NOUN: a grammatically distinct category of lexemes of which the morphologically most elementary members characteristically denote types of physical objects (such as human beings, other biological organisms, and natural or artiﬁcial inanimate objects)

No English-specific definition is given, as far as I can tell; but there's a characterization of what a definition *should* look like, similar to what Pullum says in the next-to-last paragraph here.

"A word is a noun if it can be used (by itself, without other words or quotation marks) to finish one of the following sentences:

I am thinking about __

I am thinking about a __

I am thinking about an __ "

The point is that linguists define these categories distributionally; nouns are the words that go in particular places in the sentence. So what you want, if you're trying to teach people to identify nouns, is to come up with sentences with gaps in them that can be filled by any noun (and only by a noun). This particular one will get you in trouble with pronouns ('him' is a noun, but 'he' isn't?), which gives you a chance to follow this up with a discussion about Case.

]]>It's quite normal for us to use "lies to children" in education. Primary school science classes don't offer anything resembling a correct explanation for what a solid and a liquid really are, or why they behave differently. No-one will offer an explanation that actually makes any sense until you're on a degree course. No-one worries about the problematic definition of "species" when teaching a class of twelve year olds about dinosaurs, and fortunately dinosaurs are cool so the class might not notice this oversight.

These "lies to children" are legitimate so long as the lie isn't allowed to become the dominant understanding in adult society as can easily happen. Watching a group of otherwise intelligent adults argue about the Monty Hall problem, or worse, how an aeroplane's wings work can be a little scary. One statistician or physicist doesn't stand much chance against an army of people who half-remember a simplified (well, actually just bogus) explanation from when they were at school.

]]>I think the relevant concern is sets which are (or are not) members of themselves. I think it's relatively rare to partition some sets into subsets, where there is a single subset that all of the subsets are a members of, so he's surprised that "verb" is a member of a sibling to the class it denotes.

]]>1) a mathematical function

and

2) an indeterminate symbol standing for a number

Example of (1): The sine function, typically abbreviated as "sin". This is a function in that it identifies a rule for assigning numbers: The sine of 0 is 0, the sine of pi/2 is 1, the sine of pi is 0, the sine of (3/2)pi is -1, and so on. We write these as

sin(0) = 0

sin(pi/2) = 1

sin(pi) = 0

sin((3/2)pi) = -1

The important fact is that "sin" is not a number, but a rule for assigning numbers. (For purposes of this discussion, just what the rule is, is not important; I'll just mention that the rule is explicable in geometric terms involving a circle.)

Other examples: Further functions from trigonometry, for instance, such as cosine and tangent, usually abbreviated similarly:

cos

tan

Very common is to name a function, defined by some means or other (or left unspecified), with a letter that suggests the word "function", such as

f

g

Typical way to do this: "Let f be the function defined by f(x) = 2x-1 for any number x."

Examples of (2):

x

2x-1

sin(x)

f(x)

That all seems pretty clear (assuming you have some understanding of what the sin function is, anyway): Elements of category 2 have a typical variable-designating letter (such as x), while elements of category 1 do not.

The confusion sets in when we speak, not of "the function f" (as ought logically to be done), but of "the function f(x)" (a very common locution). For instance, we might say, instead of the prolix defining statement used above, "Consider the function f(x) = 2x-1."

With such locutions so common, it's small wonder students fail to note the distinction between categories (1) and (2); in actual practice, it's just too handy to let the distinction fuzz, and mathematicians do this all the time. But being able to recover the distinction is important. How to make it clear to the students that they must be alive to this distinction is a pedagogical challenge.

I have sometimes fallen prey to the "grammarian's fallacy" and said "sin is not a noun–it's a verb." All I can say is that it seemed insightful at the time–but I cringed inwardly. I've since taken to saying something like "sin is an incomplete symbol–it doesn't have a value until you put something after it, like sin(pi) or sin(x)". The logical problem here is that sin(x) doesn't have a value (well, a numeric value), either–rather, it's something that would have a numeric value if x were given a numeric value. But I think that's not really a pedagogical problem, as calculus students are used to seeing an expression such as "2x-1" as functioning as if it were a number. Another possibility is to say, "sin is an operator", but it's not clear that carries much pedagogical weight.

There's a deal of linguistic subtlety in the conventions that have developed for mathematical expression, with various levels of formality (i.e., adherence to the logical rules) being used for different levels of communication and presuming different levels of sophistication on the part of the reader.

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