Are the inverse problems of Systems Biology impossible to solve?
The answer is mostly no, and this is not as disruptive a statement as it seems to be. 

I think Brenner has a point of course and he is not the only one who wonders whether the systems biology can create a holistic model of living things.  Still I wouldn't take this to mean we should stop doing systems biology research. 

At 18:30 brenner illustrates the inverse problem with some examples 1) can you create a drum from only the sounds that come from the drum? (no he says). 2) can you solve the molecular structure of a molecule from a diffraction problem (no he says).

Of course we can do both of these, but the point is valid as we need to have an idea of what a drum looks like.  If we did have even a glance at a drum, this lets us build a model (cylinder of diameter d, height h and with a drum head of a tension t and material properties alpha beta, etc) from the sounds is a lot of work, but it could be done.  

The problem of solving a molecular structure from diffraction intensities alone has also been solved for small molecules in the 60s I believe by so-called 'direct methods' which don't work well or proteins and larger molecules.  Still the point is that you had to have a good mathematical model of molecules and diffraction to get to direct methods, MIR, MAD and other diffraction only methods. Sorry I can't go into details here...it might bore you to death if you don't care about crystallography. 

In both cases the statement is that you need to have that model - what does a molecule look like, what does an x-ray do when it encounters a crystal? In order to get there.  Which makes total sense and I agree with.  

I wouldn't read too much into this though, because in practice this has never been done, yet Bragg started with just the diffraction pattern and drew in models from other fields of study.  Given the nature of the proof described, no scientific discovery has ever happened in a mathematical vacuum and systems biology will not either. 

Rather I would read this talk as saying that the models we are using for action are not very good yet.  What is happening is that synthetic biology (engineering approaches), experimental observation, and intensive modeling (the 'inverse' and 'forward' approaches Brenner mentions) are all happening at once and are informing each other.  

Its hard to imagine that data from inverse approaches can't be an important part of any solution to biology.  Its also looking likelier that a mathematical model alone of biology will be difficult or impossible to find also.  What is currently (IMHO) working is a framework that is both conceptual (and not entirely quantitative) as well as analytical (mathematical) and data driven.