Bob O'Hara  wrote or quoted:
> Tim Tyler wrote:
> > Anon.  wrote or quoted:
> >>Tim Tyler wrote:
> >>>Anon.  wrote
or quoted:

> >>>>Wierd.  The Hardy-Weinberg law is deterimistic: there
is no randomness 
> >>>>in it.
> >>>
> >>>The Hardy-Weinberg law is normally stated in a form that refers to
> >>>a large population where mating is random.
> >>
> >>The theorem was derived for an infinite population.
> >>
> >>>E.g. see: 
> >>>
> >>>  http://library.thinkquest.org/19926/java/tour/06.htm?tqskip1=1
> >>>
> >>>Alas, this page expresses the law in terms of an infinite
population :-(
> >>>
> >>>A disasterous error - IMO - since talking about gene
frequencies in an 
> >>>infinite population is a sign of mathematical ignorance.
> >>
> >>No, it's a simplifying assumption.  I think to accuse Hardy in 
> >>particular of mathematical ignorance deserves, well, a 
> >>non-mathematician's apology.
> > 
> > I never said the problem was at Hardy's end.
> 
> In fairness, Hardy does suppose "that the numbers are fairly large". 
> But, as we know know, in a finite population, there will be an excess of 
> homozygotes (because of inbreeding), so H-W isn't correct (but a 
> reasonable approximation if the population is large).

I suggested explaing using a limit - not a mere finite population.

> >>>Popularisers should make explicit the behaviour is what happens as
> >>>the population size tends towards infinity - and not attempt to pass
> >>>it off as an effect in an infinite population.
> >>
> >>But it is - in finite populations, you get an excess of homozygotes, as 
> >>any student of population genetics should know.
> > 
> > Any mention of gene frequencies in an infinite population is nonsense -
> > as I stated originally.
> > 
> > You can't talk about a fraction of an infinite population having
> > a trait.  You would get different results for that fraction depending
> > on how you enumerated through the population.
> 
> I don't understand what you mean, but by that argument, you can't even 
> define a fraction or a probability.

Fractions have nothing to do with infinite sets.

> > It's like claiming that half the integers are even.
> 
> Err, they are.  There are just rather a lot of them.

No, there aren't.

There are an infinite number of even numbers.

There are an infinite number of odd numbers.

Divide infinity by infinity and the result is indeterminate.

> > Such statements are total mathematical gibberish.
> > 
> > What *can* be said is that the fraction of the set of integers from
> > 0 to N that are even tends to 0.5 - as N => oo.
> 
> So what happens when N is infinity?

The fraction is undefined.

> > No serious mathematician can talk about fractions of infinite sets and 
> > expect to be taken seriously.
> 
> But they do.

No - not unless the fractions are "zero" or "one".

> It's how probability is defined as a concept.

Probability is defined as a mathematical limit, as N approaches infinity.

That uses a limit as a finite set increases in size - not a fraction of an 
infinite set.

E.g. see:

http://www.wordiq.com/definition/Probability

> I have a colleague who even wrote mathematical papers about fractions 
> of uncountable sets.

If you can show me, I should be able to tell you if they contain the 
fallacy under discussion.

Probably he doesn't do that at all - and instead uses a limit.

> Infinity is a difficult concept (I know - there are lots of it I don't 
> understand), so I think one should be cautious about making any 
> pronouncements on it unless one is sure about what mathematics
> does and does not say on the subject.

How is that relevant?

Are you suggesting I don't know what I am talking about?

That is not the case.
-- 
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 |im |yler  http://timtyler.org/  tim{at}tt1lock.org  Remove lock to reply.
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