3.3 Nonstandard interpretations (optional)

Prime factorization

Note that there are other possible interpretations of $\mathrm{prime}$ . For example, since one can multiply integer matrices,there might be a useful concept ofprime matrices.

For example: Consider only the numbers $F=\{1, 5, 9, 13, \}$ that is, $F=\{k\colon k\in \mathbb{N}\}$ . It's easy to verify that multiplying two of these numbers still resultsin a number of the form $4k+1$ . Thus it makes sense to talk of factoring such numbers:We'd say that 45 factors into $59$ , but 9 is consideredprime since it doesn't factor into smaller elements of $F$ .

Interestingly, within $F$ , we lose unique factorization: $441=9\times 49=21\times 21$ , where each of 9, 21, and 49 are prime, relative to $F$ ! (Mathematicians will then go and look for exactly whatproperty 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 usea standard interpretation, but one can consider those formulas in different, non-standard interpretations!

The poincarDisc

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 differentpostulates, and got different geometries. However, it was not clear whether these geometrieswere sound whether one could derive two different results that were inconsistent with each other.

Henri Poincardeveloped an ingenious method for showing that certain non-Euclidean geometries are consistentor at least, as consistent as Euclidean geometry.Remember that in Euclidean geometry, the conceptspointandlineare 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 interpretpoint,line, anda point is on a line, any interpretation which satisfies the axiomsmeans that all theorems of geometry will hold.

The Poincardisc is one such interpretation:pointis taken to meana point in the interior of the unit disc, andlineis taken to meana circular arc which meets the unit disc at right angles. So a statement liketwo points determine a linecan be interpreted as

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

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