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19 December 2018
I was watching Richard Osman’s House of Games recently when a game about ‘opposites’ piqued my attention. The contestants were given two ‘opposite’ clues to find an object in the desired category; the examples I saw were in category: sweets. The first clue given was ‘apple catch’ and the answer given was ‘pear drop’.
The next clue was ‘ice-cream adult’. Have you worked out the answer yet?
Has it annoyed you?
Being both a writer and a mathematician, I am particularly interested in this idea of ‘opposite’. In writing, we might use ‘antonym’ to denote ‘meaning the opposite of another word or phrase’. Looking for synonyms for the word antonym is quite fun – except there aren’t many in a standard dictionary, just ‘opposite’ or ‘reverse’1. (Side note: did you know you can suggest words for the dictionary? I came across ‘phantonym’ – ‘An informal term for a word that looks as if it means one thing but actually means quite another’ – this way. It’s currently rejected from the Collins dictionary due to ‘not enough evidence’2, although appears elsewhere3.)
I, like many people, enjoy collecting pairs of opposite words: turgid and flaccid, affray and repose, iridescent and monochromatic, mellifluous and cacophonous. But these are clearly not unique; even working within the nuance of language meaning, it is obvious that I have choices of ‘opposite’ here. Dulcet and cacophonous might be just as good; it may become simply a matter of taste. Aesthetics matter – an elegant or contrasting word or phrase might just ‘sound better’.
Mathematically, of course, we have ‘inverse’ – often colloquially called ‘opposite’ – but is this the same as the usage in the game show? Undeniably not. The hallmarks of mathematical inverse processes or functions generally involve a type of process that somehow ‘reverses’ something. Doing and undoing like this is defined by whether it restores the original thing. We can group operations in pairs of inverses, like addition and subtraction, multiplication and division, in this way. It might also be said that an inverse can be an object; we can certainly find the inverse of a number (in the sense of ‘reciprocal’) to form a pair, like 2 and 1/2. However, this is only one type of inverse (a multiplicative one) – plenty of others exist, and it also makes sense to speak of 5 and -5 as ‘opposites’ when dealing with negative numbers, for example. I would suggest that mathematically, there is no such thing as one unique ‘opposite’, but multiple possibilities based on the salient attributes that one is attending to and the purpose of finding the opposite. I can imagine a situation when 24 might be the opposite of 42 (that being about the symbols as opposed to anything else). This idea relies on symmetry, which further relies on the idea of a point or line of reference – both of which can be different, depending on the type of mathematics one is interested in. As there is no one single ‘symmetry’ in maths, or one single ‘reference point’, then there cannot logically be one single ‘opposite’.
So far, the game show question would fail based on the lack of uniqueness property – it is difficult to say definitively that there is a single ‘opposite’ of almost anything. (Can you think of something?) However, it also fails the test because almost no-one would agree that the opposite – or one of the opposites, at least – of ice cream is ‘jelly’. It’s a pair, certainly – but opposite? On what scale? We need to have a sense of which properties are being considered here. Do they actually mean ‘complement’, which also has a particular mathematical meaning that is very different?
What would you say is the ‘opposite’ of jelly? (Aside from something you are most definitely ready for?)
My first thought was a breezeblock (it’s heavy, boringly coloured, non-wobbly, and non-edible). Could you make a list of the attributes of jelly and find something that is at the ‘other’ end of each scale (we are back to symmetries and reference points)? You’d have to stop somewhere. You could say ‘foodstuff’ – for which the opposite would be some kind of ‘non-foodstuff’ – but if you settled on ‘dessert item’, your answer would be completely different. (Main meal? Starter? Savoury dish? Condiment?) I asked colleagues to create a list of their top three properties of jelly: ‘wobbliness’, ‘ability to be moulded’ and ‘sweetness’ seemed to come up as important. Based on this, breezeblock still seems a pretty solid answer. (I’m only a little bit sorry for that one.) It also passes the aesthetic test in some way, too – it’s something that would be horrid and disgusting to eat, with an element of comedy that is also present in the wibble-wobble of the consuming jelly image.
The possible process of finding an opposite is something I have mapped out below, and is surprisingly complex the more you scrutinise it. It contains some of the most important mathematical ideas I know: classification, ordering, visualising, scales, measurement, centrality, balance, equivalence.
But of course ‘breezeblock’ isn’t the only answer here – I’d encourage you to have a think about alternatives that work, too, and consider why. The diagram above suggests just some of the many decisions we are making when finding an appropriate opposite. What do you think the opposite to jelly is?
Anything that isn’t jelly?
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