I would like to thank yossarian22c for challenging me to revisit models of formal systems in which: 1+1>2.
For many years I was stuck at:
quote:I can't come up with a precise formal system that would reflect the behavior of clouds and families. All I have been able to come up with in the years since I started thinking about it is specific models that do not have the aesthetic beauty that I enjoy so much when I think of Number theory, Euclidian geometry or Spherical geometry.
The conjunction with hobsen's population growth thread helped me clarify my thoughts enough that I would like to try again.
So let's try to define a set of formal languages that expresses the growth rate of a population with a single species and two sexes.
A valid statement in this language would be of the form: (a+b)^n=c Where a, b, c and n are natural numbers. An interpretation of this language would be that: starting with "a" males and "b" females we would have "c" individuals after "n" generations. As a convention a+b=c will be shorthand for (a+b)^1=c.
Here are some axioms that I think would be good to start with:
for any a and b: (a+b)^0=c where c is the sum of the natural numbers a and b. In English: In the same generation, the number of individuals is the sum of the males and the females.
for any b: (0+b)^1=0 In English: The next generation would have no individuals if there are no males.
for any a>0: (a+b)^1=c where c=floor(b*r), r is a real number that approximates the average number of offspring per female, "*" means multiplication and floor mean rounded down. In English, as long as there is at least one male the number of individuals in the next generation is given by the formula above.
Since "c" does not distinguish between males and females we need some way to derive the number of males and females in the next generation. If "p" is the percentage of fertile females and q is the percentage of fertile males born in each generation we can define a rule of inference as follows:
If A=floor(c*q), B=floor(c*p)and s=n+m then given that (a+b)^n=c, (A+B)^m=C we can deduce that (a+b)^s=C
Note that what I defined here is not a single formal language but a set of languages that depend on three implicit numbers: r: number of offspring per female p: percentage of fertile females q: percentage of fertile males
If it bothers you that these numbers are not explicitly mentioned in the language, think about the fact that when we say 2+2=4 we are implicitly using decimal notation even though the number 10 is not explicitly mentioned. If we were to use base 3 implicitly instead the same statement would be expressed as 2+2=11 and as long as the context is clear there is no problem or contradiction between those statements.
I'll read through the whole thing in more detail when I have time but let me bring up a couple points. First you have defined things in terms of two numbers because q=1-p (assuming no one is trans gendered). Secondly if you are interesting in a model for population growth I would suggest reading up about the Fibonacci sequence. He developed it while studying the population growth of rabbits, a slightly different phenomenon but related so their may be something you find interesting there.
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q=1-p is a special case. My first draft used q=1-p, but if p is the percentage of fertile females then q<1-p is also possible. Some interesting models that would be excluded by assuming q=1-p are:
Worker ants are infertile females so q<<1-p where "<<" means much smaller.
High infant mortality rates (individuals die before they get a chance to reproduce).
Low fertility rates.
If it were not for the rounding issues, the Fibonacci sequence could be derived by using r=(sqrt(5)-1)/2. In English that would be "if the number of offspring per female was the golden ratio". English may sound more poetic but is less precise.
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