"Strong typing vs. strong testing" [OT]

[Steven D'Aprano]

On Wed, 13 Oct 2010 21:52:54 +0100, Arnaud Delobelle wrote:

> Steven D'Aprano <steve at REMOVE-THIS-cybersource.com.au> writes:
> 
>> On Wed, 13 Oct 2010 16:17:19 +0200, Antoon Pardon wrote:
>>
>>> On Wed, Oct 13, 2010 at 01:20:30PM +0000, Steven D'Aprano wrote:
>>>> On Tue, 12 Oct 2010 22:13:26 -0700, RG wrote:
>>>> 
>>>> >> The formula: circumference = 2 x pi x radius is taught in primary
>>>> >> schools, yet it's actually a very difficult formula to prove!
>>>> > 
>>>> > What's to prove?  That's the definition of pi.
>>>> 
>>>> Incorrect -- it's not necessarily so that the ratio of the
>>>> circumference to the radius of a circle is always the same number. It
>>>> could have turned out that different circles had different ratios.
>>> 
>>> If that is your concern, you should have reacted to the previous
>>> poster since in that case his equation couldn't be proven either.
>>
>> "Very difficult to prove" != "cannot be proven".
> 
> But in another section of your previous post you argued that it cannot
> be proven as it doesn't hold in projective or hyperbolic geometry.

But in Euclidean geometry it *can* be proven. What I was pointing out 
that it can't be taken for granted. Under non-Euclidean geometries, it 
can't be proven because it isn't necessarily true; under Euclidean 
geometry, there was a time when people didn't know whether or not the 
ratio of circumference to radius was or wasn't a constant, and proving 
that it is a constant is non-trivial.


> But you were claiming that the proposition "C = 2πr is the definition of
> π" was false.

Well, what is the definition of pi? Is it:

the ratio of the circumference of a circle to twice its radius;
the ratio of the area of a circle to the square of its radius;
4*arctan(1);
the complex logarithm of -1 divided by the negative of the complex square 
root of -1; 
any one of many, many other formulae.


None of these formulae are intuitively correct; the formula C = 2πr isn't 
a definition in the same sense that 1+1=2 defines 2. The point that I was 
trying to get across is that, until somebody proved the formula, it 
wasn't clear that the ratio was constant.


> Also, it is very intuitive to think that the ratio of the circumference
> of a circle to it radius is constant:
> 
> Given two circles with radii r1 and r2, circumferences C1 and C2, one is
> obviously the scaled-up version of the other, therefore the ratio of
> their circumferences is equal to the ratio of their radii:

That's exactly the sort of thing Peter Nilsson was talking about when he 
said "Most attempts by students collapse because they assume the formula 
in advance". It might be "obvious" to you that the two circles are merely 
scaled up versions of each other, but that is equivalent to assuming that 
the ratio of the circumference to radius is a constant. Well, yes, it is 
(at least under Euclidean geometry), but assuming it is a constant 
doesn't allow you to prove it is a constant -- that's circular reasoning, 
if you excuse the pun.



-- 
Steven