Historical growth looks exponential to me.

# population growth not exponential

Maybe we're at Peak Population!

Everyone on this site that are new need to grasp the message in these films. This was the very beginning of my journey of understanding to the importance of the *Exponential Function ...*

*~ VF ~*

Hi VF,

Do you feel I've failed to grasp the message or failed to appreciate the exponential function?

You might want to go back and read what I actually posted ;-)

MarkM -

yes historically population growth does look exponential. Largely because for much of the timeline you've shown it was exponential. What I said is that in recent times (last few decades) the rate of growth has slowed and can therefore no longer be described as exponential.

A key feature of exponential growth is that a small growth rate can quickly give rise to a massive change in numbers. But it's equally true that a small but continuous change in that rate can give a correspondingly large change in eventual outcome. The first part of a Hubbert curve looks exponential (I know it's not in case anyone feels like jumping on that point) but as the rate of change reduces the curve undergoes inflexion and soon peaks out. Whether population peaks, continues to grow or plateaus is of course an open question.

It never ceases to amaze me how quick people are, here and elsewhere, to assume an agenda behind what is said rather than responding to what has actually been said!

1. The rate of increase in population is now slowing which would not occur with true exponential growth, the rate of which is approximately constant.

2. Current population growth is being driven entirely by decreasing death rates in developing nations and not by increasing birth rates. This is important because although growth is baked into the cake it also means population can be projected to plateau around the 9bn mark.

Would you please provide a source for the first point? Also, with regards to your second point (which seems to contradict the first one) what difference does it make that "Current population growth is being driven entirely by decreasing death rates in developing nations and not by increasing birth rates" if the net result is a population increase?

Maybe we're at Peak Population!

INDEED...... pretty close. IF you look at the chart posted, you will see the curve going off the vertical trajectory, BUT, population is still growing at 1.6% (?), and so it still is exponential. It stops being exponential when growth rate reaches 0%. At 1%, we still have a 70 year doubling time, at 0.5% we still have a 140 year doubling time, at 0.25% we STILL have exponential growth..... it's just that the curve changes shape.

When 0% is reached, THEN we have Peak Population, and no exponential growth. But watch out...... at that stage, we may well have reached exponential shrinkage!!!

Mike

Hi VF,

Do you feel I've failed to grasp the message or failed to appreciate the exponential function?

You might want to go back and read what I actually posted ;-)

Piquod,

I fear you've failed to grasp the message, but I want to genuinely find a middle ground to work with you on this. Please bear in mind that Dr Martenson has had the Crash Course completed for over a year, with all possible chances to change information within it with supporting arguement from 'Working Joe's' to Mathematical Scientists, based on close to 1.5 million viewings.

Let me see if I can help with offering you World Population Estimates, that are about as precise as can be extrapolated.

1810 one Billion.

1930 two Billion.

1960 three Billion.

1975 four Billion.

1990 five Billion.

1999 six Billion.

2110 A whisker from seven Billion.

2050 Between 9.1 and 9.3 Billion.

Are we to nitpick over fripperies in explaining exponential growth in layman's terms to a couple of generations of mathematical Luddite, or would you say that, with global oil production in the process of a decline between 4.5 and 6.7% (IEA Figures), and with human population growth figures showing that oil and population have been marching ever upward in lock step with each other since 1930, that we've reached a peak in human population and are about to lose a few billion to famine?

You do understand that this is the actual inferred statement?

*~ VF ~*

VF, I love the way you put things....very sober swallow. I've read it, I grasp it.

And yet I read you again and ouch, it's painful. Do I have enough food?

When is the tipping point?

Hi Set

Global population growth rates and numbers are well covered in the Wiki article;

http://en.wikipedia.org/wiki/World_population

Note the comments about the recent decline in rates of growth and the population curves for different regions and the whole World.

The second point appears to contradict the first only if you focus on growth rather than *exponential* growth. I don't deny growth is still happening, what I'm saying is this can no longer be described as exponential because the rate is not constant but declining. The difference caused by the driver of growth being predominantly declining death rates comes in the extrapolation - if the trend continues then it is reasonable to extrapolate human population to a plateau as the death rate will inevitable level off. If growth were being driven primarily by birth rates at higher than required for replacement then no such plateau could be extrapolated. These outcomes diverge dramatically over periods of decades or longer and therefore have very different implications.

Hi VF

*Please bear in mind that Dr Martenson has had the Crash Course completed for over a year, with all possible chances to change information within it with supporting arguement from 'Working Joe's' to Mathematical Scientists, based on close to 1.5 million viewings.*

The crash course should be as open to criticism now as when it was first posted. Just as scientific theories are as robust as the latest test regardless of how many tests have previously been carried out. If we stop questioning something because it has survived challenge up until now then we might as well stop thinking for ourselves altogether. Sorry but as a scientist my training pretty much forces me down this line!

I do agree that the message about human population is important and that the human footprint is huge and getting worse. Maybe I am nitpicking the detail but CM made a point in the first chapter to explain the importance of the difference between fact, opinion and belief. It is the facts that provide the edifice for the whole argument and belief construct that follows. And it is a fact that population growth is no longer exponential. You may believe this doesn't make much difference but, as I've responded to Set, you can get widely divergent results depending on it. For one thing it is not even open to question that continued exponential growth is unsustainable - as someone once said only an economist would believe such a thing! But it is very much open to opinion/belief as to whether a population leveling off at 9bn is sustainable - we won't know until or if we get there.

The belief that the facts need to be dumbed down or approximated is again a complete anathema to me as a scientist. It's not difficult to present growth extrapolations based on current rate changes and such extrapolations are not hard to grasp for anyone capable of reading a graph (see Wiki entry posted previously). Without such facts being presented accurately CM is in danger of not only misrepresenting the situation but also of losing credibility in the wider community, or at least the more scientifically minded part of it. Which would be a shame because, as I've said, I completely agree that the message about our impact on the planet and its resources is extremely important.

All of my work is open to critique at all times.

Without such facts being presented accurately CM is in danger of not only misrepresenting the situation but also of losing credibility in the wider community, or at least the more scientifically minded part of it.

As a scientist, I think precision is important, and it is a trait I share - to a point.

Briefly, I would like to point out that I have (very carefully) never made any claims that human population will *continue *to expand exponentially, you might have inferred that into the presentation.

Here's the text:

In this particular case you are looking at a chart of something that historically grew at less than 1% per year. It is world population and because it is only growing at roughly 1% per year we need to look at several thousands of years to detect this hockey stick shape. The green is history and the red is the most recent UN projection of population growth for just the next 42 years.

I have drawn the very clear and completely defensible conclusion that human population *has been* growing exponentially up to now, but all I do for the next 42 years is cite the most recent data that exists from the UN. I never actually state anything about my own predictions for human population growth over that time frame (precisely because I didn't want to get bogged down in the arena of making and defending such predicitons).

However, I could also say that right now human population is *still growing* on a percentage basis every year which means that it is still growing exponentially. You've made the claim, based on extrapolating a recent change in the rate of growth, that the future will see population growth will go to zero or even negative growth which itself is a prediction - something I studiously avoid because predictions open up a whole can of logical worms, are hard to defend, and leave your critics a lot of easy territory to exploit. Once you start making predictions the next thing you know you are stuck in a sea of pesky facts.

For example, I might trot out that US fertility rates have been going up lately:

The fertility rate among Americans has climbed to its highest level since 1971, setting the country apart from most industrialized nations that are struggling with low birthrates and aging populations.

The fertility rate hit 2.1 in 2006, according to preliminary estimates released by the National Center for Health Statistics. It's a milestone: the first time since shortly after the baby boom ended that the nation has reached the rate of births needed for a generation to replace itself, an average 2.1 per woman.

Then I might note that despite this "flattening out" you've cited, that for some reason we still seem to be adding record breaking numbers of new babies to the landscape:

The journal Pediatrics has published its annual report on birth and pregnancy in the United States, and it found that

the number of babies born in 2007 — 4,317,119 — was the highest ever recorded.The birthrate rose in all age groups, including teenagers, whose birthrate had been declining since 1996. (A Guttmacher Institute report last month came to a similar conclusion.) Rates for women in their 30s were the highest since 1964, the last year of the postwar baby boom.

Births to unmarried women in 2007 increased to the highest levels ever measured, rising 4 percent over 2006. Almost 40 percent of all births were to unmarried women, and there were increases in births to single women of all races, including those identified as Hispanic.

So even though the rate might have come down some in the past, it is now headed up again in the US and we are still adding more new humans *than at any time in history* to the surface of the planet year after year. Perhaps this will decline to zero at some point in the future as you've predicted. Maybe not. Perhaps we could argue that for something to be considered "exponential" as a purist would define it in a textbook that the rate has to be constant, and not wobbling about. Okay, but what does that do for us in terms of effectively communicating the crux of the issue to the widest possible cohort of people? See? We're down a rabbit hole.

The facts, as they stand are precisely what I put in Chapter 3 and which VF has recited above. We know how fast population has grown up to now, and we can extrapolate into the near future using currently available data on lifespans and birth rates. That gets us to 9 to 10 billion by 2050.

What additional clarity, or illumination, do you see as being offered by wading into a relatively complicated prediciton about the precise mathematical characterization that we may, *or may not*, experience in 40 years?

How will this help to make a better decision today? How does this deepen our understanding or change the nature of the predicament? If it turns out that even 7 billion people are too many to sustain, what changes in our thinking by knowing that the final rate of growth that brings us to 9 billion in 2050 is not really exponential because its either falling, or rising, or doing something that doesn't quite fit the definition?

After all this discussion, are any of us any closer to accepting the nature of our predicament, our responsibility for the future, and taking concrete actions to try and make the best of it? If so, then I'll have these sorts of discussions all day long. If not, then I respectfully suggest that there are more important things to be doing.

Touche!

*~ VF ~*

Hi Chris

Firstly thank-you for taking the time to respond in such detail. Before replying to your points I should perhaps put my original post in context because I imagine it must have come across as rather confrontational, esp for a new member! Overall I think you've done an excellent job of bringing together some difficult but important issues and that's the main reason I'm here. If I appear critical it's a consequence of my scientific training and natural awkwardness to focus and criticise what I see as being weak points rather than praising the much larger volume of praiseworthy effort.

Briefly, I would like to point out that I have (very carefully) never made any claims that human population will

continueto expand exponentially, you might have inferred that into the presentation.

Fair comment although I would stick to my original assertion that the impression from the presentation is that population growth is currently as well as historically exponential. And as we all know impression is everything. The fact that the responses I got were along the lines of how important it was that I understand the exponential function (I do!) suggests that others got the same initial impression that I did.

You've made the claim, based on extrapolating a recent change in the rate of growth, that the future will see population growth will go to zero or even negative growth which itself is a prediction - something I studiously avoid because predictions open up a whole can of logical worms, are hard to defend, and leave your critics a lot of easy territory to exploit. Once you start making predictions the next thing you know you are stuck in a sea of pesky facts.

Also fair. However when you present the perils of exponential growth and include population in that presentation then you make an implicit if not an actual extrapolation in the mind of the reader - i.e. that population growth will continue to grow exponentially. Even if the small print sais otherwise ;-) Also I would say that extrapolating from an existing trend is fair play as long as you make the caveat that it is valid only if the current trends continues. Which is where we get to my main point - that extrapolation from a true (constant rate) exponential growth rate is very divergent from extrapolation from a reducing rate. They have hugely different implications for both future population levels and the perils those levels may bring.

The facts, as they stand are precisely what I put in Chapter 3 and which VF has recited above. We know how fast population has grown up to now, and we can extrapolate into the near future using currently available data on lifespans and birth rates. That gets us to 9 to 10 billion by 2050

True but there is still a big difference between 9bn and still growing and 9bn and plateaud.

What additional clarity, or illumination, do you see as being offered by wading into a relatively complicated prediciton about the precise mathematical characterization that we may,

or may not, experience in 40 years?How will this help to make a better decision today? How does this deepen our understanding or change the nature of the predicament? If it turns out that even 7 billion people are too many to sustain, what changes in our thinking by knowing that the final rate of growth that brings us to 9 billion in 2050 is not really exponential because its either falling, or rising, or doing something that doesn't quite fit the definition?

Understanding whether we are likely to plateau around 9bn or continue growing exponentially makes a potentially huge difference and how we might go about tackling the issue. If we are confident (acknowledging the uncertainty we've both highlighted) that 9bn is going to be a plateau then we can more easily and quantifiably address the question as to exactly how the Earth can sustain such numbers (I think you and I are probably in agreement that it can't but that's another debate). If the belief is that population will still be growing exponentially even at 9bn then it's a whole different predicament!

After all this discussion, are any of us any closer to accepting the nature of our predicament, our responsibility for the future, and taking concrete actions to try and make the best of it? If so, then I'll have these sorts of discussions all day long. If not, then I respectfully suggest that there are more important things to be doing.

Happy to draw a line under this discussion and move on. For what it's worth I do recognise stubborness as one of my character flaws! No offence intended with any of my comments and I reiterate my support for what you are attempting. Who knows I might post something complimentary one day ;-)

"The fact that the responses I got were along the lines of how important it was that I understand the exponential function (I do!) suggests that others got the same initial impression that I did."

But *do you...?*

As I explained in an earlier post (as Chris himself said) if the population is still growing at ANY percentage, then it is still growing exponentially. Changing the number in front of the % sign changes the current steepness of the exponential curve, but that's all..... Until we get to negative numbers of course which is when the curve starts diving, exponentially too!

Mike

Hi Mike

As I explained in an earlier post (as Chris himself said) if the population is still growing at ANY percentage, then it is still growing exponentially. Changing the number in front of the % sign changes the current steepness of the exponential curve, but that's all..... Until we get to negative numbers of course which is when the curve starts diving, exponentially too!

Sorry but that's just wrong. If you change the rate of growth then you no longer have an exponential curve. Of course whilst the sign is positive you have growth but it's not exponential. Whilst it is subject to error, an extrapolation of current trends in growth gives a plateau in population around 9bn. If growth were currently exponential then simple extrapolation would give a continued exponential curve. Big difference.

Also, whilst Chris makes a valid point about continued growth of the US population, it's worth pointing out that there are now regions of the developed World where populations have plateaud or even started to decline slightly - large parts of Europe and Japan are examples. Whilst this alone doesn't prove World population can or will plateau it does show it is at least possible. And as I've said careful analysis of birth vs death rates lends credence to the Global plateauing theory.

Just come across this interesting article in NS which talks about population trends both wrt growth rates and demographics;

*.......First, we are not producing babies like we used to. In just a generation, world fertility has halved to just 2.6 babies per woman. In most of Europe and much of east Asia, fertility is closer to one child per woman than two, way below long-term replacement levels. The notion that the populations of places such as Brazil and India will go on expanding looks misplaced: in fact, they could soon be contracting. Meanwhile, except in a handful of AIDS-ravaged countries in Africa, people are living longer everywhere.*

* *

I've got to agree with Damnthematrix. Changing the percentage doesn't suddenly make the growth non-exponential. True enough that the overall graph isn't the graph of the same function over time, which would exhibit the classic exponential curve. It's possible to draw graphs over whatever time period one desires. If one takes a small enough time period, that period will show exponential behaviour, if there is any percentage growth at all, during that period. If each period is showing exponential growth then all periods together are showing exponential growth.

To put it another way ... if population was growing 0.2% per month for three months, then that is exponential growth for those three months. If population then grows at 0.15% per month for the next three months then those three months are showing exponential growth (though the curve will be different from the previous three months). If both periods are exhibiting exponential growth, then why isn't the 6 month period showing exponential growth? Of course, the answer is that it is showing exponential growth; it's just that the curve is a concatentation of various exponential curves.

So, population is certainly rising exponentially and it's worth pointing this out. Global population growth rates had been steady or increasing since 2004 but it looks like the rate dipped down again last year. I think it's wrong to assume that population will level out and, somehow that will solve the population problem. As Chris points out, no one knows the maximum sustainable population (though there have been many guesses around 1-2 billion) but you can bet that the higher it goes, the more likely it is to reach an unsustainable level, if it hasn't already.

Sorry but that's just wrong. If you change the rate of growth then you no longer have an exponential curve. Of course whilst the sign is positive you have growth but it's not exponential. Whilst it is subject to error, an extrapolation of current trends in growth gives a plateau in population around 9bn. If growth were currently exponential then simple extrapolation would give a continued exponential curve. Big difference.

Sorry mate, but YOU are plain wrong...... you can only get a plateau at 0%.

All that happens when you lower the number in front of the % sign is you change the shape of the curve. Yes it'll take longer (maybe a lot longer!), but eventually the curve will again go skywards, ESPECIALLY as it starts from an already very large number, unlike the current one which starts (pick a date, any date) around 2000 years ago when world population was 200 million as opposed to today's 7,000 million....

I can't put it any clearer than this. If you don't get it, well you just don't get it!

Mike

Piquod12 seems to be very lonely here, with everyone telling him he is wrong.

I believe that many are reading his posts wrong, since I believe his points are perfectly true.

What he is saying is that the growth rate is dropping.

In fact if you look over the last 20 years at the birthrates it is dropping in almost every country of the world.

The important turningpoint is when the worlds fertillity rate is at or below 2.1 childs per woman, as that is the rate for constant population in the long term.

Piquod12 also mentions (but not in these excat words) that average lifespan is increasing.

If we for the sake of simplicity say that the average lifespan will increase to a certain level and then stop, then the increased lifespan will increase world population by a factor k. Lets NOT go into what exactly this factor is. The important point is that it is a constant, and not anything that leads to exponential growth.

This is assuming you believe there is a maximum age that humas can become. If you believe there will ever be a way for eternal life for humans then we are in deep trouble again.

So, even if population are still growing for the next 40 years it is a big difference if we will be at a point when population will start to decline at 2050 or if population is still growing a lot.

Just my 2 cents to show my support for Piquod12 for bringing up this subject.

Hi Mike

Sorry mate, but YOU are plain wrong...... you can only get a plateau at 0%.

All that happens when you lower the number in front of the % sign is you change the shape of the curve. Yes it'll take longer (maybe a lot longer!), but eventually the curve will again go skywards, ESPECIALLY as it starts from an already very large number, unlike the current one which starts (pick a date, any date) around 2000 years ago when world population was 200 million as opposed to today's 7,000 million....

It's possible we're arguing at cross-purposes here because I don't disagree with any of the above. What I would say is that the type of growth you describe is not exponential. This really isn't a matter for debate but simply a consequence of a mathematical definition that describes the rate of growth of a function as proportional to the function's current value and where that rate is a constant;

A quantity *x* depends exponentially on time *t* if

where the constant *a* is the initial value of *x*,

and **the constant b is a positive growth factor**, and

*τ*is the time required for

*x*to increase by a factor of

*b*:

My bold.

So whilst it is perfectly valid to say population is still growing it is not valid to describe the growth as exponential*. You can say this is nitpicking but population projections based on continued exponential growth would be very different from those based on current growth trends (i.e. plateauing around 9bn).

*Sofisteks example of exponential growth at one rate followed by exponential growth at a diferrent rate, apart from being highly unlikely, is also not an example of overall exponential growth. I'm not sure what it is an example of, if it even has a name, but you can't just stitch curves of different shape together and give the result the same label as the components.

I should also make clear (in case I've created the wrong impression) that I do not believe 9bn to be a sustainable level for our planet, especially once economic growth projections are factored in! I would also find it hard to argue with anyone claiming the current population X gdp product is sustainable. But until this has been demonstrated by the passage of time it remains in the realm of opinion. The mathematical definition of exponential growth is not.

Hi Silvervarg

Support appreciated thanks :-)

To be honest I'm not that surprised I've ruffled a few feathers on here by introducing myself with a criticism of part of CM's presentation. With hindsight it probably wasn't the most tactful approach! It's just that I believe that fundamental assumptions (whether intended or not) need to be regularly reviewed and challenged in order to make sure we don't get into group-think which ends up helping no-one. It's just too easy to get sucked into a paradigm, especially if there is an inclination already present, and filtering all new information through the same lens.

Which is why in a seperate thread I've challenged the perception of our dependence on oil by questioning how dependent we are on the biggest user - the private car. Of course that is a discussion which is much more in the realm of opinion than fact and has massively different implications depending on where in the World you live. E.g. people in the US may see private car loss it as an insurmountable issue whilst those in Europe might view it as just an inconvenience.

All worthy of discussion IMO and if it makes me unpopular then so be it. Since when have scientists been popular anyway, look at what they did to Galileo ;-)

Sofisteks example of exponential growth at one rate followed by exponential growth at a diferrent rate, apart from being highly unlikely, is also not an example of overall exponential growth. I'm not sure what it is an example of, if it even has a name, but you can't just stitch curves of different shape together and give the result the same label as the components.

Of course you can - that is what just about every growth graph I've ever seen does. Your idealised notion of exponential growth is not, in the least, important, from a real world perspective. My example of exponential growth at one rate, followed by exponential growth at a different rate is certainly not "highly unlikely"; it is a fact. Even your idealised exponential curves are a specific example (where X=Y, when the growth rate of one period is X and the growth rate of the next period is Y). All you have to do is keep shrinking the time period to get to a period where the growth rate is constant for that period. The subsequent time period can also be shrunk, until the same condition is met. Concatenate all of these time periods and, provided each period has growth, you have exponential growth throughout the whole period - just not the idealised version that you think is important.

The critical feature is that any percentage growth is on an increasing population size.

It's not a matter of rustling feathers, it's just that your criticism has no merit, regardless of how mathematically accurate it may be.

Hi Sofistek,

My example of exponential growth at one rate, followed by exponential growth at a different rate is certainly not "highly unlikely"; it is a fact

Can you give a real World example where this has occurred? I don't mean a continuously varying growth rate but where growth at one rate constant is replaced by growth at a different rate constant, i.e. two actual exponential in sequence.

Of course idealised exponential growth will not occur in nature either, there will always be some variation in the rate. But it is still valid to talk about the importance of exponential growth even if it only an approximation and to be clear to distinguish between systems that exhibit it and those that do not. **The merit of this distinction comes in the nature of reasonable projections going forward**. As I've said, the difference between an exponential projection and that obtained from a declining rate is not trivial. It is extremely pertinent to what is most likely to happen to population in the future and how we go about dealing with it.

You talk about shrinking the time interval to a point of constant growth rate, essentially taking the derivative of the curve. Yes you can do that and, yes, you can express the level of growth as a function of the population size but that doesn't make the curve exponential. Even linear growth can, at any infinitessimally small time interval, be exressed as a % of the underlying but it doesn't mean the linear growth is now exponential! Take a simple example where, say, the underlying population is 6bn and it's increasing linearly by 70mn/year (remember this is an example not my belief about what is actually happening). At the point at which the population is 6bn then you can express the growth as either 70mn/year or 70/6000 X 100% = 1.17%. The following year 70mn peolple will have been added and you can, at the specific point of year-end, once again express growth as a %, this time it will be 70/6070 X 100% = 1.15%. The numerator (top bit of the equation) is still 70 because we have defined linear grwoth. But by your argument you have exponential growth because growth can still be expressed as a % of the underlying population. It isn't, it's linear.

The critical feature is that any percentage growth is on an increasing population size

Critical in what way? If the rate is falling quickly enough then the absolute growth, exressed as a number, will also be falling regardless of the increasing size of the function - i.e. you will get plateauing rather than continued acceleration.

We seem to be on a completely different wavelength, here, piquod12.

By your reckoning, there is no real world example of exponential growth, because, taken over the whole time period of that growth there cannot be a constant rate of growth and, so, no exponential growth. If you deny this for, say, population during some period then you have to limit the time period to get that roughly constant rate. All I'm saying is that all examples of growth can be seen to be exponential over a short enough time period (which you acknowledge) so that, when talking about that growth at any particular instant in time, it will always be exhibiting exponential growth, even though you're right that it won't be a pure exponential graph over the whole time frame of that growth.

Your notion of linear growth can be as easily criticised as Chris's hockey stick because it would imply that the linear rate would continue ad infinitum, which no-one could possibly know (aside from the near certainty that the increase would vary and so not be linear). However, even linear growth would have horrendous implications somewhere down the line, since some quantity of humans (probably in the tens of millions) would be added to the planet every year. I prefer to call it exponential growth because it can more clearly demonstrate the enormity of our predicament.

This is why your, possibly valid, mathematical criticism of that one part of the Crash Course, is irrelevant.

Hi Sofistek

By your reckoning, there is no real world example of exponential growth, because, taken over the whole time period of that growth there cannot be a constant rate of growth and, so, no exponential growth.

No, I'm happy to acknowledge exponential growth (or approximation to the same) for systems that exhibit it for a relevent period of time. By relevent I mean finite (as opposed to instantaneous) and of sufficient duration to be meaningful from the pov of the system in question. I absolutely agree that human population growth has, in the past, exhibited exponential growth.

when talking about that growth at any particular instant in time, it will always be exhibiting exponential growth,

No this bit is technically wrong - you cannot actually determine the shape of a growth curve by by taking the derivative at one time point. Hence my example of linear vs exponential growth. One simply doesn't have enough information.

However, even linear growth would have horrendous implications somewhere down the line, since some quantity of humans (probably in the tens of millions) would be added to the planet every year. I prefer to call it exponential growth because it can more clearly demonstrate the enormity of our predicament.

I agree that linear growth would also eventually hit limits on a finite planet. It would take longer but yes it would get there in the end. If you like to call linear growth exponential then I guess that's your call and I'm getting to the point where I no longer believe I can persuade you otherwise.

This is why your, possibly valid, mathematical criticism of that one part of the Crash Course, is irrelevant.

I maintain that the shape of the population curve going forward is relevent. For one thing the amount of time we will have to adapt will be very dependent on the speed with which things are likely to change. Same for oil depletion - if it happens at 10%/year it will have very different consequences than at 2%/year. I'm not arguing against the seriousness of our predicament but do believe the details matter.

Anyway I'm aware that I'm starting to repeat myself here and therefore not really adding anything new. So, unless anyone has a specific question for me, I'll stop for now.

piquod12,

I get your point and, in pure mathematical terms, you have a valid point but has little value in reality unless you can show that the future will pan out in a way that has population levelling out at some figure (say, 10 billion), because, if you could do that, we'd at least have a maximum figure to deal with. However, if we're already in overshoot, it seems to me that it makes little difference whether some idealised curve will see a maximum number in 2050 or 2100 or 2200. Chris has focused on what the historical trend was and we need to pay attention to that. Forecasts for future trends are liable to be wrong (almost all predictions about the future are destined to be wrong) and population growth didn't stick to the projected trend between 2004 and 2009, so who knows what the future holds in terms of population size?

We need hard hitting messages to try to get a significant shift in thinking and to get a new strategy for a sustainable future. It seems to me that a mathematical criticism that some strict definition of a word means it is the wrong word to lose, is largely irrelevant. There is a grave story to tell and it needs to be told, without picky side threads that could dilute the message.

By the way, you're right about my use of the word instantaneous but the fact that you picked up on it shows that you tend to miss the primary message, which is what you did with this thread.

you're right it's not exponential - it's worse

Mark

In enjoyed the discussion here as there were interesting arguments from various sides which do inspire thinking..

As a physicist I tend to agree with piquod12 and Chris who could find peace with each other. Piquod12 s observation of a recent change of the rate of growth was something I also noticed with some interest when I check the numbers at Wikipedia after watching the crash course. Piqued12, I believe just wanted to help Chris fix an inaccuracy to make him less vulnerable to more venomous attacks.

I believe piquod12 is right with the mathematics and of course **a curve with changing exponential growth rates cannot be an overall exponential curve at the same time. **

Maybe this is more obvious to the non mathematician like myself if you think of a curve that changes its **linear **growth rate i.e. its "steepness".

Is the result of a linear function that changes rates (even if limited to positive numbers) linear?

No. Nobody would call such a curve as overall linear as it could describe any zigzag patterns or castle patterns or even exponential patterns. In fact,** the only thing it cannot describe is .... linear behaviour, a straight line**.

Also: Even though a changing rate "linear" function can also describe an exponential function, it would not make sense to claim an exponential function is linear.

Even simpler: it is not really useful to consider a curve to be straight even though **one can describe a curve as a straight line with a varying degree of "bendness"...**

A varying rate of an exponential curve will also be a lot of different things but certainly **not overall exponential**.

However, these are academic arguments. Mathematicians are the only people I know who can claim truth and perfection - as long as you can't question the accuracy and meaning of the data they play with.

Interestingly the moment we leave the mathematical truth behind, we can call it science

Scientist like anyone else have to live with **unavoidable errors **in our data.

Nobody here talked about the error bar in the population data. Do we really know how many Africans or Indians or Chinese people there are?

Many people in these countries do not have birth certificates or pay rent or taxes or even official names... and even if we could determine their number counting income tax returns, do all countries count their tax returns (or census forms) at the same time?

We also don t know the global population 200 years ago - the error for the data in Chris graph for that time is likely considerable. The data he got were probably already extrapolated **assuming exponential growth **until more reliable global data became available.

The graph in the previous post is a reminder that it is scientifically and mathematically incorrect (although not uncommon depending on the type of publication) to connect **uncertain data points and trying to call the resulting shape something e.g. exponential.**

The scientifically correct procedure would be (in simple terms)** to fit the data points** including their error bars (if known) with a mathematical function that would make sense while trying to keep the deviation from the data points a minimum.

Assuming reasonable error bars of the data, nobody would seriously suggest to use a linear function if we look at , say a few thousand years of human population growth. It is most likely an exponential curve that changes its rate as human fertility and mating behaviour may depend on changing external factors, climatic changes, wars, cultural changes over time.

However,** since the raw data we have are probably rather inaccurate, science tells us to fit a simple exponential function until the very last data points and further.. **

**Note that depending on the errors and the unknown true growth function the fitted exponential function would not go through the data points but generally only approximate them.**

Now we are back to the beginning of the discussion of this interesting (for some) thread. Chris' representation makes practical sense - but he could have saved himself and us some time arguing by adding error bars to his data points or at least mention their uncertainty.

Now, before someone comments that data in the last years are probably more accurate and may justify a linear fit (which is not shown in the previous posts graph due to the large x scale): There may also be higher incentives to manipulate these data (I would expect a larger and perhaps asymmetric error bar).

In a time dominated by a profit and growth driven economy that benefits a few percent of the population - can we trust the same few percent to give us accurate numbers supporting exponential growth - a threat to their "values" and financial position?

Let us rather be careful..and focus on the larger picture and common sense which is more reliable nowadays - like Chris suggests.

<a href="http://en.wikipedia.org/wiki/Exponential_growth">WikiPedia says:</a> <i><b>**Exponential growth</b>** (including exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value.</i>

My background is in ecology and dynamic systems, rather than mathematics, but when a population's growth is a function of its present size, the vernacular term is "exponential growth."

It's the self-referential bit that makes it exponential. If you say, "120,000 million people are added to the world's population each year," that's arithmetic growth. But if you say, "the world's population increases by 2% each year," that's exponential growth, assuming that the people added are not sterilized and will begin themselves to add to population in the next twenty years or so.

Is it fair to look at projected future trends and cite them as arguments against calling current growth "exponential?" <a href="http://en.wikipedia.org/wiki/Panarchy#Panarchy_in_systems_theory">Panarchy systems theory</a> says that everything goes through birth, growth, coalescence, dissolution, re-organization, and re-birth. Therefore, <i>nothing</i> physical can sustain exponential growth indefinitely, so it seems a bit disingenuous to cite the inevitable end of growth (or minor changes in growth rate) as evidence that self-replicating growth is not exponential.

The presentation leaves one with the impression that population is increasing exponentially but this is no longer the case.

2 important points;

1. The rate of increase in population is now slowing which would not occur with true exponential growth, the rate of which is approximately constant.

2. Current population growth is being driven entirely by decreasing death rates in developing nations and not by increasing birth rates. This is important because although growth is baked into the cake it also means population can be projected to plateau around the 9bn mark.

Note - I am not here making a judgement on the carrying capacity of the planet or whether, if it is ever reached, 9bn will exceed it. I am merely trying to make sure the factual information wrt population is presented correctly - ascribing current population growth as exponential is incorrect and misleading wrt exrapolations based on that assertion.