prime factorization

Note that there are other possible interpretations of primeprime. For example, since one can multiply integer matrices, there might be a useful concept of prime matrices.

For example: Consider only the numbers F = {1,5,9,13,…} -- that is, F = {4k+1 | k in N}. It's easy to verify that multiplying two of these numbers still results in a number of the form 4k+1. Thus it makes sense to talk of factoring such numbers: We'd say that 45 factors into 5*9, but 9 is considered prime since it doesn't factor into smaller elements of F.

Interestingly, within F, we lose unique factorization: 441 = 9*49 = 21*21, where each of 9, 21, and 49 are prime, relative to F! (Mathematicians will then go and look for exactly what property of a multiplication function are needed, to guarantee unique factorization.)

The point is, that all relations in logical formula need to be interpreted. Usually, for numbers, we use a standard interpretation, but one can consider those formulas in different, non-standard interpretations!

The Poincaré Disc

A long outstanding problem was that of Euclid's parallel postulate: Given a line and a point not on the line, how many lines parallel to the first go through that point? Euclid took this as an axiom (unable to prove that it followed from his other axioms). Non-Euclidean geometries of Lobachevsky and Riemann took different postulates, and got different geometries. However, it was not clear whether these geometries were sound -- whether one could derive two different results that were inconsistent with each other.

Henri Poincaré developed an ingenious method for showing that certain non-Euclidean geometries are consistent — or at least, as consistent as Euclidean geometry. Remember that in Euclidean geometry, the concepts point and line are left undefined, and axioms are built on top of them (e.g. two different lines have at most one point in common). While it's usually left to common sense to interpret point, line, and a point is on a line, any interpretation which satisfies the axioms means that all theorems of geometry will hold.

The Poincaré disc is one such interpretation: point is taken to mean a point in the interior of the unit disc, and line is taken to mean a circular arc which meets the unit disc at right angles. So a statement like two points determine a line can be interpreted as

[*] For any two points inside the disc, there is exactly one circular arc which meets the disc at right angles.

Figure 1: Some lines in the Poincaré disc, including several lines parallel to a line L through a point p.

(Note that the distance function is very different within the Poincaré disc; in fact the perimeter of the disc is off at infinity. Angles, however, do happen to be preserved.)

The critical point of his interpretation of a non-Euclidean geometry is this: it is embedded in Euclidean geometry! So we are able to prove (within the embedding Euclidean geometry) that the disc-postulates hold (e.g. we can prove the statement [*] above as a theorem about circular arcs in Euclidean geometry). Therefore, if there is any inconsistency in non-Euclidean geometry, then that could be parlayed into some inconsistency of Euclidean geometry. Thus, his interpretation gives a proof that the strange non-Euclidean geometry is as sound as our familiar Euclidean geometry.

P vs NP and Oracles

A well-known problem in computer science -- P vs NP -- asks whether (for a given problem) it is truly more difficult to find a short solution (when one exists) (NP), than it is to verify a short purported solution handed to you (P). For example, Given a set of people and how strong person is, can you partition them into two tug-of-war teams which are exactly evenly matched? Certainly it seems easier to check that a pair of proposed rosters has equal strength (and, verify that everybody really is on one team or the other) than to have to come up with two perfectly-matched teams. But conceivably, the two tasks might be equally-difficult up to some acceptable (polynomial time) overhead. While every assumes that P is easier than NP, nobody has been able to prove it.

An interesting variant of the problem lets both the problem-solver and the purported-answer-verifier each have access to a particular oracle -- a program that will gives instant yes/no answers to some other problem (say, given any set of numbers, yes or no: is there an even-sized subset whose total is exactly the same as some odd sized subset?).

It has been shown that there is some oracle which makes the problem-solver's job provably tougher than the proof-verifier's job, and also there is some other oracle problem-solver's job provably no-tougher than the proof-verifier's job.

This means that any proof of P being different from NP has to be subtle enough so that when P and NP are re-interpreted as "P and NP with respect to a particular oracle", the proof will no longer go through. Unfortunately, this eliminates all the routine methods of proof; we know that solving this problem will take some new attack.

Real-World Arguments

Finally, it is worth noting that many rebuttles of real world arguments (see also some exercises) amount to showing that the argument's form can't be valid since it doesn't hold under other interpretations, and thus there must be some unstated assumptions in the original.