Yes, but number theory only gained practical application relatively recently. Your claim was that if people in the 1500s and 1600s had had access to this number theory, we'd all be better off now. I

Sorry, poor wording on my part. I mentioned number theory initially as an area because it is the one where we most unambiguously lost Greek knowledge. But it seems pretty clear we lost many other areas also, hence why I mentioned trig, where we know that there were multiple treatises on the geometry of triangles and related things which are no longer extant but are referenced in extant works.

I'm not at all convinced incidentally by the argument that people would have just sat on the number theory. Since the late 1700s, the rate of mathematical progress has been rapid. So while direct focus on the areas relevant to cryptography might not have occurred, closely connected areas (which are relevant to crypto) would certainly be more advanced.

I'm glad that the economics faculty spend much of their time thinking of ways to fix the problems caused by the sociology faculty, but it would save everyone time and money if they all went home.

So what evidence do you have that the economists are fixing the problems created by the sociologists in any meaningful sense?

No. I'd be more impressed with Gates if he gave people cash to satisfy their own revealed preferences, rather than arbitrary metrics. But that wouldn't look as caring.

So that won't work at multiple levels. A major issue when assisting people in the developing world is coordination problems (there are things that will help a lot but if everyone has a little bit of money they don't have an easy way to pool the money together in a useful fashion). Moreover, this assumes a degree of knowledge which people simply don't have. A random African doesn't know necessarily that bednets are an option, or even have any good understanding of where to get them from. And then one has things like vaccine research. You are essentially assuming that market forces will win do what is best when one is dealing with people who are lacking in basic education and institutions to effectively exercise their will even if they had the education.

Moreover, bundling, marketing, brands and network effects are not examples of market failure.

Huh? All of these can result in total utility going down compared to what might happen if one picked a different market equilibrium. How are these not market failures?

In fact, marketing produces positive externalities.

In limited circumstances, marketing can produce positive utility (people learn about products they didn't have knowledge of, or they get more data to compare products), but I'm curious to here how marketing is at all likely to produce positive externalities.

I'll grant you that Microsoft has advantages in government contracting that they wouldn't have in a proper free market, but you should in turn admit that they have also suffered from anti-trust laws.

Yes, they have and that's ok. Anti-trust laws help market stability. The prevent the problem we've seen in the banking and auto industries of being too big to fail, and prevent the problem of bundling to force products on new markets (which again I'm quite curious to here an explanation for how that isn't a market failure).