Laura Jimenez, whose PhD explores the idea that reality is a mathematical structure, tells us more
From the properties of atoms and the realm of quantum mechanics to the formation of galaxies, mathematics proves unquestionably effective in describing and predicting physical phenomena. A recent example can be found in the detection of the elementary particle known as the Higgs boson: equations predicted the existence of the particle decades before physicists working at the Large Hadron Collider at CERN could detect it experimentally. A question that lies at the intersection of philosophy and science arises: why is mathematics so successful at describing physical reality?
Mathematics is the only reality
While we might believe that differential equations that describe, for example, the behaviour of elementary particles – the most fundamental particles in the universe – are simply tools to approach such phenomena, there is a philosophical view which states that mathematics is not just a tool that describes reality: mathematics is the only reality.
This controversial position, known as the Mathematical Universe Hypothesis (MUH) is held amongst others, by theoretical physicists Max Tegmark, author of the book ‘Our Mathematical Universe’. With this theory, Tegmark solves the problems raised in 1960 by Eugene Winger in his famous paper “The unreasonable effectiveness of mathematics in the natural sciences”. There is nothing mysterious about the success of mathematics in describing reality – according to MUH – because the universe itself is a giant mathematical object. Initially, the claim might not seem as extremeas it is. If we think that the laws of nature assumed by physicists are formulated in a mathematical language, it seems logical to believe that these laws are somehow mathematical in nature.
But Tegmark’s theory doesn’t stop here: physical concrete objects can also be reduced to mathematics. The computer screen from where you are reading this article, the chair in which you are sitting, everything that you think is real is nothing but a product of an underlying mathematical structure. “So what about me?” you might be thinking. Are people made of maths too? According to Tegmark, yes. We humans are nothing but a self-aware substructure living in a relational reality; a reality made up from mathematical relations.
Does this sound exciting? Unfortunately for Tegmark, and for all of us who are sympathetic to his theory, MUH is incompatible with what has become quite a popular view amongst the most well renowned philosophers of science: scientific anti-realism. There is a vast philosophical literature arguing against the reality not only of mathematics, but also of science. But what is exactly scientific anti-realism? In a nutshell, it is a philosophical view that regards scientific theories and mathematical equations as a mere tool that humans have created to describe their perceived reality and systematize what happens in the world. The laws of physics and the equations scientists come up with do describe the patterns and regularities in nature, but they are not a fundamental part of reality. In other words, we don’t ‘find’ the equations, but we create them.
You might be thinking that scientific anti-realists are somehow against science, but this is not the case. Scientific anti-realism argues that just because a theory is empirically or observationally adequate doesn’t mean that it is true. But what exactly does empirical adequacy mean? It is just a word that philosophers use when a theory matches the observations.
If this sounds too confusing let’s think of an example. When working with very small scales of energy, physicists can’t really know the exact location of particles in space. So what do they do? Well, if you don’t know where something is, it is reasonable to start thinking about where it could be. This is what physicists do when using models in quantum mechanics: they describe the state of a particle using probabilities, more specifically, probability functions. And of course, these probabilities are described with maths. This model is highly empirically adequate; it matches the observations and helps us make accurate predictions, but is it true?
Mathematics – just a tool?
Are these small particles truly probabilistic in nature? Are these mathematical objects, i.e. probability functions, real? For the scientific anti-realist, no. As long as the model helps with the calculations and predictions, it is good enough. And as long as mathematical objects do the job, it doesn’t matter if they are real or not. What concerns the scientist is whether a mathematical model can describe phenomena and make accurate predictions. According to the anti-realist, it is not the job of science to care about whether such a model is also in touch with the truth. As you can imagine, this perspective is completely opposed to Tegmark’s view.
Is mathematics the ultimate feature of reality or just a tool that allows us to model our limited knowledge of the world? Does the success of mathematics in science imply that there is an external mathematical structure? If not, how can mathematical modelling be so accurate in describing the world? These are some of the questions I explore as part of my PhD project. I believe that there are many unresolved issued that can only be unraveled through a better understanding of the relationship between philosophy and science.
Tegmark, M. (2014). Our Mathematical Universe (Knopf)