From: Brian D Harper <bharper@postbox.acs.ohio-state.edu>
>At 01:57 PM 8/29/00 +0100, Richard wrote:
Brian, I'm going to take the liberty of addressing just part of your post
for now. I think it will be difficult to make further progress until we
settle on what we mean by "random".
>Yes, pattern can be very subjective of course, so let me
>back up. The key issue with algorithmic randomness
>is compressibility. A string that cannot be compressed
>is said to be random.
OK. This is the crucial point. I'm using the word random in the statistical
(i.e. probabilistic) sense. Your algorithmic sense of the word is quite a
different one. It is devoid of probabilistic content. Let's be clear in
future which one we're using at any time.
>Generally speaking, a pattern is
>what one uses to compress the string. Generically, this
>kind of agrees with the everyday notion of pattern, but
>there may be exceptions. Humans seem particularly
>good at "seeing" patterns which aren't really there. Likewise,
>patterns in the sense we're speaking of here may not be
>obvious at all.
>
>You mention above there are three possibilities. Actually
>from this point of view there is a continuum of possibilities,
>however, purposeful is not included. Purpose is subjective,
>it can't be measured. The reason for the continuum is that
>the notion of randomness admits of degrees. Something
>may be very nearly incompressible but not quite. One
>can make this notion precise by introducing per cent
>compressible. Something only 5% compressible is
>nearly random. Something 95% compressible is very
>ordered.
Yes, algorithmic randomness (as you've defined it) admits of different
degrees. Statistical randomness does not, strictly speaking (although
different degrees of variability are sometimes referred to as degrees of
randomness).
>In my view, evolution fits on this scale towards the upper
>end. It is not random but highly complex.
Having just established that algorithmic randomness is a matter of degree,
what do you mean by "it's not random"? Relative to what?
>I don't know if this helps. I think our key difference is the
>process versus result distinction.
>
>Oh, I almost forgot to mention the "dilemma" which Chaitin
>solved. This is usually described something like this. Suppose
>you have the following two strings which supposedly represent
>tosses of a fair coin. 1=heads, 2=tails:
>
>(A) 0101010101010101010101010101010101010101010101010101010101010101
>
>(B) 1110101010010010101001010011000111100111100110011111000011010011
>
>Is either of the results more surprising than the other? From the point of
view
>of probability theory we cannot really say since the probability of any
>specific
>sequence is the same.
>
>What we learn from the above considerations is that if a number is
>defined to be random in view of the process by which the number is
>produced, then an obviously ordered result such as the first sequence
>above must be considered random since the random process of coin
>flipping produces that result with a probability equal to that of any
>other specific sequence.
To clarify, you are talking about statistical randomness here, right?
>In the words of Gregory Chaitin "The conclusion
>is singularly unhelpful in distinguishing the random from the orderly.
But what does Chaitin mean by random here? Since he hasn't yet introduced
his algorithmic randomness, he must mean statistical randomness. So his
objective is to detect statistical randomness.
>Clearly a more sensible definition of randomness is required, one that
>does not contradict the intuitive concept of a 'patternless' number."
>(Chaitin, G.J. (1975). "Randomness and Mathematical Proof," <Scientific
>American>, 232 (May 1975):47-52.)
>
>As might be expected from the above, the "more sensible definition
>of randomness" is Chaitin's :).
But it is *not* sensible to redefine a word (totally changing its meaning)
just because you are having difficulty detecting the presence of the
phenomenon.
Effectively, Chaitin is saying that we can't definitively detect statistical
randomness, so he defines a measure which we *can* detect, and, confusingly,
calls this randomness too. But what use is this measure, unless it's used to
detect statistical randomness? Why do we care whether a set of data is
patterned or not? And, if algorithmic randomness *is* used to detect
statistical randomness, then it is really statistical randomness that we're
interested in, and algorithmic randomness is just a marker for it.
Also, what relevance does algorithmic randomness have to our original
subject? We were talking about whether random variation and natural
selection (or evolution) are random processes. But you can't talk about the
algorithmic randomness of a process. You can talk about whether contemporary
organisms exhibit a greater or lesser degree of algorithmic randomness (i.e.
they are more or less patterned), but what's the relevance of that?
[...]
>phew, it's hard to find time for such demanding replies (i.e. those
>requiring that I think :)
Me too. Most of what passes for discussion here is just rote repetition of
much-rehearsed arguments (and I include my own stuff). It's good to have a
tough debate for a change. ;-)
Richard Wein (Tich)
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