We present a set of positions, many of them controversial, on turbulence modeling for the Reynolds-Averaged Navier Stokes (RANS) equations. After outlining what we call the “Fundamental Paradox” of turbulence modeling, we first oppose two paradigms, or philosophies. The ``systematic'' paradigm attempts to model the exact transport equations for the Reynolds stresses term by term, gradually relegating the Closure Problem to higher moments and invoking the ``Principle of Receding Influence.'' In contrast, the ``Openly Empirical'' paradigm produces models which satisfy strict constraints such as Galilean invariance, but lack an explicit connection with terms in the exact equations. The prime example is the eddy-viscosity assumption. We then explain a few in a series of perceived fallacies in turbulence knowledge, with consequences for turbulence models. We divide them into ``hard'' fallacies for which a short mathematical argument demonstrates that a statement is wrong or meaningless, and ``soft'' fallacies for which approximate physical arguments can be opposed, but we contend that a clear debate is overdue and wishful thinking has been involved. An example in the hard class is the isotropy of the diagonal Reynolds stresses. An example in the soft class is the value of realizability in a RANS model.