the usefulness of 'purely mathematical' structures

Aristotle believed that true science provides us with explanations of the causes of the phenomena it studies. For example, true science (as it relates to astronomy) should seek to explain the cause of celestial bodies’ movement through physical models, rather than just provide mathematical models. The Jewish philosopher Maimonides echoed Aristotle’s view: he claimed that while Ptolemy’s model of the cosmos could accurately predict and retrodict the trajectory of celestial bodies, it does not “tell us in which ways the spheres truly are.” In other words, both philosophers thought that Ptolemy’s model of the cosmos was not true science, and thus should not be a goal of scientific inquiry. However, I argue that purely mathematical models that serve to accurately predict future events should be a goal of scientific inquiry.

One condition for something to be a ‘goal of scientific inquiry’ is that it serves to further other scientific progress. This may sound cyclical, but let us remedy that by defining “true scientific progress” in Aristotle’s terms: progress towards something which explains the cause of certain phenomena. So, if we can show that there are examples of accurate purely mathematical or geometric models that inspire other “truly scientific” endeavors, we have shown that the model should be a goal of scientific inquiry. I will present two such examples: one hypothetical example which implements Ptolemy’s model, and one example involving Newton’s Law of Gravitation.

For our first example, let us assume that we still do not know the cause of celestial motion, and further, that we only have Ptolemy’s mathematical model to inform us of the locations of the planets in our solar system. Let us also imagine that we are trying to land a craft on Mars in order to learn more about the composition of Martian soil. Under Aristotle’s definition of science, learning about the composition of Mars is true science — if we can discover what substances Martian soil is composed of, we can gain insight into how the planet was formed, what elements were necessary for its creation, and general information that explains the cause of various related phenomena. In order to land on Mars, we clearly also need an optimal landing spot on Mars and a trajectory that our spacecraft will take, avoiding celestial bodies along the way. To deduce this information, we also clearly need predictions of the positions of these celestial bodies and their trajectories during the time that our spacecraft is on its mission, and all of this information can be found through the ephemerides that Ptolemy created based on his mathematical model. Through this hypothetical example, we can see how Ptolemy’s mathematical model can contribute to the understanding of the causes of planetary phenomena. Since this model can serve to further science, it should be a goal of scientific inquiry.

For our second example, let us turn to Newton’s laws — specifically, Newton’s Law of Universal Gravitation. The law states that “every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.” Newton’s discovery is an extremely accurate prediction of everyday phenomena, including the motion of celestial bodies and how masses act as they fall. However, Newton’s law was not applicable in some extreme cases, like regions of high gravitation near black holes. In these regions, there exist phenomena that cannot be explained by Newton’s law: event horizons (a line past which even light cannot escape) and gravitational lensing (the bending of light due to an area of high gravitation). After observing these phenomena, Albert Einstein proposed his theory of General Relativity, which scientists now agree to be the best universal explanation of gravity. However, despite the issues in his causal understanding of gravitation, Newton’s mathematical model helped perform calculations to launch spacecrafts and satellites that sought to explain various phenomena. For example, Newton’s laws were used to calculate the specific trajectory of the Voyager spacecrafts. These spacecrafts then went on to take detailed pictures of Jupiter and Saturn, which helped improve our scientific understanding of the formation of these planets. Clearly Newton’s law helped further science and thus, these mathematical models should be a goal of scientific inquiry.

Critics might argue that the pursuit of mathematical models might hinder the pursuit of an external truth. That is, they would rather spend their time focusing on the root cause of phenomena. However, in both examples above, the application of these mathematical models gave us more insight into the physical nature of the universe: both Ptolemy’s model and Newton’s law of gravitation helped us understand more about the composition of celestial bodies, and the nature of their creation. Another counterargument is that a focus on mathematical models can lead to an over-importance being placed on mathematical beauty. This leads to other, less aesthetic models not garnering enough scientific attention. For example, take the idea of supersymmetry. This is a theoretical phenomenon where each particle has its own symmetric partner particle. This idea is extremely mathematically elegant, but we haven’t seen any experimental evidence for this. Regardless, physicists spend time searching for evidence for this model rather than looking at less elegant models that may be supported by empirical evidence. This means we need to place restrictions on how we use mathematical models to conduct further scientific inquiry. We must understand that they are just models that we can use to predict and retrodict, rather than assuming that they provide a complete, causal explanation.