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Before we start, I'd like to thank listener RobocopMustang, who wrote another nice review on iTunes. Remember, you too can get your name mentioned on the podcast, by either writing an iTunes review, or sending a donation to your favorite charity in honor of Math Mutation and emailing me about it.

Anyway, on to today's topic. Recently I was thinking about various classic mathematical and philosophical paradoxes we have discussed. I was surprised to notice that we have not yet gotten to one of the most well-known classical paradoxes, the Heap Paradox. This is another of the many paradoxes described in ancient Greece, originally credited to Euclid's pupil Eubulides of Miletus.

The Heap Paradox, also known to snootier intellectuals as the Sorites Paradox (Sorites being the Greek word for heap), goes like this. We all agree we can recognize the concept of a heap of sand: if we see a heap, we can look at the pile of sand and say "that's a heap!". We all agree that removing one grain of sand from a heap does not make it a non-heap, so we can easily remove one grain, knowing we still have a heap. But if we keep doing this for thousands of iterations, eventually we will be down to 1 grain of sand. Is that a heap? I think we would agree the answer is no. But how did we get from a situation of having a heap to having a non-heap, when each step consisted of an operation that preserved heap-ness?

One reason this paradox is so interesting is that it apples to a lot of real-life situations. We can come up with a similar paradox if describing a tall person, and continually subtracting inches. Subtracting a single inch from a tall person would not make him non-tall, would it? But if we do it repeatedly, at some point he has to get short, before disappearing altogether. Similarly, we can take away a dollar from Bill Gates without endangering his status of "rich", but there must be some level where if enough people (probably antitrust lawyers) do it enough times, he would no longer be rich. We can do the same thing with pretty much any adjective that admits some ambiguity in the boundaries of its definition.

Surprisingly, the idea of clearly defining animal species is also subject to this paradox, as Richard Dawkins has pointed out. We tend to think of animal species as discrete and clearly divided, but that's just not the case. The best example from the animal kingdom may be the concept of "Ring Species". These are species of animals that consist of a number of neighboring populations, forming a ring. At one point on the ring are two seemingly distinct species. But if you start at one of them, it can interbreed with a neighbor to its right, and that neighbor can interbreed with the next, and so on... until it reaches all the way around, forming a continuous set of interbreeding pairs between the two distinct species.

For example, in Great Britain there are two species of herring gulls, the European and the Lesser Black-Backed, which do not interbreed. But the European Herring Gull can breed with the American Herring Gull to its west, which can breed with the East Siberian Herring Gull, whose western members can breed with the Heuglin's Gull, which can breed with the Lesser Black-Backed Gull, which was seemingly a distinct species from the European gull we started with. So, are we discussing several distinct gull species, or is this just a huge heap of gulls of one species? It's a paradox.

Getting back to the core heap concept, there are a number of classic resolutions to the dilemma. The most obvious is to just label an arbitrary boundary: for example, 500 grains of sand or more is a heap, and anything fewer is a non-heap. This seems a bit unsatisfying though. A more complicated version of this method mentioned on the Wikipedia page is known as "Hysteresis", allowing an asymmetric variation in the definition, kind of like how your home air conditioner works. When subtracting from the heap, it may lose its heapness at a threshold like 500. But when adding grains, it doesn't gain the heap property again until it has 700. I'm not convinced this version adds much philosophically though, unless your energy company is billing each time you redefine your heap.

A better method is to use multivalued logic, where we say that any pile has some degree of heapness which continuously varies: over some threshold it is 100%, then as we reduce the size the percentage of heapness gradually goes down, reaching 0 at one grain. A variant of this is to say that you must poll all the observers, and average their judgement of whether or not it's a heap, to decide whether your pile is worthy of the definition.

If you're a little more cynical, there is the nihilistic approach, where you basically unask the question: simply declare it out-of-bounds to discuss any concept that is not well-defined with clean boundaries. Thus, we would say the real problem is the use of the word "heap", which is not precise enough to admit philosophical discussion. There are also a couple of more involved philosophical resolutions discussed in online sources, which seem a bit technical to me, but you can find at the links in the show notes.

Ultimately, this paradox is pointing out the problem of living in a world where we like things to have discrete definitions, always either having or not having a property we ascribe to it. It is almost always the case that there are shades of grey, that our clean, discrete points may reach each other by a continuous incremental path, and thus not be as distinct as we think.

And this has been your math mutation for today.

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