What came first, the chicken or the heap?
Monday June 12, 2006 by Noam Samuel
I have in my pocket an inconsequential sum of money, something along the lines of 25 cents. Now, suppose I added a cent to it, I would then have 26 cents. In fact, one could generalize and say that if you add one cent to a small sum of money, the result will still be a small sum of money. It seems reasonable, but is it?
Suppose that on the desk next to me I have a large sum of money, 100$, let’s say (not that I ever have that kind of money, but this is a hypothetical situation). Now, if I were to remove 1 cent from it, I would still have $99.99. It would, in fact, seem reasonable to say that if I were to remove 1 cent from a large sum of money, it would still be a large sum of money, right? Wrong!
If we suppose that 1 cent is small, and that adding 1 cent to a small amount of money would still generate a small amount of money, we have therefore proven by induction that all amounts of money are small.
What is induction, you ask? Well, induction is a principle (and method of proof) that states the following: If property p(x) is true for 1, and if for all 
, 
, then p(x) must be true for all of N. In fact, induction is not exclusive to N, it can be used with any well-ordered countable set (where you replace 1 with the first item and n+1 with the successor of n).
But why does it work? Well, suppose you have a number n. Now, you know that if p(n-1) is true, then p(n) must be true, and that if p(n-2) is true, then p(n-1) is true. This can continue indefinitely, but eventually you get to p(n-[n-1]), which is also 1, for which we know p(x) to be true. Thus, like a row of dominoes, all the other numbers also fall into place, eventually showing that p(n) is true.
Back to our problem: Now, since all sums of money are small, we can assume that any large sum of money therefore cannot exist. Thus, we can show that the assumption that 100$ is a large sum of money must therefore be wrong.
But that seems quite rediculous, doesn’t it? We all know that there are large sums of money. This problem is known as the paradox of the heap, and is actually quite a lot of fun.
The original paradox stipulates that 1 grain of sand is not a heap, and that adding a grain of sand to something that is not a heap will not make it a heap. It then goes on, as you have probably guessed, to disprove the existance of heaps.
But it doesn’t just end with cash and heaps (or heaps of cash?), you can use it to show that there aren’t any bald people, that no measurment is accurate enough, or that you don’t have enough air to breathe. This paradox can even be employed to obfuscate the ancient question, “what comes first, the chicken or the egg?”
How so? Well, logic says that since there must have been a first creature that could be considered a chicken, and that the creature must have been born from an egg, and that thus the egg came first. However, if you consider that there was once a time in which no chickens existed, and if you consider that no two non-chickens will spontaneously give birth to a chicken, you can therefore prove by induction that not only did the egg not come first, but that there are no chickens!
Obviously, by now you have probably guessed that I am assuming a fallacy. In fact, you might have even found what my fallacy is, or you might have not. It’s actually quite tricky, since I didn’t put it in my list of assumptions. My fallacy is in assuming that the size of a sum of money, the baldness of a person, or the chickenhood of a creature are all restricted to the set of {true, false}. Indeed, while two complete non-chickens might not give birth to something that is a complete chicken, they can give birth to something closer to a chicken. Or, if we look at our sum of money, we can say that while adding one cent to a small amount does not turn it into a completely large sum of money, it can add largeness to it. In other word, we are using a Bayesian system.
What is a Bayesian system, you ask? A Bayesian system is one in which a property can have any real value between 0 and 1 (inclusive), and where p(x) > p(y) implies that x is more p than y. In a bayselian system, 0 is considered to be the equivalent of false, and 1 is the equivalent of true. Bayselian systems are often used in an alternative form of logic dealing with probabilities (this form is in many ways more useful than “normal” logic), but in this case we will use it for something different.
So, if to take baldness as an example, a person with a head full of hairs has a baldness of 0, whereas a person with only 100 hairs on his head might have a baldness of 0.75, and a person with no hair on his head whatsoever has a baldess of 1.
Back to the chickens. Suppose there is a creature, C, let’s call him, and suppose we want to know whether C is a chicken or not. Now, we have a list of traits (or genes) that make a creature a chicken, and it’s chickenhood is the fraction of the criteria that it matches. This, however, seems much less satisfactory for chickens than it does for baldness. Therefore, one can take a chicken range of the chickenhood of all known chicken, and arbitrarily decide to call anything within that range of chickenhood a chicken, and say that anything outside of it is not a chicken. This solution to the mound paradox is indeed useful for some purposes, but for most situation the Bayselian has more use.
But what about the question of the chicken and the egg? Which one came first? Well, if you have read the post carefully, you might have picked up the answer: “I don’t know, but I’m sure it wasn’t a heap.”