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You are walking past a phone box. (What’s one of those? you ask, if you are under thirty). It rings, piercing the still air, and you jump, looking around. It’s weird. You wonder what to do? The sound persists, shrilly. You heave the door open and lift the clunky plastic receiver. “Hello?” you say, tentatively.
The person on the line knows your name. In fact, they are known to you.
It’s a compelling scenario – such a good story that it has been the approximate plot of at least one film, and included in many lists of popular coincidences (Prof David Spiegelhalter is collecting these at the Cambridge Coincidences Collection as part of the Understanding Uncertainty project). I shouldn’t have led with that. I’ve probably lost you for a few hours now, as reading stories of real-life coincidences is pretty fascinating stuff. But if you can bear to, put a bookmark in that rabbit hole, mix some metaphors, and keep your head above water with me for just a little bit longer while we take a stroll through the fascinating cloud-cities of logical reasoning.
It’s not a coincidence that we like reading stories about coincidences. Stories about the incredibly-strange-yet-true have a powerful hold on us humans, and some of the best of them feature the human brain as the hero of the tale. The setup: something amazingly weird has happened. The development: someone with the superpower of analytical reasoning arrives on the scene to sort it all out. The barrier: someone is lying, or otherwise covering up evidence. The conclusion: our hero figures it out, unmasks the bad guy/gal/person/robot/animal/alien/entity, and we get the satisfaction of mystery solved. In other words: the classic detective story.
“Life is infinitely stranger than anything which the mind of man [sic] could invent. We would not dare to conceive the things which are really mere commonplaces of existence. If we could fly out of that window hand in hand, hover over this great city, gently remove the roofs, and peep in at the queer things which are going on, the strange coincidences, the plannings, the cross-purposes, the wonderful chains of events, working through generations, and leading to the most outré results, it would make all fiction with its conventionalities and foreseen conclusions most stale and unprofitable.”
Arthur Conan Doyle, The Adventures of Sherlock Holmes, A Case of Identity.
You, like Holmes himself, were probably one step ahead of me here in thinking of the much-loved Sherlock as the epitome of this kind of thinking. As the quote above demonstrates, not only does Holmes use reasoning like this, he often talks about it too, in a very metacognitive way – which is why I am quite the advocate of using the literature of Conan Doyle to teach logical reasoning in the mathematical classroom. (I also enjoy the irony of a fictional character talking about fictional things being stranger than fiction.)
But what is logical reasoning?
It is forgivable to consider that the only kind of logical reasoning we do in mathematics is the deductive kind, which is the basis for proving and proof. In fact, research suggests that logic is comprised of at least four different kinds of reasoning.
Deductive reasoning, the one we might have thought of first, is using valid and logical arguments from an accepted or agreed premise, or drawing conclusions as the logical consequence of conditions and assumptions (e.g., Stylianides & Stylianides, 2008).
Inductive reasoning is described as drawing logical abstractions or generalisations from individual observations, moving from the specific to the general, or making and testing conjectures (e.g., Conner et al., 2014).
Analogical reasoning is reasoning with relational patterns, as in "this is like that in this sense," or noting the correspondence between the structures of one system and that of another system (e.g., English, 2004).
Abductive reasoning is using "most likely in the face of the facts" arguments such as Occam's Razor, formulating a lawlike hypothesis, or making an inference which allows the construction of a claim starting from an observed fact (e.g., Reid, 2018).
Which of these types do you associate most strongly with the type of thinking done by Sherlock Holmes? More on that in a moment.
Looking at this lovely range of reasoning types allows us to remember that logical reasoning happens across all contexts and applications of mathematics, and helps connect them all together. In particular, ideas of probability, inference, coincidence and risk are intimately related to deductive reasoning, and should not be seen as its poor cousins simply because they are dealing in uncertainty. Stylianides and Stylianides (2008) note that often "the boundaries are faint" (p. 105) between these types of logical reasoning, and Inglis et al. (2007) suggest exploring with students the idea of qualifying different types of argument by relating them to how much uncertainty is in the picture. In the real world, certainty is so rare that comfort with – and reasoning under – uncertainty is extremely valuable and important, and not to be disdained as “less important” than the deductive reasoning which underlies ideas of mathematical proof (and the relationship between proof and certainty may be less sure than you think, in any case – see Inglis et al., 2013). Both are important, and they complement one another. In fact, our perception of maths is all the poorer if we only consider the beauty of the subject to be its propensity to provide a single, perfect answer.
If you thought that Holmes’ analytical thinking seemed like abductive reasoning, you’d be agreeing with most of the research on the subject (e.g., Patokorpi, 2007) – but interestingly, it is not clear-cut, suggesting again that the boundaries between these types of reasoning are not straightforward.
For example, try some of the puzzles below, considering what type(s) of reasoning you may be using to do so.
What has six faces but zero heads; twenty-one eyes, but zero mouths?
How do you make the number 7 even without addition, subtraction, multiplication, or division?
The du Châtelets have six daughters and each daughter has one brother. How many people are in the du Châtelet family?
On an island where everyone is either a knight (always tells the truth) or a knave (always lies), you meet two people, called Ali and Charlie. One is a knight, and one a knave, but you don’t know which.
Charlie says, “We are both knights.”
Who is the knight, and who the knave?
Helen is looking through some national data and notices that there appears to be a strong correlation between the number of ice-creams sold and the number of deaths by drowning. What is going on?
The more you take away, the more I become. What am I?
Have you ever experienced a strange coincidence, or enjoyed Sherlock Holmes’ logical reasoning? Do you have an interesting puzzle to share? You can tweet us @CambridgeMaths or comment below.