In my previous post, I discussed a conjecture that certain quantities near singularity should demonstrate power-law behavior, and based on this we can potentially predict when the singularity is supposed to happen. In this post (you can read it independently if you haven't read the first one), I explain why I think that *a priori* probability of this statement holding true is high enough to make the efforts to check it worthwhile, and I discuss how can we validate (or invalidate) it.

As I mentioned in the previous post, the Earth population was growing almost all its history till 1960^{th} as a hyperbolical growth. Later I found an excellent post by Scott Alexander, that I highly recommend on this topic. In a nutshell, the reason for the growth is the following. In the ideal situation with unlimited resources and no scientific progress, the speed of the population growth would be simply proportional to the number of people that is already here, making the growth exponential. However, there is scientific progress in real life, that can be estimated as roughly proportional to the number of researchers, which in turn is also proportional to the number of people . The speed of population growth can be estimated (in the simplest model) as proportional to the current level of scientific progress. This gives us the equations, leading to the hyperbolic growth:

There can be different reasons for the termination of this growth. First, at some point, rapid population growth becomes physiologically impossible (for example, doubling the population every day is impossible for humans). Second, we can invent a super AI, and reach the singularity. In the first scenario, the termination occurs when reaches a threshold value, and in the second, when reaches a threshold value. Notice that these two events are not connected in this model – namely, they do not have to coincide at all. There is no inner reason for it.

In our world, the hyperbolic growth terminated far before the physiological limit, because of the demographic transition – people just started to have fewer children. “Aha” – one could say – “ here you go! The model stops working, the speed of population growth is not simply linear on science, but has more complicated non-linear dependence. We need non-linear terms!”

However, adding non-linear terms is opening Pandora's box – there are too many options. And these options leave a large space for the time when singularity in science happens (in principle we can add non-linear terms that will cancel singularity at all). What can we do? Let me use an analogy with the motion of planets. Looking at them from the Earth system is super complicated. You need to calculate the Ptolemaic epicycles, and it gets very hard. However, you can go to the Sun system, and then you have simple ellipses. Can we switch our system to "Sun"?

So, let us use two hints in which direction to go. First, science singularity is apparently not far from the point when population growth terminated (comparing with the total time of the hyperbolic growth). Second, we observe the rise of computers at the termination of the hyperbolic growth of the population.

Now we can collect all the hints together and revise the model. Before our main quantity was the number of people. To explain its hyperbolic growth, we introduced in some sense hidden variable, science, that also in this model grows hyperbolically. The reason for science growth were humans, and they were the only agents, producing science… until the invention of computers. So, we can try to rewrite our model, leaving equations the same, but instead of the number of people use more general “science-producing power”. So, now we still have easy hyperbolic equations without any non-linear terms, but instead of the straightforwardly measurable number of people we have less transparent " power". If the model is correct in any sense, there should be a way to measure either "science" or "power".

The reasonable thing to do would be the following:

- Create a list of quantities that can be measured and that are potential candidates for either “science” or “power”.
- Check if at least one of them demonstrates clear power-law behavior.

For (I), I have the following ideas (I will really appreciate more suggestions and critics in comments):

- Computational power (human with some factor + computers).
- A number of scientists (CS counts ) + amount of computational power used for science.
- Amount of time (human + computer) spend on science.
- Amount of published papers + code written
- Amount of all available data (books, Internet…)

For (II), how to get it?

- They're not so many companies producing processors. They must have statistics on how much they produced over years. Maybe it is even accessible on their websites.
- No idea
- No idea
- For published papers, I think the data is available. For code, no idea.
- No idea

I will appreciate any comments/critics. If you want to go ahead and try to see what well-defined quantity like computational power gives, it is awesome – I will be glad to see results in the comments.