Basically I've managed to convince myself that there isn't a sensible mathematical definition of "real GDP" or the "price level" (also known as "GDP deflator"). Which means you can talk about nominal GDP and about growth rates and inflation rates, but you cannot really compare real GDP across countries or across  time. And it casts serious doubts on PPP. And this is in principle, without even going into the practical difficulties of gathering real data, or the fishiness of arbitrary "baskets of goods", "hedonic pricing", and other assorted suspicious stuffnonsense.

I have hinted at it here and here. There's also Keynes:

That the units, in terms of which economists commonly work, are unsatisfactory can be illustrated by the concepts of the national dividend, the stock of real capital and the general price-level:

...

Thirdly, the well-known, but unavoidable, element of vagueness which admittedly attends the concept of the general price-level makes this term very unsatisfactory for the purposes of causal analysis, which ought to be exact.

Nevertheless these difficulties are rightly regarded as 'conundrums'. They are 'purely theoretical' in the sense that they never perplex, or indeed enter in any way into, business decisions and have no relevance to the causal sequence of economic events, which are clear-cut and determinate in spite of the quantitative indeterminacy of these concepts. It is natural, therefore, to conclude that they not only lack precision but are unnecessary. Obviously our quantitative analysis must be expressed without using any quantitatively vague expressions. And, indeed, as soon as one makes the attempt, it becomes clear, as I hope to show, that one can get on much better without them.

The fact that two incommensurable collections of miscellaneous objects cannot in themselves provide the material for quantitative analysis need not, of course, prevent us from making approximate statistical comparisons, depending on some broad element of judgement rather than of strict calculation, which may possess significance and validity within certain limits.

But the proper place for such things as net real output and the general level of prices lies within the field of historical and statistical description, and their purpose should be to satisfy historical or social curiosity, a purpose for which perfect precision--such as our causal analysis requires, whether or not our knowledge of the actual values of the relevant quantities is complete or exact--is neither usual nor necessary. To say that net output to-day is greater, but the price-level lower, than ten years ago or one year ago, is a proposition of a similar character to the statement that Queen Victoria was a better Queen but not a happier woman than Queen Elizabeth--a proposition not without meaning and not without interest, but unsuitable material for the differential calculus. Our precision will be a mock precision if we try to use such partly vague and non-quantitative concepts as the basis of our quantitative analysis.

-- John M. Keynes in The General Theory of Employment, Interest and Money

(my emphasis)

The "general price level" is the "GDP deflator".

I don't know whether Keynes was thinking about the mathematical reasons I'm thinking about, but it sounds like he knew this stuff was nonsense already.

BruceMcF:

Keynes' basic argument for working in terms of nominal amounts and employment was that the ability of a person to work as unskilled labor if need be provided a connection between various specialized and more restricted skilled labor markets in terms of their renumeration as a multiple of the wage of unskilled labor, providing a quantity that could be aggregated with greater justification than the vector of the amount produced of each and every different type of final product for sale in the economy.

And now for the IM:

Colman: Is the underlying model even faintly relevant to the real world here?
Migeru: maybe what I'm doing is reducing GDP to absurdity
Colman: That's my immediate thought. ... Not that it needs much help.
Migeru: hmm. It is king of policy right now. It might be absurd but that matters little. The thing is if REAL GDP is not a state function then you CANNOT compare it across countries or across times.
Colman: Well, that's obvious to start with. In fact, most of these sumamry statistics are incomparable.
Migeru: not obvious to economic policy makers. ... So in economics what you can measure is meaningless and hwat is meaningful is unmeasurable?
Colman: I don't know about that - though it may be true - but what is measured is not comparable. Unemployment, gdp, military spend, research spending, health spending etc.
...
Migeru: apparently keynes wanted to use only nominal GDP but I don't know why other than some qualitative statements
Colman: Because real GDP is made up? Measurement errors on top of measurement errors on top of biases. Inflation is affected by political and technical choices in what is measured, GDP is "corrected" for various factors - hedonistic pricing among them, and so on.
Migeru: well, that just makes matters even worse
Colman: Absurd on the face of it.
Migeru: my argument doesn't involve arbitrary choices of baskets, or hedonic pricing or anything like that
Colman: I know.
Migeru: so that, plus practical issues of measurement just make matters worse.
Colman: GDP is, on many levels, a shit measure of anything. That it has appalling technical problems is hardly surprising.
Migeru: but it doesn't even make mathematical sense in a perfect world.
Colman: Frankly, I think it harks back to a time when it was the best they could do. GDP is easy to measure. Or approximate. From tax records.
Migere: sure, what else is the government going to use for policy? (by way of quantitative information)
Colman: The problem is that the number is, again, only slightly meaningful. So that, all things being equal (hah!) and with a sensible choice of basket, real GDP growth tells you that your economy is producing wealth faster than it was. ... Colman: But comparing 3% for one country vs 5% for another doesn't make much sense. And comparing GDP/head is meaningless. Especially since the policy discussions generally take place on the basis of provisional figures which are then revised. And US figures (broadly) are revised down, while European figures are (broadly) revised up.
Migeru: so it's all bullshit.
Colman: The way it is used is bullshit. Like unemployment.
Migeru: I'm beginning to think economics really is not at all a quantitative subject
Colman: Well, it has severe measurement problems. Rather like doing physics with only a metre stick, and without standardised metre sticks in different countries.
Migeru: no, I think it's not rather like doing physics. At all. Or maybe it's like Aristotelian physics: with entirely the wrong concepts.