All Absentees Assume Formation
Friday, September 19, 2008 2:50:00 AM
This joke was funny the first time I heard it. But why is it funny, indeed? Common sense tells us that it's impossible for there to be −2 students in a room. But the professor was somehow not surprised; to him −2 easily adds up with two latecomers and yields “nobody”. The problem is that we have no idea what negative two students look like and what properties they have, but it's natural for us to assume that they annihilate with positive two students.
Where does our “common sense” regarding to quantities come from? Natural numbers have properties (such as the possibility to increase any natural number by one) defined by a set of axioms, but why are these axioms exactly what they are? Natural numbers are called natural for the very reason that human invented, or, to be precise, apprehended them directly from the properties of the environment. One doesn't have to know math even at elementary school level to perceive empirically certain properties of natural numbers, for example, that 2 > 1.
For a long time, negative numbers were thought to not exist (and even zero took time to come into use), but as soon as at the dawn of Common era, such an extension to the set of numbers was first mentioned. It's not that anyone actually saw −2 students, but in bookkeeping of debts negative amounts of money or goods turned out to be quite conceivable. But if negative two swords, oxen or even slaves are possible, negative two students shouldn't seem something incomprehensible either.
The setting in the joke still seems absurd because nobody has ever witnessed a pair of positive one and negative one students being produced from nothing. But the very existence of a negative one student, or, to put it that way, lack of a student, is not absurd. One can even imagine that whole galaxies exist with negative quantities of stars, planets, universities and students. They probably call their half of the numeber axes “positive” and theorize about our existence, or, to be precise, the lack of us. (This is not the same as antimatter, despite some similarity. Antimatter exists in positive quantities, but this is rather about negative amounts of matter.)
By the way, while we're telling existential jokes, physicists shamelessly use negative quantities of electrons. A lack of electron, or −1 electron, is called an electron hole and used in analysis of semiconductors on par with an electron. Unfortunately, the number of studies of this kind in other areas of physics is negative with a large absolute value.
See also: Is logic empirical?
UPDATE: There we go: Negative probability.
По-русски: Всех отсутствующих построить в одну шеренгу