From: rusin@math.niu.edu (Dave Rusin) Subject: Re: Changing the Guard Date: 4 May 00 17:44:36 GMT Newsgroups: niu.tompaine Summary: [missing] In article , Gilbert Sebenste wrote: >the death >penalty is handled by God, which is currently running about 100% of >all who've lived on earth! Actually one consequence of exponential growth is that the fraction of all-who've-lived-on-earth who have died is _not_ nearly 100%, but rather stays at a fixed fraction decidedly below 100%. Let B(t) be the number of births in year t and D(t) the number of deaths. We might assume, for example, that everyone lives exactly N years (so that D(t) = B(t-N) ) and that B(t) = A e^(k t) for some A and k. Now, at time T, the number of people who have ever lived on earth is the integral of B(t) from t=-infinity to t=T, and likewise the number of deaths-so-far is the integral of D(t). OK, well just "do the math": you'll find that the total deaths so far is under these assumptions exactly e^(-N k) of the total number of people ever born. In this model k may be calculated by dividing the time it takes the population to double (about 25 years) into ln(2) (about 0.7), and N is the average lifespan (about 50 years, say), so the fraction is e^(-2 ln(2)) = 1/4. That is, this simple model would predict that only a quarter of the people who have ever lived have died so far, if I haven't slipped in the arithmetic. See? Aren't you glad we have a general education math requirement? :-) dave ============================================================================== From: morphis@niuhep.physics.niu.edu Subject: Re: Changing the Guard Date: 4 May 00 20:17:02 GMT Newsgroups: niu.tompaine rusin@math.niu.edu (Dave Rusin) writes: [quote of previous message deleted --djr] Your arithmatic looks good to me, your assumptions however ... Got me to thinking so ... gopher://gopher.undp.org/00/ungophers/popin/wdtrends/histor Year Population (in billions) Source (see bottom) 0 0.30 Durand 1000 0.31 Durand 1250 0.40 Durand 1500 0.50 Durand 1750 0.79 D & C 1800 0.98 D & C 1850 1.26 D & C 1900 1.65 D & C 1910 1.75 Interp. 1920 1.86 WPP63 1930 2.07 WPP63 1940 2.30 WPP63 1950 2.52 WPP94 1960 3.02 WPP94 1970 3.70 WPP94 1980 4.45 WPP94 1990 5.30 WPP94 1994 5.63 WPP94 2000 6.23 WPP94 projected stablization to take place in: 2200 at 11.6 billion LR This data is out of date but it was the quickest I could lay my hands on. Gross simplification: total pop years the average population (in billions) 10,000 bce to 1 bce .10 .10 * 10000/50 = 20 1 ce - 1000 ce .30 .30 * 1000/50 = 6 1000 - 1800 ce .6 .6 * 800/50 = 10 1800 - 1930 1.5 = 4 1930 - 1960 2.4 = 1 1960 - 2000 4.5 = 4 grand tot 45 bill 6.3/45 ~= .14 so around 86% of all people have died. Interesting. Durand: J.D. Durand, 1974. Historical Estimates of World Population: An Evaluation (University of Pennsylvania, Population Studies Center, Philadelphia), mimeo. D & C: United Nations, 1973. The Determinants and Consequences of Population Trends, Vol. 1 (United Nations, New York). WPP63: United Nations, 1966. World Population Prospects as Assessed in 1963 (United Nations, New York). WPP94: United Nations, 1993. World Population Prospects: The 1994 Revision (United Nations, New York) LR: United Nations, 1992. Long-range World Population Projection: Two Centuries of Population Growth, 1950-2150 (United Nations, New York). Interp: Estimate interpolated from adjacent population estimates. ============================================================================== From: rusin@math.niu.edu (Dave Rusin) Subject: Re: Changing the Guard Date: 5 May 00 07:30:48 GMT Newsgroups: niu.tompaine Gee, I didn't think anyone would think to compare this to real data! >Your arithmatic looks good to me, your assumptions however ... It's certainly true that the total number of souls to have graced the planet is the integral of B(t) up to t=T, and that of those the number who have passed on is the integral of D(t). The premise D(t) = B(t-N) may well be valid, but not for the reasons I gave: we know there's not a single fixed lifespan. The premise B(t) = A e^(k t) is certainly _false_ for global human population, according to data. But the model ain't half bad, nonetheless. The simplifying assumptions led me to conclude that the fraction to have died so far is always e^(- k N). Well, those same assumptions can also be shown to lead to the conclusion that the population is greater when a person dies than when they were born, by exactly a factor of e^(k N). In other words, those assumptions lead to what I'll call my Main Theorem: (# still alive at any time) / (# to have lived up to that time) equals (# alive when a typical person is born) / (# alive when same person dies) For example, I assumed the population doubles twice in a (50-year) life, meaning the last fraction is 1/4, which is the value I obtained for the first fraction, too. The data presented seem to indicate populations have grown much more slowly, in general, and we know lifespans have been shorter. These facts conspire to make the world population hardly larger when the typical Roman centurion died than when he was born. So in the Main Theorem, the second fraction is just less than 1, and then the first fraction is then also just less than 1. That must have been the model Gilbert S. had in mind in the first place. While the rigid assumptions I used to prove the Main Theorem are not really valid across history, I think the theorem is (at least approximately) true under much weaker assumptions. After a couple of beers, it's not hard to be swayed just by some verbal gymnastics that the Theorem is true "in general" (whatever that means). dave