"One of the foundational myths of modern science ... is that "Mathematics is the language of nature and that, therefore, if you don't speak math, you can't do science." "
You have to speak enough maths to understand the modelling process. A good example of the process is in a course description written by the Open University:
'An example of a problem that can be understood better through mathematical analysis is that of the injection of medication. A mathematical model can be constructed which predicts how the concentration of the medication in the blood changes with time after an injection. This model can then be used to answer such questions as:
how often should a specified amount of medication be administered to be effective?
how much medication should be given if the period between injections is specified?
what happens if the medication is given intravenously?'
If you can't do the maths you can't answer the questions. The best you can do is repeat what someone else says are the right answers. I think it is when you can't do the science that people slip in to the erroneous view that you can understand physics without understanding how physics relies upon many finely structured mathematical models of aspects of the world. That requires understanding what mathematical modelling is. No Royal Road.
2) "The Popperian theory of science as a Millean process of free inquiry ... is too optimistic."
Of course, the Popperian theory of science is not a theory of how scientists undertake the practice of science or of scientific research. Kuhnians have done excellent work in the study of lab life. In the Logik der Forschung and other works Popper does use historical examples of what he later calls the process of conjecture and refutation. But historical examples are not relevant to his main points, which are how science is marked out from other human activities (the demarcation problem in his terminology), and an attack on induction as a means of justifying scientific conclusions.
The first presentation of Popper's ideas had a number of weaknesses, some of which he came to acknowledge (for example, the claim that the conclusions of science come closer and closer to the truth, outlined in Conjectures and Refutations, was rigorously refuted by David Miller, who had been one of Popper's PhD students.)
William Bartley III and David Miller (in his book Critical Rationalism) have interestingly reformulated Popper's ideas.
An interesting paper of Miller's is here: http://www2.warwick.ac.uk/fac/soc/philosophy/staff/miller/dm_control.pdf
Richard