Understanding scientific laws
andrea.parisi | Published: 4 Aug 2023, 9:06 p.m.
Economics is a social science, where theory is developed using logic, a look at historical data and comparison with present data. Theory then gets formulated in a set of equations that are used to make forecasts. Samuelson and Nordhaus state, in the introductory chapter of their university textbook "Economics", that economists use the scientific approach to understand economics. Science at large has adopted this approach in order to achieve a reliable description of our world: it goes through the development of theories based on observations, and methods that aim to confirm or dis-confirm such theories. This approach has a particularly strict meaning in physics, where it takes the form of the scientific method. The latter is a methodology where observed facts are quantified through measurement, and a mathematical theory describing a relationship between measured quantities is provided and tested through experiments. It is this mathematical relationship that can be used to produce reliable forecasts. In other non-hard sciences this path cannot be followed. Take, for example, the case of historical data sets. Several historians have provided estimates of the past human population on Earth, of specific areas and cities, based on a number of methodologies which give us an idea of how crowded the world was. For instance, multiple estimates of the population of the city of Rome in the first century AD rate it at about 1 million inhabitants. However, estimates based on housing density and occupancy in Pompeii and Ostia have suggested a much smaller population of about 500,000 individuals or less. In fact, data used for these estimates include written statements and records, as well as information on burials, and other staff. The methodologies behind all this trove of information is scientific, but the estimates themselves vary, because each piece of information has to contend with its own reliability (for instance, in the case of written records they may be misinterpreted, they may contain errors or altered data for political reasons, whereas burial counts may be incorrectly extrapolated) and suggests a specific estimate. All of this leads to different estimates depending on how we think certain facts are more reliable than others. We have reasonable estimates, but we cannot provide exact relationships as we have no certainties and we cannot perform experiments to check: we cannot reproduce Rome in the 1st century AD.
Economics also uses historical data, but it typically focusses on recent years and thus, unlike the case of ancient Rome, data are well measured and reliable. However, the problem with experiments still stands: in economics experiments are not possible. Looking at historical data is not equivalent to performing experiments, and as such, economists do not follow the scientific method. Historical data do not give you the possibility of tuning the various contributions to a phenomenon which is being observed, you cannot isolate the effects of each contribution by tuning them out: historical data is given, and there is nothing you can do to change it. Historical data is hardly used in hard sciences to develop a mathematical theory: it may be used in certain contexts as a tool to verify the goodness of an independently developed mathematical theory1. Yet economic theory makes large use of mathematical relationships, and most economic laws describe the relationship between the various economic variable through mathematical formulas. The fact that it is possible to build such relationships, does not make them valid in the same way as it occurs in hard sciences. The lack of experimental testing prevents the discovery of any fundamental flaw that might be present in the theory. In alternative to experimental evidence, economics makes use of thought experiments: even in physics Albert Einstein based his theory of special relativity on gedankenexperiments, that is thought experiments. However, a thought experiment in physics and one in economics are not the same. The reason, in my view, is that physics is built on granitic foundations, while economics is not.
For a non-scientist, it is hard to understand this difference: it remains elusive even with sufficient scientific background and it is certainly not self evident. To clarify, let us consider an example of a “law” in physics. Suppose that you are driving a car on a motorway, and you keep a constant speed, say 70 miles per hour. Now if I ask you how far away will you be in 2 hours from where you are now, it is easy to work out that you will be 140 miles away — that is 70 miles each hour times 2 hours. If one writes this calculation more formally, one would need to use the following equation (equation of motion): \begin{equation} \label{eq:motion} x = v\cdot t \end{equation} where $x$ is the travelled distance, $t$ is the time interval, and $v$ is the speed (supposed to be constant). So, in practice, you take $v = 70$ miles/h, $t$ = 2h, and multiply the twos to get the final result: $x$ = 70 miles/h $\times$ 2h = 70 $\times$ 2 miles = 140 miles. Exactly what we wrote just a few lines earlier. Note that in the calculation we are also multiplying the units of measurements: miles/h $\times$ h = miles. The "h" at the numerator and denominator simplify and the result is "miles''. This multiplication of units of measurement is an aspect of dimensional analysis, which is essential to corroborate the correctness of an equation between physical quantities. Equations must be dimensionally correct to be valid: you cannot equate a time to a length. In this case, the left-hand side is a length (expressed in miles), so the right-hand side must also be a length.
So far so good: so what is the point? The point is that the above equation implies several things that are not readily explicit and I will try to list them here: I need to stress that these implied facts are crucial to view the above equation as a valid ‘law’ or ‘theory’. First, $t$ measures a time interval. A time interval is a physical quantity that is defined by how it is measured. We measure a time interval using a “chronometer” or time measuring device — a stopwatch. That defines a unique quality to the quantity ‘time interval’ with respect to other measurable quantities. We do not measure time using a balance or scale, we do not measure it with a distance metre, nor with an ammeter or a multimeter. The fact that we measure time with a stopwatch identifies the unique quality of time. We can associate a unit of measure to this physical quantity: one hour, one moon cycle, one second: you name it. Scientists have devised a universal set of units called the “International System of Measurements” that precisely defines such units in a way that if two people in two different parts of the world (and maybe different times) make a measurement of something, if that something is exactly the same, they get exactly the same measurement. The unit of time in this International System is the second, and it is defined in a way that anyone in the world can use exactly the same definition of “second” as a unit of measure. Some time in the 1800s, the second was defined as the 1/86,400th part of the mean solar day. Essentially, it was the 1/86,400th part of a day (where 86,400 comes from the fact that each day has 24 hours, each made of 60 minutes, each made of 60 seconds, giving a total of 24 x 60 x 60 = 86,400 seconds), except that days do not have exactly the same duration throughout the year because of variation of the Earth distance from the Sun during its path around it, so the “mean” solar day (meaning the mean duration of a day throughout a year) was used as reference. Once again, modern techniques led to detection of variations in the mean solar day from year to year, so a more precise definition that used modern knowledge in physics was adopted. The current definition is quite obscure: it is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the hyperfine levels of the unperturbed ground state of the 133Cs atom. Unless you are a physicist, you will hardly understand what it means, but it does not really matter as scientists know how to measure it with extremely high precision. So what matters is that we have a precisely defined and measurable unit of time, which is agnostic to where and when it is measured. In other words, this definition will provide the same result with high accuracy, provided we have the apparatus and the knowledge to perform the measurement, if we do it on the cliffs of Dover, on the river banks of the Ganges river, in the middle of the Amazon rainforest, or on a human colony on Mars!
Then we have $x$, which measures distance. A distance is a physical quantity that is again defined by how we can measure it: we measure a distance using a distance meter, not with a scale or a stopwatch. This differs completely from the physical quantity ‘time interval’ because we measure it differently. Again, we can associate to this physical quantity a unit of measure: for example one “mile”, or one “metre”. The key important aspect of this is that it must be a “universal” unit of distance. That ought to say, a unit that is defined independently of where or when we measure it. The International System of Measurements uses the “metre” as it has a well defined formal definition. Until the 1799 AD it was defined as the 1/10,000,000th of the distance between the North Pole and the Equator along the meridian passing through Paris (the famous “line of the rose” popularized by “The Da Vinci Code” novel). Through the years, the definition has changed as more and more precise measuring techniques led to the detection of variations in time of this distance (the Earth is continuously changing shape for various reasons). The latest definition is once again more obscure: it is defined as the length of the path travelled by light in a vacuum, in a 1/299,792,458th of a second. This definition requires knowing what a second is, but that does not matter as long as we have an overall coherent set of units, and we know how to measure a second with sufficiently high precision, which is actually the case. What is relevant thus is that scientists know how to measure this length with high precision, anywhere in the galaxy, today as in 100 years, so we have an accurate definition of a unit of length. Time and distance are physical quantities that have distinct qualities, as they are defined in terms of how you can measure them. So the equation \eqref{eq:motion} has meaning in physical terms because it is a relationship between measurable quantities, whose unit of measurement can be universally and unequivocally defined.
Furthermore, when scientists say that eq. \eqref{eq:motion} is a "law" what they mean is that this is something that came out of a scientific methodology: the well-defined process used in exact sciences that guarantees that the "law" has been verified to be behind observed phenomena in Nature. This process, the "scientific method", is made of a set of steps that work from gathering a set of observations, formulating a hypothesis, conducting experiments to verify or dis-confirm the hypothesis, produce predictions if the experiment were successful, verify the predictions. Only in this case one can have some level of confidence that the hypothesis can be regarded as a "law", and indeed it requires continuous verification and verification for the hypothesis to become widely accepted as true.
All the above is required to have laws that can be trusted as a solid basis for further research and for forecasting, and unfortunately, much of this is missing in economic theory. Any equation in economics has to do with money or prices, but as I will discuss in later articles, these are never properly defined. There is no notion of 'dimensional analysis' which permits verifying the correctness of an equation with respect to the quantities it deals with. Money appearing in equations is measured as the number of units of a currency, which however does not have an objective and space-time agnostic definition. The lack of these ingredients undermines the capacity to have a reliable mathematical theory and forecasts. Moreover, mathematics is often used in ways that subtly hide the lack of knowledge of certain phenomena, or the lack of a reliable experimental proof. This specific use of mathematics is widespread in economic theory, and has to do with the intertwining of the hard and soft aspects of the theory, whereby mathematics is used to extend the theory to include cases that do not conform with it, and then the result is interpreted in terms of a verbal description that feeds back into the mathematical theory, in an ever ongoing process that drives away from an objective description. Of course this does not mean that it is not possible to produce good theories: not everything is hard science. However, the current theoretical frameworks presents several issues: some (but not all of them) have been picked up by other economic schools, but most approaches fail to recognize the intrinsic issues arising by the lack of correct foundations.
- There would be much to say regarding the role of the scientific method in research. There are other domains were experimental evidence is hard to obtain: cosmologists (physicists studying the universe and its history) are in an apparently similar situation as you cannot do experiments with distant stars or with the universe at large. This has a number of consequences, among which the existence of competing theories that explain observed data and scientists behind them behaving similarly to members of competing schools, as in economics. Having said that, cosmology has to do with a physical universe that tends to behave like a clockwork mechanism. Also, it can leverage a profound understanding of physical phenomena, and an extremely high level of mathematical knowledge - a level that is simply out of reach in economics. Economists are in a weird situation, as they try to build solid scientific economic theories without the solid support of the scientific approach which, at its heart, uses experimental evidence.
