The Intermediate Value Theorem

THE INTERMEDIATE VALUE THEOREM

Another way to deal with fine lines and grey areas that isn’t fully logically determined is to acknowledge that the line is somewhere in the grey area and we don’t know where it is; we just know it’s somewhere in that area. We can put bounds on the area by pointing to one place that’s definitely below the line and one place that’s definitely above the line.

This is how you can make a batch of cookies so that everyone has their perfect size of cookie. You can start with a cookie that is definitely too small, maybe just 2g of cookie dough. Then you can make them get very gradually bigger so that each one seems about the same size as the previous one, but they’re getting imperceptibly bigger. Keep going until you reach one that is obviously definitely too big, say twice as big as your face.

Because there is essentially every size of cookie in between, this means everyone’s perfect size of cookie must be in there somewhere. This helpfully deals with the fact that my personal favourite size of cookie is smaller than most people’s. This way I can meet my own needs and other people’s at the same time, without actually having to know what anyone else’s perfect size of cookie is – I can be sure it’s in there somewhere.

This is an application of the intermediate value theorem, a theorem in rigorous calculus that maths students usually study as undergraduates. It says that if you have a continuous function that starts at 0 and goes up to some number a, it must take every value in between. What “continuous” means here is rather technical, but it basically means there are no gaps. You might argue that my cookies do not take every size, otherwise there would be infinitely many of them. That’s true, but I am using a real-life approximation to the intermediate value theorem. Really I’m saying that the whole thing is true up to a certain accuracy determined by our perceptions.

A few months ago I was talking to one of the art students at the School of the Art Institute of Chicago. She was exploring people’s perceptions of reality by creating visual illusions and trying to see if viewers would believe they were physical constructions, or think they were digital manipulations. The question was to find the sweet spot where people would be really unsure which it was. I realised that she could invoke the intermediate value theorem: make a series of pieces starting with one that was obviously a physical construction, gradually becoming less obvious until she ended with one that was obviously physically impossible so must be a digital manipulation. At some point in the grey area in between there would have to be a point where viewers were unsure whether it was real or digital. The artist would not need to know exactly where it was, and indeed it could be in a different place for different viewers. All the artist would need to know is that it is somewhere in that grey area.

In a way this is also what chocolate makers do with the percentage cocoa content in their chocolate. They make a whole range of different percentages so that all chocolate lovers will find their perfect percentage in there somewhere. I regret, however, that so many chocolate makers stop their range somewhere around 70 per cent (in fact 72 per cent seems a popular place to stop), because that does not encompass my own personal favourite spot. My grey area for chocolate is somewhere between 80 per cent to 100 per cent, depending on my mood. As a result, to get my perfect chocolate, like my perfect size of cookie, I usually have to make my own.

This is a similar principle to the one that is giving us more and more finely tuned gradations of race. In the unenlightened old days there was just “white” and “non-white”. We now talk about “people of colour” but we also talk about mixed race, although this often implicitly means mixed between white and non-white. People have been coming up with words for mixed races between non-white and some other non-white, such as blasian for black and Asian. A friend of mine calls himself “Mexippino” for Mexican and Filipino.

But should we have different words for someone who is one quarter Asian and three quarters white (who is likely to be quite white-passing), as opposed to three quarters Asian and one quarter white (who might look entirely Asian)? There are various possibilities, each of which has advantages and disadvantages. We can talk about “white people” and “Asian people” which is simple, but results in the exclusion of those who fall into neither category exactly. We can draw a line somewhere and end up creating bizarre anomalies, where someone is classified as a “person of colour” although they look white to most people. We can create increasingly many finer and finer categories taking everyone into account, but ending up with an intractable collection of extremely specific descriptors. We can unify everyone, ignore race, and declare that we’re “all humans”, as some people do when they call themselves “colour-blind”. But this discounts people’s real experiences of racial discrimination. I think that above all we should acknowledge that there are grey areas and become more comfortable with accepting them.