Does Pi terminate or never?

[AvalonXQ]

Perhaps, but that isn't what he said.

I really think that it was what he said. "Constructive proofs yield stronger and more useful theorems" -- I read his statements as explaining that you can prove more using constructive proofs, not that coming to the exact same conclusions is subjectively "better" if you do it by one method as opposed to the other.

 

[W.D.Clinger]

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.

Right. What I had in mind is the situation in which Theorem A is of the form "AC implies Theorem B."

Because it has been traditional for mathematicians to assume the axiom of choice implicitly, the traditional statement of Theorem A is exactly the same as the correct statement of Theorem B, even though Theorem B is stronger than Theorem A. That's confusing.

When a mathematician uses the axiom of choice to "prove" Theorem B, the mathematician is really proving the weaker Theorem A. A constructive proof of Theorem B proves a stronger theorem, even though the traditional statement of Theorem A looks exactly like Theorem B.

The situation has been improving, albeit slowly. Nowadays, a mathematician who intends to assume the axiom of choice throughout a paper or book is more likely to state that assumption up front. In a few areas, such as set theory or set theoretic topology, the axiom of choice may be explicit in the statement of every theorem that needs it; in most areas of mathematics, however, that degree of rigor would still be considered unusual.

 

[jsfisher]

Right....

So, then you are talking about constructivism and not merely constructive proofs.

 

[rsaavedra]

Of course it terminates, shortly after 3.14 and before 3.15

 

[Complexity]

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....

I think some confusion can be avoided by quoting a bit more of what W.D.Clinger said: "Constructive proofs yield stronger theorems, because the theorem doesn't need non-constructive assumptions..."

Let's play for a moment.

Let's say that P1 and P2 are valid proofs of theorem T.

P1 assumes X, Y, and Z. P1 is a proof of T if and only if X, Y,and Z are true.

P2 assumes Y and Z. P2 is a proof of T if and only if Y and Z are true.

So what's the big deal about assumption X?

Much depends upon what sort of assumption X is. X could be simple or deep, well accepted or controversial, rarely used in other proofs or often used in other proofs.

One the fun things about mathematics is that you get to pick which things you'll assume are true and work within the framework generated by those assumptions. As long as the set of assumptions is consistent, you're free to pick as you like.

While mathematics has been a remarkably useful tool for modeling reality, mathematics is much larger and more interesting than that, and a mathematician needn't care one whit about the results of his/her work conforming to reality. (Which makes it all the more fun when that work is used to illuminate some aspect of reality anyway.)


Some of the considerations that are taken into account when judging the quality of a proof include:

• How original is the proof?
• How unexpected is the proof?
• How much much does the proof have to assume?
• How well are the assumptions used in the proof?
• How big a step (from prior mathematics) was taken in the proof?
• What is the value of the new ideas (if any) introduced in the proof?
• What is the value of the new techniques (if any) introduced in the proof?
• How much new math does the proof open up and what is the estimated value of that new math?
• What is the effect that this proof may have on mathematics (e.g. a yawn, a ripple, a tsunami)?
• How elegant is the proof?
• How beautiful is the proof?

Some proofs are plodding and others are gems.

It isn't as easy as saying that a proof of T that assumes X is better than one that doesn't.

If X is widely accepted, the decision to assume X for P1 doesn't appear to be a problem. However if the validity of X comes into question, or if X can be proven not to be true, P1 becomes suspect or useless.

Basing the correctness of P1 on X makes P1 vulnerable in a way that P2 is not.

In this sense only, P2 is a 'stronger' proof than P1.

A somewhat different play theme:

What if an assumption R is so useful that it gets spread around a lot and is used as an assumpation in a great many proofs.

What if R is an open question, a very active area of research, but mathematicians find R likely enough to be true (i.e. low risk as an assumption) and useful enough that they keep basing proofs on it even though they know R may not be true.

What would happen if R were proven not to be true?

Each proof that has R as an assumption would immediately be junked. A panic and mad dash (chaos is opportunity) would start to salvage what was possible by trying to modify the junked proofs so they don't have to assume R. Mathematicians would be thinking, amidst the wreckage, "Cool! Now that we know R is false..."

A prime example (pun not originally intended) of such an assumption R is the Riemann Hypothesis.

This is but one of the reasons why the Riemann Hypothesis is thought to be so important - much of post-Riemann mathematics rests upon it and would be undermined if the Riemann Hypothesis were ever to be proven false.

Long answer, but I like this stuff.

 

[W.D.Clinger]

So, then you are talking about constructivism and not merely constructive proofs.

I thought I was talking about increasing the precision, rigor, and usefulness of mathematical discourse by aligning the statements of theorems more closely with the hypotheses assumed by their proofs.

Constructivism^WP is certainly relevant, but it is a philosophy of mathematics that sometimes verges on ideology. I prefer to speak of constructive or computational mathematics. Here's a nicely ironic summary of the situation from the PRL project:

Now that the dust has settled and the controversy dimmed by the passing of the main protagonists, we can see that there remain these two traditions: the old computational and the modern (so called "classical" or "ideal"). The computational tradition has received renewed attention because of the advent of modern digital computers. Fields like numerical analysis, symbolic algebra, computational geometry, computational number theory, and automated deduction have arisen fresh in the last fifty years to employ large numbers of mathematicians, many of whom are not working in mathematics departments. These fields have become vital to modern science and industry. Some of them are part of the subject of computer science.

Computing digits of pi is a recreational activity that also serves as a benchmark for progress in computational (constructive) mathematics.

 

[jsfisher]

Long answer, but I like this stuff.

Me, too.

Nonetheless, we seem to have a disconnect. With this I would agree: Constructive proofs are generally stronger than existence proofs. With this I would not agree: Theorems proven by constructive proofs are stronger than theorems proven by existence proofs.

I see people asserting the latter, and then justifying it by observing the former.

 

[Ron Webb]

(Sorry for the delaying responding, shadron. I'm still getting to know my way around this forum.)

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.

Let me repeat: the value of pi is what it is. God Himself cannot make it otherwise, any more than He can decide that 2 + 2 should equal 5 instead of 4.

Whatever you find in the expansion of pi, God did not put it there. The best He could do would be to influence the development of our language and our choice of the symbols we use to represent it, such that a particular sequence would appear to contain a message. If I found such a message, I would consider it strong evidence that some guiding Intelligence had been working throughout our history to make such a miracle appear to us.

The case of a hypothetical discovery of a rasterized circle, or any similar mathematical pattern, is somewhat different. I still could not conclude that God had adjusted the value of pi, because that is logically impossible. However, if the sequence is independent of language or culture or anything that God might have control over, I would be left without any explanation at all, beyond a mind-numbingly astonishing coincidence.

 

[Complexity]

Me, too.

Nonetheless, we seem to have a disconnect. With this I would agree: Constructive proofs are generally stronger than existence proofs. With this I would not agree: Theorems proven by constructive proofs are stronger than theorems proven by existence proofs.

I see people asserting the latter, and then justifying it by observing the former.

I don't think I've addressed the question of the relative strength or value of constructive and existence proofs.

Suppose that theorem T states that 'If set A has the jabberwocky property then a subset of A having the bandersnatch property exists.'

A proof that proves that T is true by actually constructing a subset that has the bandersnatch property is a constructive proof.

A proof that proves that T is true by showing that the non-existence of a subset that has the bandersnatch property results in a contradiction is an existence proof. It proves the theorem by showing that such a subset must exist without constructing a subset that has the bandersnatch property.

The remarks in my recent post had nothing to do with the proof technique - they touched on the notion of strength as related to proof assumptions.

I'm not at all sure what I think about the strength question and constructive vs. existence proofs. Fun to ponder upon sometime.

I freely intermix constructive and existence proofs in my work and enjoy and appreciate both. Constructive proofs may on occasion be more useful than existence proofs if one is trying to use the result of the construction, but existence proofs can be more fun by tantilizing one by promising the existence of something without showing how to get one.

 

[jsfisher]

I don't think I've addressed the question of the relative strength or value of constructive and existence proofs.

Suppose that theorem T states that 'If set A has the jabberwocky property then a subset of A having the bandersnatch property exists.'

Ok, we have a (perhaps unproven) theorem T.

A proof that proves that T is true by actually constructing a subset that has the bandersnatch property is a constructive proof.

Yes, and we now have a theorem T and a proof C.

A proof that proves that T is true by showing that the non-existence of a subset that has the bandersnatch property results in a contradiction is an existence proof. It proves the theorem by showing that such a subset must exist without constructing a subset that has the bandersnatch property.

No argument so far. Now we have a theorem T, a proof C, and a proof E.

The remarks in my recent post had nothing to do with the proof technique - they touched on the notion of strength as related to proof assumptions.

Proof assumptions. Not theorem. The theorem T remains unmorphed in any way by the methods used to proof it. We don't have a special versions of T, T_C and T_E, dependent upon proof method.

We do, however, have different theorems if we start from different foundations. A theorem T developed within ZF (one possible constructivism system) vs a similar in appearance theorem T' developed within ZFC, then sure, we could find differences in theorem "strength" and maybe declare T stronger than T', but we would be discussing different theorems.

I'm not at all sure what I think about the strength question and constructive vs. existence proofs. Fun to ponder upon sometime.

They [constructive proofs] can be more utilitarian, especially in applied mathematics (and I think that was one of W.D.Clinger's main points). You are right, though, about the pondering.

I freely intermix constructive and existence proofs in my work and enjoy and appreciate both. Constructive proofs may on occasion be more useful than existence proofs if one is trying to use the result of the construction, but existence proofs can be more fun by tantilizing one by promising the existence of something without showing how to get one.

As someone who always had far more talent for proof by contradiction -- seldom a constructive method -- I come with my own biases on what's to be preferred. Besides, I like having a Law of the Excluded Middle.

 

[This is The End]

*Minor technical point. You might have a number that's got some arglebargle and THEN starts repeating. In this case, you need to do it in two steps.

E.g. 0.66652323232323.... is 6665.232323232323..... divided by 10000.

We already know that 0.23232323....is 23/99. So 6665.2323232323.... is 6665 and 23/99. Which is
659,858 / 99.

So the original number is 659,858 / 99 * 10,000 or 659,858 / 9,900,000.

It's still a rational number.

Well, isn't it possible then for a number to have an amazing amount of "arglebargle" and then start repeating? If there isn't an arglebargle limit then could there be a number than has a near infinite amount (of arglebargle) before it starts to repeat? So then wouldn't we think it was irrational even though it isn't?

(Sorry if this was asked before.)

 

[Cabbage]

Well, isn't it possible then for a number to have an amazing amount of "arglebargle" and then start repeating? If there isn't an arglebargle limit then could there be a number than has a near infinite amount (of arglebargle) before it starts to repeat? So then wouldn't we think it was irrational even though it isn't?

(Sorry if this was asked before.)

There's no limit to the "arglebargle" other than it has to be finite in length. Not really sure what you mean by "near infinite" unless you're simply using that to mean "REALLY big (but finite)".

However, mathematicians would never claim that such a number was rational or irrational simply by inspecting a bunch of digits. Conversely, you could also ask the question, "What if the digits repeat for a 'near infinite' length, and only then change? Wouldn't we think it's rational even though it isn't?"

But these claims can only be made with a formal proof; looking at a finite number of digits (regardless of how large that finite number may be) never constitutes a formal proof. For example, the Euler-Mascheroni constant is not known to be rational or irrational. It may not "look" like it's repeating, but it may be (or may not be). Just looking at digits can't decide it either way.

 

[Slybat]

And The Noob Says.....

22/7

 

[This is The End]

There's no limit to the "arglebargle" other than it has to be finite in length. Not really sure what you mean by "near infinite" unless you're simply using that to mean "REALLY big (but finite)".

No, by "near infinite" i mean "really, Really, REALLY big (but finite)".

(As far as I remember "near infinite" is just a cute way of saying "you'll never find the end, but it's not infinite". For all intents and purposes, it is the same as being infinite.)

However, mathematicians would never claim that such a number was rational or irrational simply by inspecting a bunch of digits. Conversely, you could also ask the question, "What if the digits repeat for a 'near infinite' length, and only then change? Wouldn't we think it's rational even though it isn't?"

But these claims can only be made with a formal proof; looking at a finite number of digits (regardless of how large that finite number may be) never constitutes a formal proof. For example, the Euler-Mascheroni constant is not known to be rational or irrational. It may not "look" like it's repeating, but it may be (or may not be). Just looking at digits can't decide it either way.

OK, thanks, that's exactly what I wanted to be sure of.

 

[This is The End]

22/7

Well, it's WAY better than using 3.

 

[Slybat]

I'm not trying to be a smart-ass or anything. I am truly impressed by this discussion but 22/7 is the only finite definition of pi. Whenever possible, 22/7, when used in an equation, will always give the only accurate result. When pi is used in astro-physics one had better use 22/7 or just guess or use 3.

 

[jsfisher]

I'm not trying to be a smart-ass or anything. I am truly impressed by this discussion but 22/7 is the only finite definition of pi. Whenever possible, 22/7, when used in an equation, will always give the only accurate result. When pi is used in astro-physics one had better use 22/7 or just guess or use 3.

Huh? The only "finite definition of pi"?

22/7 is merely a good approximation for pi, nothing more. Good in the sense it is a relatively simple fraction. "When used in an equation" 22/7 never gives an accurate result, just a (reasonably good) approximate one.

 

[shadron]

Huh? The only "finite definition of pi"?

22/7 is merely a good approximation for pi, nothing more. Good in the sense it is a relatively simple fraction. "When used in an equation" 22/7 never gives an accurate result, just a (reasonably good) approximate one.

In a college level lecture in astrophysics delivered two years ago at Yale and videoed for the Yale lecture series, the prof explained that when doing astrophysical calculations, one and at most two places of accuracy is enough. Only when going up for peer review will an astrophysicist go farther (saith he). At one place of accuracy pi is 3, and the prof often used that in the class.

 

[Slybat]

I stand corrected.

Thank you jsfisher and shadron. I honestly thought that 22/7 was the only true way to express pi. I withdraw my assertions.

 

[fuelair]

Would it help for you to realise I'm merely joshin if I used alot of >>>

Never hurts, sometimes helps!!