Re: Hardy-Weinberg law

From: Anon. (bob.ohara_at_SOD.OFF.Spammers.helsinki.fi)
Date: 06/23/04 

Date: Wed, 23 Jun 2004 16:06:47 +0000 (UTC)

Tim Tyler wrote:
> Bob O'Hara <bob.ohara@nospam.helsinki.fi> wrote or quoted:
>
>>Tim Tyler wrote:
>>
>>>>>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.
>
But there are an infinite number of fractions, so they have at least
that 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.
>
If there are an equal number of even and odd numbers, then half of the
numbers must be even.

This must be true because for every even number, I can add 1 and get an
odd number. Conversely for every odd number I can add 1 and get an even
number. Hence, by the operation of adding 1, I can produce an even
number for every odd number and vice versa. Ergo, half of all numbers
are even, and half are odd.

I find this sort of proof preferable to throwing my hands up in defeat.

<snip>
>>>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".
>
Rubbish, unless you're denying the existence of fractions. Fractions
are fractions of an infinite set, because there is an infinite number of
numbers between 0 and 1 (proof: take the reciprocal of every positive
integer).

>
>>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
>
This doesn't show that probability is defined as a limit - the nearest
you get is in the section "Probability in mathematics", where they use
"one approach" to give an interpretation - essentially, the frequentist
approach. Note that when they discuss Kolmonogorov's definition of
probability as a measure, they make no mention of any limits.

>
>>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.
>
This was (I think - my copy is at home) the paper:

E. Arjas & E. Nummelin & R.L. Tweedie: Semi-Markov processes on a
general state space -theory and quasi-stationarity. J. Aust. Math. Soc.
(Series A) 30 (1980): 187 - 200.

>
>>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?
>
You're trying to argue about the use of infinity. I'm pointing out that
one should be careful when doing this. This seems relevant.

> Are you suggesting I don't know what I am talking about?
>
> That is not the case.

Your evidence for this is?

Bob 

-- 
Bob O'Hara
Dept. of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5)
FIN-00014 University of Helsinki
Finland
Telephone: +358-9-191 23743
Mobile: +358 50 599 0540
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