Everything from here on is Rosser's voice (except parentheticals, like this one):
It is of interest that there is a place in mathematics where confusion occasionally arises from a failure to preserve a careful distinction between an object and its name. This is in connection with fractions.
The fractions "3/4", "6/8", "9/12", etc., are all names of a certain
rational number, which incidentally has many other names such as "0.75",
"SQRT(0.5625)", "The integral from zero to one of 3x^3dx" (this last was
written mathematically, of course, but my HTML skills are rudimentary at
best), etc. Thus if one writes "3/4=9/12", one is making a statement
about the rational number, and names of the rational number appear in the
statement. This is as it should be. However, if one writes
"3 divides the denominator of 9/12", one is not making a statement about
the rational number, but about one of its names. Thus one should
write instead "3 divides the denominator of '9/12'."
The chance for confusion here is slight. However, one will occasionally encounter an alert youngster who wishes to know why, if "3/4=9/12," one cannot replace "9/12" by "3/4" in the statement "3 divides the denominator of 9/12" whereas it is perfectly correct to replace "9/12" by "3/4" in the statement "3 is greater than 9/12." The answer, of course, is that "9/12" actually occurs in the second statement, whereas "9/12" does not actually appear in the first statement but only a name of "9/12".
Since the first statement is incorrectly written, it appears to contain "9/12" at a point where it really contains a name of "9/12".
The alert youngster may then inquire why one cannot replace "9/12" by "3/4" in the correctly formulated statement "3 divides the denominator of '9/12'," since this also contains "9/12". The answer in this case is that the occurrence of "9/12" in the correct version of the sentence is a purely typographical occurrence, like the "s" in "sin x" or the "d" in "dy/dx". If one is given "s=d" one would not think of replacing "s" by "d" in "sin x" or "d" by "s" in "dy/dx". If one had used some other name for "9/12", there would have been no question of substituting "3/4". Thus one might have written "3 divides the denominator of the fraction got by placing a '9' over a bar and a '12' under the bar."
The gist of the matter is that, if we have a statement such as "3 is greater than 9/12" about the rational number 9/12 and containing a name "9/12" of this rational number, we can replace this name by any other name of the same rational number, for instance, "3/4". If we have a statement such as "3 divides the denominator of '9/12' " about a name of a rational number and containing a name of this name, we can replace this name of the name by some other name of the same name, but not in general by the name of some other name, even if it is a name of some other name of the same rational number.
(can you parse that? wasn't that fun?)
(skipping a bit, but that part above was the big payoff. It came up in a weird tangential way while I was reading the short story that was turned into the novel "Ender's Game" [which I loved when I read it last year--the novel is much better than the short story IMHO] which was recommended to me by the gentleman who wishes to remain anonymous. Actually, I was fiddling with some old nonstandard analysis, and I had Rosser open on my desk, and it fell open to the page with "but a rebus or a charade" on it, and I just had to share. grin!)
Meanwhile, there is one important point concerning names within our symbolic logic. Names are important constituents of statements in symbolic logic even as in more familiar languages. In using the English language, we always assume that our sentences have meaning and that the names which occur in them are the names of something. Not so in our symbolic logic. There is no requirement that a statement of symbolic logic have meaning. Consequently the names which occur in such statements need not be the names of anything. This is very convenient, since we are thus entitled to use names without first (or ever) being assured that they are the names of something. We shall amplify this point when we introduce names within our symbolic logic (see Chapter VII).
(this last bit becomes rather important. you must disabuse yourself
of the notion that names name things. it's a rookie mistake.)
I've reconsidered what might turn out to be a lengthy excursion into
nonstandard analysis. The consensus opinion is: "only a serious
weirdo would be concerned with nonstandard analysis and possible foundational
implications of the way it has been received." Instead, let's play
with some math for a while. Consider the "birthday
problem" from elementary statistics.
If you have comments or suggestions, email me at
INDEX to a few of Fred's pages