im still a little fuzzy on this math thing... i just want to think about this for a minute...
The accepted age of the earth is about 4.5 billion years?
The accepted age of life on earth places abiogenesis at about 2 billion years ago.
So... there are about 9 unique million species living on the earth.
With a linear model, you would have to find a new species approximately every 220 years in order to have the variety of life we see today.
With a binary model, using a 2^23 (slightly less than 9 million) as a base, you would have to have the number of species double every 86 million years. This is an average of 730,000 new species every 86 million year period or 1 every 115 years...
The simplest of all organisms on the earth right now has about 2000 genes to the most complex having about 25,000 genes. For the sake of math argument, lets assume the average number of genes per species is about 10,000, which we know is on the low side. How many of them must change for a new species to exist? 1, 2, 100?
Logic tells me there must be more positive transformations than there are years available, not including all the failed ones.... The rate of change is too slow for this to be solely responsible for the amount of species here today.
I'm not saying that your mathematics is the right approach in the first place, but I'm going to point out a MAJOR flaw in your approach. According to your assumption that the number of species has to double every 86 million years - that would mean that each individual species would have to evolve into two separate species over 86 million years. Your error is in doing a calculation assuming that only one species on Earth evolves at a time. i.e. okay, new form of bird, you have 115 years to evolve. Okay, the bird is done, now it's your turn little snail.
You're not honestly saying that a species couldn't possibly diverge into two different species in the course of 86 million years, are you?! And, to satisfy your computational method, that's all that would have to occur. However, according to your calculations, on average, humans should be able to observe speciation approximately once every 115 years. Thanks for answering the question why we don't see speciation occur every 45 minutes in a lab. Nonetheless, somewhere in this thread, I linked to a list of a couple dozen examples of speciation that HAVE been observed. Far more than you would predict. Now, since you're one of the more reasonable posters in this forum, I hope you can at least acknowledge the flaw in your mathematics.
Also, someone mentioned dogs in the thread. It is *not* true that the pool of genes for modern dogs is the same as the pool of genes of their canid ancestors (wolves.) There are genetic differences that are a result of mutations. And, those genetic differences give rise to a larger variety of sizes, snout length, etc., than exist in wolves. I hunted for an article that explains it well enough on a level that most people can follow.
http://www.karelianbeardog.us/kbd_science.html
Gibsons can probably chime in with how accurate this article is (if he has time.)