Dears Anshu, Tejinder,
Don't blame your limitations. The essay was required to be clear, the author should have done better, and you are generous to say it is well-written.
1. If you want to reason about the effectiveness of mathematics, to evaluate, or even to measure it, you must endow yourself with a frame of reference where you can measure it; you need a standard to which you can compare it. This means you need a larger space, containing mathematics and other things, within which you have at least a possibility to distinguish differences, and say what is more apt to what. This is the etymological sense of to explicate: ex-plicare, to unfold, to show there are different components, and those unfit explain by contrast those fit.
This cannot be done, for instance, in an ultra-Platonist stance, where mathematics is a completely pure, separated realm: With mathematics already split apart, how can you show them distinctively more effective than... what? It inhibits any sort of explanation about the effectiveness of mathematics.
2. To be able to account for the effectiveness of mathematics you must account for processes of perception and cognition, and within them, of mathematics as a specialised activity (or as the product of that specialised cognitive activity).
3. I propose a framework in which all kinds of perceptory or cognitive structures, including mathematical, come into being homogeneously. This should not be a big surprise, if you accept that you capture an important part of mathematics by defining them as the science of structures, or the science of patterns. To perceive is already to find patterns.
So asking about the effectiveness of mathematics is a bit like asking about the effectiveness of perception --any sort of our anticipations, any of our behaviours. Certainly there is some effectiveness, because that is what we live by.
Each perception is a bet, an interpretation, and those that are felt unsuccessful are strongly counter-selected, while we keep using those `correct, up to now'. Similarly we work hard to select only the very small subset of mathematics which works successfully in a given situation. In such a Darwinian context, there is no general reference frame to rate effectiveness, you can only say what your best result is, in the locally explored context.
4. Hence the amazement about the effectiveness of mathematics disappears.
5. (In fact, the mathematics we choose to write in our books, and use, are already a selected subset among a larger world of a priori possible mathematics: we want consistency, i.e., we prefer to describe structures which lend themselves to construction of other structures.
This is certainly a very small subset of the possibles, much like continuous and differentiable functions have been shown to be an infinitesimal subset of functions everywhere continuous, but nowhere differentiable, though we practically speak nearly only of the latter. Do we wonder why most of the laws we write are continuous, differentiable? Or why we require the algorithms we write to be effective, that is, to finish in reasonable time on our current computers? This is exactly the same bias. Already noted by Descartes: ``Good sense is the best shared-out thing in the world; for everyone thinks he has such a good supply of it that he doesn't want more, even if he is extremely hard to please about other things.'')
You said summarise?
