Physicists' reasoning has a clearly formalized part: many physical theories are formulated in terms of partial (or ordinary) differential equations -- which describe how the corresponding fields (or quantities) x change with time t: dx/dt=F(x) for an appropriate operator F. Once we know the equations dx/dt=F(x) and the initial conditions x(t_0) at some moment of time t_0, we can solve the corresponding Cauchy problem and find the values x(t) for all t.
Within a given theory, the actual values x(t) satisfy the given equation, however, it is well known that not all solutions to this equation are physically meaningful. A classical example of such a physically meaningless solution comes from thermodynamics: when a cup breaks into pieces, the corresponding trajectories of molecules make physical sense; however, when we reverse all the velocities, we get a physically impossible process of pieces assembling themselves into a cup -- a process that, however, satisfies all the original (T-invariant) equations. This "time-reversed" solution is non-physical because it corresponds to "degenerate" initial conditions: once we modify the initial conditions even slightly, the pieces will no longer get together. So, in a physical meaningful solution not only the equations must be satisfied, but also the initial conditions must be "non-degenerate".
There are two challenges in formalizing this idea: one is the challenge of formalizing "non-degenerate" (which can be often done by using the notions of Kolmogorov complexity and randomness) and the other is that, as we will show, the separation between equations and initial conditions depends on the way equations are presented. For example, in the standard form of the Schroedinger's equation, the potential V(r) is a part of the equation. However, we can use this equation to explicitly represent V(r) as a function of the wave function and its derivatives, and then differentiate the right-hand side by time and equate it to 0. As a result, we get a new non-linear equation (second-order in time) which is equivalent to the original Schroedinger's equation -- in the sense that every solution of the Schroedinger's equation for any V(r) satisfies this new equation, and every solution of the new equation satisfies Schoedinger's equation for some V(r). The equations themselves are mathematically equivalent, but in the new equation, initial conditions, in effect, include V(r) -- so the "non-degeneracy" ("randomness") condition must now include V(r) as well.
An even more radical example is when we consider a scalar field f with a generic Lagrange function L(f,f,i*f,i}). In this case, similarly, we can describe a new "equation" which does not contain L at all -- i.e., a field $\varphi$ satisfies the new equation if and only only if it satisfies the Euler-Lagrange equations for some L. Thus, similarly to Wheeler's cosmological "mass without mass" and "charge without charge", we now have "equations without equations". For a single scalar field, this occurs in space-times of dimension at least 4; for two scalar fields, in space-times of dimension at least 10. This may explain why the observed space-time is of dimension 4, and why the smallest space-time dimension for which a consistent quantum filed theory if possible is 10.
In our talk, we describe several similar examples and present some ideas of how to solve the original challenge.