Episode 4: Truth By Definition

Some claims can’t possibly be true, while others can’t possibly be false. On the other hand, most claims that people make could be either true or false, and we are faced with the challenge to decide. Here we explore a particular kind of truth and its role in mathematics and science.
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Transcript:
A guy walks into a bar. He sits down beside you and declares: “I always lie.” “Okay,” you reply hesitantly. It seems an odd thing for a stranger to say out of the blue, even as a way to break the ice. And, there is something odd about the statement itself. For, if he is lying, then he is contradicting himself. His statement “I always lie” would be a lie. Imagine he had said instead: “I always tell the truth.” He cannot be contradicting himself, but it still might not be true. In fact, we are suspicious of such self-affirming claims, which hardly inspire confidence.
Some statements are self-affirming in a different way, simply by being redundant. For example: “All men are males.” This is true by definition, since man and male simply refer to the same thing. The general equivalent in logic is “A equals A.” The equivalent in arithmetic is “One equals one.” These are tautologies, which are true by definition because any defined thing is equal to itself. Yet, real things (which may not be well-defined) change over time and become no longer equal to their former selves. Tautologies involve timeless definitions and not real things in time. The mathematical statement “X plus x equals 2x,” is a tautology if it is a matter of definition. In other words, if “two” is defined as the regrouping together of “one” and “one.” But such general and abstract statements as occur in mathematics are likely gleaned from actual experience with things in the world. Such experience depends on being able to identify individual “things.” Apples are clearly individual objects you can add together. But what about clouds?
While mathematical truths may be tautologies, scientific claims are not supposed to be. By definition, empirical claims cannot be true by definition. They can be true, but they can also be false, provided there is something you can discover that would help you decide. A claim with no means to test it is only speculation. The worst thing you can say about a scientific theory is that it’s not even false.
Scientific laws are generalizations of many data points gained through measurements of real systems. However, the very notion of “system” is a defined thing, an idealization that may correspond only roughly to the reality it is supposed to represent. The Solar System, for example, is now defined to include the sun and eight planets. (It used to include a ninth planet, Pluto.) But it could also include the asteroids, comets that come and go, and distant bodies far beyond Pluto. It is a simple math problem to consider the gravitational interaction of two celestial bodies. But three or more becomes very complex. The whole idea behind idealization is to simplify enough that the system can be treated mathematically.
A law that is formulated to express the observed behavior of a system is a tautology to the extent it describes the idealization perfectly. However, it only imperfectly describes the corresponding reality it is supposed to represent. The law expresses an average of observed data points, from which there are always outliers that don’t fit the curve. It’s a generalization, a rule to which there are always exceptions. The curve itself is an idealization, a line drawn through the data points after the fact. If a mathematical equation can be found for this curve, it becomes a mathematical law. It is a tautology in the sense that it is simply an alternative expression for the geometric curve. So, the idea of laws of nature is somewhat ambiguous. On the one hand, laws of nature are generalizations about the real world. On the other hand, they are imaginary human inventions, expressing claims that are true by definition.
The word ‘law’ conveys a dual meaning. In science it means an observed regularity, a pattern that can be formulated. In jurisprudence it means a decree made by an authority, such as a king or a legislature. That is obviously a human creation. Yet, from our religious heritage, we also have the notion of divine decrees, like the Ten Commandments. Because the early scientists were devoutly religious, they imagined that the laws of nature were divine decrees: God created the matter of the universe and also the laws of physics that govern over that matter. The idea of governing laws still lurks in the background of science, especially in the idea of determinism.
At the same time that the early scientists were trying to discern the divine decrees that govern the universe, they were also discovering that the universe could be understood as a machine. Ordinary machines are made by people. They have well-defined parts which interact in well-defined ways. They can be perfectly understood by those who made them. If the universe is a machine made by God, then we mortals should be able to understand it too—at least to the degree we are made in God’s image. So the thinking went. Like a clockwork or simple machine, the universe follows precise laws, which should allow humans to predict its behavior forward and backward in time as precisely as one likes.
However, it is not the real universe that is deterministic, but the idealization as a machine. The laws of physics have no power to “govern” how matter behaves, because they are no more than pithy descriptions of how it actually does behave. Where possible, these pity descriptions are expressed mathematically. Equations are deterministic because they are true by definition. They describe mathematical curves perfectly, yet they only imperfectly describe the natural world. Even today there remains some confusion among physicists about the status of laws, especially in the microscopic realm, where quantum systems seem to really be simple and no more than what we define them to be. However, the lesson learned on the ordinary human scale is that no idealization ever coincides perfectly with the real thing it describes. If that is so, even at the quantum scale reality is more complex than meets the eye.
If you have enjoyed this podcast, tune in to further episodes of “The Stance of Unknowing”, or visit my website: www.stanceofunknowing.com.
Music by John Nemy. Production by John Humphrey, Eureka Web Design.