Whether economics can be a science, and whether mathematics has a place in economics or economic science, seem to be vexed questions among heterodox economists. Having been a natural scientist for over four decades and thought hard about the nature of science and the place of mathematical models within it, I would hope to offer some clarification on these issues. After discussion, this post will be put in a permanent page.
This will be a summary. A longer discussion is given in my essay and chapter Is the Neoclassical Theory Scientific?
It is useful to break the process of science into two stages, each with two parts. Stage one is the perception of a pattern (or regularity or relationship) in our observations of the world, and the formulation of a description of that pattern. Stage two is the deduction of further implications of the pattern, and comparison of those implications with further observations of the world.
Stage 1(a). The perception of a pattern is not a logical or rational process, it is a process of perception or cognition. It is the creative part of science. For example, I might notice that a man and a dog walk past my house most days, and walk back a little while later.
Stage 1(b). The description of a perceived pattern constitutes a hypothesis. My hypothesis is that the man takes his dog for a run in the nearby park every afternoon.
Stage 2(a). An implication of my hypothesis is that if I were to look in the park after the man and dog pass my house then I should see the dog running in the park.
Stage 2(b). I can make a new observation to see if it is consistent with my hypothesis. I go to the park and see the dog chasing a stick thrown by the man. The new observation is consistent with my hypothesis. I conclude that my hypothesis is supported by the new observation.
I do not conclude that my hypothesis is proven. It is possible there is another explanation for what I have observed. For example it may be the man is going to the nearby bar for a drink, and only incidentally threw a stick for the dog when I happened to be watching.
Science is not about proof, although it is often spoken of loosely as if it is. This is because its hypotheses and theories are always subject to replacement by better theories. For example, Newton produced an excellent theory of gravity. However Einstein produced a theory of gravity that is more accurate and more general than Newton’s.
We cannot describe Einstein’s theory as right or true or proven because it might be replaced by an even better theory. Nor is it useful to say Newton’s theory is wrong, because for many applications it is still very accurate and therefore very useful.
The criterion for choosing among theories is that they are more useful or less useful guides to the behaviour of the world we observe. Einstein’s theory gives a more accurate description of the motion of Mercury around the Sun, and it describes black holes, bending light and other phenomena that Newton’s theory does not. But Newton’s theory is quite adequate for Earth-bound phenomena like falling apples and lobbed canon balls.
To re-iterate, science is not about proof or Truth or the mind of God. It is about finding descriptions of the observable world that provide useful guidance to the behaviour of the world, in whatever context we want to use them. It is a refinement of something we do every day, which is to figure out why things seem to be happening. That is why I could use such a homely example as the man and his dog.
Notice also that I have framed this discussion in terms of what we observe. I did not have to get into the issue of what might be behind what we observe, such as whether there is “really” an objective reality. I don’t think it’s a productive question.
Proof is something that occurs in the domain of mathematics and logic, not in the domain of science. If a conclusion follows logically from a premise, we say the conclusion is proven.
So how does mathematics relate to science? For a scientist, mathematics is a tool. The tool of mathematics can be used in Stage 2(a) of the process of science. In other words it can be useful to deduce the implications of a hypothesis. This would require the hypothesis to be posed in mathematical form in Stage 1(b). You can then describe the mathematical hypothesis as a model.
Mathematical models or hypotheses will always be approximations to the observable world, for two reasons. First, we can never be sure there is not a “better” (more accurate) theory (e.g. Einstein versus Newton). Second, the world is generally rather messy and we don’t know, at first, how to include all the complications.
This brings us to the issue of finding a judicious first approximation. There is no rule for this. It is part of the art, the non-rational creativity, of science, to find a description that captures much of the observed behaviour with only a few assumptions. For example, Ptolemy’s first approximation is that the planets follow circular paths around the Earth. However there are measurable discrepancies from the predictions of this model, so Ptolemy added a second level of approximation (epicycles). He also had to make other assumptions, for example about the coordination among the Sun, Venus and Mercury. In contrast, Newton gets there in one go: his inverse square “law” (hypothesis or model) of gravity gives an accurate description without further approximation. Except for Mercury, and then Einstein’s model takes care of that too.
Neoclassical economics is not science. It skipped through Stage One, perceiving a pattern and formulating a hypothesis (homo economicus, only one-on-one interactions, etc. etc.), became infatuated by the mathematics of Stage 2(a) and never got around to Stage 2(b), comparing with further observations. Walras was quite explicit that he didn’t need to test his deductions because he already knew the assumptions on which his model was built were true representations of the world. Call it hubris or call it faith, but it is not science.
Milton Friedman made a very confused attempt to articulate the idea of a good first approximation and said “Truly important and significant hypotheses will be found to have assumptions that are wildly inaccurate descriptive representations of reality, and, in general, the more significant the theory, the more unrealistic the assumptions (in this sense)”. This is so easy to misread or misconstrue it amounts to nonsense. A hypothesis base on “wildly inaccurate” assumptions will yield a wildly inaccurate theory.
Can there be a science of economics? In the sense I have described here, certainly. It is possible to discern patterns or regularities in the observed behaviour of economies, and one can deduce implications from such patterns that can be compared with further observations. However one must be humble, because economies are clearly very complex and are likely to produce surprises. One needs always to bear in mind the limitations of one’s hypothesis. All scientists need to be humble in this sense, though many forget, until reminded by critical colleagues.
Many heterodox economists seem to be averse to the idea of a science of economies, and this is understandable given the supreme arrogance and manifest nonsense that emanates from the dominant neoclassical school. However the lesson is to be a proper, and humble, scientist, rather than to forego the possibility of ever understanding economies.
Many heterodox economists also seem to have an aversion to the use of mathematics. This is understandable given that neoclassical economists have for more than a century confused their mathematical modelling with science. Again, the lesson is to know the limitations of your mathematical models and to be open to revising or abandoning them.
There are other aspects of science that I can’t discuss in any detail here. For example, a good theory does not have to predict the future course of the world, only the future results of more observations. That is why my own historical subject, the history of the Earth, can be a science. Also, falsification, as allegedly propounded by Popper, is not so simple in practice, because measurements always have some uncertainty and because mathematical models may not fully express a theory. I have had repeated personal experience of both features of alleged falsification.