If you read prof Bartlett's paper which he published on this, and watch say the Kahn academy lecture on exponential functions, you will see that it doesnt matter if the rate of population change changes [it makes calculation of the doubling time harder] so long as the change in x divided by the change in y is proportial to x , or as wiki puts it,

"The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value."

So, as long as the populion changes at a rate proportional to its current value it is exponential growth, even if that rate of change is itself dynamic.

or... from this site, http://members.optusnet.com.au/exponentialist/Compound_Vs_Exponential.htm

Is the human trait of not being able to internalise the exponential function a result of brain structure or a limitiation of our brain to quickly use base 10 maths, or for some other reason?

When I was born there were a little over 3 billion people on the planet, now there are 7 billion.

there have been about 4 billion people born in my lifetime [allowing for deaths], how long do you have to go back for the previous 4 billion births?

Hint. At the time of christ the world population is estimated at 150 million, less than half that of the USA today, about 7 times the population of Australia today. The 1 mile square slum in which 'slum dog millionaire' was set is home to one million people.

1,000,000 years is the answer. I million years ago until now there have been 4 billion births. I have read that 1 in 10 humans [homo erectus] to ever be born are still alive.

The inability of fire-fighters to internalise the exponential function is the major cause of fire fighter deaths [according to an Australian coronor investigating the Ash Wednesday fires].

I am not sure why I have picked up the stick to be the next to flog this poor dead carcase.

When I was in engineering school, we tried to "avoid" building circuits that were based on exponential formula and equations that let that "little puff of smoke" out - that is built into each and every electrical component. Some designs could be taken to "the hairy edge" (read, produce a LOT of heat when running) and still function... for a while. Those were the designs that didn't "fit" a pure exponential function. The "pure" exponential models just went BOOM quickly and were great fun!

I guess the point really is, pure mathematics can-not be nicely fit into the physical world. Don't get me wrong, without mathematics, we would not be able to do all the things that we do. (like type virtual letters on a keyboard stored only as ones and zeros)

The problem is, things look broke. Can it be denied that the world population is accelerating? If that "acceleration" has slowed, what does that mean? To use "pure" theories may not "exactly" fit, but they can enlighten us by pointing the direction that we are heading.

To argue "rate of change" proves or disproves exponential function is chasing windmills. Maybe, it is a little "fear" of pure exponential function that has some questioning Chris' characterizations. After all, if population growth is a "pure" exponential function, when will we hear or feel the "boom?"

Personally, I would like to hope the wave will appear to look more like the waveform of sound vibration from the striking of a bell where the earth is the bell and oil is the hammer that strikes and causes the crescendo spike on the graph (and overshoot) to then ring out into a more sustainable "ringing" for a period of time that is a function of the quality of the bell that can be "heard" for many millenia to come. The hard thing to swallow, is that the overshoot means that there will be a corresponding rapid decline before the steady state can be reached. We are not talking merely decibels of sound, we are talking humans.

It is clear that the bell has been struck, the real question is where are we along the spiking of the graph? Have we reached the peak? Are we near the peak? Or are we so near the pinnacle that we are in the overshoot period and can sense the decline that will surely come.

"120,000 million people are added to the world's population each year,"

Jeez...... I'm glad THAT's not occurring now...!

Hi Damn, does this address your point?

If you read prof Bartlett's paper which he published on this, and watch say the Kahn academy lecture on exponential functions, you will see that it doesnt matter if the rate of population change changes [it makes calculation of the doubling time harder] so long as the change in x divided by the change in y is proportial to x , or as wiki puts it,

"The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value."

So, as long as the populion changes at a rate proportional to its current value it is exponential growth, even if that rate of change is itself dynamic.

or... from this site, http://members.optusnet.com.au/exponentialist/Compound_Vs_Exponential.htm

Term traditionally used for positive population growth at a constant rate per period (graphically depicted by the exponential curve). The defining feature of exponential growth is compound interest. Thus a population which experiences positive population growth at variable rates of compound interest can also be said to be growing exponentially. For negative population growth referExponential Shrinkage.Does anyone have an answer to this question?

Is the human trait of not being able to internalise the exponential function a result of brain structure or a limitiation of our brain to quickly use base 10 maths, or for some other reason?

I am 50 years old.

When I was born there were a little over 3 billion people on the planet, now there are 7 billion.

there have been about 4 billion people born in my lifetime [allowing for deaths], how long do you have to go back for the previous 4 billion births?

Hint. At the time of christ the world population is estimated at 150 million, less than half that of the USA today, about 7 times the population of Australia today. The 1 mile square slum in which 'slum dog millionaire' was set is home to one million people.

1,000,000 years is the answer. I million years ago until now there have been 4 billion births. I have read that 1 in 10 humans [homo erectus] to ever be born are still alive.

The inability of fire-fighters to internalise the exponential function is the major cause of fire fighter deaths [according to an Australian coronor investigating the Ash Wednesday fires].

Well,

I am not sure why I have picked up the stick to be the next to flog this poor dead carcase.

When I was in engineering school, we tried to "avoid" building circuits that were based on exponential formula and equations that let that "little puff of smoke" out - that is built into each and every electrical component. Some designs could be taken to "the hairy edge" (read, produce a LOT of heat when running) and still function... for a while. Those were the designs that didn't "fit" a pure exponential function. The "pure" exponential models just went BOOM quickly and were great fun!

I guess the point really is, pure mathematics can-not be nicely fit into the physical world. Don't get me wrong, without mathematics, we would not be able to do all the things that we do. (like type virtual letters on a keyboard stored only as ones and zeros)

The problem is, things look broke. Can it be denied that the world population is accelerating? If that "acceleration" has slowed, what does that mean? To use "pure" theories may not "exactly" fit, but they can enlighten us by pointing the direction that we are heading.

To argue "rate of change" proves or disproves exponential function is chasing windmills. Maybe, it is a little "fear" of pure exponential function that has some questioning Chris' characterizations. After all, if population growth is a "pure" exponential function, when will we hear or feel the "boom?"

Personally, I would like to hope the wave will appear to look more like the waveform of sound vibration from the striking of a bell where the earth is the bell and oil is the hammer that strikes and causes the crescendo spike on the graph (and overshoot) to then ring out into a more sustainable "ringing" for a period of time that is a function of the quality of the bell that can be "heard" for many millenia to come. The hard thing to swallow, is that the overshoot means that there will be a corresponding rapid decline before the steady state can be reached. We are not talking merely decibels of sound, we are talking humans.

It is clear that the bell has been struck, the real question is where are we along the spiking of the graph? Have we reached the peak? Are we near the peak? Or are we so near the pinnacle that we are in the overshoot period and can sense the decline that will surely come.

That's the $64,000 question.

C.