Qualitative-Quantitative Reasoning: Thinking Informally About Formal Things

Qualitative-quantitative reasoning is the way we think informally about formal or numerical phenomena. It is ubiquitous in scientific, professional and day-to-day life. Mathematicians have strong intuitions about whether a theorem is true well before a proof is found - intuition that also drives the direction of new proofs. Engineers use various approximations and can often tell where a structure will fail. In computation we deal with order of magnitude arguments in complexity theory and data science practitioners need to match problems to the appropriate neural architecture or statistical method. Even in the supermarket, we may have a pretty good idea of about how much things will cost before we get to the checkout. This paper will explore some of the different forms of QQ-reasoning through examples including the author's own experience numerically modelling agricultural sprays and formally modelling human-computer interactions. We will see that it is often the way in which formal and mathematical results become useful and also the importance for public understanding of key issues including Covid and climate change. Despite its clear importance, it is a topic that is left to professional experience, or sheer luck. In early school years pupils may learn estimation, but in later years this form of reasoning falls into the gap between arithmetic and formal mathematics despite being more important in adult life than either. The paper is partly an introduction to some of the general features of QQ-reasoning, and partly a 'call to arms' for academics and educators.