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How many mathematicians does it take to change a lightbulb?

To consider the problem, we must first determine what mathematicians are. Loosely, they may be defined as a subset of the greater set of "humans" - which must now also be defined. To choose an arbitrary classification, people could be a Hilbert space. Unfortunately, the only appropriate choice for the "inner product" operation between two people is poor for two reasons. First, it is clearly not commutative - what A does to B is not the same as what B does to A, and therefore we lose useful mathematical properties. This could be forgiven - after all, many useful cases are non-commutative. However, the second impropriety in this theory is significantly more challenging. The inner product ⟨a,b⟩ (pronounced a fuck b) certainly is not well-behaved (c.f. Kama Sutra), and can create an infinite amount of trouble (c.f. Oedipus), therefore making this description of humanity mathematically inconvenient. We must therefore assume that mathematicians do not fuck - or more accurately, if they do it does not directly influence the problem at hand.

Having ruled out Hilbert spaces, we may by analogy rule out all vector spaces. Instead, we should treat mathematicians and humans as a field - mathematicians are added all of the time, sometimes (in tragic circumstances involving battles with can-openers, mad scientists, and horrid jokes) they are subtracted. Division is possible, if only on the cellular level - which is good, for as we have assumed that mathematicians do not fuck, division would be necessary in order to multiply. Having loosely proved that mathematics is a field, we consider different fields by reductio ad absurdum.

Clearly, it is possible to have a non-integer number of mathematicians; the circumstances are unpleasant to contemplate, but supremely attainable by application of slicing implements (see references, paying special attention to the actions of Alferd Packer, Jeffrey Dahmer, and so on). However, this case may be safely neglected under the assumption that these postulated non-integer amounts of mathematicians would not survive long enough to change a lightbulb - and if such a one survives, he or she may be defined as a whole survivor, reducing the problem to integers.

Now that we have moved from infinite function spaces through the real numbers, and reached the integers, it seems that our answer is nigh. To deal with zero: zero mathematicians can change no lightbulb, because the product of zero with anything is nothing - and thus no lightbulbs are changed. The next point is perhaps less obvious, and in fact depends on intensive algebraic computations, which have been omitted for the sake of space. Negative mathematicians cannot change light bulbs! This is true if for no other reason than depression, but has been formally proved by suicide (Boltzmann, 1906, Ehrenfest, 1933).

We are left with the positive integers, the counting numbers: we shall deal with them with the natural numbers' best friend, proof by induction. Clearly, one mathematician can change a light bulb - I have done it myself, though the great master Gauss no doubt could have done better. (Quibbling about historical eras proves nothing; time axes are arbitrary.) We therefore assume that some number, k, of mathematicians may change a lightbulb. If k mathematicians can change a lightbulb, can k+1? Naturally - there is nothing preventing the k+1th mathematician from merely observing, and so by induction, any positive integer number n of mathematicians may change a lightbulb. There only remains the problem of boundary conditions: we may not allow a countably infinite number of mathematicians, because they would, even in their natural emaciated state, require an infinite amount of mass, something impermissible in an assumed finite universe.

∴ Any finite positive integer n of mathematicians may change a light bulb.

Pedantic corrections welcome, threats of violence because of mental pain not welcome, but not unusual either. This was written because the punchline needed a joke, and it got an arrogant attempt at a pastiche of proof instead.

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