That would have been a good idea.
Well that justifies it somewhat, because the point here isn’t to define “set,” but rather to begin a discussion of what we can do with certain particular sets. It seems like you’re ignoring the real pedagogical aim just for the sake of complaining. The actual, true concept of set can be dealt with when it’s necessary. You don’t give children much credit by assuming that they won’t be able to handle that.
True, and you’re hitting on an important point: In math, things are their definitions.
The concept ‘set’ has a set of absolute rules, called ‘axioms’, which define it. Anything that follows those rules is a set. There are no further qualifications. Therefore, if you can show that some given object follows the rules for being a set, you can use all of the machinery of set theory to prove statements about it which must be true, because they follow validly from the definition of what a set is. In fact, a good portion of the work of mathematics is proving that certain objects of interest obey known rule sets so therefore we’re able to prove interesting statements about them.
Anyway, ‘objects in a set must be related to each other’ is not one of the axioms in any formal definition of a set. Therefore, it isn’t relevant to being a set.
On the flip side, sets in mathematics are very often defined in terms of ‘every object from this pre-existing set which obeys this rule’, and the rule given imposes a certain relationship on the objects in the new set. That’s a pragmatic issue, though; the set {a mosquito, my libido} is perfectly valid, if perfectly uninteresting from a mathematical standpoint.
So Frylock is getting annoyed that the book is imputing more to sets than there really is in a formal sense. In a formal sense, sets as intellectually vacuous as a selection of random words are well-formed and valid in every sense. OTOH, taking your view of Math As A Human Activity, such sets are rare in practice and all sets people actually care about do have something more to them.
My reaction was: the statement’s true. Mathematicians are interested in patterns and things with similar properties. There certainly is a set {the sky, the number seven, Sherlock Holmes}, but it wouldn’t interest a mathematician much. But the set of prime numbers would.
Is it just me, or did anyone else start hearing Frylock’s posts in Sheldon’s voice?
Neither will they read it in such a way as to impede their understanding of the real mathematical concept of “set” when the time comes for them to study that. It is certainly not flatly and explicitly saying that things can only be grouped into a mathematical set if they are similar. When they come to study sets properly, the students are not going to think “Oh, but my old textbook said that we only put similar things into sets (a notion that never would have occurred to me otherwise!), so what they are telling me now about arbitrary sets can’t be right.”
Pedagogically this is, at worst, harmless. Nobody is actually going to be misled by it. You have now admitted that it it is not actually false. Really, what is your problem? You are being pointlessly pedantic, and even there you have been out-pedanted by Thudlow Boink.
Sets certainly aren’t required to have similar properties, so it fails as a rigorous definition, but I think that if the examples they gave weren’t kind of bad, it wouldn’t be a terrible way to put it. Sure, I can make a set {1, 3, -10, 200, 7, 11.5, 5+2i, apple} in which the members have absolutely no relationship to each other, other than being members of a set. But most sets have a point to them, the vast majority of sets are made for a reason – {-2,1} may seem random, but it’s actually the set of roots for (x+2)(x-1)=0.
I don’t think there are many real cases, except for (ironically) school problems teaching set operations where you have a completely random set with no relation to anything just sitting there.
So yeah, not everything that fits the formal definition fits the comparison, but I think most thing do and it’s a fair way to introduce it as long as later on you tell them the tiny bit of information that makes it complete.
Granted, a good rebuttal might be that there are some things that are more or less defined by belonging to the same set, rather than the set being defined by their similar properties.
I did.
Indeed, mundanely pedantic as well. I just had to share.
Anyway, you and I disagree about the pedagogical value of the passage. I think people would be misled by the passage, but how am I to prove it? Iunno. I also think that the statement colloquially implies a falsehood, even though it doesn’t strictly imply a falsehood. How am I to prove it? Iunno.
Is there some way we can arbitrate between our positions?
Yes. There are sets such as the Integers, the Rational Numbers, and the Real Numbers where they are, in fact, grouped based on a common property. There are other sets you can encounter, such as the arbitrary set of (4, 22, 99, 332, 93322) specified on page 45, question 23. The second type isn’t “not a set” because there is no obvious common property, but mathematicians use sets to make groups that make sense intuitively, for example because the numbers have common mathematical properties, because the numbers relate to data/measurements that has been gathered from a common source, or for some other legitimate, non-arbitrary reason.
It appears the passage I quoted is found in several books.
I don’t know which I was looking at–I just saw a single photocopied page.
How did this single page come to your attention?
It was sitting there on a table next to a copying machine while I was making copies of something else. Why do you ask?
Dude, pick one.
What?! We’re not even talking about sets?! So let me get this straight. You’re worried that students will take literally something that’s only implied about something that no one’s talking about and the students don’t know exists in the first place?
Yeah…I think you’re making too much of it.
I don’t mean the same thing by “literally” and “strictly” in the quoted comments. “Literally” in that quote has more to do with the references of terms, while “strictly” in that quote has more to do with logical implications.
I don’t understand the above. They know sets exist, because the passage (as quoted in the OP) discusses them, and (on a colloquial reading) apparently defines them. But the apparent definition is a bad one. We’re definitely talking about sets–for example, the set of real numbers, the set of natural numbers, and so on. The passage colloquially implies that all sets are like that–fairly natural seeming sets that it makes intuitive sense to put together. I’m saying that is misleading.
I’m not making much of anything of it. I didn’t go posting to GD or the Pit.
The “likewise” comment is a bit weird.
Assuming we can build sets as we like, we can build true “sets are like this grouping” sentences, as we like, as follows.
On the other hand, maybe it was a comment that everything is similar.
Also, the “geologists divide the history of the Earth into eras” bit isn’t strictly wrong, but to me doesn’t really reflect a good understanding of the difference between absolute and relative dating in geology. What would be more accurate is that geologists sort rocks into relative time units (including eras) and then when possible relate those to absolute dates in Earth history. Before radiometric dating, geologists still used basically the same framework of geologic eras, periods, etc despite not having any idea how much time was actually being represented.
I would have just gone with
Just popping in to say I like Jragon’s version.
Curiosity.