This is something I debated a long time whether or not to actually write anything about, mainly because the subject matter is not something that interests me, but the situation and story behind it is certainly something worth thinking about. Neverthless, this will likely be far too many words on a matter that deserved much less.
Thinking about population models, a relatively basic model would be to assume that the rate of growth of a population is proportionate to the size of the population. That is, at a given time t, for some characteristic constant k,
That constant encapsulates a lot of information about the population, such as gestation rates, reproduction rates, etc.
Skipping the mechanics of actually solving it, you get the following,
Where P0 is the population size at time t = 0.
Now, this solution allows us to answer an interesting question, given a population – how old is that population? How long must it have been in existence to reach the population size it is at now? The model is predictive forwards and backwards, and we can project when the population was at a given size.
We assume that the population had to start at some small value, say P = 1 for an asexually reproducing organism, P = 2 for a sexually reproducing organism, and P = 3 for a hedonistic, liberal organism. We’ll assume sexual reproduction. Then we simply solve for the time at which the population would’ve been that size.
We can then say that the age of the population is whatever time it is currently, minus the time given above. Which is all very interesting.
“Well, Fox,” I hear you say, “that’s all well and good, but where’s the proverbial beef?”
A movie was recently released, Expelled: No Intelligence Allowed, purportedly about the suppression of Intelligent Design advocates in the scientific field. There are also Nazis. All this has been dealt with before.
Keeping my omniscient eye on the goings on, I noticed in the forums on the IMDB, someone made the claim that math disproves evolution. On further inspection, this person claimed to be a former algebra teacher. When teaching population models like the one above, he claimed that he would take his class to the library and let them pick sources to find two world population data points, from which they could solve for the ‘age’ of the human population through the above exercise. And that in all the years of doing this, no one had ever gotten more than that humans had been around for 25,000 years. And this was by a student deliberately searching for data to give the greatest age possible. Picking their own sources! Clearly, the scientifically established age of Homo Sapiens of 250,000 years must be incorrect. Evolution must be wrong – we’ve proved it using math!
…
It’s hard to know where to begin with this. So I’ll start with the math, since that’s what I’m good at.
To begin with, he is right. If you take that model, and two data points, you will find that humans have been around a few thousand years, at best.
Which is entirely meaningless, since the model is wrong.
The flaw is a simple one – the simple exponential is, in this case, wrong. At the very least, in complete. Think about all the assumptions that go into into the simple model I described. We assume that the rate of change of the population depends only on the characteristic constant, and the size of the population (linearly) at a given time. Think about everything this disregards. For example, environment. A given environment may only have the resources to sustain a population of a given size. There’s no regard for outside influences. War, famine, disease. There’s no regard for culture. Some countries right now have a negative rate of population change, as their population is, for whatever reason, just not reproducing fast enough. And all these factors can change over time.
Factors such as these can produce slight changes in the mathematical model that produce drastic changes in behavior. For example, if the reproduction of the population depends on interactions between individuals, for example, sex, the population might better be modeled as
Which factors in the chance of a given individual bumping in to any other individual. A simple change to the model, which perhaps more accurately reflects the behavior of a sexually reproducing organism. However, this particular change causes P(t) to effectively become infinite at a certain point in time. This is clearly not realistic. It can be balanced though, giving a finite population at all times, by introducing a death rate.
And again, this more accurately models certain populations, relating what’s known as a ‘carrying capacity’, or how large a population the environment is able to support.
The point is that each of these small modifications to the model necessary to realistically represent the behavior of a given population can cause drastic changes in the behavior of the model. And while any one of these models might be accurate and predictive near the range of data you fit it to, over the long term they can all produce radically different results. And when you consider the extent to which the parameters and behavior of a population can change over time, especially on the time scales we’re dealing with, no one model will be especially accurate over those extended periods.
When you’re dealing with something as complex as human interaction with each other and the environment, using such a simplistic model is just -wrong-, and any conclusions you derive from it, all the more so.
Which brings me to the next point.
That, as a -math teacher-, the poster should have known this. At the very least, he should’ve been exposed to different models aside from the exponential model, and realized hey, maybe these other models better represent mankind. He should’ve known this, but evidently didn’t, and as such, who knows how many kids came away from his class with a poor and damaged understanding of not only the history of mankind, but of math and population models in general?
The arrogance and the ignorance in the post was just astounding.
And that’s what gets me most about this, I think. No matter what you think, know, or believe, the fact is that this man’s logic was completely -wrong-, and he forced his inanity on who knows how many classes of kids. In matters of math and science, there can be right and wrong, true and false, and this is most definitely one of those cases.
Great post! It is quite grating that someone (clearly ignorant of population modeling methods) would put so much stock in a simple model.
A take home exam I once did had a question about modeling the human population and making a prediction of what it would be in the year 2050. I can imagine the prof would have laughed had I used an exponential model.
Comment by Ben — April 23, 2008 @ 1:16 pm
[...] interesting. However, others have risen to the task over the past few days. Check out Foxy’s obliteration of some very bad anti-evolution crank-math, and Jason Rosenhouse has his entertaining review of [...]
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