There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot defined "fractal" as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a shape made of parts similar to the whole in some way." Still later, Mandelbrot proposed "to use "fractal" without a pedantic definition, to use "fractal dimension" as a generic term applicable to "all" the variants".