When most people think of mathematics, they usually think of algebra, geometry, calculus, differential equations, and that is pretty much it. While there is nothing wrong with this analysis, the truth of the matter is, those subjects are on the bottom of the totem pole of mathematics. There is so much more to mathematics. Topology, for example, is the study of topological spaces. Topological spaces could be anything from the real number line to the 11 dimensional shape of our universe that some theoretical physicists claim it to be. In general, mathematics is the study of patterns. This allows us to do remarkable things, and one example is to “see the unseen”.
Consider the history of black holes. Einstein did not believe they existed because they were too “mathematical” and couldn’t arrive at their existence intuitively. There were other physicists who disagreed with him. Clearly Einstein was wrong here, but how was the debate settled? Black holes have such a large gravitational pull that we cannot see them. They do not even let light escape from their grasp. So without being able to physically observe them, how can we conclude that Einstein was wrong? Well, that is where math comes in. The laws of conservation can tell us many things, and one of them is if we put 10 gallons of water through a hose and only get 9 gallons on the other side we know there must be a hole somewhere in the hose. This is how physicists and mathematicians can “see” black holes. If we observe 10 particles going through a selected area and only 3 emerge on the other side we know there must be a hole somewhere in that area. I am a bit of a nerd so things like this are amazing to me. We can look at a piece of paper with symbols and numbers and literally see a black hole in those symbols and numbers. (it turns out the equations of general relativity hold true under the extreme conditions of black holes and hence, they are the strongest evidence that Einstein’s theory is true)
The powerful tool of mathematics possesses a limitless ability to discover patterns in the world, and this would also include patterns of economics. It is because of the powerful nature of mathematics to discover patterns in the world that I find it bothersome when certain economists act as if it is “silly” to use mathematics in economics. Some economists even argue that those who use mathematics in economics are not doing “real economics.” Mathematical economics is relatively young, and advancements to the methods are being improved constantly. There have been 6 (correct me if I am wrong) mathematicians to win the Nobel Prize in economics for game theory. These methods are being applied to various areas of economics, especially oligopolies, with very good results.
This isn’t to say all we need is mathematics, far from it. Good theory is always the most important part of an economic paper. All I am arguing is that mathematics can be a useful tool to complement the theoretical portion of economics.
It took mathematics to prove one of the greatest physicists was wrong about his disbelief in black holes. Will those who denounce mathematical economics come around if there is a truly significant advancement in economics achieved mathematically?