There's an interesting discussion going on over at David Glasner's excellent Uneasy Money blog, dealing with what Keynes had to say in the General Theory about saving and investment and how that fed into his theory of the rate of interest. This is an area which we tend to take very much for granted, and it's one of several points of theory with regards to which you could argue that we have discarded the economics of the General Theory and simply slapped the label "Keynesian" onto what is really AC Pigou's macro model.
Keynes is quite clear about S and I in the General Theory: each is the gap between income and consumption, and since each equals Y-C, they must be equal. They are, however, behaviourally different: Investment is a choice variable in an intertemporal optimization problem, but saving is not - it is consumption which is the choice variable in the household's intertemporal utility maximization problem. Keynes had not forgotten Frank Ramsey's work on saving, he did see the individual household as solving an intertemporal optimization problem - see the chapters in the General Theory on the propensity to consume - , it was just that he took the view that, especially in the short run, income was the predominant determinant of saving. Further, whatever decisions the individual household might be making, in the aggregate saving had to be equal to investment by definition, and the equality was not driven by saving.
The important thing, to Keynes, was what this meant for the determination of interest rate. Since S was always identically equal to I by definition, the interest rate could not be a price which equated S and I, so it must be determined somewhere else in the economy. This tied into his criticism of the way standard texts of his time treated interest rate policy - if the rate of interest, r, was determined by the S(r) and I(r) schedules, how was it that the central bank could influence r? What did monetary policy affect, S or I, and note that in the classical model this effect would have to involve a shift of whichever of the curves it was operating on, not a movement along it (or them).
Keynes' answer to this was that the rate of interest was determined by portfolio choices; not the level of saving but the way individuals allocated their savings across financial assets. The default choice of holding savings was in the form of money (in the General Theory, money pretty broadly defined) and here Keynes agreed with Irving Fisher that the fact that money is, by definition, something which confers complete liquidity on its holder means that there is a liquidity return for holding cash (including certain types of bank money) and this liquidity return creates a certain preference for liquidity. The rate of interest, then, depends on how strong this preference for liquidity is and what kind of return it takes to get people to put part of their savings (the level of which in the aggregate is determined by income) into non-monetary, interest bearing assets.
Keynes knew, by the way, that there was more than one interest rate - in the General Theory he simply assumed for simplicity that the spectrum of rates across duration and risk of assets was stable, so that a change in the short term rate would serve as an index of the shift in the whole spectrum of rates. He also assumed that what mattered for economic purposes was the long term rate of interest - that was what drove Investment spending - but he held that the bank rate could influence the long term rate in a stable manner. Hawtrey argued that it was the short term rate itself which mattered and further that, if you looked at data over a Century of Bank Rate, you'd find that the relation between long and short rates wasn't so stable after all.
When we consider what Keynes had to say about S=I it's important to remember that he was trying to counter three theoretical arguments. There was the classical one which we're familiar with, to the effect that S equalled I because the interest rate made it so. But in addition to that, there was Hawtrey's argument that I equalled S plus any new bank credit which the banks created to finance investment that period. And, ironically, there was Keynes' own model from the Treatise on Money, according to which (using what he later admitted was a non-intuitive definition of income) investment could differ from saving. In the Treatise, the central bank could influence the market rate of interest (exactly how was rather taken for granted) and Keynes defined a natural rate of interest as the rate which would make S = I (using to the definitions of the Treatise). So he had to convince people that his own earlier writings had been wrong. Thus what he needed to do if he wanted to convince people of what was really important, i.e. his monetary theory of the rate of interest, was to convince them that S had to equal I by definition.
There's much more in the Uneasy Money post, in particular about Keynes' comments on Irving Fisher's concept of the real rate of interest, but commenting on that'll wait for another post.