mildly relevant: growth rates, as mentioned by pauΙ graham.

~from "startup = growth" by pauΙ graham

The phase whose growth defines the startup is the second one, the ascent. Its length and slope determine how big the company will be.

The slope is the company's growth rate. If there's one number every founder should always know, it's the company's growth rate. That's the measure of a startup. If you don't know that number, you don't even know if you're doing well or badly.

When I first meet founders and ask what their growth rate is, sometimes they tell me "we get about a hundred new customers a month." That's not a rate. What matters is not the abolute number of new customers, but the ratio of new customers to existing ones. If you're really getting a constant number of new customers every month, you're in trouble, because that means your growth rate is decreasing.

yes, a basic notion, but one not immediately obvious to the unambitious. a constant growth rate implies an exponential growth, which is more than many people expect out of their endeavors.

in general, i like how graham is willing to quantify his conclusions. a rough computation, for example, explains why what we observe about startups seems different from why they should exist:

If you judge by the median startup, the whole concept of a startup seems like a fraud. You have to invent a bubble to explain why founders want to start them or investors want to fund them. But it's a mistake to use the median in a domain with so much variation. If you look at the average outcome rather than the median, you can understand why investors like them, and why, if they aren't median people, it's a rational choice for founders to start them.

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i also noticed this observation of graham's, though meant for startups, might also be relevant for us mathematicians to revisit:

You'll generally do best to follow that constraint wherever it leads rather than being influenced by some initial vision, just as a scientist is better off following the truth wherever it leads rather than being influenced by what he wishes were the case. When Richard Feynman said that the imagination of nature was greater than the imagination of man, he meant that if you just keep following the truth you'll discover cooler things than you could ever have made up.
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What you're looking for initially is not so much a great idea as an idea that could evolve into a great one. The danger is that promising ideas are not merely blurry versions of great ones. They're often different in kind, because the early adopters you evolve the idea upon have different needs from the rest of the market. For example, the idea that evolves into Facebook isn't merely a subset of Facebook; the idea that evolves into Facebook is a site for Harvard undergrads.

on a related note, my prejudice of startups is generally that of hyper-capitalists, fueled by powerful greed, and willing to use science and technology to do so (i.e. the dark side of the force).

the feeling i've been getting though, from the quants and startup acquaintances i've met, is that these just happen to be hard-working, quantitatively-minded people who are making usable products efficiently. they could have easily have been scientists or engineers, but working in this kind of business offers better rewards with the same interesting (and fulfilling) kind of work ..

.. or, as graham writes:

Starting a startup is thus very much like deciding to be research scientist: you're not committing to solve any specific problem; you don't know for sure which problems are soluble; but you're committing to try to discover something no one knew before. A startup founder is in effect an economic research scientist.