Mathematical History

Harys Dalvi

December 2021

I found the following equation in a paper:

$$C = - \sum_{i=1}^{N} P(i) \sum_{j=1}^{N} P(j | i) \log P (j | i)$$
What was this paper about? Conditional probability? Entropy and statistical thermodynamics? Some other field of math or physics?

In fact, the paper was about deciphering the Indus script from thousands of years ago. But it did borrow ideas from both conditional probability and statistical thermodynamics. Such methods are actually quite common now in deciphering ancient scripts: this paper uses the same idea for symbols used in Iron Age Scotland.

The Indus script

When you think about applied mathematics, history is probably not the next field that comes to mind. You might think of physics, computer science, engineering, or chemistry.

But these days, it feels like math is extending into every field. You can't learn economics without supply and demand curves. Finance and psychology are full of data and statistical analysis. There's an entire subfield of mathematical biology as our understanding of biological systems improves. So why not mathematical history?

History is interesting in itself, but it also has a lot of questions that are useful and practical. What can Easter Island teach us about the effect of population size and resource use on a society? What does the Industrial Revolution mean for how new technology can reduce poverty? How does the Islamic Golden Age relate to global scientific collaboration and exchange of ideas in the Information Age?

These are just a few important questions from history that math, especially statistics, can help us to answer more thoroughly.

Small Example Question

This is the kind of question I imagine you might find for homework in a mathematical history class.

Collapse of States

You are given the durations of various empires in history [1].

  1. Use software to construct a probability distribution for these durations. What type of distribution is it?
  2. Based on this data alone, given that the United States has lasted for about 250 years since 1776, in which year is the United States most likely to collapse?
  3. Give one historical or statistical reason why this probability distribution may give a biased estimate for the collapse of the United States.
  4. I took the data into Python and plotted it using matplotlib. Here's the histogram I got.

    Histogram for number of years an empire will last

    To me, this looks like an exponential distribution.

    Now, in principle, we can evaluate the following integral to determine the year \(Y\) in which the United States is most likely to collapse. $$Y=1776+\int_{250}^{\infty} kt e^{-k t} \, dt$$ Take a moment to think about how strange that is: an integral telling you when the United States will collapse! However, since the data is discrete, it's easier to just analyze the data directly than to try to extract a value of \(k\). We can simply look at all the empires that lasted at least 250 years and find the average duration of those. This gives 550 years. So the year in which the United States is most likely to collapse based on our model is 1776+550=2326.

    Of course, this isn't a perfect model. One issue I can think of is that the list generally includes empires and dynasties, which might last shorter or longer than relatively democratic countries such as the United States. Another is that many of these empires were from long ago, and there might be characteristics of the modern era that make states last for more or less time.

    Big Example Questions

    I think there's a lot of potential for this idea of mathematical history. Here are some cases where that might be useful.

    Industrial Revolution

    The Industrial Revolution is a great example of the impact of technology on society. It had many positive impacts: poverty was greatly reduced, and new technology increased quality of life. On the other hand, it led to a global divide between industrialized and non-industrialized countries, and increased polution which was bad for health and the environment.

    The Industrial Revolution is neither the first nor the last instance of technology drastically changing society. A lot of the questions about the positive and negative effects can benefit from a mathematical approach: we might want to know if there was a statistically significant change in political systems or political stability in industrialized nations. Math can help quantify how even or uneven the reduction of poverty was both in individual societies and around the world.

    We might also want to look at the Industrial Revolution in the context of other big changes in history, like the development of agriculture, and try to quantify these changes. We can also look at how frequent these changes are over time.

    Imperialism and Global Influence

    The best known and largest example of imperialism is European imperialism in the early modern period. However, there are a lot of examples of imperialism before that, like the Mongol empire. Since then, soft power in today's world powers arguably has some similarities.

    It might be interesting to quantify the effects of global influence on both conquering and conquered nations. These effects would be functions of many variables (sounds like multivariable calculus) like the amount of direct or local rule, geographical distance between countries, or time period. Looking at these functions could tell us something about soft power today and the conflict between the US and China.

    Future of Mathematical History

    I don't think much thought has been given to the idea of mathematical history yet, so it'll take some time to develop a proper theory and apply it to important questions. But Isaac Asimov's idea of psychohistory from Foundation is similar. In the real world, some cool sources like Our World in Data have articles (like this one) that I would say fall under mathematical history. I'm curious to see how this idea develops in the future, and I think a lot of important results can come out of it.


    1. List of empires (Wikipedia) ^
    1. Entropic Evidence for Linguistic Structure in the Indus Script (RPN Rao et al., 2009)
    2. Pictish symbols revealed as a written language through application of Shannon entropy (Lee, Jonathan & Ziman, 2010)