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.
Consider the following dialogue (p. 112) with an illiterate peasant named Nazir-Said:
The following syllogism is presented: ‘There are no camels in Germany. The city of B. is in Germany. Are there camels there or not?’ Subject repeats syllogism exactly. So, are there camels in Germany? ”I don’t know, I’ve never seen German villages.” Refusal to infer. The syllogism is repeated. ”Probably there are camels there.” Repeat what I said. ”There are no camels in Germany, are there camels in B. or not? So probably there are. If it’s a large city, there should be camels there.” Syllogism breaks down, inference drawn apart from its conditions. But what do my words suggest? ”Probably there are. Since there are large cities, there should be camels.” Again a conclusion apart from the syllogism. But if there aren’t any in all of Germany? ”If it’s a large city, there will be Kazakhs or Kirghiz there.” But I’m saying that there are no camels in Germany, and this city is in Germany. ”If this village is in a large city, there is probably no room for camels.”
Not to deny that this is an example of someone who does not think abstractly—I agree that it is—but there’s also a Gricean interpretation: From the subject’s point of view, if the experimenter already knows that there is literally not even one camel in all of Germany including all of its cities and villages, then the experimenter would not ask whether there are camels in a specific city in Germany, therefore the experimenter must mean, “There are no camels in Germany not counting the cities”, or “There are extremely few camels in Germany”.
That doesn’t wash given the dialogue, rather than a single question. A single question might reasonably elicit a Gricean answer to a different question, but repeated questioning on the same point?
Mathematical induction using the first step as the base case is valid. The problem with the horses of one color problem is that you are using sloppy verbal reasoning that hides an unjustified assumption that n > 1. If you had tried to make a rigorous argument that the set of n+1 elements is the union of two of its subsets with n elements each, with those subsets having a non-empty intersection, this would be clear.
The problem with the horses of one color problem is that you are using sloppy verbal reasoning that hides an unjustified assumption that n > 1.
I’m not sure what you mean. I thought I stated it each time I was assuming n=1 and n=2.
In the induction step, we reason “The first horse is the same colour as the horses in the middle, and the horses in the middle have the same colour as the last horse. Therefore, all n+1 horses must be of the same colour”. This reasoning only works if n > 1, because if n = 1, then there are no “horses in the middle”, and so “the first horse is the same colour as the horses in the middle” is not true.
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.
...
What on Earth went wrong here?
The problem was that your ultimate conclusion was wrong. It is not in fact the case that “mathematical induction should start with the second step, not the first.” It’s just that, like all proofs, you have to draw valid inferences at each step. As JGWeissman points out, the horse proof fails at the n=2 step. But one could contrive examples in which the induction proof fails at the kth step for arbitrary k.
I don’t think I ever got to my “ultimate” conclusion (that all of the operations that appear in step n must appear in the basis step).
I was trying to use this example where the proof failed at n=2 to show that it’s possible in principle for a (specific other) proof to fail at n=2. Higher-order basis steps would be necessary only if there were even more operations.
In The Society of Mind, Marvin Minsky writes about “Intellectual Trauma”:
One of Freud’s conceptions was that the growth of many individuals is shaped by unsuspected fears that lurk in our unconscious minds. These powerful anxieties include the dread of punishment or injury or helplessness or, worst of all, the loss of the esteem of those to whom we are attached. Whether this is true or not, most psychologists who hold this view apply it only the the social realm, assuming that the world of intellect is too straightforward and impersonal to be involved with such feelings. But intellectual development can depend equally upon attachments to other persons and can be similarly involved with buried fear and dreads. [--] By itself, the failure to achieve a goal can cause anxiety. For example, surely every child must once have thought along this line:
Hmmmm. Ten is nearly eleven. And eleven is nearly twelve. So ten is nearly twelve. And so on. If I keep on reasoning this way, then ten must be nearly a hundred!
To an adult, this is just a stupid joke. But earlier in life, such an incident could have produced a crisis of self-confidence and helplessness. To put it in more grown-up terms, the child might think, “I can’t see anything wrong with my reasoning—and yet it led to bad results. I merely used the obvious fact that if A is near B, and B is near C, then A must be near C. I see no way that could be wrong—so there must be something wrong with my mind.” Whether or not we can recollect it, we must once have felt some distress at being made to sketch the nonexistent boundaries between the oceans and the seas; What was it like to first consider “Which came first, the chicken or the egg?” What came before the start of time; what lies beyond the edge of space? And what of sentences like “This statement is false,” which can throw the mind into a spin? I don’t know anyone who recalls such incidents and frightening. But then, as Freud might say, this very fact could be a hint that the area is subject to censorship
If people bear the scars of scary thoughts, why don’t these lead, as our emotion-traumas are supposed to do, to phobias, compulsions, and the like? I suspect they do—but disguised in forms we don’t perceive as pathological. [---]
This seems to fit the anecdote very well—your interlocutor could not find a fault in the reasoning, noticed it led to an absurdity, and decided that this intellectual area is dangerous, scary, and should be evacuated as soon as possible.
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.
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.
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.
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.
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).
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.
Of course you can’t rigorously prove something that’s not true.
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.
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.”
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.
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?
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.
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.
Why didn’t you drop the “horses” example when it tripped him up and go with, I dunno, emeralds or ceramic pie weights or spruckels, stipulated to in fact have uniform color?
I suspect that I lost the second person way before horses even became an issue. When he started picking on my words, “horses” and “different world” and “hypothetical person” didn’t really matter anymore. He was just angry. What he was saying didn’t make sense from that point on. For whatever reason, he stopped responding to logic.
But I don’t know what I said to make him this angry in the first place.
Leaving aside the actual argument, I can tell you that there exist people (my husband is one of them, and come to think of it so is my ex-girlfriend, which makes me suspect that I bear some responsibility here, but I digress) whose immediate emotional reaction to “here, let me walk you through this illustrative hypothetical case” is strongly negative.
The reasons given vary, and may well be confabulatory.
I’ve heard the position summarized as “I don’t believe in hypothetical questions,” which I mostly unpack to mean that they understand that hypothetical scenarios are often used to introduce assumptions which support conclusions that the speaker then tries to apply by analogy to the real world, and that a clever rhetoritician can use this technique to sneak illegitimate assumptions into real-world scenarios, and don’t trust me not to sneak in assumptions that make them look stupid or manipulate them into acting against their own interests.
I don’t know if that’s a factor in your case or not, but I have found that once I trigger that reaction, there’s not much more I can do… they are no longer cooperating in the communication, they are just looking for some way to get out. If I press the point, I merely elicit anger and defensiveness and a variety of distractors.
The best way around this I’ve found so far, and it’s only hit-or-miss, is to avoid the stance of “here let me show you something” altogether.
I am a lot more successful if I adopt the stance of “I am thinking about a problem that interests me,” and if they express interest, explaining the problem as something I am presenting to myself, rather than to them. Or, if they don’t, talking about something else.
At the risk of sounding like Robin, the fact that this is successful leads me to believe that at least sometimes, what’s really going on is that I’ve stepped on some status-signaling landmine, and the reaction I’m getting actually translates to “I refuse to cede you the role of instructor by letting you define the hypothetical.”
And suggesting that this might be what’s going on works about as poorly as you’d expect it to were it what’s going on. Of course, that’s precisely what makes status-signaling a fully generalizable counterargument, so I take it with a grain of salt.
“I refuse to cede you the role of instructor by letting you define the hypothetical.”
You know, come think of it, that’s actually a very good description of the second person… who is, by the way, my dad.
I am a lot more successful if I adopt the stance of “I am thinking about a problem that interests me,” and if they express interest, explaining the problem as something I am presenting to myself, rather than to them. Or, if they don’t, talking about something else.
This hasn’t ever occurred to me, but I’ll try it the next time a similar situation arises.
“No. Just an example. Lies propagate, that’s what I’m saying. You’ve got to tell more lies to cover them up, lie about every fact that’s connected to the first lie. And if you kept on lying, and you kept on trying to cover it up, sooner or later you’d even have to start lying about the general laws of thought. Like, someone is selling you some kind of alternative medicine that doesn’t work, and any double-blind experimental study will confirm that it doesn’t work. So if someone wants to go on defending the lie, they’ve got to get you to disbelieve in the experimental method. Like, the experimental method is just for merely scientific kinds of medicine, not amazing alternative medicine like theirs. Or a good and virtuous person should believe as strongly as they can, no matter what the evidence says. Or truth doesn’t exist and there’s no such thing as objective reality. A lot of common wisdom like that isn’t just mistaken, it’s anti-epistemology, it’s systematically wrong. Every rule of rationality that tells you how to find the truth, there’s someone out there who needs you to believe the opposite. If you once tell a lie, the truth is ever after your enemy; and there’s a lot of people out there telling lies.”
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 might find enlightening the part of the TED talk given by James Flynn (of the Flynn effect), where he talks about concrete thinking.
Hah! I thought of the exact same thing before I saw your comment: the interviews by Luria with Russian peasants where the peasants refuse to abstract in any way. Shalizi provides an example http://vserver1.cscs.lsa.umich.edu/~crshalizi/weblog/484.html :
Not to deny that this is an example of someone who does not think abstractly—I agree that it is—but there’s also a Gricean interpretation: From the subject’s point of view, if the experimenter already knows that there is literally not even one camel in all of Germany including all of its cities and villages, then the experimenter would not ask whether there are camels in a specific city in Germany, therefore the experimenter must mean, “There are no camels in Germany not counting the cities”, or “There are extremely few camels in Germany”.
That doesn’t wash given the dialogue, rather than a single question. A single question might reasonably elicit a Gricean answer to a different question, but repeated questioning on the same point?
Mathematical induction using the first step as the base case is valid. The problem with the horses of one color problem is that you are using sloppy verbal reasoning that hides an unjustified assumption that n > 1. If you had tried to make a rigorous argument that the set of n+1 elements is the union of two of its subsets with n elements each, with those subsets having a non-empty intersection, this would be clear.
Induction based on n=1 works sometimes, but not always. That was my point.
I’m not sure what you mean. I thought I stated it each time I was assuming n=1 and n=2.
In the induction step, we reason “The first horse is the same colour as the horses in the middle, and the horses in the middle have the same colour as the last horse. Therefore, all n+1 horses must be of the same colour”. This reasoning only works if n > 1, because if n = 1, then there are no “horses in the middle”, and so “the first horse is the same colour as the horses in the middle” is not true.
The problem was that your ultimate conclusion was wrong. It is not in fact the case that “mathematical induction should start with the second step, not the first.” It’s just that, like all proofs, you have to draw valid inferences at each step. As JGWeissman points out, the horse proof fails at the n=2 step. But one could contrive examples in which the induction proof fails at the kth step for arbitrary k.
I don’t think I ever got to my “ultimate” conclusion (that all of the operations that appear in step n must appear in the basis step).
I was trying to use this example where the proof failed at n=2 to show that it’s possible in principle for a (specific other) proof to fail at n=2. Higher-order basis steps would be necessary only if there were even more operations.
In The Society of Mind, Marvin Minsky writes about “Intellectual Trauma”:
This seems to fit the anecdote very well—your interlocutor could not find a fault in the reasoning, noticed it led to an absurdity, and decided that this intellectual area is dangerous, scary, and should be evacuated as soon as possible.
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.
Why didn’t you drop the “horses” example when it tripped him up and go with, I dunno, emeralds or ceramic pie weights or spruckels, stipulated to in fact have uniform color?
I suspect that I lost the second person way before horses even became an issue. When he started picking on my words, “horses” and “different world” and “hypothetical person” didn’t really matter anymore. He was just angry. What he was saying didn’t make sense from that point on. For whatever reason, he stopped responding to logic.
But I don’t know what I said to make him this angry in the first place.
Leaving aside the actual argument, I can tell you that there exist people (my husband is one of them, and come to think of it so is my ex-girlfriend, which makes me suspect that I bear some responsibility here, but I digress) whose immediate emotional reaction to “here, let me walk you through this illustrative hypothetical case” is strongly negative.
The reasons given vary, and may well be confabulatory.
I’ve heard the position summarized as “I don’t believe in hypothetical questions,” which I mostly unpack to mean that they understand that hypothetical scenarios are often used to introduce assumptions which support conclusions that the speaker then tries to apply by analogy to the real world, and that a clever rhetoritician can use this technique to sneak illegitimate assumptions into real-world scenarios, and don’t trust me not to sneak in assumptions that make them look stupid or manipulate them into acting against their own interests.
I don’t know if that’s a factor in your case or not, but I have found that once I trigger that reaction, there’s not much more I can do… they are no longer cooperating in the communication, they are just looking for some way to get out. If I press the point, I merely elicit anger and defensiveness and a variety of distractors.
The best way around this I’ve found so far, and it’s only hit-or-miss, is to avoid the stance of “here let me show you something” altogether.
I am a lot more successful if I adopt the stance of “I am thinking about a problem that interests me,” and if they express interest, explaining the problem as something I am presenting to myself, rather than to them. Or, if they don’t, talking about something else.
At the risk of sounding like Robin, the fact that this is successful leads me to believe that at least sometimes, what’s really going on is that I’ve stepped on some status-signaling landmine, and the reaction I’m getting actually translates to “I refuse to cede you the role of instructor by letting you define the hypothetical.”
And suggesting that this might be what’s going on works about as poorly as you’d expect it to were it what’s going on. Of course, that’s precisely what makes status-signaling a fully generalizable counterargument, so I take it with a grain of salt.
You know, come think of it, that’s actually a very good description of the second person… who is, by the way, my dad.
This hasn’t ever occurred to me, but I’ll try it the next time a similar situation arises.
“No. Just an example. Lies propagate, that’s what I’m saying. You’ve got to tell more lies to cover them up, lie about every fact that’s connected to the first lie. And if you kept on lying, and you kept on trying to cover it up, sooner or later you’d even have to start lying about the general laws of thought. Like, someone is selling you some kind of alternative medicine that doesn’t work, and any double-blind experimental study will confirm that it doesn’t work. So if someone wants to go on defending the lie, they’ve got to get you to disbelieve in the experimental method. Like, the experimental method is just for merely scientific kinds of medicine, not amazing alternative medicine like theirs. Or a good and virtuous person should believe as strongly as they can, no matter what the evidence says. Or truth doesn’t exist and there’s no such thing as objective reality. A lot of common wisdom like that isn’t just mistaken, it’s anti-epistemology, it’s systematically wrong. Every rule of rationality that tells you how to find the truth, there’s someone out there who needs you to believe the opposite. If you once tell a lie, the truth is ever after your enemy; and there’s a lot of people out there telling lies.”