He wasn't wrong, but I also think this an example of "professional blindness", where experts in a topic can lose the ability to imagine the POV of a layman/novice.
I was thinking this was parody for a moment, because the "bad" formulations (0) and (2) seem obviously easier to understand than the "good" formulation (1).
I needed some attempts to parse what (1) even wants to say. (Minimum, maximum and average of what? I imagine the counts per unique item in the bag, but it doesn't say that anywhere)
It's also not obvious at all that (1) is true, so you'd need to see at least a proof.
But what's true is that (1) is more versatile and easier to apply to new problems - it's just harder to teach. So maybe the right solution would have been to start with (0) and (2), then show how those actually imply (1) - and then go on using (1) as a tool, like the quadratic formula.
It's a bit like "If I have two apples and get three more apples, I have five apples" is easier to understand for someone learning addition than "2+3=5", but you still don't want to spend the rest of your life imagining metaphorical apples whenever you have to calculate something.
Yeah, it may seem like a "better" (because stronger) conclusion to the author that if you have more pigeons than pigeonholes you must have more than one pigeon in a hole even if negative or irrational numbers of pigeons. But you're pretty much only invoking the pigeonhole principle in discrete math, where "more than one" means "at least 2".
Yeah, I think that was something that irked me about the "general" formulation: It suddenly brought in an average, i.e. a real, even though the "common" formulations only dealt with integers. This may be more general but made reasoning and understanding harder.
I think overall it's a good habit to go through your proofs again, and remove unnecessary double negations, and make proofs 'more constructive'.
Reductio ad Absurdum makes coming up with proofs easier (you have one more information to use, you can work from both ends of the problem and try to make 2 half long proofs meet, instead of one normal long), but in the end it is often unnecessary, you can remove it, and your proof reads better.
I don't share his view that Generalized Pigeon-hole Principle makes the normal version unnecessary. The normal version is used a lot in the form it is formulated.
The Pigeonhole Principle is merely the Law of Excluded Middle with more than two 'slots'.
Proof by Contradiction relies on the Law of Excluded Middle and is considered inferior than direct or by induction, so there is no elevated status given to these.
He wasn't wrong, but I also think this an example of "professional blindness", where experts in a topic can lose the ability to imagine the POV of a layman/novice.
I was thinking this was parody for a moment, because the "bad" formulations (0) and (2) seem obviously easier to understand than the "good" formulation (1).
I needed some attempts to parse what (1) even wants to say. (Minimum, maximum and average of what? I imagine the counts per unique item in the bag, but it doesn't say that anywhere)
It's also not obvious at all that (1) is true, so you'd need to see at least a proof.
But what's true is that (1) is more versatile and easier to apply to new problems - it's just harder to teach. So maybe the right solution would have been to start with (0) and (2), then show how those actually imply (1) - and then go on using (1) as a tool, like the quadratic formula.
It's a bit like "If I have two apples and get three more apples, I have five apples" is easier to understand for someone learning addition than "2+3=5", but you still don't want to spend the rest of your life imagining metaphorical apples whenever you have to calculate something.
Yeah, it may seem like a "better" (because stronger) conclusion to the author that if you have more pigeons than pigeonholes you must have more than one pigeon in a hole even if negative or irrational numbers of pigeons. But you're pretty much only invoking the pigeonhole principle in discrete math, where "more than one" means "at least 2".
Yeah, I think that was something that irked me about the "general" formulation: It suddenly brought in an average, i.e. a real, even though the "common" formulations only dealt with integers. This may be more general but made reasoning and understanding harder.
I think overall it's a good habit to go through your proofs again, and remove unnecessary double negations, and make proofs 'more constructive'.
Reductio ad Absurdum makes coming up with proofs easier (you have one more information to use, you can work from both ends of the problem and try to make 2 half long proofs meet, instead of one normal long), but in the end it is often unnecessary, you can remove it, and your proof reads better.
I don't share his view that Generalized Pigeon-hole Principle makes the normal version unnecessary. The normal version is used a lot in the form it is formulated.
The Pigeonhole Principle is merely the Law of Excluded Middle with more than two 'slots'.
Proof by Contradiction relies on the Law of Excluded Middle and is considered inferior than direct or by induction, so there is no elevated status given to these.