Here's a decent summary of the Aggregation Problem:
In neoclassical economics, a production function is often assumed, for example,
Q = A f(K, L)
where Q is output, A is factor representing technology, K is the sum of the value of capital goods, and L is the labor input. The price of the homogeneous output is taken as the numéraire, so that the value of each capital good is taken as homogeneous with output. Different types of labor are assumed reduced to a common unit, usually unskilled labor. Both inputs have a positive impact on output, with diminishing marginal returns.
In some more complicated general equilibrium models developed by the neoclassical school, labor and capital are assumed to be heterogeneous and measured in physical units. In most versions of neoclassical growth theory (for example, in the Solow growth model), however, the function is assumed to apply to the entire economy. This view portrays an economy as one big factory rather than as a collection of a large number of heterogeneous workplaces.
This vision produces a core proposition in textbook neoclassical economics, i.e., that the income earned by each "factor of production" (essentially, labor and "capital") is equal to its marginal product. Thus, with perfect product and input markets, the wage (divided by the price of the product) is alleged to equal the marginal physical product of labor. More importantly for the discussion here, the rate of profit (sometimes confused with the rate of interest, i.e., the cost of borrowing funds) is supposed to equal the marginal physical product of capital. (For simplicity, abbreviate "capital goods" as "capital.") A second core proposition is that a change in the price of a factor of production will lead to a change in the use of that factor – an increase in the rate of profit (associated with falling wages) will lead to more of that factor being used in production. The law of diminishing marginal returns implies that greater use of this input will imply a lower marginal product, all else equal: since a firm is getting less from adding a unit of capital goods than is received from the previous one, the rate of profit must increase to encourage the employment of that extra unit, assuming profit maximization.
Piero Sraffa and Joan Robinson, whose work set off the Cambridge controversy, pointed out that there was an inherent measurement problem in applying this model of income distribution to capital. Capitalist income (total profit or property income) is defined as the rate of profit multiplied by the amount of capital, but the measurement of the "amount of capital" involves adding up quite incomparable physical objects – adding the number of trucks to the number of lasers, for example. That is, just as one cannot add heterogeneous "apples and oranges," we cannot simply add up simple units of "capital." As Robinson argued, there is no such thing as "leets," an inherent element of each capital good that can be added up independent of the prices of those goods.
Based on this, we require a unit of account apart from money that represents "an inherent element of each capital good". Labor values seem to be most the logical unit of account for this task. And in order to prove that they can be a valid unit of account the Transformation Problem must be resolved, which is what this paper is all about. (The author sets out to mathematically prove that labor values and prices of production are proportional if we incorporate the entirety of labor that goes into the process. He argues that this should include the direct labor involved in production, the indirect labor involved in replenishing the non-human factors of production as they are degraded/depleted from utilization in the process of production, and the "super-indirect" labor involved replenishing the non-human factors of production that are used to create the MoP in the first place. Wright says that Marx and his critics (both bourgeois and Marxist economists) all made the same mistake - omitting super-indirect labor from calculation process, which resulted in disproportionality between (mis)calculated labor values and prices of production.)
By using labor values in this manner, we have essentially endorsed an objective theory of value.
As for the legacy and impact of the Cambridge Capital Controversy...
"It is important, for the record, to recognize that key participants in the debate openly admitted their mistakes. Samuelson's seventh edition of Economics was purged of errors. Levhari and Samuelson published a paper which began, 'We wish to make it clear for the record that the nonreswitching theorem associated with us is definitely false. We are grateful to Dr. Pasinetti...' (Levhari and Samuelson 1966). Leland Yeager and I jointly published a note acknowledging his earlier error and attempting to resolve the conflict between our theoretical perspectives. (Burmeister and Yeager, 1978). However, the damage had been done, and Cambridge, UK, 'declared victory': Levhari was wrong, Samuelson was wrong, Solow was wrong, MIT was wrong and therefore neoclassical economics was wrong. As a result there are some groups of economists who have abandoned neoclassical economics for their own refinements of classical economics. In the United States, on the other hand, mainstream economics goes on as if the controversy had never occurred. Macroeconomics textbooks discuss 'capital' as if it were a well-defined concept — which it is not, except in a very special one-capital-good world (or under other unrealistically restrictive conditions). The problems of heterogeneous capital goods have also been ignored in the 'rational expectations revolution' and in virtually all econometric work." (Burmeister 2000)