Re: OU Maths Degree Standards

On the Thursday 29 Jun 2006 12:12 pm, Dr A. N. Walker wrote:
In short, "show" is an existential statement, whereas prove is a universal
statement.
Eg. "show" x=2 is a solution to
x+2 =4
which means substitute 2 for x in the equation and evaluate LHS, RHS and
"show" that they are equal. That is, give an "x" such that the equation is
true.

Whoa! We clearly have linguistic problems here. In the
version of [British] English that I use, and that I'm pretty sure
every UK university maths department and examining board uses,
firstly what you seem to want might usually be called "verify";
but "show", "prove", "explain why" would be acceptable, and the
"model solution" in all cases would be something like "if x = 2
then x+2 = 2+2 = 4, QED" [except in an advanced logic module
where you might be expected to take many lines showing that you
can substitute for "x", that "2+2 = 4" and so on].

Errrhh: What you proved is the converse of the statement. Ie.
If x=2 then x+2 = 4 is the converse of if x+2=4 then x=2!

Secondly, it doesn't mean "give an x such that ..."; the
"x" is already given.
If you go back to the history books, Omar Khayyam, and Viete explicitly
state that in solving algebraic equation, first you *assume* that x is
known and the you reason forwards until you find x, and then you reason
backward to the actual equation involved, this is why the iff is important.
Hence the difference between
if x=3 then x^2=9 (no iff)
x=2 iff x+2=4 (iff)

Prove "x=2" is a solution means
x+2 =4 iff x=4-2 iff x=2.
No, around here it means exactly the same as your previous
question, except that the students would be worried by the word
"prove", and if I were moderator or external I would insist it be
changed to "verify" or "show".

Well, in that case we have an agreement in that the word show is ambiguous.
It can either mean verify or prove and our disagreement is that I take it
to mean verify and you take to mean proof.

Anyway I am going to have leave this discussion as I have a conference to go
to - but in true style of always leaving the audience wanting me, here is
my review of "The nature of proof", which was a Royal Society event
including 3 Fields medallist which I wrote for the London Mathematical
Society

http://www.lms.ac.uk/newsletter/333/333_06.html.

and so I hope that gives you some idea that I do know what I am talking
about when it comes to logic and maths!
Nim.
.

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