The ledgers are not the source of truth. No: this is a journaled filesystem; the transaction log is the ultimate source of truth. When you detect faults you recover from the journal, if something isn't completely written into the journal then it does not exist. The zero-sum property exists on every individual transaction and each transaction has an ID and the ledgers point at these IDs for auditing purposes.
So why have the zero sum property? Well, for one thing, it creates a uniform access model, I can't just credit my account, I have to debit someone else's account and they can have rules that might prohibit me from doing so. By carefully setting up these accounts you can also do what's sometimes called "behavior anomaly detection."
So for instance normal financial cards kind of work by opening up your entire wallet to a cash register and asking the cash register to pull out exactly the amount that you owe, understandably this might not be desirable if you are tracking some in-game-gold transactions in an internet game. You can get as elaborate as you want but think for example of a quick-consistency-check rule saying that “accounts starting 4xxx (player balances) never transfer directly to each other, instead users are expected to put the exact sum of money plus a little 5% padding into one of their 5xxx accounts and then give someone else a token permitting them to withdraw from that account." Stuff like that. A developer tries to code something up that doesn't go through this process and runs into errors in testing and has to conform their code’s behavior to the less risky process so that nobody is ever opening their entire wallet to a griefer.
Double entry basically just means that for every transaction on your books, you are showing where the money/asset/liability came from and/or where it's going. If you enter that you paid $100 for the AWS bill, you need to indicate from what account that money came from. So you would debit AWS Expense and you would credit your Wells Fargo Checking Account. With modern software, this basically just amounts to entering the expense and specifying which account it came from.
Doble entry accounting is very similar to a checksum in that sense. It provides error detection. If you make a mistake in one of those two places you will now immediately. As opposed to single entry, where you can carry that error indefinitely until somehow you catch it.
Wikipedia tells us that double-entry bookkeeping is first attested in the late 13th century.
But commercial institutions larger than the personal-trust threshold are attested as far back as written history goes.
[citation needed]? I don't necessarily doubt you but I'd like to see an example. The only thing I can really think of would be governments, but that was definitely personal trust (mixed with a little bit of threat of summary execution if the king didn't like you).
The smallest amount of tribute listed there is more than five and a half tons of silver. If we make the assumption that each of these districts was handled by a different person, we already have a staff of 20 people plus the king.
(Not to mention, the large majority of Achaemenid taxation was not collected in this form. It was collected in kind and stored in an empire-wide system of distributed warehouses. I assume metal tribute was centralized. If you think the warehouse administrators - who inventory the goods and are responsible for making payments out of them - are "responsible for money", you're probably adding at least a couple dozen more people.)
By contrast, https://www.theofficialboard.com/org-chart/exxonmobil suggests that the top two levels (including the board) of Exxon total 26 people, or (excluding the board as a level, but still including the CEO) 24 people. The CEO has 13 direct reports.
So if the argument is that ancient kingdoms were below the personal trust threshold because the king had a low - and therefore manageably trustworthy - number of direct reports, it appears to be the case that our largest corporations today are also comfortably below the personal trust threshold, so there was never any need to exceed the threshold and in fact we never have.
On the other hand, if we think Exxon is beyond the personal trust threshold by virtue of its geographic extent (huge!), or its total number of employees (huge!), or its total number of employees touching money with the potential opportunity to steal that money (still huge!)... I think we have to say the same thing about ancient palaces, and really about ancient temples. A temple might "only" employ a few hundred people, but that's more people than you can personally trust. An important temple employed a few thousand people.
In a business I hand you my money and trust that you will hand me back my profits.
A ancient government put someone in charge and demanded tribute. Failure to produce tribute resulted in an army showing up to collect tribute. It was so common as to be expected that the administrator, called a satrap, would collect excess tribute and live a lavish lifestyle. The king sent spies to root out the worst of the abuse, but it was an uphill battle.
Governments can be run this way because efficiency is not particularly essential to their profitability. But efficiency is required for commercial enterprises. It is not enough to collect money and let your subordinates enrich themselves. You want to track all the money and not let your subordinates cheat the enterprise to their personal profit.
> But efficiency is required for commercial enterprises. It is not enough to collect money and let your subordinates enrich themselves. You want to track all the money and not let your subordinates cheat the enterprise to their personal profit.
The first two sentences are obviously false; you want to track all the money, but you have no particular need to do so as long as some of the money is making its way to you.
However single entry bookkeeping is FRAUD prone. Just change one number and your theft is hard to track down!
That is why the adoption of double entry bookkeeping was critical for allowing commercial institutions to outgrow a size where owners could individually trust all who were working for them with access to money.