# TRC #279: Infinity + Placebo Revisited + Was Canada Colder Than Mars Episode #279 of TRC arrives with a whole bunch of numbers.   First Elan tries to explain infinity and why it is important to skeptics.    Next, Darren revisits placebo and finally Adam looks at whether Canada was recently colder than Mars.

If you like the show, please leave us a review on itunes. SHOW NOTES

Infinity

Wikipedia – Infinity Symbol

Wikipedia – Infinity

Wikipedia – Infinite Monkey Theorem

Wikipedia – 1/2 + 1/4 + 1/8 + 1/16 + ⋯

The Reality Check Episode 200 Comments

Math Is Fun – Infinity

Math Is Fun – Limits

Canada as cold as Mars?  Not quite, eh? – CBC

Cheer up Winnipeg, your temps could be worse – @MarsWxReport

Ottawa is somehow colder than Mars – Ottawa Sun

Other

Effective Altruism Meetup

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### 27 Responses to TRC #279: Infinity + Placebo Revisited + Was Canada Colder Than Mars

1. Yves says:

Hi Reality Checkers,

I was just listening to your segment on “infinity” and I think it was quite well done. Maths on a podcast cannot be the easiest thing to attempt. A few comments though:

-It is not entirely correct to talk about infinity as “a” concept, since it pops up all over the place in mathematics (and the sciences that use maths) and it doesn’t necessarily mean exactly the same every time. Whenever you see infinity in a mathematical expression, it is important to check exactly what it means in that particular context.

-At some point Elan calls “infinity” a number. That is definitely worth trying to avoid, because it easily leads to wrong conclusions. For example, if infinity were a number, then “infinity minus infinity” would equal zero, which it is not (what “infinity minus infinity” is, if it is intelligible at all, again depends on the context).

-In the discussion about the monkeys typing away for an infinite time, Adam had a problem with the phrase “almost surely”. In the field of probability theory, however, this is rigorously defined terminology. Something is said to happen “almost surely”, if it has a probability of happening equal to one. So “almost surely” is exactly the right phrase for the situation you were discussing.

Thanks for the show!

• Adam says:

Thanks. I was not aware of the term “almost surely” meaning anything. It sounds like an oxymoron on the surface which is why I called it out.

• Yves (good, that means at least TPM is over) says:

Yeah, mathematicians are good at taking words that mean one thing in ordinary speak and attaching a precise, but sometimes different, meaning to them for maths purposes. 🙂

2. Fouridine says:

Found the info on Decibo effect (is that how it’s spelled?) interesting but couldn’t find anything googling it (possibly cos I got the spelling wrong). Do you guys have more info on that?

• Adam says:

This is the essay that was in the book:
http://www.edge.org/response-detail/11708

It may be the only actual information on the topic. The terrible spelling pretty much guarantees it will never catch on as a term.

3. Eric says:

I was so pleased to hear the piece on the placebo effect, especially about the studies that cast doubt on its existence. This is one of those issues where skeptics do not all fall into one category. Some seem to want to accept a powerful placebo effect (despite the lack of good evidence) as a way to refute b.s. alternative medicine. I sympathize with that desire but some alt med snake oil salesmen have turned that around and claimed that all they are doing is invoking a powerful placebo effect. Skeptics need to take a stand on this issue that’s based on the very best science available. I think when they do they will realize that what we call the placebo effect is really a whole lot of different effects, almost all of which are the result of experimental protocol, some of which you mentioned here, and that it isn’t something you can just invoke with acupuncture or whatever nonsense you’re peddling. Thank you. You’ve always been one of my favorite skeptic podcasts but after so many excellent segments on tough subjects, you’re definitely at the top of my recommended listening list.

4. Yves (A different one!) says:

The whole “0.9999… = 1” debate frustrates me so much. For such a long time I felt like I was the only sane person in the world! Whenever I was presented any kind of “proof”, I could always find a fatal flaw in it.

It turns out there are, in fact, two internally consistent ways of dealing with the problem. Technically, neither is more correct than the other one and all “proofs” are correct with either one or rely subtly in some kind of circular reasoning or limitations in the decimal notations (or both!).

1. There is no such thing as an infinitesimally small number (ɛ = 0)
Another way explaining this is that any infinitely small number (let’s call it ɛ) is actually 0. This is the actual way real numbers are defined in mathematics. Personally, I don’t like it. But, it is internally consistent and for all intents and purposes, it works because in the real world we cannot differentiate the “true” answer from one that is infinitely close.

If the difference between 1 and 0.9999… is infinitely small (1 – 0.9999… = ɛ) and ɛ = 0. Then 0.9999… = 1. Simple!

Let’s see what happens with the infinite series presented in the show if we recreate it by starting with “1/2 + 1/2 = 1” and expanding it:
1/2 + 1/2 = 1
1/2 + 1/4 + 1/4 = 1
1/2 + 1/4 + 1/8 + 1/8 = 1

1/2 + 1/4 + 1/8 + … + 1/∞ + 1 / ∞ = 1

Now, remember that there is NO SUCH THING AS A INFINITELY SMALL NUMBER! (Or rather there is no difference between it and 0)

Thus:

1/2 + 1/4 + 1/8 + … + 1/∞ + 1 / ∞ = 1
1/2 + 1/4 + 1/8 + … + 1/∞ + ɛ = 1
1/2 + 1/4 + 1/8 + … + 1/∞ = 1

So we have shown that our sequence must be equal to 1.

2. There IS such a thing an infinitely small number (ɛ = 1 / ∞)
The reason I could not understand the whole debate was because I though real numbers worked like this. Well, technically, it’s only false if you use the mathematical definition of a real number. But you can just as easily imagine an alternate number system where this is true.

Let’s call our infinitely number ɛ and define it as ɛ = 1 / ∞. Thus 1 – ɛ = 0.9999…

Using this, we can show that some earlier proofs where the result of circular reasoning. Let’s use the infinite series again:

1/2 + 1/2 = 1
1/2 + 1/4 + 1/4 = 1
1/2 + 1/4 + 1/8 + 1/8 = 1

1/2 + 1/4 + 1/8 + … + 1/∞ + 1 / ∞ = 1
1/2 + 1/4 + 1/8 + … + 1/∞ + ɛ = 1
1/2 + 1/4 + 1/8 + … + 1/∞ + ɛ – ɛ = 1 – ɛ
1/2 + 1/4 + 1/8 + … + 1/∞ = 0.9999…

If we compare this to the way we were using the series earlier, we can see that the use of the series as proof that “0.9999… = 1” relies on circular reasoning! The result of the series is equal to 1 only because we already assume that “0.9999… = 1”! This is also true of all “proofs” that rely on limits.

Unfortunately, essentially anyone trying to show that “0.9999… = 1” does not understand that their “proofs” mean nothing unless they correctly explain how real numbers work. Most importantly, THAT INFINITELY SMALL NUMBERS DON’T EXIST in the real number system.

This link explains it pretty well:
http://en.wikipedia.org/wiki/0.999…#Infinitesimals

• Yves (the first one again) says:

Sure, you can come up with different number systems and different definitions of what 0.999… is supposed to mean, etc, but I think the common assumption in everyday life, is that, unless stated otherwise, we’re working with the ‘standard’ real numbers. I also assume, for example, that you are using ‘standard English’ in your comment, but it could just as well be something that looks like English, but should be interpreted completely differently in order to convey your intended meaning properly. However, in the absence of any indication that I should interpret your words in a non-standard way, I am going to assume standard usage. Same with the real numbers.

So, while it is definitely advisable to clearly define your terms when talking maths, there is always a bunch of assumptions you are going to make implicitly about what certain concepts mean, unless you want to start out every discussion about maths by a five hour explanation of how you’ve built up your system from a set of axioms. A valid and possibly interesting approach, but hardly suitable for a podcast like this. Or indeed, for a comment on this site, because even in your comment, in which you go into some detail to explain your ideas, you also leave key concepts undefined, such as what you mean by 1/infinity.

Personally, I think the main idea that needs to be explained clearly, before one can even seriously try to attempt convincing someone that 0.999… = 1 (or isn’t, if you prefer alternative number systems), is what one even means by 0.999… I agree with you that one has to be careful not to *define* 0.999… explicitly to be equal to 1, but I am not sure why you think that limit based proofs are circular? For example, defining 0.999… in terms of a geometric series (as also explained on the Wikipedia page you linked to) and then showing that that series is convergent and has value 1, sounds perfectly fine to me? Am I missing something?

5. Yves (Number 2) says:

Proofs based on limits are circular because they give a different answer depending on which system you are using. Using a limit means that the result of a statement converges toward a number as a variable approaches a certain value.

For example, using our favorite series:

limit (x -> ∞) ∑ (i = 1 -> x) 1/i = 1

Now, this statement is true whether we use real numbers or our imaginary system that allows infinitely small numbers (the limit does not give the actual result of the equation but a value of convergence).

But it is not true if we remove the limit:

ɛ = 0 -> ∑ (i = 1 -> ∞) 1/i = 1
ɛ = 1 / ∞ -> ∑ (i = 1 -> ∞) 1/i = 0.9999…

In one case, the result is equal to the limit (when ɛ = 0), but in the other case it is not. THE ACTUAL VALUE IS ONLY EQUAL TO THE LIMIT IN CASES WHERE 0.9999… = 1.

Let’s take the following (horrible, horrible) proof:

0.9999… = ∑ (i = 1 -> ∞) 9/10^i = limit (x -> ∞) ∑ (i = 1 -> x) 9/10^i = 1

See where the problem is? If 0.9999… is NOT equal to 1, then the second equality becomes incorrect. Our proof only works if the conclusion is valid! It’s the very definition of a circular argument.

Besides, while the result of a limit is usually equal to the actual result, we should never assume it is always true. In fact, it’s false in at least one case:

limit (x -> 0) 1/x = ∞, but 1/x ≠ ∞

• (And back to the first) Yves (again) says:

I still don’t completely understand your point. It seems to hinge on a difference between what you denote by ∑ (i = 1 -> ∞) and limit (x -> ∞) ∑ (i = 1 -> x). Can you give definitions of both these expressions? To me (but that may be a standard real number system bias) both of these mean the same (and are only meaningful if the limit in the second case converges).

Anyway, I agree with you that the truth value of 0.999…=1 depends on the choice of number system and that choosing your number system, determines the truth value (or, I suppose, possibly the undecidability) of it. Now, the converse is not true (and I’m going here by your Wikipedia link, because I am not well at home in the intricacies of different number systems): prescribing 0.999…=1 does not determine which number system you are working in (for example, according to the Wikipedia page, the equality is also true in the dual numbers system). So the acceptance or rejection of 0.999…=1 is not equivalent with choosing your number system.

Now, given a choice of number system (and a choice of logic, and a choice of axioms), we still need to define what we mean by 0.999…,or more generally, how we interpret the decimal notation of numbers, and what we mean by = in this context. If I make all the standard choices (standard real numbers, standard mathematical logic, standard ZFC set axioms, standard interpretation of = within the field of the standard real numbers), then I think it is also standard to define the decimals in terms of a series (fwiw, Wikipedia also thinks so: http://en.wikipedia.org/wiki/Decimal_representation):

0.abcd… := a/10 + b/100 + c/1000 + d/10000 + …

which converges since each term has absolute value bounded strictly away from (and less than) one, and so the series can be bounded by a geometric series with ratio 10, which can be proven to converge.

Now, given all these choices (and we have to make these choices one way or the other, in order to be able to interpret the statement 0.999…=1), none of which are circular, I think (?), we can prove 0.999…=1, for example using the known value of converging geometric series (as explained under your Wikipedia link). So, what about this is circular? You say, “Our proof only works if the conclusion is valid!” Well, of course the proof only works if the conclusion is true (I’m not sure what you mean by “valid” here. Arguments or formal formulas can be valid, not conclusions, in the way I know the word). If you could prove something that is not true, then your logic would be inconsistent or problematic at best.

Btw, as an aside, the series with summands 1/i diverges, both for i natural numbers, or for i real-valued (assuming that you then interpret the sum as an integral). You probably mean a different series.

• Yves (The clone... OR AM I?) says:

For this I’ll have to refer to my first post. See what happened to both series depending on what system we use? If we use real numbers, then the result of the sum is 1. But, if we use um… not real… or something… numbers, the result is 0.9999…

This is because even if the series converges toward a number, it theoretically never reaches it. Once again, if you look at how I expanded ½ + ½ to recreate the series, I was left with a dangling 1/∞. In order to eliminate it, I had to to subtract it from both sides, thus changing 1 into 0.9999… So why does the series equal 1 in the real number system? well um… because 0.9999… = 1, or at the very least, for the exact same reasons. So, if I don’t already accept that 0.9999… = 1, then I also won’t accept that the result of the series is 1, I’ll argue that it is 0.9999… My objections to both will be the same.

Yet, no matter what system you use, the result of the *limit* of the same statement is equal to 1. There’s no arguing to be done here. Thus, whether or not the result of a limit is the true result (assuming the series converges) is entirely dependent on if 0.9999… = 1.

So let’s do our proof step by step:
1. 0.9999… = ∑ (i = 1 -> ∞) 9/10^i
This is correct, we are simply representing our decimal as a series.

2. ∑ (i = 1 -> ∞) 9/10^i = limit (x -> ∞) ∑ (i = 1 -> x) 9/10^i
Hey! That’s only true if you already accept that 0.9999… = 1! You can’t use an equality that is only true if your conclusion is also true in order to prove that same conclusion.

So the reasoning of this proof (and those that use something similar) is circular!

• Yves (uncloned, afaik) says:

But what do you mean by ∑ (i = 1 -> ∞)? Apparently not limit (x -> ∞) ∑ (i = 1 -> x), but what then? I cannot parse point 2 in your post, without knowing what you mean by that notation. How do you make rigorous the notion of an infinite sum, if not as the limit of some converging series? And why would the definition of an infinite sum pre-necessitate the declaration that 0.999…=1?

Let me give you an example what your argument sounds like to me (possibly because I misunderstand it) and let me do that by taking the discussion away from 0.999… Consider the following situation.

1) I want to show that 3+2=5 (in the normal natural number system).
2) Assume, I already know that 1+1=2 and 3+1=4 and that addition in the natural number system is associative.
3) Additionally, assume I know that 4+1=5.
4) Thus I conclude that 3+2 = 3+(1+1) = (3+1)+1 = 4+1=5.

Now, would your objection be that this is circular reasoning, because, given 2), my assumption in point 3) is only true if and only if 3+2=5? (After all, if we know 3+2=5 and 2) is given, then 3+(1+1) = 5, hence (3+1)+1 = 5, and thus 4+1 = 5.)

To put it in other words, it sounds to me like you are arguing that if statements A and B are equivalent and you prove B, assuming A, then that is circular, because A is equivalent to B and thus you are, in a sense, proving B, assuming B? Surely this cannot be what you mean? But this is what it seems to me you are saying. But, as I started out with, it all seems to hinge on what your limit-less definition for an infinite sum is. So perhaps my understanding of your argument changes once you tell me that.

As a side issue, since I am not familiar enough with working in non-standard number systems, I also do not know if limit (x -> ∞) ∑ (i = 1 -> x) 9/10^i = 1 still holds true in such systems. (Do these non-standard numbers which include infinitesimals still form a metric space or topological space? If so, which limit concept follows from that? How is addition defined when infinitesimals are involved? For example, is epsilon+epsilon = 2 epsilon or epsilon + epsilon = epsilon?) It’s not really important for the argument at hand, I think, so I’ll take your word for it for now that this limit still is well-defined and equal to 1. Of course, strictly speaking this 1 is not the same 1 as the standard-real-numbers 1.

• Yves (Begun, the clone wars have) says:

∑ (i = 1 -> ∞) means the sum of all versions of a statement where i equals from 1 to infinity. Thus ∑ (i = 1 -> ∞) 9/10^i = 9/10 + 9/100 + 9/1000 + ….

You have to remember that using is a limit does *not* give you the result of a statement, it give you the number towards which the *tendency* of the result.

Let’s say we use the series “1/2 + 1/3 + 1/4 + ….”. We can express this series as the sum of all instances of 1/n where n is any values between 2 and infinity or ∑ (n = 2 -> ∞). (I’m bastardizing the notation here because I’m using text and not a whiteboard or something)

We are lucky here that there are ways of finding the actual result of the sum (as demonstrated in my first post). But, what if there was no way of knowing the result? Well, we could still use a limit. Using a limit means we are not finding the true result (as there might be none, such as 1/0) we are finding towards which value does the result of the statement tend to approach.

Thus “limit(x -> ∞) ∑ (n = 2 -> x) 1/n = 1″ means as x approaches infinity, the result of the sum approaches 1”. It doesn’t matter if you believe the “true” result is 0.9999… or 1, the result of the limit doesn’t change as it is a *tendency*.

Now it’s true that other number systems might result in a different result for our limit. But here we are only interested in two systems, and the result is the same in both.

As for your the circular proof argument, the equivalent in your analogy would be if “3+(1+1) = “(3+1)+1” was true if and only if “3 + 2 = 5” was true. Or in sense that addition can only be associative if 3 + 2 = 5.

More specifically, I was saying that limit (statement x) = (statement x) (for series can converge) only and only if 0.9999… = 1. So if your proof includes something like that, it can’t work, because that part of the proof is only correct if the very thing you are trying to prove is true.

• Yves (good, that means at least TPM is over) says:

You wrote:
“∑ (i = 1 -> ∞) means the sum of all versions of a statement where i equals from 1 to infinity. Thus ∑ (i = 1 -> ∞) 9/10^i = 9/10 + 9/100 + 9/1000 + ….

You have to remember that using is a limit does *not* give you the result of a statement, it give you the number towards which the *tendency* of the result.”

I know what a limit is, but what I do not know is what the sum of infinitely many terms is, *unless* I’m allowed to define it as the limit of a convergent sequence of sums with finitely many terms. But apparently that is not the definition you are going with, so could you just give me the exact, rigorous, definition you are using for “∑ (i = 1 -> ∞) a(i) ” (where the a(i) form a sequence of either standard-real-numbers or non-standard-real-numbers)? By now surely we have scared off any people not interested in the minutiae of maths, so we don’t need to be shy about throwing out one rigorous definition. Perhaps that will clear things up.

This is the crux of your argument, because without a rigorous definition of “infinite sum” there is no way of determining if it is true that “(infinite sum)=(limit of sequence of finite sums) if and only if 0.999…=1”.

• Yves (Nooooooooooooooooo!!!) says:

I see your point. In fact, articles on Wikipedia have a tendency of dropping the limit seemingly on a whim.

Why is it dropped so often? Because if you assume (correctly) that 0.9999… = 1 the result of the limit of a convergent series *must* be equal to the “actual” value (without the limit) and they are thus equivalent.

But what if they aren’t equal? Then obviously, it would be important to specify if the limit is used or not!

What does the limit tell us? It tells us that the result *approaches* a certain value, but technically, it can never reach it. No matter how far you go in the series, the closer you get, but you can *never* reach the result of the limit.

Let’s use the sequence I used in my first post (Oops! I said it was the one presented in the show, but it’s actually different):

∑ (n = 1 -> x) 1/2^n = 1/2 + 1/4 + 1/8 + … + 1/∞

We can actually find the answer of this sequence without using limits! We do this by starting with 1/2 + 1/2 and then expanding by replacing the last value with a sum of its halves:
1 = 1/2 + 1/2 = 1/2 + 1/4 + 1/4 = … = 1/2 + 1/4 + 1/8 + … + 1/∞ + 1/∞

Now obviously the sequence does not actually end, but strangely enough we know that the smallest value in the sequence *must* be 1/∞. So if we subtract that on both sides:

1/2 + 1/4 + 1/8 + … + 1/∞ + 1/∞ – 1/∞ = 1 – 1/∞
1/2 + 1/4 + 1/8 + … + 1/∞ = 0.9999…

So the result is a number that is infinitely smaller than 1. That’s pretty much what our limit predicted. In the world of 0.9999… = 1, infinitely small numbers don’t exist, so we simply say the answer is 1, the same as our limit. So it turns out that in the world of real numbers.

But what if I do accept infinitely small numbers? Obviously, if you believe that 0.9999… ≠ 1, then you must accept them. There’s now a fundamental difference between using a limit and not using it. In the Epsilon world, where ɛ = 1/∞, the result of the limit of the sequence remains 1, but if we solve it without the limit, we get 0.9999…

All of a sudden, the sum with and without the limit are no longer equivalent!

Thus, until you can prove to me that 0.9999… = 1, then I cannot accept the sum of a convergent series is equal to the limit of the same sum. If your proof relies on that, then there’s no way you can convince me as I’ll simply say it’s circular reasoning.

• Yves (let's go back and add extra effects to our previous posts) says:

You still haven’t given a definition of your infinite sum. You cannot use induction to go from a finite sum to an infinite sum (perhaps you can with transfinite induction, but I’m completely unfamiliar with its details), and I’m also still not sure how addition works when infinitesimals are involved, so I’m still not sure what your infinite sum *exactly* means. Why are you avoiding giving a rigorous definition?

In fact, it seems like you *define* 0.999… to be 1-epsilon, in which case of course (per definition of your epsilon) 0.999… is not equal to 1.

You wrote: “Thus, until you can prove to me that 0.9999… = 1, then I cannot accept the sum of a convergent series is equal to the limit of the same sum.”

No, this is backwards. The sum of a convergent series (at least in standard real analysis) is *defined* as the limit of the approximating finite sums. We do not need to know the equality 0.999…=1 to define that; that equality is a consequence of that definition (together with the definition of 0.999… in terms of a convergent geometric series). There is nothing in that definition that pre-necessitates 0.999…=1. It’s like saying that we need to assume that 3+2=5 *in order to* define standard addition, because with some non-standard addition, we might get 3+2 not equal to 5.

Now, I’m perfectly willing to believe that there might be a generalized notion of infinite sum, which reduces to “limit of a convergent sequence of approximating finite sums” in the case of standard real analysis, but not in the case of your particular non-standard number system. However, then I still would like to know (in an exact mathematical way) what this generalised concept is. Curiosity, you know.

Now, even if all this is the case, and there exists some generalised setting in which the statement “0.999…=1” is equivalent to the statement “the value of an infinite sum is equal to the limit of a convergent sequence of finite approximating sums”, that still would not make those proofs for “0.999…=1” circular, would they? It would depend on whether you take “0.999…=1” as one of the defining characteristics of your particular system, with “the value of an infinite sum is equal to the limit of a convergent sequence of finite approximating sums” as a consequence, or the other way around.

Since there are ways of defining (synthetic approach) or constructing (Dedekind cuts) the real numbers that, I think, do not presuppose 0.999…=1 (the Cauchy sequence approach on the other hand, might actually presuppose this), I think we can just start from the standard real numbers, define infinite sums in terms of limits, and then deduce 0.999..=1 without circular reasoning.

(Link to ways of defining or constructing the reals: http://en.wikipedia.org/wiki/Construction_of_the_real_numbers)

• Yves (No new hope) says:

“The sum of a convergent series (at least in standard real analysis) is *defined* as the limit of the approximating finite sums”.

This is not quite right, the sum of a convergent series is *equal* (not defined) by the limit of the approximating finite sums. And is only ONLY true if 0.9999… = 1.

Let’s use the “∑(n = 1 -> x) 9/10^n” series:

First off, the limit of the series is equal to 1, this is true no matter if 0.9999… equals 1 or not.

But if I don’t use the limit, do I get the same answer? Once again, it *is* possible to get the result of the sum without resorting to limits:

So the function of the finite series is “f(x) = ∑(n = 1 -> x) 9/10^n” where x is a non-zero positive integer.

Let’s create a new function g(x), based on f(x) that is always equal to 1:
g(x) = f(x) + 1/x = 1

This also means that:
f(x) = g(x) – 1/x ->
f(x) + 1/x = 1

Woah! Holy simplification Batman!

But wait! If you *always* have to add something to the result of the series for it to be equal to 1, how can it actually be equal to 1? Well, it all depends on if… well… if 0.9999… = 1…

If 0.9999… = 1, then
f(∞) + 1/∞ = 1 ->
f(∞) = 1 – 1/∞ ->

f(∞) = 0.9999… = 1 OR f(∞) = 1 – 0 = 1

Depending on the approach: either with “0.9999… = 1” or “1/∞ = 0”. It doesn’t matter, if one is true, the other one is also true.

But if 0.9999… ≠ 1, then there is no escape! The result of the infinite series *cannot* be 1. It must, at the very least be infinitely close to it! (something not allowed in real numbers)

Conclusion: the result of a infinite convergent series is equal to the limit of the sum of the approximate finite series *only* if 0.9999… = 1. At the very least, for the series used in the proof. Thus the proof:

0.9999… = ∑ (i = 1 -> ∞) 9/10^i = limit (x -> ∞) ∑ (i = 1 -> x) 9/10^i = 1

cannot work unless you already presuppose that 0.9999… = 1!

• Yves (the saga ends... or does it) says:

And still you haven’t given a rigorous definition of what an infinite sum is supposed to be. We’re just running in circles now, so I think this is going to be my final response, unless that definition shows up, because else there isn’t anything new to say, it seems.

If f(x) = ∑(n = 1 -> x) 9/10^n, then f(x)+1/x does not equal one, f(x) + 1/10^x does. I’m also not sure what exactly you claim to have simplified by doing this (or by calling it g(x)), because the point you then proceed to make (“you *always* have to add something to the result of the series for it to be equal to 1”) is only true for finite values of x. I have no idea if it is true or not for the infinite sum, because you have not defined how to make sense of an infinite sum (besides claiming that this particular one is equal to 1-epsilon).

• Yves (The infinite series strikes back!) says:

1. Oops, yes I meant 1/10^x and not 1/x. This doesn’t change much about the proof… except that 1/x should be replaced with 1/10^x and 1/∞ with 1/10^∞ (which is equal to 1/∞ anyways).

2. I don’t have to define what the what an infinite sum is, because you already now. It’s the sum… of an infinite number of items, the same way a finite sum as a sum of a finite number of item. Except this time, the endpoint is infinity. The properties of infinite series are identical to those of finite series, they are just trickier to use since we obviously can’t manually add an infinite number of items. You are stuck confounding the sum itself with *one* method with which you can find the result. I, in the meantime, am arguing that the method works, but only if you already accept that 0.9999… = 1.

And this is besides the point, because the proof I have shown doesn’t even rely on what the properties of an infinite series are supposed to be (except for the fact that it *is* a series) or how you’re supposed to calculate it, by using something we both know very well how to use: a finite series.

All that matters is that:
1. “∑ (i = 1 -> x) 9/10^i + 1/1o^x = 1” is true for all non-zero positive integers values of x. We can prove this with little difficulty by treating it as a finite series (x=1) and then proving by induction that it is true for all higher integers, including infinity.

2. If x = ∞ then the finite series becomes an infinite series, but this does *not* change the veracity of the statement as x is still a non-zero positive integer.

Even without knowing how to do an infinite sum we found the result!

So stop asking for the “definition of an infinite sum” you already know what it is and you don’t even need it for the proof to work.

• Yves (... and misses) says:

” I don’t have to define what the what an infinite sum is, because you already now.”

No, I don’t know, or I wouldn’t ask.

“It’s the sum… of an infinite number of items, the same way a finite sum as a sum of a finite number of item. Except this time, the endpoint is infinity.”

That makes no sense to me. Perhaps it does to you, but not to me. There is no “same way” when going from finite concepts to infinite ones. The history of maths shows that if you are not very careful in these situations, things can and will go wrong. If you are familiar with the rigour necessary for mathematical definitions, then you should be able to understand why this doesn’t cut it. If you are not, then I advise you to look into it and try to reformulate your argument rigorously.

“You are stuck confounding the sum itself with *one* method with which you can find the result.”

No, I’m not. I’m stuck at not getting your generalised definition of infinite sum, which is supposed to be equivalent to the limit definition (or method, if you prefer) in standard real analysis, but will reduce to something else for non-standard real analysis.

“And this is besides the point, because the proof I have shown doesn’t even rely on what the properties of an infinite series are supposed to be”

How can a proof not depend on the definitions of the concepts involved?

“We can prove this with little difficulty by treating it as a finite series (x=1) and then proving by induction that it is true for all higher integers, including infinity.”

I agree with this, up to (but not including!) those last two words. Standard mathematical induction does not allow you to make the jump from “this holds for any positive integer value of x” to “this holds for x=infinity” (whatever “x=infinity” is exactly supposed to mean here, which I don’t know, because the lack of definition).
Perhaps there are other concepts of induction out there that I am not familiar with and which allow you to make this jump to infinity somehow, but they definitely would have to require more setup than just the normal standard induction argument as you present it here. In short, I don’t agree that your conclusion follows from the argument you presented here and I don’t see any obvious way to fix the argument (in large part, because the conclusion involves an undefined entity).

“If x = ∞ then the finite series becomes an infinite series, but this does *not* change the veracity of the statement as x is still a non-zero positive integer.”

Does your x also live in some kind of non-standard number system, because in the world of the standard positive integers, “infinity” is not a number? So I also disagree with your conclusion that whatever held for the finite case, can just simply be assumed to hold as well for the infinite case.

“So stop asking for the “definition of an infinite sum” you already know what it is and you don’t even need it for the proof to work.”

Okay (because it’s clear you will not give it anyway).
No I don’t (how would you know what I know?).
Of course you do (in any proof you need to know the definitions of the concepts involved).

• Yves (Return of... whatever) says:

“That makes no sense to me. Perhaps it does to you, but not to me.”

What’s so hard about imagining an infinite sum of terms? How many way are there of describing this?

And once again, it *doesn’t matter*.

“Does your x also live in some kind of non-standard number system, because in the world of the standard positive integers, “infinity” is not a number?”

I’m assuming here you meant: “Does your x also live in some kind of non-standard number system? Because in the world of the standard positive integers, “infinity” is not a number.”

Of course infinity is a number! It only has an undefined value and it’s agreed that it doesn’t hold a *single* value (after all, 2∞ or ∞/2 is still infinity). That doesn’t mean you can’t use it or even figure out some properties!

For example, in the realm of integers, if I have b = 2a, then I can safely assume that 2 is an even number. But what if a = ∞? Then b = 2∞ = ∞. Yet, we can still claim that b is an even number. Why because its infinity is not the same as a’s infinity and doubling an integer *always* gives you an even number. Even if that number is infinity!

As defined on the on the show, infinity is simply an unknown number that is always bigger than any specific number you can conjure up.

“in any proof you need to know the definitions of the concepts involved”

Bzzzt! Wrong! As long as your train of logic uses known concepts, you can still figure out the properties of unknown or vague concepts. Take the “this infinity must be even” example. Infinity does not have a known value, yet I was able to deduce that a specific example of an infinity (namely the one in the “b” variable) was even! Why? Because no matter the value of a, b has to be even.

You are in fact faulting me for not defining concepts *I’m not even using*.

Take this new example:
f(x) + 1/x = 9

What do I know of f(x)? Nothing. Is it a series? a limit? an integration? a fixed list of numbers? We don’t know. Yet, I can give you it’s value for any value of x (except 0)!

This is exactly what I did in my proof: I concentrated on known properties of the series and I went around those I didn’t.

First the result of the series is quite clear when it is finite, this allows me to figure out some of it’s properties. Namely, that s(x) + 1/10^x = 1 (where s(x) is the series) for all non-zero positive integers.

“But wait!”, you say, doesn’t it behave differently when if becomes an infinite series? Nope! Any properties of a series remains the same whether it becomes finite or infinite. I don’t *need* to know what an infinite sum is supposed to be because I’m concentrating of the properties of the *series*, which, once again, *do not change if it becomes infinite*.

So what’s so special about infinite series? Well, it’s simply that you can’t calculate it manually as it’s impossible to manually add up an infinite number of terms. The rest stays the same!

Now, contrary to what you believe, in the integer system, infinity IS a number, we just don’t know exactly what number it is, thus we aren’t sure of all it’s properties. This is no different than if I had an unknown variable.

Thus if I say x = ∞, then the statement becomes “s(∞) + 1/10^∞ = 1”, the difficulty remains in how I treat the “1/10^∞” part. And it depends on if 0.9999… = 1 or not.

If 0.9999… = 1 and 1 – 1/∞ = 0.9999…, then 1/∞ = 0. And 1/10^∞ = 1/∞ = 0. So finally “s(∞) + 1/10^∞ = 1” -> “s(∞) + 0 = 1”

If 0.9999… ≠ 0 and 0.9999… + 1/∞ = 1, then “s(∞) + 1/10^∞ = 1” -> “s(∞) + = 1” -> “s(∞) = 1 – 1/∞” -> “s(∞) = 0.9999…”

• Yves (putting it in different hands?) says:

Clearly, this isn’t working. I keep asking for rigorous definitions, you keep telling me they’re not needed. So let me give it one last shot, but in a different way: Do you have any references (journal papers, maths books?) for the maths you’re using? If I’ll find the time I’d like to read up on it.

6. Yves (... and misses) says:

Btw, thanks for the discussion, namesake! Too bad we’re stuck on some of the details, but it was fun nonetheless.

7. Eric Tergerson says:

Ahh…………. such serendipity. much wow.

Speaking of the edge.org, and their Annual Question:

This year’s was, “What Scientific Idea Is Ready For Retirement?”

The answer from Max Tegmark: “Infinity”

I’ll have to add, I totally agree. Infinity seems like a silly cop-out, and an artifact of math that works while describing a more immediate world, but comes to wrong conclusions about the very big and very small. Newtonian physics worked great for our observable world before giving way to relativity-based, and quantum-based theories.

(scroll to the bottom for the article)

8. David says:

Wow after that spirited discussion from Yves and Yves of the sigma sigma sigma fraternity, my comments are going to sound downright frivolous!

First, on the topic of altruism, Douglas Hofstadter wrote about this, and the evolution of the superrational cooperator, in his Scientific American articles, later published as the book, “Metamagical Themas”. The basic idea was that, if a lot of people assume that they are representative of a group of like-minded individuals, and act accordingly, good things can happen that won’t happen otherwise. One example from recent history might be a large group of airline passengers attacking a much smaller number of armed hijackers. Any individual might not survive, but the group will surely prevail, so long as enough have that “superrational cooperator” mindset. Yay, Todd Beamer!

Hofstadter also wrote a truly life-changing, mind-bending and just plain delightful treatment on infinite series, including the vital feature that each step in the regression takes less time than the last, in a wonder piece called “Little Harmonic Labyrinth”. It’s one of the dialogs between the chapters of his book, “Godel, Escher Bach: An Eternal Golden Braid”. Just to whet your whistle, when a character asks a genie for a wish, the genie has to check with “GOD” to see if it’s okay (because it’s a wish for wishes!). GOD is a recursive acronym for “GOD Over Djinn”, so each genie must consult with the set of all genies over him (or her — they alternate like even and odd numbers!), in order to make a request of “GOD”. But of course there is no GOD, just the infinite sequence of genies above any given one. Fortunately each genie speaks twice as fact, with a voice exactly one octave higher than the one before, so the request makes it all the way to “GOD” and back in finite time (precisely one “moment”, in fact). Great fun.. read ’em if ya got ’em. I’ve read that book at least four times, and the particular dialog many more times than that. I suppose I could read it an infinite number of times if each pass took half the time of the previous one.

I’ll end this frivolity with a joke. In the mid 1990s when Intel unleashed some processors with defective divide algorithms on the masses, they got some pretty bad press for it. A popular joke was, “Intel: where quality is job 0.9999999”. I suppose in light of this podcast, it wasn’t a slam after all!

9. Ian says:

Hahahaha, I’ve made the TRC with my comment? I know I’m slow in catching up on the episodes (I listen to podcasts from their beginnings, rather than listening to current ones and then going through the archives), but now that I did catch up to this one it made my day.

For the 0.99999… = 1 thing, the fact that it doesn’t really affect policy or anything is part of what I meant by the scope and scale. The arguments got the same though. You have math on one side, neozenoists on the other, and round and round in circles it goes, with the latter slipping through by refusing to un-fuzzy the definitions they’re using, as seen here in the comments above. 0.(9) is the sum of an infinite series, the sum of an infinite series is by definition the limit of a sequence of finite sums of the form 0.9999… , and that is by definition 1. If you don’t agree, you have to disagree with one of these points, and that means you have to build separate mathematics of your own that doesn’t enrich our ability to calculate things at all, only makes you feel better. Hey, that sounds like a certain something I mentioned before. That alternative mathematics would also have to deal with this:
0.9999… = x, where x is some number which is totally not one
9.9999… = 10x, where x is still totally not one
9.9999… – 0.9999… = 10x – x, where x remains totally not one
9 = 9x, where x is, nothing having changed about it, still totally not one
1 = x, where x remains still totally not on…. wait a minute. But alas, the ride never stops.

As for infinity, meaning ∞, is it a number? Ehhhhhhhh… it’s mostly a placeholder for a concept and something of a notation convention, rather than a number. If you want to know of some interesting things about infinity as a number, you want the aleph hierarchy. Consider this… if we have a set of things, how many things are in there? We’d use a number to describe that. |x| is the number of things in the set x, whereas {} enclose the list of things that make up a set.
|{apple, pear, plum}| = 3
|{pomegranate, car}| = 2
|{}| = 0
|{dog,cat,different dog,parrot}| = 4
|{tree,5,moon,giraffe}| = 4
|{1,2,3,4,5,6}| = 6

So hey, if N is our natural numbers (1,2,3,4,5,…), how many are there? Well… |N|. But how much is that? “Infinity”, I hear you say! Very well. How about if we include 0 in our natural numbers? Are there more of them now than before?
“Yes! All the numbers that were in there are still there, and there’s another one on top of that! Clearly there’s more!”
“No! It was infinity, it’s still infinity! That’s the same!”

Hmm… yeah, one of these has to be wrong. How about if we include negative numbers, creating integers (I)? Same two arguments apply. Hmm… how about rationals (Q), which are fractions made of integer over integer? Still the same two arguments apply. …how about if we take all real numbers (R), then? Still the same two arguments apply.

But if at least one of these were right, then either |N| = |N+{0}| = |I| = |Q| = |R|, or |N| < |N+{0}| < |I| < |Q| < |R|, since the "good" argument would have worked every time. And yet, in actuality…
|N| = |N+{0}| = |I| = |Q| < |R|

Yeeeep. There's the same amount of natural numbers as there are rationals, but there's more real numbers than the rest of those. Both of these natural and kinda compelling arguments are wrong, heh. For a short overview of how we can know that…

Imagine we have one set of numbers on the left, and one set of numbers on the right. We can't count them manually because they're infinite sets, so we have to use some tricks. Let's imagine we can tie strings between the numbers, with one end on the left and and one on the right. One number can only have one string tied around it though. If we can make it so that every number has a string attached to it, then every single number has a very specific partner in the other set that this string is tied to – and that in turn will mean there's the same amount of numbers in each set. The amount in both is the same as the amount of strings used, after all.

So let's take N and N+{0} and do this – tie every x from the left with x-1 on the right. Will anything on the left not have a string attached? No. Will anything on the right not have a string attached? No. Would anything on either side have two strings or more attached to it? No. So hey, same amounts in both.

How about, say… N and I. We tie:
1 with 0
2 with 1
3 with -1
4 with 2,
5 with -2
6 with 3
7 with -3
8 with 4
9 with -4
and so on (in other words we tie every x on the left with [x/2]*(-1)^(x+1), for anyone wanting a formula rather than an intuition). Is there any number on the left or right we won't get to? No. Will we tie any twice? No. Same amount!

A similar construction (which requires some drawing to be best explained, so I'll skip it – but basically you consider these fractions as two integers, you draw axes, start at (0,1) and draw a path spiralling out, tying the rationals to naturals as they come along) can be made for N and Q, again proving that |N| = |Q|. Things get interesting though once you try to do it for N and R. We can try, but we'll fail. In fact, there's proof that it's impossible, which goes roughly like this (I won't write it fully, it's elegant but a bit complex and anyone interested can find it):
1. Let's assume that somehow we've successfully done it. Somehow. Doesn't matter how. All numbers in R now have a string attached to them, connecting every single one of them each to some number in N.
2. Ohcrap, it turns out that it's possible to use those pairs of Ns and Rs to make a real number that can't possibly have a string attached to it. And it should, because of the assumption in #1.
3. Since we didn't assume anything about the method, just that it's possible, and that was enough to have a contradiction, it's therefore impossible.

So yeah. There's no "infinity" on its own. There's different infinities, and they have different sizes, and they work not quite like our natural assumption would go, heh. The number describing the amount of numbers in N is called aleph zero (or small omega), while the number for amount of numbers in R is called aleph one (or continuum). It's possible to go up towards aleph 2, aleph 3, aleph 4 and so on, even to aleph aleph 0 and other silly things like that, but that is a story for another time and one I don't feel quite qualified to tell, heh.

• Ian says:

Oh, also… about the strong negatively loaded words used by me, I should apologize. I’m from the cranky and crotchety school of skepticism and I’m glad there’s guys like you to be the public face of it instead, heh. I suppose I’ll break with that character for a moment and say “keep up the good work”.

Though if I’m already writing this, I might as well finish some other thought. Good idea with bringing the Zeno’s paradoxes, since 0.(9) = 1 is the modern version of them.

“An arrow gets launched at Achilles, but he’s not afraid. He knows that to reach him, the arrow will have to 0.9 (in other words, 90%) of the distance between him and the archer, and then there’ll still be distance and he won’t be hit. After that it’ll keep going, but it’ll have to cover 90% of the remaining distance, and it’ll still not have reached him. And then another 90%, and another, travelling 0.9999999… part of the distance but never actually reaching him” – obviously the steps the arrow takes are true and yet it hits him. Let’s add a twist though… how about instead of going 90% of the remaining distance, the arrow started by going a quarter of the way and then in each successive step half as far as in the step before? “Achilles has heard of this ‘infinity’ business, but he knows that even after an infinity of steps the arrow will only be half way through to getting to him, so he feels ever more invincible”. This shines even more light on this problem, I feel.