Re: Reproductive Selection

On Dec 19, 11:02 pm, Treus <treusd...@xxxxxxxxx> wrote:
leland.mcin...@xxxxxxxxx wrote:
What I really want here is a function of a continuous variable, since
it makes things easier. Critters are discrete, but are spaced through
time so we can use a function of time as a continuous approximation.
Approximating discrete systems with continuous ones is common enough
-- think of fluid mechanics, which approximates the discrete particles
of a fluid with a continuous idealised fluid.

The way to think of this is as a one dimensional curve twisting
through n-space, parameterised by time. We can think of the critters
as discrete points all along the continuous curve. Exactly how the
discrete critters are spaced along the curve is matter of how time
was parameterised (ie. if each unit time step is a year, then we might
have only a few critters over each unit of time, but if we are taking
each unit time step as 100 years then we will have many critters
packed into each unit of time). Since how time parameterises the curve
is arbitrary for us, we can essentially assume our discrete critter
points on the curve are as densely packed as we like (since if they
aren't, we can just reparameterise the curve to pack them more
densely).

Not to make too much of a peripheral point, but why use critters at
all? Why not show continuity of reproductive type with respect to time
directly? (It's fine as is.)

Well if you're happy with that, I certainly have no problem. I was
including critters as actual discrete points so as to provide some
model for the discrete granularity that will exist in practice. Of
course, as I then went on to point out, that shouldn't be a problem
for our model since we can simply reparametrise as required to
eliminate any problems introduced by the apparent granularity. If
you're happy though, we can skip that step, and simply take our
function as a function of time and ignore discrete critters. We can
always re-introduce it later -- see below.

Okay, good to know I have that much sorted. Now, my understanding is
that you are effectively claiming that reproductive selection, if it
is an active force, is going to prevent species evolving truly novel
or different reproductive types. That is, given a starting point (t=0
on our curve) there are going to be some points in our n-dimensional
reproductive type space that are "unreachable": we can't both satisfy
the reproductive selection constraint, and also be able to have our
curve pass through/get to these unreachable points. Is that a fair
representation of what you are claiming?

Yes, I think so.

Okay, then I think we have a problem. The reproductive selection
constraint, that for a given epsilon there is a delta such that for
any t1 and t2 with |t1-t2| < delta we have d(r(t1),r(t2)) < epsilon,
simply says that r(t) is a uniformly continuous function that defines
a continuous curve in the n-dimensional space of reproductive type.
Now uniform continuity is not a sufficient requirement to bound a
curve; that is, in a path connected space, given any two points in the
space, there exists a uniformly continuous function that passes
through both points. Reproductive selection, as a constraint, does not
make any part of the "reproductive-type space" fundamentally
unreachable. In fact there actually exist "space filling curves" that
reach every single point in the space on a single curve.

There are a few ways you could try and salvage unreachability. The
most immediate is to simply claim that reproductive-type space is not
path connected. That is a very different claim, on which reproductive
selection has no bearing, and for which you would need to devise
significant justification (i.e. a whole new argument). Worse still for
this approach, it then behooves you to demonstrate that some existent
life-forms lie on separate path-connected-components. That is
certainly non-trivial.

Another approach is to note that we do, at least, have that fact that
a uniformly continuous function is bounded over any finite interval.
This might seem promising at first, since we do have a finite time
interval to work over, however we are only assured the existence of
*a* bound; there are no constraints on the bound. That means that our
uniformly continuous curve can reach any point that is *some finite
distance* from our starting point (within the path-connected-
component); we are only prevented from reaching *infinitely* distant
points, which aren't really relevant for your problem. Ultimately this
approach is probably a non-starter.

The third approach is to build on the previous point by re-introducing
the discrete granularity of critters. In principle the rate at which
successive generations are produced will provide a upper bound on how
we can reparameterise the curve over a finite time interval, and at
least give some bound on how far from the source point the curve can
have reached *within the current time interval*. Note that under this
argument every point of reproductive-type space is still reachable,
its just that, at least in theory, some points might not have been
able to have been reached *yet* (though of course they can be reached
at some time in the future). Now, however, you have the same sort of
issue as with path connectedness: to make the argument it behooves you
to show that some life-form has a reproductive-type that lies outside
the space of points that are reachable inside of the current time
interval. Again, this is highly non-trivial.

So, to summarise, your argument, as it stands, gives no reason to
assume that any part of reproductive-type space is unreachable in
principle. This means your only option (if you wish to pursue this
sort of line) is to at least show that some points may be unreachable
in practice; that is, you will have to forego your theoretical
argument and instead build an empirical argument.

On that front there are a few points I would like to suggest are worth
considering. The first is that life already seems relatively well
organised as far as relative reachability is concerned: generally
speaking the more distinct the reproductive systems of organisms, the
longer ago they branched apart in the evolutionary tree. Thus the most
radically different systems, such as fungi compared to plants,
compared to animals, all separated early; less radical differences
(such as egg-laying reptiles, vs. live-young bearing mammals) are
later splits, with descreasing degrees of difference as we go on (for
example, birds and reptiles, which split much later than reptiles and
mammals, both lay eggs). This isn't a hard and fast rule (I'm sure the
biologists here can come up with a vast array of examples where it is
inexact), but the fact that it exists at all is a significant pointer
toward the reachability of the diverse reproductive systems we see.

Another point I would like to make is not to get drawn into the
illusory temptation of improbability arguments. These are the sort
that claim that diverging wandering curves couldn't credibly have come
up with something as diverse and distinct as X and Y in the time
given, or that the odds of managing to have a curve reach a point like
X from any common starting point are just astronomical. You have to
keep in mind that evolution is massively parallel, with literally
billions upon billions of curves all branching out at once -- many of
those curves go nowhere, but with enough curves all flowering from
every point along the way, enough will lead somewhere interesting.
More importantly, as long as some curves are heading off, they have to
get *somewhere*; prior probabilities just don't matter.

The final point is that "there's not enough time" arguments actually
need solid empirical grounding. We have a model that can certainly in
theory explain the diversity of life, and the nature of the diversity
(as noted above) stacks up remarkably well with that model, so without
any evidence to the contrary, it is reasonable to assume the model is
sufficient. If you want to claim it isn't sufficient you'll actually
have to pony up a reason: empirical data on rates of change, time
available, etc. The burden of empirical proof is squarely on those
claiming insufficiency.

.

Relevant Pages

  • Re: Reproductive Selection
    ... Critters are discrete, but are spaced through ... Approximating discrete systems with continuous ones is common enough ... The way to think of this is as a one dimensional curve twisting ... The continuity argument you made states what I ...
    (talk.origins)
  • Re: Reproductive Selection
    ... Approximating discrete systems with continuous ones is common enough ... as discrete points all along the continuous curve. ... Why not show continuity of reproductive type with respect to time ... most immediate is to simply claim that reproductive-type space is not ...
    (talk.origins)
  • Re: Ruler and Compass in Mathematics
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  • Re: Ruler and Compass in Mathematics
    ... Drawing a curve or line is useful. ... its discrete". ... I have argued that there is nothing in mathematics or out of mathematics that can quantify a continuity, ...
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  • Re: Ruler and Compass in Mathematics
    ... contnuous curve or straight line between the facts that are given. ... Drawing a curve or line is useful. ... its discrete". ... Maybe your complaint is that continuity is not well-defined. ...
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