And we’re off…
Aren’t the first two examples of sets, too?
Constellations are a set of stars; geological eras are a set of events.
I don’t get it. Are you implying that the other sets of groupings depend more on subjective imposition of patterns on things by the human mind, while mathematical sets just “exist” – and that this passage fails to make that distinction?
I think the point is that constellations predate modern astronomy, and, obviously, nobody is literally moving either stars or geological time periods around and grouping them together, which is what the phrase “put similar items in the same place” normally implies. (In fact, I’m not even sure what the “similar items” would be in the geology example.)
I’d be surprised if that were Frylock’s beef, although I agree with you that the writer could have used a better word than “places” for all three – “groups,” maybe?
Yes–I think failing to make that distinction from the get-go is likely to confuse more then help when things get more technical on down the line.
IMHO you are taking the phrase too literally. However, if this is a textbook for kids it could leave some with the wrong impression. ‘Put in the same place’ means categorizing, putting things in the same categorical place, which may not be well understood by some. I wouldn’t call it misleading and false, so much as ‘out of place’ in a math textbook.
Actually, I think it’s a pretty cool way to begin, to show that math is a realm developed by humans, for humans, just like other realms of thought, and that mathematicians are (in part) spurred on my the human urge to classify things, just like botanists, linguists, and political scientists, even if the classification systems they come up with tend to be, in certain ways, more “objective” than the sorts the others come up with.
But mathematicians still have to decide what’s important, what problems they should spend their energies working on, and that’s bound to be embedded in culture and history. Denis Wood describes this phenomenon well in The Power of Maps, in the context of how cartographers who make “objective” maps like topographic quad sheets are still making all kinds of human choices about what to include and how to display it.
How is any student of any age, to whom this is wholly new material, supposed to take these statements other than “literally”?
Someone reading this passage is going to come away thinking that there is no set {the sky, the number seven, Sherlock Holmes}.
I don’t see the pedagogical value of implying that sets are the kind of things that feel natural when collected together. I can see the value in starting out by discussing those kinds of sets, but I would think you must almost immediately correct any impression you may be giving that sets must “feel natural.” Otherwise, it’s going to come as a shock later on. The shock can be obviated by making things clear from the outset.
If you are taking college level math and don’t understand that the stars weren’t physically placed into constellations, then you should be taking some other courses.
I think you’ve misunderstood me. I’m not saying the passage implies that stars are physically placed into constellations. I’m saying it implies that sets are collections of objects for which it feels natural, or at least for which there’s a good reason, to think of them together as a group.
Ahhh…I think I see now. (Math is not my strong suit, sorry). In a way, you’re worried about almost the opposite of what I said. In math, apparently, a “set” has a formal definition which does not involve the elements in the set necessarily having any common property, other than someone chose to group them together.
What I said about math being partly about classifying things by common properties, then, is still true – “all rational numbers”, “all positive even integers”, etc. – and that’s what the author seems to be trying to convey. BUT, they probably shouldn’t use the word “set” to describe such groupings, if I understand what you’re getting at.
There’s not even the “someone choosing to group them together” element when it comes to defining what sets are. A set, literally, is just any collection of elements. (There are actually some strictures you have to put in there to make it clear there’s no such thing as a set of all sets, but the reasons for that go far beyond the purpose of this thread! (and I am not up to reminding myself about those technicalities…))
A set is any collection of elements–but I think a person reading this textbook will come away thinking a set is a collection of elements that all have some important or interesting relationship with each other.
It’s fine in the sense that those are indeed sets. It’s just that the passage gives the impression that all sets are in some sense “natural” collections. But that is importantly not part of what must be true of something in order for it to be a set.
Some sets are. As I said before, it’s kind of out of place in a math textbook, too estoreric for a subject dependent on precision and accuracy. I wouldn’t call it misleading or false though, just not informative.
The stars in a constellation are not “similar”. Astronomers don’t put stars into constellations. They are not really concerned with those so much.
How would you do this better, in an introductory text, without making things utterly confusing from the get-go? This seems entirely justified to me, in pedagogical terms. Being over-precise, up-front is just going to alienate and confuse a student. It is perfectly legitimate, and usually a good idea, to introduce a technical concept at first by analogy to something familiar, even if the analogy is not exact, and then to make it more precise by adding qualifications and clarifications after they have a grasp on the general idea.
By not even mentioning sets. (I didn’t quote enough of the context of the passage, though, to make it clear that talk of sets isn’t necessary at this point in the text. The larger passage is about the differences between natural numbers, integers, rational numbers and real numbers.)
If sets must be mentioned at this point, then I’d add a sentence that says something like “(In fact, absolutely any collection can be considered a set, even if the members of the collection don’t have similar properties.)”
Not particularly precise or confusing.
But like I said, my own approach would be not even to introduce sets at this point. I can’t think of any good reason to introduce the concept of a set unless at that very point in the course the arbitrary membership possibilities within sets are themselves something that will need to be highlighted.
There is nothing false about this statement. It doesn’t say that they only place objects with similar properties in sets.
So, I don’t think you have a legitimate gripe—but I do sympathize with your watching out for things like this.
(I used to wonder whether I should be bothered by all the math books that described Calculus as “the mathematics of change and motion” or some such wording. Sure, that’s what Calculus is almost always used for, but it’s not how it’s defined, and it’s possible to give a pure mathematical description of Calculus without ever once referring to something changing or moving. I figure it’s analogous to defining a pencil as something that is used to write and erase with. That’s what a pencil is for, but a pencil is still a pencil even if nobody ever writes anything with it.)
It’s definitely misleading, and I think colloquially implies the missing “only” hence my “false, really,” but you’re right it’s not false when read strictly. But a student won’t read this strictly.