November 23, 2025
By Stephen Stofka
I forget how old I was when I learned that there were numbers between the numbers, that 1.333 was between 1 and 2, for example. We lived on the east coast near the ocean. Learning that there were more decimals between 1 and 2 than all the grains of sand on all the beaches in the world astounded me. A brush with infinity. My dad worked for Lockheed Aircraft doing sheet metal fabrication and worked with decimals all day. “The prices in the grocery store are simple decimals,” he told me. “Take away the dollar sign and you have a decimal.” To me, it was a revelation, a connection between what we learned in school and the adult world outside school. Math was a skill, a mind tool.
This week I want to examine two axes. The first is an axis of countability, from discrete to continuous. The concept is familiar to students in computer science, math and statistics, but I want to use that axis to help me understand some philosophical and historical ideas. Discrete data is countable, like the number of students in a class. Continuous data is not countable, like a person’s height. At first, this might seem confusing, because we do measure height. Parents may mark their children’s increasing height on a door jamb, but those are marks we make of a continuous process of growth.
Like the marks of height on a door jamb, I want to contrast countable things as they appear in a continuous field of related data. For instance, we can count the number of days but time itself is continuous. We can always discover a smaller unit of time until we get to Planck time, the smallest unit of time current theory allows us to count (Source). We borrow money for a car. We can count that, but we can’t count the use of that car, the utility we will get from the car.
Most of us are familiar with the drawing of the Mandelbrot fractal that I included at the beginning of this essay. At first glance, the protuberances or spikes on the blob seem countable and the image seems defined, but as I zoom in, I see that the Mandelbrot fractal is replicated inside of each protuberance. I can keep zooming in and the fractal appears much like it did when I was zoomed out.
At the opposite ends of another axis are law and custom. Laws and regulations are defined rules. We can identify when a law was made, a regulation written. Customs, on the other hand, are behaviors and understandings that evolve over time. Marriage and funeral ceremonies help bind a community together. Trading goods and services helps us manage our daily needs. As a society becomes more organized, these customs become institutions. People write laws to codify the customs.
A law becomes law on a certain date using a specific number of words. It has definition like a whole number, but the meaning of that law is not so definite, like the real numbers that lie between the whole numbers. In a 1965 decision in Griswold v. Connecticut (1965), the Supreme Court struck down an old Connecticut law that banned the use of contraceptives by married couples and forbade medical personnel from providing information about birth control. The decision was based on a right to privacy not explicitly stated but contained within the “penumbra” of the guarantees of several amendments to the Constitution (Source).
Justices and scholars of different persuasions liked the result of the decision but not the reasoning. Former Justice Ginsburg thought the justification was too vague and that the court should not have restricted the right of privacy only to married coupled. She thought that the court should have based its decision on the equality principle in the 14th Amendment (Source). In a 1992 case Planned Parenthood of Southeastern Pa. v Casey, the majority of the Supreme Court held that the right to privacy had a better grounding in the 5th Amendment’s Due Process clause (Source).
Conservative justices who favor textual analysis do not like penumbras. They favor a just the text approach as though the text of a law provided some definite boundaries that separated the judiciary from the legislature. Sometimes the text of the law gets in the way of a desired conclusion and a textual analysis simply ignores the text as Justice Scalia did in the 2008 Heller decision. His reasoning dismissed a third of the words in the Second Amendment and overturned two centuries of law and custom. A law and its interpretation often follow separate but related paths.
Textualist justices prefer a narrow interpretation of the word liberty in the 5th and 14th Amendments. Richard Epstein (2014, p. 323) disagrees, making a case for an interpretation of liberty to include a broad liberty of contract. What does that mean? We enter informal contracts when we buy and sell goods in the marketplace. We enter employment contracts when we begin working for an employer. We enter a contract when we get married. He advocates limits on government intruding on these liberties which are grounded in custom, in the very nature of being human.
Last week I wrote about many of the variables that economists must infer from the data. Many of these inferred variables are rates, like the natural rate of interest, or a potential rate of growth, or a natural rate of unemployment. In these cases, the word “natural” is used as Adam Smith and other people of the 18th century used the word, as usual, common or average. If an unemployment rate averages 5% over ten years, that 5% becomes a benchmark. Economists focus on any deviations above and below that benchmark. The 5% is like a pencil line on a door jamb to mark a child’s growth. Each month personnel at the Bureau of Labor Statistics try to estimate unemployment but it is only an estimate, a mark within a continuous process. People leave jobs and start new jobs every minute of the day.
The economist John Maynard Keynes attributed the business cycle to an imbalance of “animal spirits.” While this sounds very much like Galen’s medical theory involving humors, Keynes recognized that investor sentiment is a continuous process. We can mark the day that the stock market dropped 10%, for example, but that is a discrete event that occurred within a field of investor sentiment. Because sentiment is continuous, it does not suddenly turn unless in reaction to an act of war or some other catastrophic event. When sentiment reaches a particular threshold, investors buy or sell. They react to the buy and sell decisions of other investors. The buying and selling are discrete events but the animal spirits that provoke those events is continuous.
What are discrete customs or cultural institutions? Rituals or holidays like Christmas and Thanksgiving come to mind. A people or a culture define the beginning and end of a holiday period. If it is May 10, it is not Christmas. What is continuous is the preparations for Christmas. Shortly after Christmas, stores begin to plan for the following Christmas. First comes the analysis of what was sold and not sold, the excess inventory or items out of stock during the holiday season. Was there a problem with the displays, the layout of the store or the merchandise? Were customers not directed properly to popular items? Did the store managers misjudge the popularity of some items? Were there problems with the wholesale distributors? Once again, we see a discrete event like Christmas operating in a field of continuous activity.
We can count the letters of the alphabet like we can count whole numbers. We use ten symbols like 1, 2, 3 and so on, adding to them and rearranging them to count to infinity. We have 26 letters and we can do the same as we did with numbers, adding letters to form the names of columns in a spreadsheet or words in a book. The letters and numbers are symbols, formed by darkening some of the points of a page that has an uncountable number of points. Again, we see this phenomenon of the defined, like the number 1, as little more than a mark in a field of white that is not countable.
Historians study events, marking their beginnings and endings with other events. World War 2 began when Hitler invaded Poland and ended when he committed suicide as the Soviet army entered Berlin, the capital of Germany. Like a law, the interpretation of a war defies easy analysis because war emerges from a continuous stream of perceived injustices, human vanities and vices. This provides a deep pool of research material for historians as they try to unravel the causal threads before, during and after a war.
We define people, animals, plants and things with a set of facts that act as a boundary. Anatomically, a horse has four legs, and a long head, and long legs, and one hoof per leg, and so on. Like the drawing books we had as kids, these facts are the dots, the boundaries of the definition of a horse. But that doesn’t answer the question What makes a horse? Those are amorphous qualities that each of us might see differently. Gentleness, agreeableness, patience and pluckiness come to mind. A horse is its well defined physicality and its less defined qualities, as well as the feelings that a horse evokes in me.
I began this essay with a story from grade school. Finally I jumped off the philosophical cliff, concluding that everything that is bounded swims in a pool of the boundless. Everything that exists is an instantiation of a continuous process, an emergence from some sequence of circumstances. What do you think? Have a good turkey day and I hope to see you next week!
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Photo by Shino Nakamura on Unsplash
Epstein, R. A. (2014). The classical liberal constitution: The uncertain quest for limited government. Harvard University Press.
Innocent Investor 