# Refute The Carter Hypothesis (Statistics & Probability)

I’m reading a (fiction) book just now that makes much use of the “Carter Hypothesis”. Apparently this is not a fictional hypothesis but has actually been seriously proposed.

The plot of this book rather depends on a large percentage of the world accepting this hypothesis. It is called ‘statistically valid’ and plenty of otherwise intelligent characters in the book fail to find a problem with it.

I, however, am calling it total BS, and find its acceptance has ruined an otherwise good book. So, what’s the SDMB verdict? Why has this hypothises gained any credence at all, either within this book or in reality?

It goes something like this:

We must accept that our lifetime occurs somewhere during the period of all human life. Human population growth is expodential, at no point in time has more people been alive. Therefore our lifetime currently lies close to the mean of all human lifespans. But, if we were to look into the future, with continuing expodential growth, we would find our lifetime becoming closer and closer to being the very first lifetimes, with more and more lifetimes occurring after us. Pretty soon our lifetime will be within the first 1% of all lifetimes.

That, it is argued, is very unlikely. Far more probable is that we lie somewhere near the mean which, as we’ve already established is very much towards the peak of population growth. Therefore, statistically speaking, we are likely to be close to the end of all human life.

The argument’s due to the cosmologist Brandon Carter and is one of several anthropic-type arguments he’s proposed.
While Carter’s respected, I don’t think this particular argument has gained much credence. Someone else (and I can’t remember who, though I should) published an article in Nature a few years back developing essentially the same argument in terms of some Ultra-Copernican Principle: our vantage point shouldn’t be special, so we should roughly be in the middle of human history. But, from my memory of conversations at the time, nobody took this terribly seriously. A bit of fun, at best.
In fact, I’m never quite sure to what extent Carter proposed it merely to be provocative.

As usual, Barrow and Tipler’s The Anthropic Cosmological Principle (Oxford, 1986) is the place to read more about these sorts of arguments. And much else besides.

Perhaps you could expand a bit on the point of the Carter Hypothesis, because right now I don’t see why I should care (i.e., I don’t see why this should make any difference at all in my life, or how it could be significant in the plot of a novel).

I assume by “expodential” you mean “exponential”?

Well only in that were it true humankind is going to hit a mighty disaster in the near future. Whether you care is a matter of opinion on how good this would be. People in my book are, generally speaking, very much against the idea.

Typo

My immediate reaction to the hypothesis was that if it is true for my lifetime, it was equally true for every lifetime since the year dot. But it hasn’t caused doomsday yet.

I still don’t get it. Why does it require some mighty disaster?

Incidentally, the real world aside, the laws of physics would eventually stop us from increasing exponentially. Given the volume that a human takes up, and how fast we’re increasing, we’d eventually get to the point where it’s physically impossible to increase without exceeding c. It’s something like 6000-10,000 years in the future.

It sounds like little more than the typical, Malthusian doomsday predicted by Erlich and his ilk in 80s, and pretty roundly debunked now. Populations have periods of exponential growth–it’s not continuous.

Of interest:

http://www.greylabyrinth.com/Puzzles/puzzle106.htm

and the followup discussion:

http://www.greylabyrinth.com/Forums/Forum4/HTML/000280-2.html

Thanks for the links daniel. It would seem that the whole thing is more of a logical/philisophical paradox, not intended to be taken literally. Which makes it all the more disappointing that it’s being used as a serious plot device .

It’s not unrealistic to suggest that a large number of people can be taken in by pseudoscientific nonsense, that happens all the time. But you’ll have noticed that the nonsense that people will swallow tend to be positive ideas. You will be cured by my touch, your grandpa lives on on the “other side”, you do have special powers, you can tell the future, you can get something for nothing.

But most are usually far more reluctant to believe pseudoscience that tells them something bad.

1. Human population growth is not simply exponential. The rate of growth varies over time, and the number of people currently alive versus the resources available to sustain them plays a part in this rate. In fact in many industrialized countries reproduction rate is so low that population is almost level or even shrinking. Thus there is no reason to believe that population growth will not stabilize, and thus no reason to believe we are statistically growing exponentially closer to either end of the overall human population.

2. This argument would be true for all generations, not just ours. Thus the first humans would also have to accept being near the end of the human race as well. All this probability extrapolation really demonstrates as far as I can tell is that we’re getting closer to the end of humanity with each generation, which is a truism.

3. Human lifespan is not a constant. Therefore it is not legitimate to guess at the temporal end of humanity by extrapolating from the percentage of humans who have lived already from the group of humans who will ultimately exist.

Those are the problems that occured to me at least.

The problem with Carter’s hypothesis is that it uses a shaky definition of the word “random”.

Assume for the sake of argument that the human race will eventually die out (even if it’s only at the Big Crunch, or heat death of the universe or whatever). Then the total number of people, past and future, is finite. Then if some omniscient being chooses a human being at random from all those who have lived or ever will live, and if human population growth is truly exponential (which, as laigle points out, is a shaky assumption), then yes: that randomly chosen person is much more likely to live at the end of human history than at the beginning or middle.

But Carter is not an omniscient being, and he’s not choosing a person at random from the set of all possible people. He’s choosing a person who is alive at the time he makes his argument. This is completely different from choosing a person at random, and totally fubars the argument from a mathematical perspective.

This, really, is the central problem with the Carter hypothesis. The fact that most humans might exist around the averaged midrange of human existence does not trump direct observation.

If Carter’s hypothesis is valid, then I must logically conclude that I live in Asia, not Canada, since it fact it is vastly more likely that a person would be Asian than Canadian. I must conclude that I don’t really own a computer, since it is extraordinarily unlikely that any given human owns a computer. And yet, I do seem to have a computer in front of me, and that’s the CN Tower I see out my windows.

Yes, but, as you might surmise, it is hard to observe the end of existence and say, “Well shoot, it was right!”

Orbifold, I’m not quite sure I follow. All that we know is that we are born at some time in the overall existence of the human race.

Would you agree that, if every person who ever lived stated, “I am in the peak of population growth”, statistically, most humans would be right?

So now I say to you, “I am in the peak of population growth.”

Help!

Yes, that’s correct. (Assuming that the human species will eventually die out and that population grows exponentially in the meantime, blah blah blah…)

But that statement has a very particular meaning. Specifically, if we pick one person at random from the set of all past and future people, then that person very probably lives near the end of human history.

If we pick a person in any other fashion, then we can no longer draw the same statistical conclusion.

There’s the rub. You have not been chosen at random from the set of all past and future people to make that statement. You have been chosen (if we can even call it that) from a more limited set of people, namely the people who are alive now. By only asking people who are alive now, we’ve ignored part of the sample space: namely the people who live in the future. The probability that the person we’ve selected (you) lives near the end of human history is therefore not the same as the probability for a person randomly selected person from all of human history.

To get geeky for a moment, Carter is describing a statistical experiment: pick a human being and measure where that person lives in history. The output of such an experiment is a random variable X; Carter is claiming that the average value of X is “near the end of history”, and that the deviation is low. But Carter isn’t performing the experiment correctly, or at least not in a way that would result in the conclusion he’s claiming. Which is not suprising, because in order to do so he’d require a time machine.

I understand that this could be significant, but alas I don’t see how it impacts this case. Likely my obtuseness rather than anything else. It seems the statistical likelihood of my statement’s truth does not change with respect to how I am selected. It’s factual truth would, of course.

Well, consider an extreme case. Suppose I give you an urn with 10 red balls and 90 blue balls. If you pick a ball at random, the probability of that ball being blue is 90%. But if you look into the urn and pick a red ball at random, the probability of that ball being blue is 0%. The statement “this ball is blue” is still either true or false for each particular ball, but the probability of that statement’s truth for a “randomly chosen” ball depends on just how randomly that ball is chosen.

Hmm! Well, that certainly makes sense. Forgive me for beating this into the ground, though… but the selection method of the Carter Hypothesis is only not selecting people who have already lived then died. So this excludes people that we know couldn’t have been living in the end of days. So perhaps if the analogy was that there were three colors, blue for people already dead, red for people currently alive, and green for those that were to live in the future, and we’re color blind and can’t distinguish red and green (but can tell blue)… what color are we likely to pick? Is that a closer situation?

But of course, to make such an experiment and actually put the balls in, we would have to know the exact thing we’re using the experiment to find out. Hmm. So, if it is done right, it can’t be done, but if it is done wrong, we’re all gonna die. That about right?

Pretty much. The balls-and-urn isn’t really analogous to Carter’s hypothesis; I was just using it to demonstrate how the sampling method can affect probabilities.

I always thought of this sort of argument in the opposite way. For instance, the odds are extremely poor (or at least they may be) of intelligent life developing given a starting condition of the universe, namely some bizarre “quantum soup” of fluctuations. So I might ask, “why should I exist when the odds are so low,” but to do so is like a fish asking why it’s in a fish bowl. If it weren’t it would be unable to ask anything at all. So I ought not be surprised that I exist since I find myself existing.

I thought that was the “anthropic principle” referred to above.

For all the statistic-based arguments people throw around, I’d prefer the Rosencratz and Guildenstern example to any of them. Just because the last 500 flips produced heads doesn’t change the 1:2 odds of the next one being heads. Statistics are fine for single events, but extrapolated to multiple events they seem to produce silliness like the theory in this OP.

I have heard of this principle. I confess to know little about it.

However, when I was very young about 8-10 I remember seeing a program on TV about ancient civilizations. I remember being confused and asking my dad how old human civilization was. He said about 6000 years.

Seriously, I was stunned. I remember thinking that it couldn’t be right. It seemed so wrong. Human history should have been really ancient like millions or tens of millions of years.

It still seems wrong to me. If the human race really is to last a long time and colonize the galaxy then I should have been born with a huge history behind me and human population many magnitudes greater than it is.

Finding out I am 6000 years from the start does seem ominious to me. I know some people have to be at the 6000 year mark but…

However, I do not know if what I feel has any relation to reality.