Ok, I was confused by "Platonic" which I thought you were using to refer to intrinsic as opposed to subjective value. Thanks for the clarification.

Subjective value is indeed among the core assumptions of neoclassical economics. The problem is that a whole lot of stuff that economists would like to be able to do (due to both theoretical and ideological interests) is automatically ruled out by this assumption. Reification of "real" values is one way how they try to square this circle, since it enables them to introduce intrinsic value in all but name into their theories.

Putting aside what mainstream economists believe, when you say "meaningless" or "fundamental arbitrariness", do you mean for example that there is no way, even in principle, to compare the marginal utility of a dollar in 1950 with the marginal utility of a dollar in 2010? Is it due to the standard interpersonal comparison of utility problem, or something else?

Yes, you can see this as a corollary of the general problem of interpersonal utility comparison. (Although even if interpersonally comparable utilities are granted and known, you need additional strong assumptions to get rid of all the degrees of freedom that make the choice of index arbitrary.) But these are all different ways of looking at the same problem, namely the problem of intrinsic vs. subjective value.

This is not to say that every attempt to compare the value of money in different places and times for some particular purpose is meaningless, but whether a given attempt is meaningful depends on the context and the sort of comparison used. To guarantee soundness, such comparisons should be justified on a case by case basis by demonstrating that the conclusion indeed follows from the particulars of the way comparison is done. What is definitely unsound is defining a general-purpose “real” value of money and then using it as de facto intrinsic value, without any reflection on how exactly its definition connects to the concrete problem at hand.

To put it another way, do you think the mainstream economics community should be more aware of problems with the theory and practice of price indexes and perhaps allocate more resources to solving them, or do you think they are just not solvable, and an entirely new approach is needed?

Your question seems to assume the existence of a real scientific community in economics, of the sort that exists in natural sciences. However, the problem is that the economics profession has always been deeply intertwined with politics, government bureaucracy, and broader ideological controversies, and as with other social sciences, many of its basic theories and concepts were invented to support an ideological agenda, not as part of a true scientific endeavor. Moreover, many questions in economics have real immediate implications in terms of power, wealth, and status -- to take a pertinent example, entitlement payments by the government are often linked to price indexes, so the question of how they should be defined is not just theoretical, but of immediate financial interest to many parties. Clearly, it would be naive to expect that such questions will be treated with a pristine scientific approach.

In this situation, it’s unrealistic to try to identify and fix the problems and biases in economics (and other social sciences) on a case by case basis, since the real problems are much more general and fundamental. Of course, the existing body of knowledge in economics is far from being entirely worthless, but separating the wheat of true insight from the chaff of ideological delusion and dishonesty, let alone establishing a real epistemologically sound science in place of what exists now, would be a very radical project.