"Strong typing vs. strong testing" [OT]

[Antoon Pardon]

On Wed, Oct 13, 2010 at 07:31:59PM +0000, Steven D'Aprano wrote:
> 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".

Your missing the point. You started talking about non-euclidean geometries
as an argument against the notion that pi was defined as the ratio of
the circumference and the diameter. But in non-euclidean geometries
the equation doesn't hold. So either you think non-euclidian geometries
matter and in that case you should have questioned the equation or
you accept that the context was euclidian geometries and in that case
non euclidian considerations don't matter.

> > 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.

Your confusing the concept with its specific numerical value. It's not
uncommon in mathematics to give a name to a number that is defined in
a specific way, without knowing its numerical value.

> 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:
> 
> ?? = 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.

So? That the ratio of the circumference and the diameter of a circel was
constant was proven a long way before people had the tools to calculate
that ratio to very high precision. They did that by noting that the
ratios of the circumference of a regular polygon to the diameter of the
inscribed and outscribed circle were constants and converged to each
other as the number of sides increased.

So there is no problem defining pi as the ratio between the circumference
and the diameter of a circle even if one has only very crude approximations
to the numerical value of that ratio.

-- 
Antoon Pardon