February 27, 22012
to be posted on nobabies.net
Michael P. H. Strumph
Centre for Integrative Biology and Bioinformatics
Imperial College London
London SM7 2AZ
Dear Michael Strumph:
I was mightily impressed with your paper (Michael P. H. Stumph and Mason A. Porter Critical Truths about Power Laws SCIENCE vol. 335 no. 6059 February 10, 2012 page 665) about power laws.
I almost missed it. I thought, “Power laws. My interest is in fertility. This has nothing for me.” But some providential thought said, “This is basic. And you are always doing the best when you go to basics.” Lucky old me.
As you point out, a power law is a relationship between two variables, such as the metabolic rate of a kind of animal and its size, that holds constant over a long range, say more than one order of magnitude, a graph of the observations will describe either a straight line or a curve described by some exponent other than one. Obviously “size” could be length, surface area or weight and in each case the exponent would be different but a power law would still hold. Then you drop in your conversation stopper. Although power laws are adduced in many situations, by and large they are not supported by evidence or not supported by mechanism. Oops.
But it’s intuitively appealing that power laws should hold. As a child I had trouble understanding that scaling did not work. I mean if I looked at a diagram of how a car worked, it all seemed logical. And I had no thought that a car built on the same scale as the picture might not work at all. Nor might a car work that was a thousand feet long using the identical design.
Then one day my father, in a nostalgic mood, described seeing an oversized model of a typewriter in action, built by some typewriter factory. The folly was much higher than a man is tall, and Daddy described with enthusiasm watching the great keys swing up very slowly, one by one, to strike the enormous platen. I asked why they moved so slowly, and without hesitation he remarked that if they had come up in the time it takes an ordinary key to strike, the machine would have been demolished on the first stroke. That puzzled me immensely.
Then as a teenager I was working one summer helping build an astronomical radio-telescope. It was an array – actually two arrays – of wires supported by tall two-by fours. (Do you have two by fours in England? It’s a board that measures roughly two inches by four inches in cross section.) The researchers had bought an extra one, assume that we ham fisted helpers would probably break one, and when it was evidently no longer needed they told me to throw it over into some nearby brush to get rid of it.
In my youthful vigor, my own physical strength seemed to me to be the most important issue in the world so instead of setting it by I literally threw it. It was seventeen feet long but I got a pretty good toss on it. When it came down and hit it broke. I was stunned. A two by four was iconic for “that which cannot be broken by human strength.” But it broke. Sure, it was only scaled up in length, but I was amazed it was so fragile. Intuitively things ought to scale, but they don’t.
I would look at an insect and marvel that it was not just a small sheep. The design was completely different. Why? A sheep is a good design.
But if mechanisms do not scale, then surely power laws should hold. There should be enormous numbers of them. But you have demolished the notion.
So just for fun, I wonder if you would consider the biggest, scariest power law of them all. Call it the Law of Malthus. I just made that up. Let’s say you take 1,000 healthy young mice of mixed sex and keep them safe, well fed, comfortable, uncrowded, clean, well watered in cages with optimal lighting, temperature and interesting things to play with. Assure that there is random mating. I don’t know exactly what a mouse generation is, but it has to be a month or two. Let’s say you can get ten generations in a year. So to begin with you should get lots of little mice. Keep track of their fertility. If their numbers become too high, you eliminate mice at random when you take out the old generation. Each generation you let the size of the new generation increase by 7%. Add cages as needed to keep the environment optimal. So in a year the population should have about doubled. After ten generations you have about a million mice in there. If so, you will conclude that a population of mice will grow exponentially over three orders of magnitude. There is your power law as slick as can be. Carry this out long enough and eventually it will get scary if you are disposed to be at all scared of big numbers.
Thomas Malthus was the first one who said population will increase exponentially, at any size, for ever and ever amen, halleluiah. Well until the environmental limit is reached anyway.
The most eloquent apologist for the Law of Malthus was I reckon a man named Nicholson, who raised a lot of blowflies off in Australia. At one point I read a list of over a dozen reasons that we must accept the law. Glancing over them, they appeared to fall into two categories: “It’s obvious” and “Everybody believes it.” You see a teensy weensy little problem I think: no evidence.
And yet evidence could be easily produced by the method I just described. Sure it would take ten years. But think how important it is. Think about how The Law is essential to everybody’s thinking. Surely it wouldn’t be as expensive as oh, say, a manned mission to Mars, or finding the Higgs bozon, or sorting out “dark energy.” And it would be a lot more important to people. It would just be a few years and a few million dollars.
So you could do it. Just make sure that the random mating rule is never violated.
There’s only a small problem. It wouldn’t work. Within a year you would find fertility falling until it went below replacement. Shortly thereafter the absolute numbers would decline until they were in two digits; the population would probably go extinct.
Yes, that’s what I said. Population growth does not follow a power law. Oh indeed you can get millions of mice, but you need to change the structure. Keep them in enough small populations and indeed you can fill the galaxy. But the single random mating population, great for a couple dozen mice, is no more viable for thousands than say making a sheep by scaling up the design of an ant. It won’t work.
As you no doubt ascertain, I haven’t done the experiment. But I do have evidence about what to expect. Here is a link to it:
Along with the evidence there is an explanation of why nature should have done us such a dastardly deed. I’m working on the mechanism, but that will take at least another year, assuming I ever get started on it. I can let you know what I have already worked out.
Of course everybody knows that the developed world has had fertility below replacement for decades. And I know what you’re thinking: “Doom and gloom. Doom and gloom. The developed world dies out taking technology with it. Sure, sure. The world is left with ten billion people and the technology to feed only two billion. Agriculture collapses. Production and distribution go down the siphon. There’s nothing to eat but bush meat and then long pork.”
And I say, “If people can’t take a joke, who would want to keep them around
This letter is just for fun. But I thought you might like to know that there is another power law that is nothing but smoke and mirrors.
Let me know what you think.
M. Linton Herbert MD
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