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Mathematics is really cool, especially in the way that it is applied to the natural world. Math is the language of physics, chemistry, biology, and economics.

Math, as taught in most high schools, is taught with little regard for the actual applications for which it is designed. And, admittedly, some mathematical concepts are taught almost exclusively for their theoretical uses instead of their practical applications.

In nature, we can find a place for most everything in mathematics. If you took math in high school to a reasonable level, you’ve probably heard these terms before: addition, multiplication, exponents (powers), roots, sines and cosines, and logarithms. While you might not have understood them, scientists use every bit of math in numerous applications.

In science, mathematics is our greatest tool.

One of the most dire mathematical functions we stare down is the exponential. The exponential is written as e^x or exp x. Its graph looks like this:

The exponential function appears in quite a few places in nature. Most often it presents itself in the solutions of differential equations: equations in which the rate of change of a time-varying quantity depends on the quantity itself. For example, money grows exponentially. If you have a lot of money you can make money quickly, and if you don’t have a lot of money, you can’t.

A particularly scary example of exponential growth is world population. Here are a few estimates of the population of the world at various points through the past couple centuries.

Year	World Population
1750	629,000,000
1800	813,000,000
1850	1,128,000,000
1900	1,550,000,000
1950	2,400,000,000
2000	6,070,581,000
now	6,653,000,000

(Source for values up to and including 1950: United States Census Bureau Historical Estimates of World Population. Summary values used combining multiple models.)

Just looking at the data table, you probably won’t recognize this as an exponential. That is, until you see its graph:

There are a LOT of factors that go into estimating population. Global pandemics, available food supply, and climate are some biggies.

We can’t easily figure out all the factors that went into making this chart. But, we know that this data already happened. There will still be global pandemics, a similar food supply, and so it’s not out of the question to find a graph that closely matches the above graph. This ends up being not so hard, and with a few clicks you can even do it in Excel!

Here’s a graph of the actual population between 1750 and 1950 (thin blue line), and the exponential curve that best fits this data (thick black line).

Pretty close. But this assumes that the fundamental conditions governing population growth don’t change.

Let’s pick a nice date in the middle: 1850. What was medical science like in 1850? Cocaine and opium were used for medicinal purposes. The germ theory of disease was not yet widely accepted. Bloodletting was somewhat common as a way to “cure” disease. The basic concepts of physiology had been established, but medical care didn’t resemble the medical care available today.

Today we have safer workplaces and less child labor. Childbirth, while not perfect, is by-and-large a pretty safe process. In 1850, about one birth in a hundred resulted in the death of the mother — now that number is about one birth in ten-thousand. Finally, we have much better hygiene — soap is widely used and widely available.

We have substantially extended the lifespan of humans. Naturally, this will cause the population to go up. As there are more people around, there are more fertile people around, and so the population will increase at a faster rate.

Let’s try to fit the exponential curve all the way from 1750 to today.

The graph agrees with us. Our population growth is even surpassing the rate of recent population growth. Why is this frightening? As a society, we’re doing great!

Yeah, for now. But consider three things:

The food supply

The food supply is a function of arable land. More people means less land for farming, and more resources used to support people that could have been used for farming. While food supply and population aren’t automatically inversely proportional, there’s reason to believe that we are overfarming the land. This means that we can at best hope for the same amount of food per acre. If global warming continues as predicted, low-lying land rich with moisture will flood causing good farmland to become good swampland.

Poverty and pandemics

As the rich get richer the poor get a LOT poorer. Poverty is rampant, especially in the “developing” world. Because of poverty and AIDS, the average lifespan in Botswana has decreased from 61 years in 1987 to 38 years in 2003. As we continue to develop as a society, the divide in wealth will prove catastrophic for the “have-nots”. Many other nations, especially in sub-Saharan Africa, will follow in Botswana’s sad footsteps.

Peak oil

It’s very likely that we’ve reached “peak oil”, the point at which the petroleum we’re pulling out of the ground has leveled off and is starting to decrease. This will be tragic not just because of the disruption of the American lifestyle, but that the increases in food production per acre of farmland has been aided by petroleum-based fertilizers. Food production decreases as a result of the peak oil phenomenon.

We’re reaching a pivotal time, in that we’re straining the boundaries of what the earth can provide for us. The citizens and governments of the world need to band together to reduce fossil fuel usage, decrease the effects of global warming, and reduce overpopulation to extend the exponential and slow our population growth sanely. If we don’t, we’re going to have it done for us and it’s not going to be pretty.

Tags: chicken little, exponential, growth, math, mathematics, peak oil, population

This entry was posted on Wednesday, February 27th, 2008 at 9:18 pm and is filed under Physics Is Phun. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.