Mathematics, rejuvenation, and immortality walk into a bar…

If a fully rejuvenated person was hit by a train at full speed, I can promise you they would stand the same pathetically low chances of ever being reassembled into a single, barely functional piece as any non-rejuvenated person of any age. Keeping that in mind, if anyone tried to sell me rejuvenation as ‘immortality’, rest assured I would demand to see the manager right away.

On a different yet unexpectedly related note, if I had a nickel for every time I heard or read something along the lines of ‘death is inevitable because probability’, I could donate so much money to LEAF the IRS would start thinking they’re a bit too well off for a charity.

Oh, and with the rest of the money, I could buy LinkedIn and pay someone to finally give it a user interface you can look at without your eyes bleeding.

That being said, no matter how hard rejuvenation advocates strive to make a clear distinction between rejuvenation and immortality, nearly everyone keeps cheerfully swapping them around, largely convinced that two absolutely incompatible statements hold true at the same time, namely

  1. rejuvenation implies immortality, and
  2. probability grants you will die at some point, rejuvenation or not.

But that’s okay, really—I mean, people hold contradictory beliefs all the time. What pisses me off is that statement ‘b’ is false.

Immortality buys the first round

You might be thinking I must be a few fries short of a happy meal, because after all, statement ‘b’ is only a special case of the more general statement ‘Given infinite time, anything that is possible, however unlikely, will eventually happen’. This is false too, by the way—if a special case is false, then the general case isn’t true either.

To be fair, among people who think both statements ‘a’ and ‘b’ are true, probably many intend immortality in a weaker sense than I do. To me, ‘immortal’ is somebody who cannot die, no matter what; to them, ‘immortal’ is just someone who doesn’t die of their own accord past a vaguely defined ‘best before’ date. I think using the word ‘immortality’ for the latter is wrong and misleading, but since surely most people will keep using it no matter how much I whine about it, I could agree to call ‘strong immortality’ the former and ‘weak immortality’ the latter. Weak immortality is thus what some people call ‘functional immortality’, ‘biological immortality’ or ‘amortality’—again, the property of a creature whose body doesn’t fall apart by default with the passing of time.

Quite frankly, I think strong immortality is not possible, because it implies that one’s chances of ever dying are exactly zero; or, if it is possible, I think it’s so far off that it’s not worth holding your breath.

Some are convinced the laws of probability say that, inevitably, the odds a living creature will die go up with the passing of time, and from this premise, they conclude rejuvenation (or weak immortality, if you really want to use the i-word) is a pointless goal: Sooner or later you will die anyway, so why bother with rejuvenation?

Well, apart from the fact that staying biologically young and healthy is a worthy goal regardless of how long you’ll live, as I was saying above it’s not true that probability makes your death inevitable: In fact, under certain circumstances death may become more and more unlikely with the passing of time. Whether or not we can produce these circumstances is a different matter, which we will also look into; my point here is debunking the myth that death is certain no matter what because probability says so. Another way of putting it is that if your goal is to never die, you don’t need strong immortality: You can be mortal and yet never die, so long as your death is a vanishingly unlikely event.

Mathematics takes over the open mic stage

Maybe you’re used to think of probabilities in terms of percentages, but in statistics you generally use numbers between 0 and 1 instead, which is perfectly equivalent. Intuitively, a probability of 0 is 0%, 1 is 100%, 0.5 is 50%, and so on. That’s how we’re going to do it here.

We want to prove that death is not necessarily certain even given infinite time; in other words, we want to show that your probability of death over an infinite time can be less than 1. To do so, we need an expression for your probability of death over an infinite time; more precisely, we are going to need an expression for your probability of death over an infinite number of years, starting from the first year of your existence, which we say is year 1.

As you probably already know, nobody dies twice; in order for you to die in year 10, for example, ten distinct events must all happen: You must live through (or, equivalently, not die in) years 1, 2, 3, 4, 5, 6, 7, 8, 9, and you must die in year 10.

On any given year, say year n, you have a certain overall probability of dying that takes into account your probability of death from all causes: accidents, murder, heart attacks, etc. All of these add up to a number between zero and one, which we call dn, as in ‘chance of death in year n’. This is your chance of dying in year n on the assumption you have not died before; in other words, dn is the answer to the question: ‘Assuming you did not die in years 1, 2, 3, …, n – 1, what is your chance of dying in year n?’. Notice that this question is not the same as ‘What is your chance of not dying in years 1 through n – 1 and dying in year n?’ The reason is that the former assumes you already made it through the first n – 1 years, and asks what are your chances of dying in year n in such case; the latter makes no assumptions and asks what are your odds of making it through the first n – 1 years and not making it through the nth year.

The answer to the second question is the probability that n events all happen, which is calculated as the product of the individual probabilities of each event.

Clearly, in a given year you either die or you don’t; these two events represent 100% of our options, so their probabilities sum up to 1. Thus, if dk is your overall chance of dying in year k, your chance of not dying in year k is 1 – dk.

Now we can calculate your chance of not dying up until year n – 1 and dying in year n; we call it Dn and it is

Dn = (1 – d1)×(1 – d2)×(1 – d3)×…×(1 – dn – 1dn.

For example, we have D1 = d1 (because you didn’t exist before year 1 and therefore couldn’t die before then), and D4 = (1 – d1)×(1 – d2)×(1 – d3d4.

Each Dn is the chance of a specific combination of events happening, namely your living through years 1 to n – 1 and dying in year n. Obviously, only one of these event combinations can happen, because as said you don’t die twice. For example, you either live through years from 1 to 5 and die in year 6, or live through years 1 to 10 and die in year 11, but not both.

The question ‘What is the chance that you will ever die, given infinite time?’ can thus be restated in a more verbose, but more accurate, way: ‘What is the chance that you will die in year 1, or live through year 1 and die in year 2, or live through years 1 and 2 and die in year 3, …’ and so on, ad infinitum.

While you multiply together the probability of individual events to find out the probability that they will all happen, you add together their probabilities when you want to find out the probability that only one of them happens. (Technically, from the result you should subtract the probability that any combinations of them happen, but in our case this probability is exactly zero, because as said death can happen only once, so there’s nothing to subtract.) We call this chance D, and it is

D = D1 + D2 + D3 + … + Dk + Dk + 1 + …

all the way up to infinity. There is a more convenient way to write the sum of infinitely many terms, which is this:

So, instead of writing D as the long sum in the middle, we express it as the Σ (‘sigma’) thingy on the right. They mean the exact same thing.

Alright, so now we have an expression for your probability of dying over an infinite time, D, and we want to show it can be less than 1. This might seem absurd to you: How can an infinite sum be less than 1? Shouldn’t it always be infinity? No. A sum of infinitely many terms is called a series. If a series happens to add up to a finite value, it is said to be convergent; if it blows up to infinity, it is said to be divergent. I know it seems strange; welcome to maths.

In order to get to the desired result, let’s first notice that the series D is actually always smaller than another series. Remember that D is just the sum of infinitely many Dk’s, and each Dk is equal to the product

(1 – d1)×(1 – d2)×(1 – d3)×…×(1 – dk – 1dk.

All the terms in these products are larger than zero and at most 1, which means that when we multiply something by any of them, this something won’t get any larger; if anything, it could get smaller. In other words, the term

(1 – d1)×(1 – d2)×(1 – d3)×…×(1 – dk – 1dk

isn’t any larger, and is possibly smaller, than

(1 – d2)×(1 – d3)×…×(1 – dk – 1dk,

and if we iterate this reasoning, we see that ultimately

(1 – d1)×(1 – d2)×(1 – d3)×…×(1 – dk – 1dk

isn’t any larger, and is possibly smaller, than dk. This means that D1 is no larger than d1, D2 is no larger than d2, and so on. Therefore, the sum of all Dk’s (that is, D) is no larger than the sum of all dk’s. In symbols,

which means your probability of ever dying is always smaller than (or equal to) the sum on the right. If that sum could ever be smaller than 1, we would be done. Can it?

Note that the sum on the right is the sum of all your overall yearly probabilities of death—that is, the sum of your probability of dying in year k on the assumption you safely made it through all years until year number k – 1, for each k. Suppose that, thanks to technology, the elimination of ageing, and so on, we were able to reduce these probabilities each year in a very specific way, that is according to the formula

where p is any number larger than 2. (You’ll understand why in a moment.). What would happen then? Let’s just plug the expression for dk in the expression for D and we’ll find out.

Remembering that the term on the far right is just a seriously long sum, we can rewrite it like this:

So, ultimately we get

Now look at the sum on the far right again—only the sum, forget 6 divided by πp. That is a known convergent series, and it adds up to π2 divided by 6. This means the following:

Now you see why I required that p be larger than 2: This way, p – 2 is always larger than 0, which in turn means that πp – 2 is always larger than 1, and thus the whole fraction is smaller than 1. Therefore, D, that is your probability of dying over an infinite time, is less than 1. Furthermore, the larger p is, the smaller D will be, which means that if we were able to make your yearly overall probability of death decrease faster and faster as time passes, your chance of ever dying would become smaller and smaller, though it would never be exactly zero. This means that, over an infinite time, you might die, but it would not be very likely.

On top of that, if we assume you safely made it through 5000 years, then your probability of ever dying past that point would be only

which is clearly less than D, as it is only part of it. More generally, if you make it through the first n years, your probability of ever dying past year n is

The stuff above is called the tail of a series. In particular, this one is the tail of the series D, which we know is convergent, since whatever it adds up to, it is less than 1 divided by πp – 2. So it is the tail of a convergent series. The tail of a convergent series has an interesting property: as n grows larger, the tail tends to zero. In other words: Under these circumstances, the longer you lived, the closer to zero your chances of ever dying would get (though they would never be exactly zero).

This, mind you, is true if we can make it so that your yearly probability of dying decreases each year according to our formula—or, more generally, in any other way such that the sum of all these probabilities is a series converging to less than 1; and this, make no mistake, is no small if.

Go home maths, you’re drunk. Rejuvenation will take care of you.

Before we send maths on its way, it’s important to notice that our assumption of a decreasing overall yearly probability of death is crucial to the reasoning. If we assumed this probability was constant, for example, then no matter how small it was, you’d be royally screwed, as your probability of ever dying over infinite time would then be exactly 1. Indeed, say that each year you have a probability of dying of d, of which we require to be just any number larger than zero and at most 1. Then, your probability of living through years 1 to n – 1 and dying in year n would be

Dn = (1 – d)×(1 – d)×(1 – d)×…×(1 – dd,

where the term (1 – d) appears n – 1 times, ultimately yielding

Dn = (1 – d)n – 1×d.

At this point, D would be

where we factored d out just like we have done before in a similar case. Now, the series on the right is called a geometric series, and if (1 – d) happens (as it does) to be smaller than 1, then the whole thing will add up to 1 divided by 1 – (1 – d). That is,

As I was saying, royally screwed. You can imagine that, if your yearly overall probability of dying is not a constant but it increases instead, then you’re even more screwed—and this, by the way, is exactly what happens because of ageing. In this very moment, as things stand, your probability of dying goes up as time passes. Sorry about that.

But let’s not get all doom and gloom about it, and rather, let’s focus on another question. We established that, if we can make your overall yearly death probability decrease fast enough, then death is no longer a certainty and can even become increasingly unlikely. So, can we accomplish this? Hard to say.

The good news is, if we managed to eradicate age-related diseases, if nothing else your probability of death wouldn’t go up by default with the passing of time. I’ve done some digging on the Internet and I found a few informal calculations about how long the average lifespan would become if ‘natural’ causes of death were eliminated altogether; estimates vary, although they all talk about centuries. In particular, this article on Gizmodo discusses a statistical simulation according to which if all ‘natural’ causes of death were taken out of the equation and the only remaining causes of death were things like accidents, murders, etc, then the average lifespan (in the US) would be 8,938 years. (There’s also a fabled calculation apparently made by an insurance company, or something like that, according to which the average lifespan would be ‘only’ about 800 years; I say fabled because I’ve heard about it several times, and I was told about it by more than one person, but no one had the source and I wasn’t able to find it. If you happen to know where I can find it, please let me know, and I’ll update this post.)

Now, 8,938 (or even 800) years is a bloody long time, but it is not ‘forever’; however, you have to take into account that the fine chaps who made the simulation did not take into account how much we might be able to reduce the chances of ‘non-natural’ death over the course of almost 9000 years, and they didn’t do it for a very good reason: They don’t know if, when, and how much the chances of specific death causes are going to decrease. Taking wild guesses at it would introduce far too much uncertainty in the calculations to make them any useful. Regardless, it is perfectly safe to assume that over the course of such long lifespans, we’ll have figured out a way to make cars safer, for example, and thus reduce the chances of dying in a car accident. To be perfectly frank, that’s already happening with self-driving cars, and it is quite possible that within a couple hundred years cars will have been replaced by something else entirely.

Similarly, we can expect other possible causes of death will be either mitigated or eliminated altogether, pushing the average lifespan up and up while you still are alive—though, meanwhile, entirely new causes of death might appear. We are nothing but extremely complex pieces of biological machinery, and the more complex something is, the more numerous the things in it that can go wrong. (And yes, even if we became non-biological machines, technology through and through, as some people would like, we could still be destroyed or irreparably damaged. Could, but not necessarily would.)

On top of that, we can’t know for a fact if an unexpected world war, or a pandemic, will break out and push up everybody’s chances of death, or on a grander scale, we don’t know for a fact if the heat death of the universe will happen or not, or if we’ll have figured out ways around it by then.

In my opinion, the hard bit will not be making our death chances decrease fast enough; the hard bit will be making sure they keep decreasing fast enough forever without any significant oscillations. If the general trend of your death chances over an infinite time is a negative one (i.e., overall your chances keep going down all the time), relatively short periods of time during which your odds of death stay constant or even go up can be okay—if these odds aren’t too high to begin with. For example, if you had a 0.5% chance of death each year for just five years and then the chance went down again, most probably (but not certainly) you’d safely make it through those five years alive and you could look forward to a very long life. If this period of constant death chance was much longer than five years, your odds would be worse. However, there’s no way to know for a fact that we’ll be able to maintain a general downwards trend indefinitely, so, sorry, neither I nor anyone else can promise you with 100% certainty that you will never die, if that’s what you were hoping for. Then again, death isn’t necessarily the mathematical certainty some people would like to pass it for, which is all I wanted to show with this article, and just in case someone decides to interpret it the wrong way, let me stress again what this article doesn’t do. It

  • does not claim strong immortality is possible;
  • does not prove you (or anyone else) will never die/live forever;
  • does not prove that we will necessarily be able to constantly reduce anyone’s death chances enough to make their death over an infinite time extremely unlikely. It’s possible, but not certain;
  • is not a great discovery or anything like that. What you just read is basically a simple Calculus 101 exercise.

I also care to emphasise that I’m not flawless or always right, so if a fellow mathematician spotted a mistake in my reasoning, I’d be happy to correct it. Meanwhile, be careful out there, and try not to make your chances of death any larger than they have to be—so don’t browse LinkedIn for too long, ’cause bleeding eyes aren’t good for you.

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4 thoughts on “Mathematics, rejuvenation, and immortality walk into a bar…

  1. “And this secret spoke Life herself to me. “Behold,” said she, “I am that which must ever surpass itself”
    -Thus Spoke Zarathustra-

  2. Thank you for the nice article. An interesting consequence of your calculations
    is that we do not need a technological singularity in order to achieve indefinite
    lifespans. In fact, we may not even need constant growth rates of the economy.
    Let us assume that the economy has to grow by an factor of x if we want to
    decrease the probability do die next year by the same factor. Of course this
    assumption may be wrong. But if it is true, we achieve weak immortality even if
    the economy grows only quadratically instead of exponential. Compared to what
    most transhumanists think what will happen, quadratic growth is a fairly
    conservative requirement.

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