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Applied Mathematics

An elementary argument to predict the future, exactly half the time

Crossroads of two futures, imagined by MidJourney.

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The ability to predict future events lies at the heart of physics. Ever since its inception, a key goal of the discipline has been to forecast the future state of a physical system based on current knowledge. In my work on stochastic and quantum systems, it’s clear that a system’s current state doesn’t always deterministically map to a future state.

Rather than diving into equations of motion today, we’re going to explore a handy risk analysis tool. Regrettably, I’m unaware of the official term for this method, or where it’s most commonly applied. So, if you’re familiar with this approach, please enlighten us in the comments. This method bears some resemblance to the doomsday argument and employs the Copernican principle, both of which we’ll touch on later in this post. I recently brought this line of thinking up when visiting Tim Byrnes at the NYU-ECNU Institute of Physics: the discussion was fun enough that I figured its worth presenting it here.

Our initial premise is straightforward and can be encapsulated in assuming the existence of a finite lifecycle. We’re interested in predicting the endpoint of a lifecycle. This suggests that the entity in question – for example, a university or any other institution – has a finite lifespan with a clear beginning and end. We further presume that this entity currently exists and that we know its inception date. It’s intuitive to illustrate these premises on a timeline, as follows:

Given our knowledge of the inception time which we set to 0, we’re aware of the total elapsed time, T_0. Our goal is to predict the endpoint of the current lifecycle, based on this information, which we’ll refer to as the unknown total time, or end time T.

Let’s make a reasonable assertion along the lines of the Copernican principle. There’s nothing special about any particular time or T_0, the total elapsed time. You’re either in the first half or the second half of this unknown total interval. Each of these options carries a probability of \frac{1}{2}.

If you can accept that, then it’s equally reasonable to assume that, with a probability of 1/2, you’re in the middle interval, as illustrated below:

To estimate T, given the total elapsed time T_0 and the fact that this assumption holds true 50% of the time, we consider the scenarios in which T_0 is either at the start or the end of the middle interval. These scenarios provide upper and lower bounds, as illustrated below:

So, we have two scenarios that represent extreme points. Either T_0 is at the start of the middle interval or the end of the middle interval. As we know the total elapsed time T_0, we can assume that the lifecycle will not exceed 3 T_0, and it will extend into the future for at least another T_0/3.

I hope this provides a clearer understanding of this concept, and I look forward to your input and questions in the comments below.

The doomsday argument a probabilistic argument that suggests, assuming you’re a random member of the set of all humans born, there’s a higher probability that we’re in the middle of the distribution of all humans that will ever live. Based on current population estimates, this implies that the end of the human species may occur sooner than we might intuitively expect. It was proposed by Brandon Carter in 1983, and is based on a form of anthropic reasoning. It was championed by philosopher John A. Leslie and was independently discovered by J. Richard Gott.

Named after Nicolaus Copernicus, the Copernican principle posits that Earth and its inhabitants aren’t in a unique or privileged position in the universe. In the context of the Doomsday Argument, it’s used to justify the assumption that a person could be considered a random sample from the set of all humans that have ever lived and will ever live.

The evident problem with this line of reasoning is that there is no way of knowing if we are in the middle interval or otherwise. We also assumed that all time points are equivalent.

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