Marty,

Brent wrote:

On 21 Jul 2009, at 23:24, Brent Meeker wrote:

>
> Take all strings of length 2
> 00             01                   10               11
> Make two copies of each
> 00      00      01      01      10      10      11      11
> Add a 0 to the first and a 1 to the second
> 000    001      010   011      100   101   110      111
> and you have all strings of length 3.


Then you wrote

> I can see where adding 0 to the first and 1 to the second gives 000  
> and 001 and I think I see how you get 010 but the rest of the  
> permutations don't seem obvious to me. P-l-e-a-s-e  explain,  Best,
>
>                                                                               
>                                                                               
>                                                                     m 
> . (mathematically hopeless)  a.
>
>


Let me rewrite Brent's explanation, with a tiny tiny tiny improvement:


Take all strings of length 2
00
01
10
11
Make two copies of each

first copy:
00
01
10
11

second copy
00
01
10
11

add a 0 to the end of the strings in the first copy, and then add a 1   
to the end of the strings in the second copy:

first copy:
000
010
100
110

second copy
001
011
101
111

You get all 8 elements of B_3.

You can do the same reasoning with the subsets. Adding an element to a  
set multiplies by 2 the number of elements of the powerset:

Exemple. take a set with two elements {a, b}. Its powerset is {{ } {a}  
{b} {a, b}}. How to get all the subset of {a, b, c} that is the set  
coming from adding c to {a, b}.

Write two copies of the powerset of {a, b}

{ }
{a}
{b}
{a, b}

{ }
{a}
{b}
{a, b}

Don't add c to the set in the first copy, and add c to the sets in the  
second copies. This gives

{ }
{a}
{b}
{a, b}

{c}
{a, c}
{b, c}
{a, b, c}

and that gives all subsets of {a, b, c}.

This is coherent with interpreting a subset {a, b} of a set {a, b, c},  
by a string like 110, which can be conceived as a shortand for

Is a in the subset?   YES, thus 1
Is b in the subset?   YES thus  1
Is c in the subset?    NO thus   0.

OK?

You say also:

> The example of Mister X only confuses me more.

Once you understand well the present post, I suggest you reread the  
Mister X examples, because it is a key in the UDA reasoning. If you  
still have problem with it, I suggest you quote it, line by line, and  
ask question. I will answer (or perhaps someone else).

Don't be afraid to ask any question. You are not mathematically  
hopeless. You are just not familiarized with reasoning in math. It is  
normal to go slowly. As far as you can say "I don't understand", there  
is hope you will understand.

Indeed, concerning the UDA I suspect many in the list cannot say "I  
don't understand", they believe it is philosophy, so they feel like  
they could object on philosophical ground, when the whole point is to  
present a deductive argument in a theory. So it is false, or you have  
to accept the theorem in the theory. It is a bit complex, because it  
is an "applied theory". The mystery are in the axioms of the theory,  
as always.

So please ask *any* question. I ask this to everyone. I am intrigued  
by the difficulty some people can have with such reasoning (I mean the  
whole UDA here). (I can understand the shock when you get the point,  
but that is always the case with new results: I completely share  
Tegmark's idea that our brain have not been prepared to have any  
intuition when our mind try to figure out what is behind our local  
neighborhood).

Bruno



http://iridia.ulb.ac.be/~marchal/




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