Two recent threads have come together to raise my sensitivity
to the issue of unclear language.
1) Consider the assertion "the universe is closed". Does that
mean
a) The universe is a closed set?
b) The universe is a closed manifold?
The two statements (a) and (b) are not equivalent, yet both
can be shorthanded by the seemingly precise, seemingly
mathematical assertion "the universe is closed". Hmmmmm.
2) Consider the statement "the checkerboard is not black".
There are two ways to interpret that:
a) The checkerboard is not (everywhere black).
b) The checkerboard is everywhere (not black).
The meaning depends on where you stick the "everywhere".
Statement (a) is true but (b) is false, yet both can be
shorthanded by the seemingly innocuous statement that the
checkerboard is not black.
2) Suppose the initial temperature was 30 and the final
temperature was 60 in some units. Suppose the temperature
in fact changed stepwise, in three steps of 10:
60 _____
50 _________|
40 ______|
30 _______| time --->
3a) Consider the statement "the temperature is constant".
By convention we interpret that to mean the temperature
is everywhere constant. I think it is reasonable to
insist on that interpretation ... but I sympathize with
intro-level students who may not have been born knowing
this.
3b) Things get even uglier when we consider the statement
"the temperature is not constant". How are we to interpret
that? It depends on where you stick the "everywhere":
a) The temperature is not (everywhere constant).
b) The temperature is everywhere (not constant).
Again, (a) is true but (b) is false. You could argue that
"constant" is a technical term and is /defined/ to mean
"everywhere constant", so that (a) is the only acceptable
interpretation ... but students certainly weren't born
knowing that, and even if they saw it defined that way
once, they could well have forgotten it. This would be
an example of negative transference from non-technical
usage to technical usage.
==========================
Imprecise language isn't just a problem with intro-level
students; I see it with when reviewing papers for Phys Rev
Letters. The author writes a sentence that (presumably) is
clear to him ... but I can't figure out what it means. An
illustration is:
Suppose we have an object X of type Y.
X is round.
Does this assert
a) For /all/ X of type Y, X is round.
b) For /some/ X of type Y, X is round.
Statement (a) uses the universal quantifier "for all",
while statement (b) uses the existential quantifier "for
some". The problem is that both of them are sometimes
shorthanded in ways that don't look ambiguous, but are.
It seems that natural language entices and almost forces us
into Manichaean fallacies: Not black (in one sense) can be
misinterpreted as not black (in the other sense). That is,
anything that is not everywhere black "must" be everywhere
not black.
========================================
So, there is a problem here. I don't claim to fully understand
the problem. Actually I suspect it is a cluster of loosely-
related problems. I don't have much in the way of constructive
suggestions for what to do about it. Here are some partially-
baked thoughts:
-- Requiring people to say "the universe is a closed manifold"
(as opposed to the "the universe is closed") seems unduly
burdensome.
-- Requiring people to say "everywhere constant" (as opposed
to "constant") seems unduly burdensome.
++ I guess it never hurts to remember that technical language
is different from vernacular language, and that students
weren't born knowing it. It's a foreign language to them.
It is easy to forget how big a problem this is.
++ Sometimes you have to explain more than once that "constant"
is defined to mean "everywhere constant". If we meant piecewise
constant we would have said "piecewise constant".
++ OTOH it doesn't pay to get too wrapped up in terminology.
Ideas are primary and fundamental; terminology is secondary.
Terminology is important insofar as it helps communicate the
ideas.
++ Sometimes it helps to say things twice, using two different
ways of saying the same thing. There is some chance that one
way will be clear even if the other isn't.
++ Sometimes graphics help. Say it once in words, and say the
same thing again graphically. A graph of speed increasing
smoothly from 30 to 60 seems trivial, but it might have
circumvented the confusion about what was meant by "constant"
and "not constant". It might be worth re-running the quiz
to see how well the students do with graphical versions of
the question. Maybe two versions of the question, one smoothly
monotonic and one piecewise constant. This might help sort
out some of the conceptual and/or terminological issues.