Imagine there is a leech stuck on your body sucking blood. You will now have to eat more to replenish the blood. Here is the kicker though, the leech gets bigger as it sucks out more blood out of your body and therefore it will start draining more blood from your body.

You go to your doctor and he says take these pills they will make your body grow bigger. As the leech gets bigger you must get bigger which means you must take more pills. If you stop taking the pills, the leech will outpace your body growth and your body will suffer from anemia despite having more blood in your body than the average healthy human.

Suddenly everyone gets a fetish for being as large as an possible, it is good for your health, nobody ever questions whether they should just get rid of the leech and get their body to a normal height.

The problem with the leech analogy is that we can't actually tell what part of society is "the leech".

Society and economics can be best described not as a human with a leech on it, but a writhing mass of leeches all sucking each others' blood. In other words, mutual parasitism. What parts of the economy can be "gotten rid of" - as in, the people who run them being socially marginalized or worse - depends solely on what you value; rather than any objective criteria.

The historical examples of zero-growth societies are rather bleak. They were closed, feudalistic societies that relied upon strict classes with disenfranchised peasants or slaves at the bottom. This is the most optimal strategy for exploiting a static resource economy[0]. Liberal democracies are tied to economies with some level of growth to them, because there needs to be new treasure to chase in order to allow a continued changing of the guard and the deposition of elites.

[0] Related note: most historical examples of witch trials in Europe were ways to punish people with newly-acquired wealth. Zero-growth was the oxygen people breathed back then; and anyone with new wealth was looked at with suspicion. Did they steal it? Did they sign a deal with the devil?

The distinction between parasites and symbiotes needs to be made here. Too many in society are parasites, despite portraying the impression they are symbiotes. I'd posit entire industries fall into that category.

Every living organism that doesn't have photosynthesis may be seen in some way as a parasite.

Every individual live in something like an ecosystem. I would argue that society is part of the ecosystem for humans.

Whether or not (or in what cases) one individual should be forced to give up resources to another will be a never-ending discussion. Should government money be used for development aid abroad? For social welfare domestically? Are you responsible for caring for your parents? Your sibling? Your children? Is a mother responsible for providing her biological resources to an unborn child?

We may never come to permanent, full agreement on most of these questions.

"in some way" sure, but in lots of other ways not. For the sake of discussion let's pose one way in which we could define the distinction as a purely utilitarian metric - does an individual contribute more to society than they take out. The problem is that it seems to me that many individuals are professionally parasitic. That is their jobs, which they carry out diligently and conscientiously, don't contribute anything useful. As such they are purely drains on society.

I would put much of the finance industry in that bracket, which is essentially a rent seeking industry (albeit in various convoluted ways). It's doubly bad in finance because it sucks huge amounts of talent from areas where the individuals could actually contribute usefully.

Is there a mathematical proof that that is the optimal strategy for zero growth resource economies?

No, because there is no mathematical proof for any assertion in economics. You cannot prove economics true.

However, historical counter-examples of zero-growth democracies are limited. We might look to the Greeks or Romans as examples of democracy in the antiquities; but there's two problems with that:

1. Their democracies were limited, with no universal suffrage and very low citizenship rates.

2. They had economic growth anyway. Rome in particular was as addicted to military conquest then as America is to oil today.

Selectorate theory[0] would argue this point in the opposite direction; i.e. that strict dictatorships squander economic growth. Political and economic franchise are inherently linked: people who cannot vote will have their livelihoods taken away from them, while those who produce the economy's goods will eventually demand political power themselves. Democracies are ultimately the means by which we distribute political power among the people, and that is itself a reflection of the distribution of economic production. If you give the people political power, the economy can't help but to grow as they are no longer being shackled.

But I think there's an even simpler argument than that. If the economy is zero-growth, then the only way for an individual to improve their living conditions is to make someone else's worse. This turns the economy into a zero-sum game of who can monopolize everything the fastest, which spikes income inequality, which makes it easier to disenfranchise voters, and eventually results in the ossification of social and political structures (since you shut up everyone that you said didn't matter).

[0] Excellently summarized by CGP Grey here: https://www.cgpgrey.com/blog/rules-for-rulers based on a book called The Dictator's Handbook

Is the leech the welfare system?

It’s certainly the best ladder I’m aware of that we’ve discovered- you climb it, by and large, by providing value to other people. A little value to a lot of people, usually. And because you are producing new value, you don’t provoke a conflict by reallocating the value away from the previous people.

Most older ladders were providing a lot of value to a few existing rulers, and concentrated the utility-allocating aspects of society in the hands of a few. And much of the value being provided was through violent seizure.

Eventually these ladders have a top. The 20th century is a mere aberration in growth in human timescales. It's easy to say this is the best - but we're simply biased - it was the best, for us, for a time, largely due to hegemony. The cracks are rapidly showing. Eventually knew value can't be created out of massive scientific advancements (low hanging fruit) we've had. New value will come from others (labor cost reductions to the far east, for example.)