This paper shows that because growth models in the tradition of Solow's and Romer's are framed in terms of production functions, they are equally subject to a criticism developed by, among others, Phelps Brown (1957), Simon (1979a), and Samuelson (1979). These authors argued that production function estimations are flawed exercises. The reason is that the series of output, labor, and capital stock used are definitionally related through an accounting identity. Consequently, the identity predetermines the estimates that regressions yield. We show that the identity argument helps demystify two illusions in the literature: (i) finding the Holy Grail: total factor productivity is, by construction, a weighted average of dollars per worker and a pure number (the rate of profit or the rental rate of capital); and (ii) the possibility of testing: if estimated properly, production function regressions will yield: (a) a very high fit, potentially an R2 of unity; and (b) estimated factor elasticities equal to the factor shares, hence they must always add up to 1. We illustrate these points by discussing a series of well-known growth accounting exercises and models directly derived from production functions. They are merely tautologies. We conclude that we know substantially less than we think about growth and that many of the discussions in the growth literature are Kuhnian puzzles that only make sense within the neoclassical growth model paradigm.