As noted in the epigraph, a map was long ago seen as the map of the map, the map of the map, of the map, and so on endlessly. This recursive perspective on maps, however, has received little attention in cartography. Cartography, as a scientific discipline, is essentially founded on Euclidean geometry and Gaussian statistics, which deal respectively with regular shapes and more or less similar things. It is commonly accepted that geographic features are not regular and that the Earth's surface is full of fractal or scaling or living phenomena: far more small things than large ones are found at different scales. This article argues for a new paradigm in mapping, based on fractal or living geometry and Paretian statistics, and – more critically – on the new conception of space, conceived and developed by Christopher Alexander, as neither lifeless nor neutral, but a living structure capable of being more living or less living. The fractal geometry is not limited to Benoit Mandelbrot's framework, but tends towards Christopher Alexander's living geometry and is based upon the third definition of fractal: A set or pattern is fractal if the scaling of far more small things than large ones recurs multiple times. Paretian statistics deals with far more small things than large ones, so it differs fundamentally from Gaussian statistics, which deals with more or less similar things. Under the new paradigm, I make several claims about maps and mapping: (1) the topology of geometrically coherent things – in addition to that of geometric primitives – enables us to see a scaling or fractal or living structure; (2) under the third definition, all geographic features are fractal or living, given the right perspective and scope; (3) exactitude is not truth – to paraphrase Henri Matisse – but the living structure is; and (4) Töpfer's law is not universal, but the scaling law is. All these assertions are supported by evidence, drawn from a series of previous studies. This article demands a monumental shift in perspective and thinking from what we are used to in the legacy of cartography and GIS.