Replies

Thank you for the time and effort you took to address my rather lengthy post. I appreciate your endeavor to respond in a polite and informative manner, unlike many posters on such threads as this. Therefore, in the same spirit, permit me to present some counters and ask some questions concerning your post.

Unfortunately, such will again lead to a lengthy post… My apologies, again. To avoid your having to “flip” back and forth to previous posts, I have color-coded the responses… my original in blue, your response in green and the current counters or questions in black.

As I understand “Darwinism,” it is based upon inductive reasoning.

No, it isn't. It's based on empirical science, and rests upon a foundation of evidence.

em·pir·i·cal adj.

a. Relying on or derived from observation or experiment: empirical results that supported the hypothesis.
b. Verifiable or provable by means of observation or experiment: empirical laws

I am unaware of any experiment even suggesting “evolution” beyond that involving bacteria or other very simple life forms. Even those are apparently subject to some controversy as to whether they establish the concept or not…To wit, see below:

From post 20 on this thread by Getready:” Bacterial resistance to antibiotics has been shown to be due to transmissible genetic snippets, or loss of information causing less susceptibility to certain natural chemical compounds or their congeners. (no new structures are developed in order to cause resistance...just alterations of existing one… i.e. antibiotic pumps, or bacterial pore population as examples, inability of certain antibiotics to attack certain molecular structures, or expression of enzymes which destroy the antibiotic.. (some organisms remain susceptible to the same antibiotics...whatever change there might be, it is very, very limited, and does not represent a change from lets say a bacterium to a PPLO, or a fungus, or even a prion (yet)

“Whenever an event occurs, at least one sufficient condition is present and all the [...]

Yeah, yeah, get to your points.

Sorry, but I have found it necessary in posts on these threads to be very specific in attempt to avoid extraneous wanderings concerning unstated premises or explanations.

The proponents of “Darwinism” currently propose that the mechanism (in my limited understanding of the subject) for evolution is a purely random “mutation”

There's no requirement that it be "purely random". Many mutations are decidedly non-random, in fact.

Perhaps, I am misunderstanding you. There is no other “naturalistic” assumption other than “pure randomness” available as to how mutations can occur. Consequently, if mutations are not “purely random,” the implication is that something or someone is directing them. As this is exactly the argument for intelligent design, I do not think that is what you intended to say. Care to expand on this?

of existing an organism which is both heritable and beneficial in terms of natural selection to the enhanced survival and/or reproduction of future generations of that organism.

No, non-beneficial mutations also play a part, and neutral mutations can fix in the genepool through genetic drift alone.

Again, perhaps, I am misunderstanding you. Is not the basic tenant of “Darwinism” that natural selection “chooses” those individuals which have mutations that enhance their ability to survive and reproduce for continuation of their genetic material? Is this not the theory’s posited mechanism by which more complex organisms appear? It may be true that “genetic drift” would cause change, but without benefits in terms of natural selection, the same mechanism, “genetic drift,” would “erase” the change or replace it with another, would it not?

in particular, with relation to the “Cambrian Explosion:”

Stated factors/conditions:

1. Mutations must exist randomly.

Wrong.

As noted above (I will repeat for your convenience): “Perhaps, I am misunderstanding you. There is no other “naturalistic” assumption other than “pure randomness” available as to how mutations can occur. Consequently, if mutations are not “purely random,” the implication is that something or someone is directing them. As this is exactly the argument for intelligent design, I do not think that is what you intended to say. Care to expand on this?”

2. Mutations must be heritable.

If they're going to make any long-term evolutionary difference, right -- mostly. Non-heritable mutations play a small but significant role in evolution (via horizontal transfer, etc.)

3. Natural selection pressures must determine that a particular mutation provides a benefit to the organism.

Oversimplified, but close enough for a discussion like this. And again, you're leaving out genetic drift. Plus you've skipped things like mutations which by themselves may be non-beneficial, but when shuffled together with another non-beneficial mutation via sexual recombination may provide some advantage -- it's a mistake to think of a given mutation as providing a specific advantage *in isolation*. Furthermore, mutations which are advantageous in one setting may be non-advantageous in another, etc etc.

Please see my discussion below on probabilities concerning multiple beneficial mutations occurring simultaneously.

Unstated factors/conditions:

1. Mutations can only be classed into the following categories: a) none; b) benign: c) beneficial: or d) detrimental.

See above -- even the same mutation can shift from category to category depending upon other factors.

If your point is that a benign mutation may at a later point become a beneficial one, I will concede the point if you will agree that there is an equal probability that it could also become a detrimental one.

2. The sum of mutations in categories a), b) and c) must significantly exceed category d) or the organism will become extinct prior to the production of a “better adapted” next generation.

If you mean "in a specific individual", I'll buy that.

Is there any case where a heritable characteristic is not passed to the next generation except by “individuals” in the preceding generation? If this is so, do not all mutations, their probabilities and possible survival benefits in terms of natural selection have to be considered on an “individual” basis?

(note: this condition disallows excessive mutagens which puts an absolute upper limit on mutation rates which,

Correct.

Again, there is a strong implication associated with this point. If there is an absolute upper limit on the possible mutation rate, then the probability of more than one, simultaneously, becomes the mathematical product of that rate times itself. Similarly, the probability of multiple, simultaneous beneficial mutations, which occur at a much smaller rate than mutations in general, also becomes a mathematical product of a small number making an extremely small probability for such. Therefore, for all practical considerations, the accumulation of beneficial mutations in any organism is restricted to those that occur in successive generations. This successive accumulation means that benign mutations that could later become beneficial must survive many, many generations in the face of genetic drift and “negative” selection.

when combined with the requirement for purely random occurrence, negates the likelihood of multiple beneficial mutations occurring simultaneously or successive mutations occurring excessively rapidly.)

If you mean in a particular individual, good enough.

Perhaps I am missing something. Is there any other way to consider this point?

3. Beneficial mutations must occur at a frequent enough rate to accommodate the time frame estimated for new organism appearance from fossil record. (note: this condition mandates the presence of sufficient mutagens which puts an absolute lower limit on mutation rates.)

Okay.

Yet, again, there is a implication here. If there is an absolute lower limit on mutation rates, the theory of punctuated equilibrium becomes tenuous at best and, more likely, completely unsupportable.

4. A sufficient number of generations must occur within the fossil record time frame to make the beneficial mutation present in enough individuals for production of a sufficient population size to generate the next beneficial mutation within acceptable mutation rates.

Uh.. You need to clarify this more, it's ambiguous in a lot of ways.

Sorry, I failed to communicate well here by trying to cram too much into a single sentence. Let try again:

By a “sufficient” population size I intended to point out that, statistically, a minimum number of individuals are required at a certain mutation rate within a limited time to produce a mutation. In turn, even more individuals are required at that same mutation rate and limited time for the occurrence of a “beneficial” mutation in terms of natural selection. Assuming such a “beneficial” mutation had occurred in a succeeding generation, that generation’s successors would, likewise, also have to reach a minimum size population, statistically, for the “next” beneficial mutation to occur. For “beneficial” mutations to accumulate, the required population size, the required number of generations to reach the appropriate size after the appearance of each beneficial mutation, the time between generations and the mutation rate act multiplicatively to decrease the probabilities of additional beneficial mutations within a limited time.

5. For the appearance of successive (in time) species in the fossil record having new/better capabilities, beneficial mutation must make the organism more biologically complex that its parent.

Nope! Just different. It can even be simpler -- and in a lot of environments, simpler is better. But if your point is that if evolution *can't* ever increase complexity, then living forms today would still be awfully simple and not like what exists today, I'll buy that. The evolutionary scenario for producing today's living things doesn't "work" unless evolution can indeed result in increased complexity.

Agreed. There is a strong implication with this point. Bacteria have the ability to produce large populations of individuals with very short time periods between generations and thus are potentially subject to many “beneficial” mutations. Consequently, by “Darwinism’s posit, bacteria would arguably increase their complexity to the point of becoming another organism. However, to my limited knowledge, experiments with bacteria have never produced anything but other bacteria even after many thousands of generations.

6. The time between generations must be short enough to accommodate the time frame estimated for new organism appearance from fossil record.

You're throwing around a lot of undefined terms like "short enough" without actually getting specific...

Sorry, again, I failed to adequately communicate trying for brevity. By “short enough” I was referring to the time allowed in the fossil record between the appearances of different organisms. This would be the minimum time allowed for a beneficial mutation to appear and be propagated to the next generation and successor generations become large enough to sustain the mutation’s presence and, then, grow large enough for the next beneficial mutation’s appearance, etc. In short, the total time for required for the accumulation of enough beneficial mutations between generations for a new organism’s appearance in the fossil record.

8. Natural selection pressures must determine that a particular mutation provides a benefit to the organism at rate that prevents the mutation from disappearing due to genetic drift or other phenomena.

I think I know what you're trying to say here, but you're not saying it right.

Sorry, I cannot think of a better way to say this. Perhaps you could “reflect” your understanding and let me try again on that basis.

9. Natural selection pressures must not be so great as to cause organism extinction before the organism has produced an adequate number of generations with the beneficial mutation to make the beneficial mutation widespread enough to ensure its survival.

When speaking of "negative" selection pressure, yes, but "positive" selection pressure can't be "too great".

If I understand your point, here, I must disagree. As I understand it, natural selection can only offer one “pressure:” that which enhances the survival and/or reproduction of an organism having a beneficial mutation. By “Darwinian” posits, if there is no benefit to a mutation, will it not, most likely, eventually change or disappear due to genetic drift?

10. As some simpler “parent organisms,” e.g., sharks, continue to exist alongside “descendant” organisms, Natural selection pressures must not be so great as to cause organism extinction or the simpler “parent organism” cannot appear simultaneously with its more complex “descendant” organism.

Not really a problem, actually, since the "daughter" species may have moved into geographic isolation, or into a niche non-competitive with that of the "parent" species, etc. And again, it's a mistake to presume that the parent species must necessarily be "simpler" than the offspring species.

Perhaps, I can concede your point concerning “niches.” Nonetheless, there are apparently numerous situations where the “parent” organism is in competition with the “successor” organism in the same niche. If, in fact, the successor organism had an advantage by natural selection, it should completely supplant the “parent” organism, should it not? Otherwise, what is the advantage?

If the successor organism is not more complex than the “parent” organism, then how can the posit of “Darwinism” that more complex organisms sprang from less complex ones hold?

The list of “necessary” factors could potentially go quite a bit further. However, the above number, alone, establishes that “Darwinism,” as it is postulated, must overcome some very near impossible odds.

WOW, a mathematical claim without any math! Try again, son.

Alright, “dad.” Any situation less than certainty carries a probability of less than one, i.e., a fraction. Any fraction multiplied time another fraction yields a smaller fraction. If there are a number of probabilities that must occur sequentially, then each must be multiplied by the other. For example, the probability of heads of tails on a fair coin’s toss is 0.5. However, for the same coin, the probability of two heads (or tails) in succession is 0.5 times 0.5 or 0.25, i.e. a smaller number. Similarly, the probability of 7 heads (or tails) in succession is 1.52 x 10^-5. Is the math sufficiently displayed for you, now?

The probabilities associated with each “necessary” factor are multiplicative with the probabilities of each additional factor.

Horse manure! You have in no way established this. And you can't establish it by just "I say so, that's why".

Perhaps, you did understand that each condition to which I referred carried a probability of less than one. Consequently, it was not I who “just said so”… The rules of mathematics did.

Therefore, even if the probability associated with each factor were only a single decimal place

"If". Feel free to actually support such an assumption.

Again, as an example, 0.1 times 0.1 equals 0.01. Sufficiently supported?

the resulting product yields a number with a tremendously large negative exponent.

Faulty conclusion due to being based on unsupported premises and numbers pulled out of a hat instead of from real-world testing.

Unless you wish to assert that on the basis of” real-world testing,” the probability of a beneficial mutation is a certainty for any randomly selected individual of any given species, then the premise is not “unsupported.”

Additionally, there are those factors with many more than one decimal place such as beneficial mutation rates which have a negative exponent greater than 6. The implication is that the “millions” (i.e., a positive 6) of years allowable even with “millions” of individuals (i.e., another positive 6) within a species in the Cambrian fossil record cannot account the appearance of between 17 and 34 animal phyla attributed to that time frame.

Again, you've provided no real math, just made up numbers followed by a non-mathematical conclusion ("can't!"). Sorry, but that just doesn't cut it.

Again, as an example, 0.00001 times 0.00001 equals 0.0000000001. Does this example “cut it?”

Furthermore, you've overlooked quite a few things in your simplistic "Cambrian" scenario:

1. Mutation rates are *per nucleotide* *per copy*, if I recall correctly. Quick, Einstein, how many nucleotides in the genome of a single organism? You've presumed it's per individual, per generation. Wrongo. In fact, I don't have it at hand right now, but one study determined that every individual human has on the average *four* new mutations that are unique to him (or her), having occurred in the mother/father germ line which led to his conception.

My comments were based upon a rate of mutation in organisms of "one mutation per locus per 10⁵ to 10⁶ gametes."(Campbell, 1990, p. 445) Calculations must consider the total number of possible loci, the probability that any particular gamete will fertilize/be fertilized by another gamete, the probability that a resulting zygote will be viable, the probability that the mutation is benign or beneficial, the probability that any resulting individual organism resulting from maturation of the zygote will be fertile, the probability that the organism will survive, the probability that the organism will reproduce, the probability that the succeeding generations will pass on the mutation unaltered, etc.

2. You use "million" as your "population size", but that's a *very* small number for most species. Do you have any idea how many, say, Green Crabs there are in the ocean? A million is a drop in the bucket.

Actually, I thought I was being generous with this number. The situation of a beneficial mutation appearing in any given individual is very low. Even assuming that the mutation appears and is passed to the next generation, the size of that first generation having that mutation is limited. Similarly, each new generation must pass the beneficial mutation to successor generations unaltered, etc. In short it will take a finite period of time for the population to reach a million. In fact, I suspect that the population size must be much greater than one million for the probability of a beneficial mutation to occur.

There a great many green crabs in the ocean, now. However, the operative question is how many were there when they first appeared?

3. You mention the Cambrian radiation evolving from "a species", but even leaving aside the obvious that the Cambrian explosion likely resulted from advances in *multiple* species, and buying your oversimplified scenario that it arose from *one* preCambrian species, you're still overlooking the fact that after the first bifurcation, now there are *two* species within which the further mutations can arise, and so on as the clades further split in turn.

Your point is well taken. However, my point was not with the number of species or clades. Rather, my point was that each branch, a beneficial mutation had to appear, be propagated, and pass through the next branch, etc.

4. You have provided zero, zip, nada estimate of how many mutations (at least on the order of) would be a necessary number to result in the Cambrian evolutions. Sort of *need* that for your calculation, don't you?

If my above arguments have been insufficient at this point to illustrate my point, it would require a doctoral dissertation for me to provide more detail. Sadly, I haven’t quite got the time for that.

5. The actual time over which the Cambrian radiations occurred is at least 100 million years, not just on the order of "millions" as you assert.

I am willing to admit that I may be misinterpreting what I have read. However, according to my information, on which I could be mistaken, the Cambrian radiations occurred over a period of approximately 17 million years approximately 570 million years ago. Nonetheless, even if your number is the more accurate, it only increases the positive exponent by 2. Such an increase does not seem, in my estimate, to alter the overall probabilities I have been discussing significantly.

6. You mention "between 17 and 34 animal phyla" as if that's a large number to evolve in 100 million years, but a "phylum change" back in that era was little more than what we'd write off as a "species change" now (although in another half billion years a few of today's "species differences" may have resulted in enough subsequence divergence to be recognized, *then*, as a "phylum split" -- it's all relative to where you're measuring from). For example, today vertebrates are hugely different than invertebrates, thanks to a half billion years of subsequent evolution and divergence, but back in the early Cambrian the difference between the "vertebrate phlyum" and the "invertebrate phylum" was little more than a worm with a notochord versus a worm with a neural net -- hardly a huge amount of genetic change.

Even if the differences were small, the probabilities of that many splits, given the mutation rate discussed above makes the issue still on the “very low end of probable” in my view.

Given the length of this post, with your concurrence, I will terminate here rather that continue.

[No, it isn't. It's based on empirical science, and rests upon a foundation of evidence.]

em·pir·i·cal adj.
a. Relying on or derived from observation or experiment: empirical results that supported the hypothesis.
b. Verifiable or provable by means of observation or experiment: empirical laws
I am unaware of any experiment even suggesting “evolution” beyond that involving bacteria or other very simple life forms.

By specifying only "experiment", you've left out the "observation" part (i.e., the gathering and testing of evidence by other than just "experimental" means). Don't do that.

But as to your being "unaware" of such, there are countless thousands of experiments and observations which support every aspect of evolutionary biology. What aspect of evolution would you like me to show you empirical evidence for? You'll find dozens of different examples in the links I've already posted on this thread.

Even those are apparently subject to some controversy as to whether they establish the concept or not…To wit, see below:

Oh, puh-lease... The "loss of information" goofiness is from Dembski, who's rather an idiot (details upon request), which is probably the reason he's a favorite of creationists. And Dembski is quite simply wrong. Furthermore, the evolution of *new* metabolic pathways has been observed for example, contrary to #20's implication that bacterial evolution has just been a matter of "losing" function. See for example this link which was *already* posted on this thread by me: http://www.jbc.org/cgi/reprint/279/40/41259.pdf.

There is no other “naturalistic” assumption other than “pure randomness” available as to how mutations can occur.

Sure there are.

Consequently, if mutations are not “purely random,” the implication is that something or someone is directing them.

Depends on how you define "directed", I suppose. For example, since the 1960's it has been known that environmental stresses on some organisms causes them to trigger shuffling of certain parts of their genomes. This is not "purely random", and it is "directed" in a sense, but not in the sense of sitting down and thinking, "hmm, how do I want to alter the genome today..."

As this is exactly the argument for intelligent design, I do not think that is what you intended to say.

The problem with the "argument for intelligent design" is that it's not terribly familiar with the fact that there are a wide range of possibilities between "purely random" and "intelligently designed". For a trivial example, natural selection is neither "purely random" (nor "purely *non*random), nor is it "intelligent" or "design", and yet it still brings "order from chaos".

If there is an absolute upper limit on the possible mutation rate, then the probability of more than one, simultaneously, becomes the mathematical product of that rate times itself.

What you keep missing is that there's no requirement that useful mutations happen "simultaneously".

Your "product of the rate" calculation assumes that the ONLY way to get two useful mutations together in a population is to have them originate *TOGETHER* in the very same individual. This is entirely incorrect.

For example, one of the mutations ("A") could occur in Green Crab #304,392,673 out of a 400,000,000 population in generation #3829, and then the other mutation ("B") could occur in Green Crab #286,293,869 out of another set of 400,000,000 crabs in generation #4728. As both mutations get passed around in the populations of generations #4729 and later, they then subsequently manifest together in many individuals. (And while almost a thousand generations later may seem like a long wait, keep in mind that even for a species that has only one generation a year, that thousand years for two "required" mutations to occur is still a tiny fraction of the time the Cambrian explosion took to happen -- it's 1/100,000th of the "quick" Cambrian timespan.)

You're calculating the odds of "Joe Smith wins the lottery two weeks in a row", when the actual case is "someone, somewhere, wins the lottery most every week, what are the odds that two of them will run into each other at some point?"

Similarly, the probability of multiple, simultaneous beneficial mutations,

*NOT* "simultaneous"...

which occur at a much smaller rate than mutations in general, also becomes a mathematical product of a small number making an extremely small probability for such.

*NOT* a "mathematical product...

Therefore, for all practical considerations, the accumulation of beneficial mutations in any organism is restricted to those that occur in successive generations.

Bingo!

This successive accumulation means that benign mutations that could later become beneficial must survive many, many generations in the face of genetic drift and “negative” selection.

You make genetic drift sound like a destructive process only. It's also likely to "boost" a neutral mutation in the population. And there's no "negative selection" on neutral mutations.

If there is an absolute lower limit on mutation rates, the theory of punctuated equilibrium becomes tenuous at best and, more likely, completely unsupportable.

Huh? Does not follow. Run realistic real-world numbers and you'll find it's not a problem. People *have* analyzed these things, you know.

For “beneficial” mutations to accumulate, the required population size, the required number of generations to reach the appropriate size after the appearance of each beneficial mutation,

Say what? "Reach the appropriate size"? What's this about expanding populations, and why?

the time between generations and the mutation rate act multiplicatively to decrease the probabilities of additional beneficial mutations within a limited time.

You're trying really hard to justify that "Multiply" key on your calculator again. And again, I think your reasoning is faulty, but in this case I'm not even sure *what* you're trying to say, so I can't tell what the correct mathematical model for it would really be.

Bacteria have the ability to produce large populations of individuals with very short time periods between generations and thus are potentially subject to many “beneficial” mutations.

Correct.

Consequently, by “Darwinism’s posit, bacteria would arguably increase their complexity to the point of becoming another organism.

< INcorrect. As I've already stressed, evolution does *not* "strive for greater complexity". Yes, it sometimes produces greater complexity in its exploration of fitness space, but contrary to popular impressions, evolution is *not* a walk up a "ladder" of "progress". Increased *fitness* is not synonymous with increased *complexity*.

Furthermore, "bacteria" covers an *enormous* range of types of organisms. The complaint that some are "still bacteria" -- when evolution has produced many different *kinds* of bacteria -- is as specious as seeing evidence that, for example, dinosaurs evolved into birds, but shrugging and saying, "so what, they're still vertebrates..."

Furthermore, today's bacteria are *far* more "complex" than the bacteria of, say, a billion years ago. The fact that they're "still bacteria" doesn't mean that they haven't changed tremendously.

However, to my limited knowledge, experiments with bacteria have never produced anything but other bacteria even after many thousands of generations.

And exactly how much evolution would one *expect* to achieve in only "many thousands of generations" of lab-sized populations?

And before you try, don't bother comparing numbers of bacteria versus numbers of animals and trying to draw any direct comparisons. Animals have two advantages which allow them to evolve faster with smaller combinations -- sexual reproduction, and the greater phenotypic plasticity that comes with multicellularism.

This would be the minimum time allowed for a beneficial mutation to appear and be propagated to the next generation and successor generations become large enough to sustain the mutation’s presence and, then, grow large enough for the next beneficial mutation’s appearance, etc.

Huh? Again with the "growing populations"... Why? Evolution works just fine in populations that remain constant from generation to generation.

[When speaking of "negative" selection pressure, yes, but "positive" selection pressure can't be "too great".]

If I understand your point, here, I must disagree. As I understand it, natural selection can only offer one “pressure:” that which enhances the survival and/or reproduction of an organism having a beneficial mutation. By “Darwinian” posits, if there is no benefit to a mutation, will it not, most likely, eventually change or disappear due to genetic drift?

"Negative" selection is that which differentially hinders survival or successful reproduction, like famines and droughts for example. "Positive" selection is that which gives certain individuals greater survival or reproductive success, like taller giraffes getting more food than their shorter-necked brethren for example, or most forms of sexual selection.

While you are correct in that "negative" selection can't be too "strong" or else it can wipe out a population before it can adapt, but "positive" selection pressure can be arbitrarily strong without endangering the population being selected.

If, in fact, the successor organism had an advantage by natural selection, it should completely supplant the “parent” organism, should it not? Otherwise, what is the advantage?

If still in direct "competition", yes it probably will supplant the "parent" species.

If there are a number of probabilities that must occur sequentially, then each must be multiplied by the other.

And again, you're incorrect in assuming that they *must* occur "sequentially" or "simultaneously". No, they do not. And not even "sequential" probabilities are correctly multiplied together unless they are *required* to occur *together* in sequence (e.g., mutations #456 and #457), which is *not* the case with mutations facilitating evolution. As previously explained, they can occur quite separated in time and location.

For example, the probability of heads of tails on a fair coin’s toss is 0.5. However, for the same coin, the probability of two heads (or tails) in succession is 0.5 times 0.5 or 0.25, i.e. a smaller number. Similarly, the probability of 7 heads (or tails) in succession is 1.52 x 10-5. Is the math sufficiently displayed for you, now?

Sure, but it's still wrong for the process you're trying to model.

Furthermore, I was complaining not only about the lack of equations in your post, but your lack of "real numbers" -- you were pulling some numbers from a hat, and just waving your hands and calling others "not too large", "not too small", etc. If you're going to do a mathematical argument, DO THE MATH. Use the best real-world numbers available for the parameters you're using in your model, or best estimates with appropriate error-bars.

You can't just say, "this is a factor and this is a factor, and I'll bet if we combine them, then WOW, that sounds pretty unlikely to me."

Consequently, it was not I who “just said so”… The rules of mathematics did.

Ahem. The "rules of mathematics" only give the correct answer if you build your model correctly and use correct values for your parameters.

Again, as an example, 0.1 times 0.1 equals 0.01. Sufficiently supported?

No, since I asked you to support your estimate that "0.1" was a realistic figure for the parameters you were using. And again, simple multiplaction is the wrong way to figure the odds you're trying to work out.

Again, as an example, 0.00001 times 0.00001 equals 0.0000000001. Does this example “cut it?”

Again, no.

Even if the differences were small, the probabilities of that many splits, given the mutation rate discussed above makes the issue still on the “very low end of probable” in my view.

Do the *full* model, especially with realistic estimates of population sizes, total mutation events, number of mutations necessary, etc., and you'll find it's entirely within the realm of reasonable possibility.

Check out the links later in the post you were replying to. They discuss the results of similar issues where *actual* rates of accumulated mutations in populations and *actual* numbers of mutations required to effect morphological change were measured, and found *well* within the rate required to support the evolutionary change which is seen in the fossil record, and which would be required to produce the diversity of modern life that we see.

So if your model's conclusions demonstrate that such results are impossible in the real world, well, it's your model that's wrong. And I've already identified a number of serious false assumptions in it (primarily your requirement that mutations happen *together*).