At 08:04 PM 09/26/2000, you wrote:
>Ralph
> >I find it interesting that the following relationship is true:
>
> >1 cubed = 1 squared
> >1 cubed + 2 cubed = (1 + 2) squared i.e. 9=9
> >1 cubed + 2 cubed + 3 cubed = (1+2+3) squared i.e. 36=36
>
> >Anyone who wants to carry this out farther will find that it
> >continues in this vein. Now many people would (and do)
> >see design in this. They regard this as the mathematical
> >equivalent of Paley's watch. Numerology is very popular.>
>
> >But we have a pretty complete picture of the history of
> >mathematics. It was not designed so that this pattern,
> >and others like it, would work out. These patterns are
> >in there by chance. The number system was not
> >designed with them in mind.
Chris
Though I agree with the *point* you make here, I think the word "chance" is
problematic in this context. It's chance from our point of view that
mathematics just *happened* to work out this way, but it's *inherent* in
the basic concept of quantity that this relationship obtains, so, in a
logical sense, it's not chance at all. Given the nature of quantity, we
cannot change this result, even if we wanted to design things so as to be
different (we can, of course, make arbitrarily many systems of a *similar*
nature that have different implications, but then the primary underlying
idea would be something other than the concept of quantity).
>Bertvan:
>You seem confident that numerical relationships were not designed. Do you
>consider this particular relationship more due to chance than others? In
>any case, numerical relationships always existed, didn't they? Even before
>there were human minds to understand them?
Chris
Yes, even before (or independently of) *God's* mind and choices (were God
to exist).
Bertvan
>Do mathematical systems exist
>independtly of the human mind or were they created by human intellignece?
Chris
A mathematical system as a complex set of relationships among facts would
"exist" regardless of minds or not. But, a mathematical system as a complex
of *ideas* would, of course, require a mind in order to exist (though a
book presenting a mathematical system could continue to exist and to
implicitly continue to "contain" such a system even after it's author was
dead).
Bertvan
>Complex biological systems, on the other hand, came into exisence. At one
>time, they did not exist on earth. I don't know if we could claim
>mathematical relationships "created" in the same way.
Chris
No, they're not. But, the point is that these relationships are *also* not
*designed*, whether by God or anyone else. Thus, complex, rational systems
*can* exist as a complex of factual relationships, even though they are
*not* designed. Similar conclusions can be drawn from logic.
More importantly, they can be drawn from physics and other sciences.
Physics *implies* complex, rational systems as possible structures. It
also, along with mathematics and information theory, implies that simple
things can be made into complex things a little at a time. Thus, while each
letter Shakespeare wrote down was a small addition to his works, the entire
body of his work was achieved by just such small steps. Note that I'm not
claiming that Shakespeare's work was not designed, but only that it was
built as a large number of small steps.
That Shakespeare's work as a whole, is complex is hardly to be debated.
What's left to claim? Ah: That the *same* small steps, if they occurred
randomly, along with trillions of others, would not lead to the production
of such a system. But, it would. Here's how: Assume that the entire body of
Shakespeare's work is N letters, numerals, spaces, and punctuation marks
long. Then, to generate the works of Shakespeare, all we have to do is
generate *every* string of such letters, numerals, etc. that is
theoretically possible. Since Shakespeare's work *is* both possible (it
exists) and N such characters long, Shakespeare's work must necessarily be
at least *one* of these strings.
This is true even though it is complex and, some would say, rational, and
even a kind of system.
The same considerations apply to any complex, rational system. Just
generate *all* possible assemblages of the requisite size and with the
requisite *simple* parts, and, voila!, that complex, rational system will
be among them. It *can't* not be, unless it is simply logically impossible
for it to exist at all, unless it is a logically contradictory thing.
Nature does not, of course, try *every* possible variation, except maybe at
the start or in very narrowly defined classes of cases. But, it can try
trillions and trillions of them, increasing size and complexity of some a
little at a time, as "good" variations happen to occur. This means that we
don't have to save *every* variation that is produced. We need only "save"
the "rational" ones (the ones that have attributes that make them and their
continued existence compatible with their environments and their internal
structure) and reproduce it enough times, because, *eventually*, other
small changes will occur that increase complexity while preserving rationality.
Thus, even if we don't get the biological equivalent of Shakespeare's
works, we *will* get complex, rational systems as long as quite a few are
*possible* (and, obviously, a great many *are* possible) and they are not
specifically selected out or selected out by extinction of the gene pool.
There is little else that can happen as long as some of these small steps
toward more complexity are *also* of survival value.
It occurs to me that you may be thinking that complex, rational systems are
numerically very rare among the set of all *possible* systems of the same
parts. But, this is not true. As you can see from what little biological
diversity there is, the number of possible complex, rational systems is
very large indeed. And that's *not* counting the complex, rational systems
that were nevertheless unable to remain viable. People are complex,
rational systems (in a certain narrow sense of "rational"). Yet, there are
*billions* of them. And they are all *different* (as far as we can tell),
despite being complex, rational systems. Thus, evolution does not have to
wait for just exactly *one* variation that will increase complexity while
preserving rationality; it can settle, at any given time, and even in the
case of any given organism or mated pair of organisms, for any of possibly
many *millions* of variations. Even organisms as simple as bacteria *can*
survive and prosper with any of a large number of possible variations.
Remember, for evolution to work in this way, it does not have to achieve
its complex, rational systems as it has apparently done so, because there
were billions or trillions of *alternative* complex rational systems it
could generate instead. For example, perhaps with the switching of a couple
of base pairs 3.8 years ago, *our* place would be occupied by some creature
that was radically different and yet also *similar* ecologically. Just as
there are billions of possible human beings, there are billions of other
possible complex rational systems that could be generated but that haven't
been.
Put another way, while a great many biological organisms that exist may be
complex and rational, by no means do *all* logically possible complex
rational biological organisms exist. If you ignore what *doesn't* exist in
your attempts to understand why what *does* exist has the overall
"statistical" attributes it does, you will almost *certainly* arrive at
false conclusions. This is because the "sample" of complex biological
systems we see is an *extremely* biased sample. Why? Because it shows us
*only* the ones that survived. Even fossils show us only the ones that (in
general) either survived or were the immediate offspring of ones that did.
Thus, the same kind of bias that you would have if you just examined
*today's* life on Earth and ignored *past* life on Earth is made *vastly*
worse when you ignore the variations that *never* survived, or that never
even *occurred* (because the organisms that would have been their parents
did not survive). If you fill out the tree of variations with all the ones
that no longer exist and/or that *never* existed, you will have four types
of branches, "twigs," and leaves:
1. Those that are the examples of life we see today.
2. Those that *once* existed.
3. Those that could have existed insofar as their structure would have
allowed them to continue existing (at least for a while) if they had
occurred to begin with.
4. Those that were *not* rational (biologically) and which therefore could
not continue to exist because of internal biological defects. This would
include all possible DNA variants that could not "instruct" the molecules
in a cell in order to cause the cell to divide, or that could not even
serve as the information-basis for the survival of the cell itself. It
would include all the DNA that simply made no biological sense, regardless
of its complexity (or even of a certain organizational rationality).
Of all complexity in organisms or genotypes, you will have the following
classification:
1. Complex rational systems that survive or that survived for a while in
the past.
2. Complex rational systems that get excluded after being generated, for
whatever reason.
3. Complex rational systems that do not even get generated, because
variations are not sufficiently random and/or the populations of parent
organisms is not large enough to ensure complete coverage of all possible
variations.
4. Complex *irrational* systems that can't survive biologically.
The largest category by far is the latter. The second largest is the third.
The smallest is the first, but even *it* has contained millions of species,
all built on one basic information molecule type (DNA). If we include
information molecules *other* than DNA, or radically different ways of
using DNA, the third category might be beyond normal calculation. The
fourth category would be even larger still, but, no matter how we slice it,
the number of possible (biologically coherent) complex rational systems
(i.e., organisms of distinct species) is still an astronomical number.
Can random variation produce complexity by cumulative adding to existing
not-so-complex structures. Yes. Can it produce complex *rational*
structures this way? Yes, because it produces so many attempts and because
there are so *many* possible complex rational biological structures; out of
billions of instance of variation, it only needs *one* to increase
complexity (slightly) while preserving biological rationality (at least
enough for the resulting organism to survive and reproduce). With a large
enough population and with enough generations, millions upon millions of
variations get applied.
With sexual recombination, variations can occur independently in one
population and then end up together in all members of a later population of
descendants. If each of these variations increases complexity a little,
both together will normally increase complexity somewhat *more* than either
one separately. In a large enough population, many novel variations may
occur in each generation and be recombined in many millions of ways with
other genes in later generations, many of them increasing the overall
complexity of the organism (and/or the genotype) considerably. Evolution
overcomes the "six monkeys" effect by producing large numbers of variations
and filtering so that the ones that don't work are excluded from further
participation. All further variations are applied *only* to genotypes that
have *already* survived and which are therefore biologically rational. And
then, *those* variations are filtered. And, at least in principle, many of
the *surviving* variations could be increases in complexity without a
decrease in biological rationality (perhaps even an *increase* in
biological rationality).
Thus, *unless* there are special circumstances, special factors that
*prevent* evolution from building complex systems, the complexity of the
requirements of real world survival all but *guarantees* that some
organisms that actually exist will be very complex indeed. And, all the
ones that actually continue to exist will be basically biologically
rational (because no others *can* survive).
Basically, it's mathematically *extremely* unlikely that there would be no
evolutionary paths to complex, biologically rational organisms, as long as
there is a large population of slightly simpler organisms (or pre-biotic
"thingies") prior to each of a large number of generations. Complex,
biologically *rational* organisms are a subset of complex organisms. Since
complexity can be easily achieved by merely adding more random material to
existing DNA, the only question left is whether it can be biologically
*rational*. Mathematically, random changes *must* occasionally be
biologically rational. If those occasional events are perpetuated via
genotype reproduction, then, eventually *more* biologically rational
variations will randomly occur and "stick" with the prior ones, until a
large system of such small changes is built up.
If you still claim otherwise, please tell us *how* it could be otherwise.
If you choose randomly from a pile of jigsaw pieces to add to the one
you've already started with, eventually you will *happen* to hit upon one
of the correct next pieces to fit with the one you've already got. You can
do the entire puzzle this way, with no more intelligence needed than the
mechanical ability to recognize when a piece fits the pieces already in
place. That's what evolution does. It picks a variation and tries to put it
with the pieces already present. If it "fits," it is generally kept. If it
clashes, *other* variations are tried until one *does* fit. Just as a
jigsaw puzzle with a billion pieces would be a complex and rational system
(as could be determined by looking at the picture on the front of the box,
if we had a box big enough for a billion pieces of a jigsaw), and yet could
easily be put together by enough repeated trials of randomly selected
pieces, so naturalistic evolution just keeps "trying" variations and
keeping the ones that fit (i.e., that work). Because evolution is a
branching process, not all evolution is toward complexity, but *some*
definitely *is* toward complexity.
So, if you *still* claim that it won't work, can you try to tell us why
not? Do you *deny* that a *complex* and rational (i.e., coherent,
well-fitting-together pieces) jigsaw puzzle (or an even more complex
three-dimensional puzzle) could be put together by the process I described
above?
Actually, the biological case is better than the jigsaw puzzle, because
there's only *one* way to correctly assemble a jigsaw puzzle that
reconstructs the original picture. But there may be many trillions of ways
to make complex biologically rational organisms, counting *all* possible
paths from the very simplest to any specified *level* of complexity. There
seem to be *many* organisms at the level of complexity of a dog, for
example, including many *different* dogs. But only *one* good reproduction
of the original picture in the average jigsaw puzzle. And yet, a blindly
mechanical process that has only the "smarts" to *recognize* (not
intelligently *choose*) correct pieces (and even then only when it tries to
put each piece in place) can build a jigsaw puzzle of nearly any imaginable
complexity. So, *why*, with so many *more* functional possibilities, would
you deny that complex living organisms could not be constructed a little at
a time by "trying" *many* variations?
This archive was generated by hypermail 2b29 : Wed Sep 27 2000 - 03:54:41 EDT