# "Strong typing vs. strong testing" [OT]

Steven D'Aprano steve at REMOVE-THIS-cybersource.com.au
Wed Oct 13 21:31:59 CEST 2010

```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".

> Since by not reacting to the previous poster, you implicitely accepted
> the equation and thus the context in which it is true: euclidean
> geometry. So I don't think that concerns that fall outside this context
> have any relevance.

You've missed the point that, 4000 years later it is easy to take pi for
granted, but how did anyone know that it was special? After all, there is
a very similar number 3.1516... but we haven't got a name for it and
there's no formulae using it. Nor do we have a name for the ratio of the
radius of a circle to the proportion of the plane that is uncovered when
you tile it with circles of that radius, because that ratio isn't (as far
as I know) constant.

Perhaps this will help illustrate what I'm talking about... the
mathematician Mitchell Feigenbaum discovered in 1975 that, for a large
class of chaotic systems, the ratio of each bifurcation interval to the
next approached a constant:

δ = 4.66920160910299067185320382...

Every chaotic system (of a certain kind) will bifurcate at the same rate.
This constant has been described as being as fundamental to mathematics
as pi or e. Feigenbaum didn't just *define* this constant, he discovered
it by *proving* that the ratio of bifurcation intervals was constant.
Nobody had any idea that this was the case until he did so.

--
Steven

```