Re: Randomness

On 17 Nov 2005 04:26:00 -0800, "John Bode" <john_bode@xxxxxxxxxxx>
wrote:

>The word "random" gets tossed around a lot on t.o., and I just want to
>make sure I'm understanding and using the term as intended.
>
>An event is random if previous events do not affect or predict future
>events. A coin toss is random because previous tosses do not affect or
>predict the next toss (assuming a fair coin, anyway). Tossing 50 heads
>in a row does not change the probability that the next toss will be
>tails.
>
>Genetic mutations are random because previous mutations do not affect
>or predict future mutations (e.g., a deletion of a nucleotide does not
>change the probability of a future deletion in the same region).
>
>Selection is not random because previous selections *do* influence
>future selections. The composition of the population changes with each
>generation, which changes the probability of a particular trait being
>passed on.
>
>I've noticed that some Creationist arguments seem to confuse randomness
>with probability, notably the "tornado in a junkyard" analogy.
>Chemical reactions are random, but not all possible reactions are
>equally probable.

You are right; "random" is a word that seems to have very different
meaning to different people. It is important to clarify exactly what
it is supposed to mean. Unfortunately, I think you are on the wrong
track in your interpretation.

The word "random" is used in three very different contexts:
1) to mean that the outcome of a process is somehow
"uncertain" in contrast to "deterministic"
2) to mean that the outcome of a process is not influenced by
other outcomes, including previous outcomes of that process
3) to mean that the chance (whatever that means) of getting
one particular outcome from a process is exactly the same
as the chance of getting any other outcome.

Your definition corresponds to notion 2. You confound the problem by
asserting (incorrectly) that people confuse randomness with
probability (notion 1) and even attribute that confusion to
Creationists to emphasize just how wrong it is. Unfortunately,
mathematicians use the notion of random exactly in the sense of notion
number 1. The Creationists are right in that notion. They are just
wrong in how they calculate probabilities and apply probability
notions.

Mathematically, a random process is one whose outcome is a point in a
"sample space" (with certain mathematical properties of space-ness)
associated with a "probability" (a certain notion of a measure on the
sample space with specific mathematical properties). Something is
then "random" if it is associated with the outcome of a random
process.

Technical note: OK, that really corresponds to a discrete random
processes, more properly a single instance of a discrete process. A
continuous random process is one that at each point in time (a
parameter defined over the real line) generates a point in the sample
space.

That roughly corresponds to notion number 1: the outcome of the
process is not exactly defined (determined) but can only be described
as a set of probabilities over a set of possible outcomes.

Notion number 2 corresponds to the notion of independence of random
processes (or trials). There are many random processes that do not
demonstrate independence. Mutations are, indeed, not independent
because the occurrence of one mutation alters the possibilities
(sample space) available for future mutations. Selection is not
random because changes in gene frequencies from generation to
generation are divided into two conceptual camps: random (called
drift) and deterministic (called selection). So selection is the
non-random component. In reality, you can't separate the two. Notion
number 3 is a common misconception -- people say a loaded die or an
unfair coin is no longer "random". Wrong. It is random; it is just
that the probability distribution is no longer uniform; different
outcomes have different probabilities. In fact, real evolution always
is the combination of selection with drift. That is, the allele
frequencies in the next generation are the result of a random process
but certain outcomes have a higher (perhaps much higher) probability
than others. The non-uniformity in probability is called selection.
Mathematicians don't care about non-uniform probabilities (notion 3).
In fact, having equal probabilities is a very special case of the
general sample space/probability measure concept. And mathematicians
have well developed ways of treating dependence between trials or
between times for a continuous process (notion 2).

A very separate question is how all this abstract mathematics relates
to physical processes in the real world. This is an issue of
philosophy more than of science or mathematics. Is the physical
universe completely deterministic or is it random (that is, has a
random component)? Most people think that the Newtonian universe was
deterministic while quantum mechanics introduced the
randomness/non-determinism, the "dice" that God may or may not play
with. I don't know the physics, but I do know that physicists invoke
the name "Bohm" to indicate that the answer is not all that clear cut.
See the article on "causal determinism" in the Stanford Encyclopedia
of Philosophy (http://plato.stanford.edu/entries/determinism-causal/).

The simple fact is that major areas of the working out of the physical
universe are extremely well described by the abstract mathematical
notion of random processes. If the model fits, then it is random; the
philosophy be damned (sorry, Wilkins). There are many reasons why
things behave randomly even in a completely deterministic universe,
even without invoking the Heisenberg principle of incomplete
knowledge. The real world is a "statistical ensemble" (another nice
mathematical abstraction from the world of probability theory) of
particles far too numerous to enumerate and describe individually.
And even if we could almost completely describe everything, most
interesting systems are sufficient chaotic (again, a precise abstract
mathematical concept) to preclude prediction. And even if we could
describe everything exactly, the finite limits on computation render a
prediction impossible in a reasonable time (the lifetime of the
universe). So, as a very practical matter, events like throwing dice
or shuffling cards or generating lottery numbers or a very large
number of scientific processes must be treated as random processes.

Evolution is the result of a random process because the mathematics of
random processes well describes what happens whereas deterministic
ones do not. Mutations have very strongly random aspects. Whether
they are completely independent or whether all mutations have the same
probability is irrelevant. Allele changes from generation to
generation have both random (drift) and deterministic (selection)
aspects or, alternatively, have non-uniform probabilities. Ecological
and geological and astronomical events such as earthquakes, forest
fires, volcanic eruptions, asteroid impacts and the like, not to
mention simple changes in climate, can be considered to occur at
random intervals and locations and magnitudes. However, to say that
evolution is a random process does not in any way mean to say that you
can not make specific predictions or statements of a very
deterministic nature about many aspects of the outcome of the process.
There are also many aspects of evolution that are well described in
deterministic terms. It is simply that evolution includes random
components.

Note: the errors of the "blind watchmaker" or the "tornado in a
junkyard" probability calculations are of an entirely different
character. They are in calculating and applying probabilities, not
just in trying to use probabilities.









.

Relevant Pages

  • Re: Cantorian pseudomathematics
    ... Chas Brown wrote: ... You can even check other facts told by probability theory, but I cannot see a *mathematical* reason why a die should give a random value: I cannot even define random better than ... set theory to think about all that; and mathematicians didn't wait for ... That's probably why potential infinite was considered enough, ...
    (sci.math)
  • Re: Cantorian pseudomathematics
    ... > set theory to think about all that; and mathematicians didn't wait for ... >>> using assumptions made by basic probability theory (i.e. that true ... Infinite sets are equivalent. ... You might want to be prepared to correctly argue that. ...
    (sci.math)
  • Re: Laws of Mathematical Probablilities -The Old Testament containsover three hundred prophecies
    ... >>> Mathematicians generally consider any event with a probability of less ... > inanimate matter, when it takes certain forms, such as that of crystals. ... > does not seem possible to apply the laws of probability calculus to the ...
    (talk.origins)
  • Re: probability (was Re: EEQT)
    ... in which case they have nothing to do with conventional probability ... >formal items the mathematicians use with the appropriate notions ... >know about ensembles than the definition I gave. ... >measure theoretic foundations; they would be hard pressed to remember ...
    (sci.physics.research)
  • Re: The Theory of Evolution is Mathematically Irrational Round 2
    ... The probability of having a male vs. a female child is 50:50. ... principle of the mutation and selection phenomenon. ... beneficial mutations were to occur as a function of population size. ...
    (talk.origins)