Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

234 michael hallettSuppose now that α and β are in the field; we can then consider the complexnumber α + iβ, and these elements give rise to the complex extension of thefield. In fact, such an element will be represented by an appropriate powerseries built on t where the coefficients are imaginaries. If α + iβ is in thecomplex extension, then so will e α+iβ be.Hilbert now defines an analytic geometry as follows. Points are definedby pairs of coordinates (x, y), where x and y are elements of T. Straightlines will be given by linear equations in the usual way, and so the usualincidence and order axioms will then hold, as well as the Parallel Axiom;the Archimedean Axiom, of course, will not hold. This leaves just thecongruence axioms to consider, and, as one might expect, this is where theart comes. First, two segments are said to be congruent if parallel transportcan shift the line segments so that they start at the coordinate origin, andthen (keeping one of the segments fixed) one of them can be rotated intothe other by a positive rotation through some angle θ. The two ‘movements’,parallel transport and rotation, are of course, properly speaking, transformationsof the plane onto itself. What will be congruent to what will thendepend on the definition of the transformation functions; parallel transportis trivial, so the key matter is the transformation function corresponding torotation.This function is based on the following trick. The coordinate (x, y) canbe coded as a single complex number α = x + iy in the complex extensionof T. The function e α can then be used to form another complex number,which can then be decoded to form a new coordinate (x ′ , y ′ ). Suppose (x, y) isthe coordinate of one end of a segment whose other end-point (perhaps afterparallel transport) is at (0, 0). Suppose the segment is positively rotated throughan angle θ + τ, whereθ is real (and could be 0) and τ is infinitesimal. Thenthe new coordinates are given by the formulax ′ + iy ′ = e iθ+(1+i)τ · (x + iy)Clearly (0, 0) transforms into the coordinate (0, 0).³⁷Segments can be assigned length acccording to the following procedure. Asegment on the x-axis is said to have length l when one of its end-points liesat the origin and the other has x-coordinate ±l. Any other segment has lengthl if one end can be shifted to the origin by parallel transport and the other endcan be rotated into (±l, 0) by the rotation function or where (±l, 0) can be³⁷ As Hilbert makes clear, it can be shown that given any line segment and any point in the plane, arotation can be found so that the rotated line passes through that point.