Mathematics: a surprisingly natural fit for science
But only when we look closely at the structure of scientific knowledge
Across most human cultures over time, daily life revolved around rituals to please or placate the various gods believed to control the world. These rituals were necessary as the gods, and the world, seemed capricious and inconsistent. In other words, the world, as most people have experienced it, isn't orderly or predictable. So the idea that we might be able to use mathematics, a way of thinking dominated by logic and order, to explain the world is highly unintuitive and surprising. And yet, modern science has been largely built on mathematical techniques.
However, this historically surprising development is not because mathematics has some special insight into the nature of reality. Instead its methods have turned out to be a perfect fit for the structure of scientific knowledge. It is less that the world is inherently mathematical and more that mathematics is well calibrated tool for the construction of scientific knowledge.
The structure of scientific knowledge
It is all tied to the particular structure of scientific knowledge and its relationship to reality. As noted previously, science builds knowledge via an interplay between theories (including hypotheses) and reality. We, humans, come up with a theory (or model) that we think explains how some aspect of reality works. Good scientific practice then systematically, normally via experiments, tests the theory against what actually happens. Ideally the focus is on trying to find ways in which the theory fails.
The dynamic interplay between abstraction or theory and reality defines all human knowledge. Science has refined explicit methods that take advantage of this structure, and established a norm of rigorous, systematic testing.
So theories are central to the structure of scientific knowledge. But what are they made of? It might seem like an odd question, but the answer isn't too difficult. Scientific theories are built from ideas, or concepts, that seek to explain the world.
So, for example, our modern theory of gravity is built from the idea that any two physical bodies with mass will attract each other, and that the attraction depends on the mass and the distance between the bodies. Or modern genetics is based on the idea that all physical life has core information that is inherited by offspring. Obviously, scientific theories are more complex than this and have many more concepts involved. But the basic building blocks are ideas and concepts.
When we construct a theory from concepts, we need to rigorously explore the consequences of the concept, including any non-intuitive ones. Scientific theory building therefore involves being systematic about understanding the implications of the concepts involved. Hopefully, if you read my previous post about mathematics, this is sounding very familiar. In it, I noted that
Mathematics is the systematic exploration and elucidation of the inevitable (or logical) consequences of different foundational concepts.
In other words, the basic structure of mathematics involves exactly the same logical process as scientific theory building. It is therefore unsurprising that mathematics has turned out to be useful in constructing scientific theories.
Why is mathematics so helpful?
While the structure of scientific knowledge means that mathematics is likely to be useful for theory building, there are a couple of different reasons why it has been particularly effective.
For a start, many of our foundational concepts for mathematics have been taken from observations of the real world. It stands to reason that the systematic exploration of these concepts is highly useful for scientific theories. For example, the core concept behind counting and arithmetic arises when we observe what happens to objects in the world. If objects in the real world behave in a way that corresponds to the concepts we have derived, then all the consequences we have identified through mathematics will apply in the world.
Mathematics sometimes takes core dynamics at play in the world, isolates them as concepts, and explores them in an artificial environment away from the complexity of reality. This is often highly productive as the full range of consequences of concepts is often easier to identify when dealing with them in isolation from the complexity of the world. Then, to the extent the dynamics identified hold universally in the world, the resulting mathematics will also apply.
A more surprising strength of mathematics is that it is famously unconcerned with real world applicability. This means that a lot of work has gone into taking interesting concepts and exploring their consequences - entirely for the sake of it. So, for example, mathematicians decided to look at weird non-Euclidean geometries (where parallel lines can touch) and take seriously the idea that -1 has a square root (and so we get what are known as imaginary numbers). The remarkable thing is that the resulting theories, based on apparently nonsensical concepts, are used in important scientific theories. General relativity is built on non-Euclidean geometry and imaginary numbers are used in electrical theory and engineering.
However, this success is not because mathematicians have some deep insight into the nature of reality. Instead it can be explained via the ongoing trial and error involved in scientific practice.
Reality often surprises us as it doesn't conform to our initial expectations or concepts. We often need to try out a range of different concepts to see which fits what we know of reality. As mathematicians spend their time playing with all sorts of concepts and theories without worrying about applicability, they end up building out a wide range of interesting structures and theories and so have lots of ideas that scientists can try out. Some of them turn out to be useful in scientific theories. In other words, mathematics supplies some of the creativity necessary to develop better scientific theories.
If mathematics matters mostly as a source of structured creativity for theory building, this changes the importance of mathematics in scientific practice. It isn't that mathematics describes reality in some special way, so mathematical approaches are not inherently better than other approaches. As we should expect, if the underlying concepts a type of mathematics is built on don't describe reality, then no amount of mathematical sophistication can help rescue a theory. In particular, the increasing reliance on statistical techniques is likely a distraction rather than helpful in the search for better theories. Instead, we need to focus on getting the core concepts right - no matter how counter-intuitive they may be.
Interestingly, the experimentation and play involved in mathematics also explains why science fiction has been a productive source of scientific and technological ideas. Often we need to explore concepts at a remove from our understanding of reality to really tease out the consequences and see how they work. Mathematics is the most rigorous way we can do that, but it isn't the only one.