Re: Linford's Fallacy


"Emmanuel Goldstein" <religionisacrutch@xxxxxxxxxxx> wrote in message
news:1006123821.23337.1254872141907.JavaMail.root@xxxxxxxxxxxxxxxxxxxxxxxx
So, which statement do you not agree with?

This one:

"While it is true that flipping out any given sequence of heads/tails is
equally probable, it is not true that the probability of having all heads
is the same >as the probability as having any mixture of heads and tails."

It's technically true but it's incomplete and it doesn't imply what he
thinks it does. He's singling out a string of all heads and ignoring the
fact that you >can make the same argument on any string, for example, a
string of THTHTHTHHHTHTHTHHTHTTTHT.... and so on.


Yes, but you still don't get it. He could have made his entire argument
about that string too... but then he would have to use different categories.
The probabilities would all be the same and all his logic would be the same.

You say it's incomplete but it's only incomplete because you are adding your
own idea. If you read exactly what he said then it's all true and logical.
You basically agree with this but then go to say it's doing something you
don't like.

He's claiming that a string of heads is so rare that it will never happen
while also failing to understand that ALL strings are equally rare. If you
single >any string out to find, the longer the string gets, the closer the
odds you will find it approach zero.

No, I don't think so. He is just using that one as an example. You are
reading into that he is treating it as special. He has to talk about one
string and just pulls one out of his ass. It is special in that it is easily
recognizable. Your best argument is that he doens't do the same with tails.
But obviously he is just demonstrating his idea. If he said that the heads
string is the only special string then that is different. He does not do
this but just uses that string and hence you think he is singling it out. He
as too! Because he has to demonstrate the problem. That makes it special in
some sense but he does not imply that it is the only special string.

For example, If I'm demonstrating what primes are and I use the number 43 as
an example then it doesn't mean it is necessarily any better than all the
other primes... I just had to choose at least one to give you an example.
Sure it's best that I give you all the primes but then that doesn't help you
if you don't know what a prime is.



The original context that started this argument was the fact that all
electrons are identical. I claim that if we are selecting from all
possible universes, >we have just as much odds of hitting the universe
with identical electrons as any other. It's only the fact that you start
looking for it that particular ?>outcome that seems rare because they are
ALL rare.


You can't prove that electrons are indistinguishable or not by mathematics.
We can easily distinguish them by their locations as long as we know their
locations for all of time. We can also make a distinction such as free
electrons and bound electrons.

You say that we are selecting from all possible universes and we have equal
chance of getting a universe with identical electrons as ones that are not
identical?

First, this is fallacious unless you know that the number of universe with
identical electrons are equal to those that are not. The number of sequences
of all heads is not the same as the number of mixed sequences. Since we
cannot determine such things about all the universes we cannot make such a
statement.

Second, If there were such a universe with distinguishable electrons then
they probably wouldn't all be called electrons. After all, we have other
negatively charged particles that we don't call electrons.

Eventually, in any mathematical analysis, you are going to have to take some
axioms or make some assumptions. What are your assumptions here? That each
universe is indentical? In that case then they either all have
indistinguishable electrons or they don't. Are we going to assume that all
our electrons are indistinguishable? etc...

Unless you start from a clear set of axioms you can't argue logically.


Are you going to play this game?

No but the mistake that he is making is in thinking that this game is
somehow special. There is nothing special about all heads. If you pick any
>particular string to bet on, it's the EXACT SAME poor odds. That's my
point. He keeps claiming that his argument is somehow proving that a
string of >all heads is special. It's a very subtle mistake.

Mathematics does't treat them as special... But humans do. the sequence of
all heads is special. I can make it special just by saying I think it is
special. I can also come up with some mathematical way to make it well
defined. "Special" is not a mathematical term...

He did prove they are special. Infact you are making the subtle mistake.
Special is relative.

You keep thinking he is proving that heads is special compared to every
other arbitrary sequence. He is only proved it is special relative to
"mixtures". In fact it is not hte only one though as tails is also just as
special as the heads. In fact, in the original post, I do not see any use of
the term "special".

We can make it mathematically precise:

Special sequence: Given a binary sequence, a sequence is "special" if it
consists of only 1's.

In this case the special sequences are "rare" and "special".

Or we could define it this way

The entropy of a binary sequence A_k is sum(A_k)/N.

Here we see that the string of heads has the highest entropy and hence we
could say that is special. In fact this isn't as arbitrary as the above. Of
course I'm attaching special meaning to such sequences and they are
"special" with regards to the entropy definition.

Again, special is relative. You are using a different "special" than he is.






.

Relevant Pages

  • Re: Coin tossing guessing strategy...
    ... You need a precise string in exactly one order, ... with respect to sequences of n flips, is to note that if some sequence ... In fact, assuming a fair coin, all 10 coin sequences are equally ... heads and tails. ...
    (sci.math)
  • Re: Coin tossing guessing strategy...
    ... You need a precise string in exactly one order, ... with respect to sequences of n flips, is to note that if some sequence ... In fact, assuming a fair coin, all 10 coin sequences are equally ... heads and tails. ...
    (sci.math)
  • Merry Christmas!
    ... dictionaries and every variant possibility has a separate "word" entry. ... The byte string of the "word", whose length is specified by a four ... match is found for a source byte sequence in the dictionary. ...
    (rec.arts.sf.written)
  • Merry Christmas! Linux RULES! New applications to develop!
    ... dictionaries and every variant possibility has a separate "word" ... Each entry in the dictionary contains: ... The byte string of the "word", whose length is specified by a four ... addresses whose entry is selected by the first byte of the sequence. ...
    (comp.os.linux.misc)
  • Re: user defined function that converts string to float
    ... > I need user defined function that converts string to float in c. ... initial, possibly empty, sequence of white-space characters (as ... point character, then an optional exponent part as defined in ... then a nonempty sequence of hexadecimal digits ...
    (comp.lang.c)