It is impossible for an accounting identity to have any causal implication or explanation

Paul Krugman, in an apparent attempt to be too generous to the argument he's taking apart, gets something fundamentally wrong about accounting identities:

Here’s what happens: you start with an accounting identity, in this case savings = investment, and treat it as a causal relationship – savings => investment – imagining that this excuses you from the need to lay out a mechanism for this alleged causation. 

The immediate thing Fama should have asked himself, even if completely ignorant of the history of macroeconomics, is why the causation necessarily runs from savings to investment. Why not the other way around? 

Let's stipulate that if the sun comes up tomorrow at 6:30, it will also be the case that tomorrow at 6:30 the sun will come up.  Does the sun coming up tomorrow at 6:30 cause, at 6:30, the sun to come up, or is it the other way around?  Neither.  It makes no sense to say either causes the other when they're the same event.

There are interesting questions around causality in economics.  If the government spends money today, will this cause a private business to hold back on investment?  Maybe, maybe not, but what's important is to understand why no accounting identity can possibly shed any light on this question.  The problem is that now, unlike in the sun example above, we are dealing with two events, and these events are separated by time.  But accounting identities are too absolute to tell us anything about causally linked economic transactions, because they hold over any span of time, including spans of time that include just the cause and those including just the effect.  So even if you find a real causal relationship between some economic transactions, no accounting identity could have required that causal relationship to exist.

This can be stated even more strongly: every single transaction that occurs in the economy preserves the accounting identities all by itself, regardless of what other economic transactions occur.  Therefore, accounting identities only constrain the nature of individual transactions; any conceivable correct or incorrect causal story about how different economic transactions relate to each other will always be consistent with all accounting identities because the individual transactions must be.

Let's work through an example of how savings = investment is preserved in practice.  Let's say I'm considering buying a $5 loaf of bread which is made of ingredients that cost a bakery $3 to acquire.  In the scenario where I buy it, the bakery profits $2, which, because that profit can't immediately be spent will become $2 in savings.  If I don't buy, I save $5, and the bakery has unsold inventory that cost $3 to make, so it has invested $3.  (Yes, when a business makes a good no one wants to buy and ends up throwing it away, that's an example of "investment", why do you ask?)  In the scenario where I buy, there is $3 less saving and $3 less investment.  So did the fall in savings cause the fall in investment or vice versa?  It makes little sense to ask when only one thing has happened.

But won't my decision to consume or to save affect my future decisions to consume or save?  Won't the bakery's profit affect their decisions to consume, save, or invest?  Probably, but the accounting identity does not require this; this single transaction preserved S=I all by itself, and if other future transactions are caused by this they will also need to preserve S=I all by themselves.  My decision to buy or not buy a loaf of bread does not cause some imbalance in the identity that must be corrected by some other economic transaction; no corrective forces in the economy come into play to balance S=I over the long run because S=I can't be violated for even a nanosecond.