Re: Bruno's mathematical reality

```On 12/29/2013 7:45 PM, Jason Resch wrote:
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```

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On Sun, Dec 29, 2013 at 6:58 PM, meekerdb <meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote:
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On 12/29/2013 3:49 PM, Jason Resch wrote:
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```

On Sun, Dec 29, 2013 at 5:42 PM, meekerdb <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:

On 12/29/2013 2:08 PM, Jason Resch wrote:
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```

On Sun, Dec 29, 2013 at 4:51 PM, meekerdb <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:

On 12/29/2013 1:28 PM, Jason Resch wrote:
```
```

On Sun, Dec 29, 2013 at 2:25 PM, meekerdb <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:

On 12/29/2013 5:56 AM, Bruno Marchal wrote:
```
```
On 28 Dec 2013, at 22:23, meekerdb wrote:

```
```                On 12/28/2013 4:09 AM, Bruno Marchal wrote:
```
```                For a long time I got opponent saying that we cannot generate
computationally a random number, and that is right, if we want
generate only that numbers. but a simple counting algorithm
generating all numbers, 0, 1, 2, .... 6999500235148668, ...
generates all random finite incompressible strings,
```
```
How can a finite string be incompressible? 6999500235148668 in
base
6999500235148669 is just 10.
```
```

You can define a finite string as incompressible when the
shorter
combinators to generate it is as lengthy as the string itself.
This definition is not universal for a finite amount of short
sequences which indeed will depend of the language used (here
combinators).

Then you can show that such a definition can be made universal
by
adding some constant, which will depend of the universal
language.

It can be shown that most (finite!) numbers, written in any
base,
are random in that sense.

Of course, 10 is a sort of compression of any string X in some
base,
but if you allow change of base, you will need to send the base
with
the number in the message. If you fix the base, then indeed 10
will
be a compression of that particular number base, for that
language,
and it is part of incompressibility theory that no definition
exist
working for all (small) numbers.
```
```
Since all finite numbers are small, I think this means the
theory
only holds in the limit.

Brent

Brent,

It is easy to see with the pigeon hole principal.  There are more 2
digit
numbers than 1 digit numbers, and more 3 digit numbers than 2 digit
numbers, and so on.  For any string you can represent using a
shorter
string, another "shorter string" must necessarily be displaced.  You
can't keep replacing things with shorter strings because there
aren't
enough of them, so as a side-effect, every compression strategy must
represent some strings by larger ones.  In fact, the average size
of all
possible compressed messages (with some upper-bound length n) can
never
be smaller than the average size of all uncompressed messages.

The only reason compression algorithms are useful is because they
are
tailored to represent some class of messages with shorter strings,
while
making (the vast majority of) other messages slightly larger.
```
```
A good explanation.

Thanks.

But just because you cannot compress all numbers of a given size
doesn't
imply that any particular number is incompressible.

That is true if you consider the size of the compression program to be
of no
relevance.  In such a case, you can of course have a number of very
small
strings map directly to very large ones.

So isn't it the case that every finite number string is
compressible in
some algorithm?  So there's no sense to saying 6999500235148668 is
random,
but 11111111111111 is not, except relative to some given compression
algorithm.

Right, but this leads to the concept of Kolmogorov complexity. If you
consider
the size of the minimum string and algorithm together, necessary to
represent
some number, you will find there are some patterns of data that are more
compressible than others.  In your previous example with base
6999500235148668, you would need to include both that base, and the
string
"10" in order to encode 6999500235148669.
```
```
But that seems to make the randomness of a number dependent on the base
used to
write it down? Did I have to write down "And this is in base 10" to
show that
6999500235148668 is random?  There seems to be an equivocation here on
"computing a number" and "computing a representation of a number".

A number containing regular patterns in some base, will also contain regular
patterns in some other base (even if they are not obvious to us),
compression
algorithms are good at recognizing them.

The text of this sentence may not seem very redundant, but english text can
generally be compressed somewhere between 20% - 30% of its original size.
If you
convert a number like "55555555555" to base 2, its patterns should be more
evident
in the pattern of bits.

```
```         For the majority of numbers, you will find the Kolmogorov complexity
of the
number to almost always be on the order of the number of digits in that
number.  The exceptions like 1111111111 are few and far between.
```
```
111111111 looks a lot messier in base 9.

base 10: 1111111111111111111
base 9: 7355531854711617707
base 2: 11110110101101110101101010110010101111000100011100

In base 9, there is a high proportion of 7's compared to other digits. In
base 2,
the sequence '110' seems more common than statistics would suggest.  In any
case,
the number is far from incompressible. It takes only 9 characters to
represent that
19 digit number in Kolmogorov complexity "1r19inb10" = "1 repeated 19 times
in base
10", in my encoding language.
```
```
So you are agreeing with me that to cite a specific number and say "That
number is
random." is meaningless.

```
I agree with that in the sense of "random as unpredictable", but I disagree in the sense of "random as uncompressible". Some numbers are objectively not compressible, just like some shufflings of a deck of cards are uncompressible, because the shortest possible description of the ordering of the cards requires more information to describe than merely giving the list itself. So it is with a number and its digits.
```
```
I think we're talking past each other. What you're calling a number is what I'd call a string of digits. I can understand a string of digits being incompressible...but the number it names has many representations. To say a number is incompressible?? There's an old joke proof in mathematics that every number is interesting, otherwise there would be a smallest uninteresting number - which would peforce be interesting. It seems that interesting means something like "has a short description".
```
Brent

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