Thinking Outside the Right-Angled Triangle: How Irrational Numbers Changed a Paradigm

In mathematics, we distinguish rational from irrational numbers. In very simple terms, rational numbers can be expressed as a ratio of two other whole numbers (Hosch, 2009). For example, the number 1 can be represented as the ratio 1/1, the number 0.125 as the ratio 1/8, and √225 as 15/1, therefore they are all rational. Conversely, it is impossible to express an irrational number as a ratio of two whole numbers. Examples are π (“pi”) and ϕ (“phi”, ironically nicknamed “the golden ratio”), and the square root of 2. No matter how much we would like it to, π will never exactly equate to 22/7, or 355/113, and we can spend our entire lives writing out the non-repeating and infinite decimals of 3.1415926…

It is important to know that there was a time when mathematicians thought there was no such thing as irrational numbers (Heilbron, 2001). This belief was prominent in the 5th and 6th centuries, back when Greek philosophers were considered living legends, when geometry was pretty much all that Ancient maths had to offer, and when Pythagoras of Samos strolled around the city of Croton, teaching his circle of followers the famous equation c2 = a2 + b2. Pythagoreanism refers to his philosophy that the entire universe can be expressed in rational numbers (among other conjectures, Thesleff, 1999).

For some time, the Pythagorean followers lived their happy philosopher-lives immersed in the study of right-angled triangles and other geometrical shapes, until Greek philosopher Hippasus came about and accidentally made an outrageous discovery (Clegg, 2004). The legend goes that he was on a sea voyage when his innocent, scientific curiosity drove him to play around with the Pythagorean theorem and apply it to a right-angled triangle with legs of length 1. The equation easily solved, Hippasus found that c2 = 12 + 12  , therefore c = √2. But oh dear! Hippasus struggled with expressing √2 as a ratio, because its decimal places are infinitely aperiodic. His mathematical proof revealed that √2 indeed cannot possibly be rational (and because I am a good story-teller, I will not bore you with the proof here. Please find a comprehensible version at https://youtu.be/sbGjr_awePE). The legend ends with an astonished Hippasus who shares the finding with his peers, who promptly throw him overboard and into the sea.

Regardless of the legend’s truth content, there is truth to the fact that Pythagoreans heavily disliked the notion of irrational numbers, and the same applied to negative numbers, which they nonchalantly neglected. If you wonder why, we might want to understand the mindset that Ancient mathematicians shared, and the Pythagorean theorem is a pretty representative example, because it was derived from a geometrical shape (a right-angled triangle). Back then, geometrical shapes were the most elementary subject of mathematical study (Heath, 2015), likely so because they are easy to visualise and easy to work with, arithmetically speaking. Because the entire concept of numbers was so closely tied to visualisations of geometrical shapes, you can now imagine why Pythagoreans had trouble dealing with irrational numbers (‘How could a hypothenuse’s length not be a ratio?’) and negative numbers (ever seen a triangle with side length -3?).

Today, we could call the Greek’s geometrical lens a “paradigm”. In philosopher Thomas Kuhn’s definition, a paradigm is a widely accepted and prevailing notion that provides the framework for how we see the world and how we conduct science (Kuhn, 2012). The assumed inseparability of maths and geometry was like a worldview that dictated which findings were worthy of acceptance and further study and which were not. Now, it can be argued that paradigms have their advantages and disadvantages. On the positive side, paradigms simplify communication because we can conduct science based on shared assumptions (ibid.) – within Pythagorean maths, there was no need for weary discussions about the importance of geometry, mathematicians could get straight to work. On the negative side, paradigms can act like blinders that make us neglect observations that oppose our shared assumptions. Kuhn’s historical work reveals a long list of examples where scientific progress stagnated because people saw the world through a paradigmatic lens and failed to notice that this lens obstructed their vision (ibid.). The discovery of irrational numbers is therefore fascinating not only because it added to our pool of knowledge, but because it was one of the cracks in the geometrical worldview that allowed science to move forward.

If we understand that paradigms shape scientific inquiry, we might wonder which paradigms prevail in psychology and how they shape psychological research. Take for example one of the most fundamental definitions within psychology that a mental disorder is “any condition characterised by cognitive and emotional disturbances, abnormal behaviours, impaired functions, or any combination of these” (APA Dictionary of Psychology, n.d.). The phrase ‘characterised by’ implies the following: A mental disorder is an existing, abstract entity that manifests itself in a range of symptoms. It is assumed that first, a person has a disorder, and the symptoms that follow are a reflection of the underlying disorder. But what if we switch it around? How about we consider the possibility that the symptoms come before the disorder? Borsboom (2017)²1 proposed the idea that first, certain behaviours occur in a person, that these behaviours reinforce each other until the person is in a disordered state, and that we can then summarise this state as a ‘mental disorder’. This idea is referred to as ‘network approach to psychopathology’, because its main objective is to rephrase mental disorders as a network of human behaviours and the environmental factors that cause or reinforce those. 

But why should we care which one came first, disorder or symptom? Because in the first case we would want to tackle the underlying cause in order to lessen the symptoms, while in the latter case, we would like to work on the symptoms in order to make the person exit the disordered state (ibid.). What is more, a paradigmatic change in the way we view mental disorders could help us in treating mental disorders in a more humane way. More specifically, there are a number of arguments that criticise the classic DSM-5 definition of mental disorders; for example Gambrill (2014) proposed that our casual use of diagnostic labels leads us to neglect the sociocultural circumstances that lead to disordered behaviour. She also proposed that there is an overreliance on using medication to eradicate “the thing”, the mental disorder, as if living conditions like poverty and other politico-economic stressors were irrelevant to mental health. Indeed, the paradigmatic alternative of the network-approach could help redirect our focus to a more holistic definition of mental disorders, potentially leading to more effective treatments that tackle both the individual and their environment. In the similar notion, Andersen (2010) criticised the clinical practice of treating the DSM like the code of law rather than a guideline. He pointed out that a strict categorisation of people into disorders is not reflective of reality, where people can suffer from severe distress without exactly meeting diagnostic criteria. Isn’t that oddly reminiscent of the Pythagoreans, who simplified the world into geometrical shapes (like us, who simplify people into diagnostic categories) and who were blind to the reality of irrational and negative numbers (like us, who might be missing out on better alternatives to model mental disorders)?

We can wonder if we will experience a paradigmatic revolution in psychology soon. Instead of embracing the network approach fully, psychologists seem to be in the wary process of evaluating and assessing it, with proponents for both the old and new model of psychopathology (e.g. Bringmann & Eronen, 2018; Bringmann et al., 2022). In fact, Kuhn (2012) described that paradigm changes often happen gradually, almost unnoticeably, until one day, we look back at the past and think “Silly us, look at what we used to believe!”. While reminiscing about the past, we often consider our ancestors stupid or ignorant, blind to the wisdom that was so unmistakably staring into our faces all along – but I personally think that this is what makes scientific research worth pursuing: It is the hope that one day, we will look back at ourselves, knowing that now, we are a little wiser and more open to changing our worldviews for the sake of ideas that carry us forward (preferably without any shipwrecks).<<