes by contradiction.
Suppose sqrt(5) is rational, and it is equal to p/q where p and q are both (positive) integers. Then simplify this fraction to p'/q' where p' and q' have no (prime) factors in common.
Then we have: sqrt(5) = p' / q' and when we square that equation, and move q' to the other side, we get
5 * q' * q' = p' * p'
The main theorem of number theory states that factorization into prime numbers is unique. So p' * p' must contain a factor 5, and therefore p' must contain a factor 5 and therefore, p' * p' must contain at least two factors 5. That, in its turn, implies that q' * q' must contain at least one factor 5, and therefore q' contains a factor 5. However, we started out with saying that p' and q' had no common factors, and we have now established they both have a factor 5. Contradiction.
Ergo, sqrt(5) is not rational.

I actually followed that. Are you a math teacher? If you aren't then you should be. Mine all sucked.

The idea of a system of weights and measures in base-phi is even less practical than imperial weights and measures with its haphazard factors of 3, 12, 14, 16 and what have you.

I have several impractical ideas. Perhaps you'd like to subscribe to my newsletter.

 

[jsfisher]

Constructive proofs yield stronger theorems

A semantic quibble: A theorem is no stronger nor weaker because of how it is proven. However, a constructive proof for a theorem can provide a useful technique for exploiting the theorem. Given an existence proof, for example....

 

[AvalonXQ]

A semantic quibble: A theorem is no stronger nor weaker because of how it is proven.

I don't think he means that Theorem A is "stronger" if it's produced by one proof as opposed to another. I think he means that with a constructive proof we can often get Theorem B instead, which implies Theorem A but not vice versa -- hence B is formally a stronger theorem than A.

 

[jsfisher]

I don't think he means that Theorem A is "stronger" if it's produced by one proof as opposed to another. I think he means that with a constructive proof we can often get Theorem B instead, which implies Theorem A but not vice versa -- hence B is formally a stronger theorem than A.

Perhaps, but that isn't what he said. On the other hand, had he stated it as you did, well, then, that would have been fodder for a separate thread and debate.