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ArisKatsaris comments on How to Not Lose an Argument - Less Wrong
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Most of the comments in this discussion focused on topics that are emotionally significant for your "opponent." But here's something that happened to me twice.
I was trying to explain to two intelligent people (separately) that mathematical induction should start with the second step, not the first. In my particular case, a homework assignment had us do induction on the rows of a lower triangular matrix as it was being multiplied by various vectors; the first row only had multiplication, the second row both multiplication and addition. I figured it was safer to start with a more representative row.
When a classmate disagreed with me, I found this example on Wikipedia. His counter-arguement was that this wasn't the case of induction failing at n=2. He argued that the hypothesis was worded incorrectly, akin to the proof that a cat has nine tails. I voiced my agreement with him, that “one horse of one color” is only semantically similar to “two horses of one color,” but are in fact as different as “No cat (1)” and “no cat (2).” I tried to get him to come to this conclusion on his own. Midway through, he caught me and said that I was misinterpreting what he was saying.
The second person is not a mathematician, but he understands the principles of mathematical induction (as I'd made sure before telling him about horses). And this led to one of the most frustrating arguments I'd ever had in my life. Here's the our approximate abridged dialogue (sans the colorful language):
Me: One horse is of one color. Suppose every n horses are of one color. Add the n+1st horse, and take n out of those horses. They’re all of one color by assumption. Remove 1 horse and take the one that’s been left out. You again have n horses, so they must be of one color. Therefore, all horses are of one color.
Him: This proof can't be right because its result is wrong.
Me: But then, suppose we do the same proof, but starting with on n=2 horses. This proof would be correct.
Him: No, it won’t be, because the result is still wrong. Horses have different colors.
Me: Fine, then. Suppose this is happening in a different world. For all you know, all horses there can be of one color.
Him: There’re no horses in a different world. This is pointless. (by this time, he was starting to get angry).
Me: Okay! It’s on someone’s ranch! In this world! If you go look at this person’s horses, every two you can possibly pick are of the same color. Therefore, all of his horses are of the same color.
Him: I don’t know anyone whose horses are of the same color. So they’re not all of one color, and your proof is wrong.
Me: It’s a hypothetical person. Do you agree, for this hypothetical person—
Him: No, I don’t agree because this is a hypothetical person, etc, etc. What kind of stupid problems do you do in math, anyway?
Me: (having difficulties inserting words).
Him: Since the result is wrong, the proof is wrong. Period. Stop wasting my time with this pointless stuff. This is stupid and pointless, etc, etc. Whoever teaches you this stuff should be fired.
Me: (still having difficulties inserting words) … Wikipe—…
Him: And Wikipedia is wrong all the time, and it’s created by regular idiots who have too much time on their hands and don’t actually know jack, etc, etc. Besides, one horse can have more than one color. Therefore, all math is stupid. QED.
THE END.
To the best of my knowledge, neither of these two people were emotionally involved with mathematical induction. Both of them were positively disposed at the beginning of the argument. Both of them are intelligent and curious. What on Earth went wrong here?
^One of the reasons why I shouldn’t start arguments about theism, if I can’t even convince people of this mathematical technicality.
You didn't actually prove that n+1 horses have one color with this, you know, even given the assumption. You just said twice that n horses have one color, without proving that their combined set still has one color.
For example consider the following "Suppose every n horses can fit in my living room. Add the n+1 horse, and take n out of those horses. They can fit in my living room by assumption. Remove 1 horse and take the one that’s been left out. You again have n horses, so they must again fit in my living room. Therefore, all horses fit in my living room."
That's not proper induction. It doesn't matter if you begin with a n of 1, 2, 5, or 100 horses, such an attempt at induction would still be wrong, because it never shows that the proposition actually applies for the set of n+1.
.... The first n horses and the second n horses have an overlap of n-1 horses that are all the same color. So first and the last horse have to be the same color. Sorry, I thought that was obvious.
I see your point, though. This time, I was trying to reduce the word count because the audience is clearly intelligent enough to make that leap of logic. I can say the same for both of my "opponents" described above, because both of them are well above average intellectually. I honestly don't remember if I took that extra step in real life. If I haven't, do you think that was the issue both people had with my proof?
I have a feeling that the second person's problem with it was not from nitpicking on the details, though. I feel like something else made him angry.
You need to make this more explicit, to expose the hidden assumption:
Take a horse from the overlap, which is the same color as the first horse and the same color as the last horse, so by transitivity, the first and last horse are the same color.
But why can you take a horse from the overlap? You can if the overlap is non-empty. Is the overlap non-empty? It has n-1 horses, so it is non-empty if n-1 > 0. Is n-1 > 0? It is if n > 1. Is n > 1? No, we want the proof to cover the case where n=1.
That's exactly what I was trying to get them to understand.
Do you think that they couldn't, and that's why they started arguing with me on irrelevant grounds?
And the point that I am trying to get you to understand, is that you do not need special rule to always check P(2) when making a proof by induction, in this case where the induction fails at P(1) -> P(2), carefully trying to prove the induction step will cause you to realize this. More generally you cannot rigorously prove that for all integers n > 0, P(n) -> P(n+1) if it is not true, and in particular if P(1) does not imply P(2).
Sorry, I can't figure out what you mean here. Of course you can't rigorously prove something that's not true.
I have a feeling that our conversation boils down to the following:
Me: There exists a case where induction fails at n=2.
You: For all cases, if induction doesn’t fail at n=2, doesn’t mean induction doesn’t fail. Conversely, if induction fails, it doesn’t mean it fails at n=2. You have to carefully look at why and where it fails instead of defaulting to “it works at n=2, therefore it works.”
Is that correct, or am I misinterpreting?
Anyways, let's suppose you're making a valid point. Do you think that my interlocutors were arguing this very point? Or do you think they were arguing to put me back in my place, like TheOtherDave suggests, or that there was a similar human issue that had nothing to do with the actual argument?
To butt in, I doubt your interlocutors were attempting to argue this point; they seem like they were having more fundamental issues. But your original argument does seem to be a bit confused.
Induction fails here because the inductive step fails at n=2. The inductive step happens to be true for n>2, but it is not true in general, hence the induction is invalid. The point is, rather than "you have to check n=2" or something similar, all that's going on here is that you have to check that your inductive step is actually valid. Which here means checking that you didn't sneak in any assumptions about n being sufficiently large. What's missing is not additional parts to the induction beyond base case and inductive step, what's missing is part of the proof of the inductive step.
Your hindsight is accurate, but more than just recognizing the claim as true when presented to you, I am trying to get you to take it seriously and actively make use of it, by trying to rigorously prove things rather than produce sloppy verbal arguments that feel like a proof, which is possible to do for things that aren't true.
This is accurate, and related, but not the entire point. Distinguish between a proof by mathematical induction and the process of attempting to produce a proof by mathematical induction. One possible result of attempting to produce a proof is a proof. Another possible result is the identification of some difficulty in the proof that is the basis of an insight that induction isn't the right approach or, as in the colored horses examples, that the thing you are trying to prove is not actually true.
The point is that if you are properly attempting to produce a proof, which includes noticing difficulties that imply that the claim you are trying to prove is not actually true, you will either produce a valid proof or identify why your approach fails to provide a proof.
No, your interlocutors were not arguing this point. Their performance, as reported by you, was horribly irrational. But you should apply as much scrutiny to your own beliefs and arguments as to your interlocutors.
The case of two horses is special here because the sets 1..n and 2..n+1 don't overlap if n+1 = 2, and not because of some fundamental property of every induction hypothesis, but that -- along with some arbitrary large n, and maybe the next case if I'm using any parity tricks -- is one of the first cases I'd check when verifying a proof by induction.
The case of P(n) -> P(n+1) (i.e., the second part of the induction argument) that fails is n=1. (In other words n+1 = 2).
The second part of the induction argument must begin (i.e., include n >= n0) at the value n0 that you have proven in the first part to be true from 1 to n0. In this case n0 = 1, so you must begin the induction at n = 1.
I have edited my comment to avoid this confusion.
You're right, of course. I was trying to describe the flaw in the set-overlap assumption without actually going through an inductive step, on the assumption that that would be clearer, but in retrospect my phrasing muddled that.
I'll see if I can fix that.