DanielMcLaury

I wouldn't call the fact that pi is irrational an insight of "early mathematics." We knew that sqrt(2) was irrational around 500 BC if not earlier. We didn't know pi was irrational until around the time of the American revolution.

Multiplication of real and complex numbers is typically defined by starting from repeated addition, extending this notion to rationals, and then extending that notion to reals by taking limits.

How exactly are you going to present multiplication of real numbers axiomatically without essentially including an axiom that bootstraps everything from repeated addition?

I suppose you can try defining the reals as "the unique complete ordered field" or the complex numbers as "the unique algebraically closed field of characteristic zero with cardinality c," but I don't think either of those are pedagogically useful to someone who is still learning what multiplication is.

This statement is correct, but doesn't address the problem with the OP's calculation.

This fund is up basically entirely on the strength of TSLA being up 700%. OP is basically considering two possibilities:

1. TSLA stock is a driftless geometric Brownian motion with a volatility matching that of the general market, and happened to get a 700% return purely by chance, or

2. The fund manager, due to his exceptional skill, knew that TSLA was going to be up 700%.

The OP is rejecting option (1) and then concluding that option (2) must be the case.

Of course in reality neither is the case and the OP's calculation is totally irrelevant.

They may not have "a mental model of statically typed languages," but they sure "agonize" when there are eight different incompatible datatypes that represent a datetime and the library functions they want to use don't even specify which of these they take or return.

That's not necessarily the case in general. If you had an algorithm which, given a billion dollars' worth of specialized lattice reduction ASICs, could break 2048-bit RSA in a few months' time, then for all practical purposes this would mean that 2048-bit RSA was broken -- there are plenty of state actors who would drop that kind of money in a heartbeat -- but that doesn't mean that some guy who came up with it would be able to do a demo on his laptop.

(Not saying that the above has anything to do with this paper in particular.)

From the article:

"Its long/short equity fund gained an astounding 274 percent, thanks in large part to a 700 percent surge in the price of Tesla’s stock, which accounted for 37 percent of Worm’s publicly traded equities portfolio at the end of the third quarter."

This means that, outside of his TSLA position, the rest of his portfolio made about 25%. In a normal year that'd be impressive, but 2020 was a year where SPY was up 15% and there was insane volatility.

So basically this guy gets decent-to-good performance on 3/5 of his portfolio and put the other 2/5 into a blind gamble which turned out to pay off. The chances of that happening by luck aren't "4.82e-18."

"He took big, and early, stakes in both Amazon and Tesla — in 2012 and 2016, respectively."

Amazon was founded in 1994 and had its IPO in 1997 during the first dot com boom. Did they do any research for this article at all?

One subject that's pretty useful for programmers to understand reasonably well is numerical analysis, even if it's at a fairly basic level. It's amazing how quickly you run into problems in practice as soon as you try to do anything with floating point numbers.

I didn't carefully read through the section on modular arithmetic, so maybe this is already there, but one thing worth noting that bites a lot of people is that using modular arithmetic for integer representations means that addition, subtraction, and multiplication work the same for signed and unsigned integers. This means that you can typically write arithmetic expressions involving these operations without worrying too much about whether things are being regarded as signed or unsigned at intermediate steps. As soon as you introduce division into the mix, though, the whole thing falls apart, and you have to be very careful about how the language is interpreting each subexpression.

I think in a lot of cases this is sort of like asking "how can I use an abstract factory to improve my COBOL code?"

Like, if your code isn't already written in terms of classes, you probably can't just jump in and use OOP design patterns to do anything.

Similarly, you may need to start from a reasonably functional codebase to apply most functional programming concepts.

This is why affine schemes as a thing never really took off until Grothendieck, despite Noether and Krull noting some of their interesting properties decades before. If you're just using affine schemes you don't get anything you don't get just by doing commutative algebra.

However, once you start gluing affine schemes together, you start getting things that represent functors you can't represent if you're working over the category of commutative rings alone.

To be fair that's just vanilla English orthography.