When you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind.
  - Kelvin’s Dictum

Yes, and when you can express it in numbers your knowledge is of a meagre and unsatisfactory kind.
  - Frank Knight

Radical quantification is a technique for dramatically simplifying complex problems with no obvious angle of attack. It works by picking just one quantity that illuminates the essence of the problem (or at least gives you a good running start). In a sense it’s a form of dimensionality reduction– like doing principal component analysis and then looking only at the first component.

Illustrative example

To this day my first encounter with this technique is the most brilliant use of it I’ve ever seen– it’s William Nordhaus’s 1996 paper Do real-output and real-wage measures capture reality? The history of lighting suggests not (Nordhaus went on to win the Nobel Prize in economics in 2018 for his work on climate change). In the paper he attempts to answer this question: how much richer is an average American in 1992 than an average American in 1800? He explains why it’s an incredibly difficult problem:

The estimates of real income are only as good as the price indexes are accurate. During periods of major technological change, the construction of accurate price indexes that capture the impact of new technologies on living standards is beyond the practical capability of official statistical agencies.

His solution? Reduce the problem to one dimension: how many hours of work did it take to produce one lux of light? Light is an especially clever metric because the technology has been with us for half a million years, has always been in demand, and has continually tracked technological advancement. Nordhaus then goes on to do incredible detective work to determine the cost of light over time; the whole paper is worth reading, but here is the conclusion:

In terms of living standards, the conventional growth in real wages has been by a factor of 13 over the 1800-1992 period. For the low-bias case, real wages have grown by a factor of 40, while in the high-bias case real wages have grown by a factor of 190.

More examples

A more recent example of radical quantification: How Superhuman Built an Engine to Find Product/Market Fit. In this post Rahul Vohra, Superhuman’s CEO, explains his methodology for measuring product/market fit, and running his whole company to maximize this metric. This use of radical quantification is incredible because conventional wisdom is that product/market fit is a qualitative endeavor. It also reminds me of How to Measure Anything: Finding the Value of Intangibles in Business.


Another example use of radical quantification to answer a qualitative question: “how important will self-driving cars be as a technology?” There is of course no way to know, but in Artificial intelligence and the modern productivity paradox: a clash of expectations and statistics Brynjolfsson et al get a handle on it by comparing the reduction in automotive jobs to the total number of jobs in America:

According to the US Bureau of Labor Statistics, in 2016 there were 3.5 million people working in private industry as “motor vehicle operators” of one sort or another (this includes truck drivers, taxi drivers, bus drivers, and other similar occupations). Suppose autonomous vehicles were to reduce, over some period, the number of drivers necessary to do the current workload to 1.5 million. We do not think this is a far-fetched scenario given the potential of the technology. Total nonfarm private employment in mid-2016 was 122 million. Therefore, autonomous vehicles would reduce the number of workers necessary to achieve the same output to 120 million. This would result in aggregate labor productivity (calculated using the standard BLS nonfarm private series) increasing by 1.7 percent (= 122/120). Supposing this transition occurred over 10 years, this single technology would provide a direct boost of 0.17 percent to annual productivity growth over that decade.

Discretizing a continuous quantity

One interesting variation of this approach is to take a continuous quantity and discretize it. Paul Graham does this in Life is Short to get a visceral sense of how short life really is:

Having kids showed me how to convert a continuous quantity, time, into discrete quantities. You only get 52 weekends with your 2 year old. If Christmas-as-magic lasts from say ages 3 to 10, you only get to watch your child experience it 8 times. And while it’s impossible to say what is a lot or a little of a continuous quantity like time, 8 is not a lot of something. If you had a handful of 8 peanuts, or a shelf of 8 books to choose from, the quantity would definitely seem limited, no matter what your lifespan was.

Tim Urban does something very similar in Your Life in Weeks. He creates a life calendar (which you can buy) where each row has 52 columns (weeks), and there are a total of 90 rows (about the average life expectancy). Looking at the calendar gives you a visceral sense of how long your life really is.


A different example of discretizing a continuous quantity is the idea of an inferential step by Eliezer Yudkowsky. Consider the question of “how much more knowledge does Alice have over Bob?” This is a vague and difficult question; to answer it you have to figure out what exactly knowledge is and how to quantity it. But an inferential step is a specific concept or an abstraction that Alice can explain to Bob. And it’s much easier to count the number of inferential steps than to measure knowledge.

This post is part of the Creative Library– a series of clever techniques that simplify solving complex problems.