Coached by a former Wall Street bond trader who studied the opposition and set up a pipeline that produces Superstar Mathletes:
“You wouldn’t grab a kid in ninth grade who’s never played football and expect him to be a great high-school football player,” he said. “For most of these kids, this is their football.”
Mr. Frazer’s insight was to connect four levels of education: The kids he scouts in elementary school develop in middle school, compete in high school and take specialized classes from college professors that he brings to Buchholz’s campus. As soon as the system was in place, the team started winning and never stopped.
It turned out there was value in putting a bunch of smart kids in the same room: They feel empowered to make each other smarter.
Many of the gifted kids in his program have parents who work at the nearby University of Florida and push to get on Mr. Frazer’s radar. Others he finds on his own. He tracks down test scores of students in his district, follows the data and recruits high achievers. Some who were discovered by his spreadsheets have since graduated from the Massachusetts Institute of Technology with math degrees and landed on Wall Street themselves.
The mathletes who try out for the team and make the cut are combined into one class section and fly through competitive algebra, geometry and calculus during the school day. Mr. Frazer essentially bends the rules to move faster through harder material and pack more than two years of math into one school year. “I cover everything the state wants me to cover,” he said. “But there is no restriction on covering extra material.”
This (beautifully formatted and well-paced-and-delivered and surprisingly sparsely attended) talk by Gabriel Lebec on the fundamentals of Lambda Calculus is one of my favorite talks ever.
The premise is that there are enchanted forests which contain many (or sometimes very few) talking birds. Smullyan dedicated the book to Haskell Curry - an early pioneer in combinatory logic and an avid bird-watcher. The birds, which I suppose represent the combinators, have an interesting characteristic: Given any two birds A and B, if you call out the name of B to A it will respond by calling out the name of some bird to you.
This bird whose name A calls when you call B is denoted as AB. Once you have several birds in place, a single call can cascade around the forest with each call following rules depending on who produces it.
The very first bird we are introduced to is the Mockingbird whose characteristic behaviour is that whatever name you call to the Mockingbird, it will reply as if it is the bird whose name you called. This is denoted:
Mx = xx
For any bird x we can say that Mx (the result of calling x to a Mockingbird) is the same as xx (the result of calling x to a bird of type x). It really does mock other birds! And what’s more, the existence of the Mockingbird, in combination with various others, unlocks some really fascinating group behaviour from these birds.
And!
Soon we discover that birds have certain properties: The can be fond of other birds, they can be egocentric if they are fond of themselves. The can be hopelessly egocentric if they only ever talk about themselves. There are happy birds, normal birds, agreeable birds and many others. We also meet other types of birds with specific properties - the Lark, the Kestrel, Sage birds, Bluebirds, aristocratic birds, Eagles, the list goes on and on. Luckily there is a Who’s Who list of birds in the back to keep track.
This is the city of Madras The home of the curry and the dal Where Iyers speak only to Iyengars And Iyengars speak only to God.
I’d read this years ago some place and forgot where. Thought it would be in some Religious Studies textbook back from when I was (briefly) a Religious Studies major. Nope! It was the great Paul Erdős!
“Erdős said he’d modelled it after this ditty about the privileged New England families famously known as the ‘Boston Brahmins’.”
This is good old Boston The home of the bean and the cod Where the Lowells speak to the Cabots And the Cabots speak only to God.
Being a long and informative post that leads to “Use /dev/urandom” and features a quote by DJB and a list of computationally secure PRNGs. Cached here.
The argument here being that, since the ‘general form’ of the conjecture is undecidable, TypeScript’s type system is undecidable. How does one even think of doing these things?
I love Typescript, but it isn’t nearly ambitious enough. It would be vastly improved with an --extremelyStrict flag enforcing that your Typescript code is free of side-effects; that is – no Javascript code is generated at all. Real programmers do all of their computation within the type system. Otherwise, they can’t be sure their program will work in production and should be duly fired.
When you end up paying the price you bid (“first price”), you have a strong incentive to lie about how much you’re willing to pay. Suppose there’s an item for sale that you’d be happy paying up to $1,000 for if necessary, but of course you’d rather pay less. In a first-price auction, if you bid $1,000 and you lose. Well, someone else was willing to pay more than you were willing to, so that’s OK, but if you win, you know that nobody else offered that price and you’d be slapping yourself for not going for $950 and saving a little. Or, who knows, maybe there are really few buyers and you later discover that the second person was only valuing the item at $600? Damn, you could have walked away with it for $610!
[. . .]
In a second-price auction, there’s no reason for you to do that. You can simply say exactly the maximum price you are willing to pay, and there’s never any advantage for you in saying anything else:
You don’t want to post a higher bid since you might be forced to pay it, and you don’t want to do that.
You don’t want to post a lower bid since you might lose the item for no good reason at all.
You’ll end up paying exactly what it takes to win the item: one dollar or one cent more than the next person’s maximum bid.
So, a second-price mechanism encourages everyone to bid truthfully, and the item ships to the person who really values it at the highest price. It’s the best outcome for the seller and as good an outcome for everyone else as they could wish for.
Incidentally, note that this is exactly what happens in ordinary public auctions (“going once, going twice… sold!!”) Everyone walks in with an idea of how much they’re willing to pay, and they keep bidding one dollar more than the current price until they hit their max—but they’re never forced to reveal their max and what they end up paying is just one dollar (or penny, or whatever) more than the second-highest bidder.
Numbering is done with natural numbers. Let’s take zero to be the smallest natural number1. For the sequence (2, 3, 4, … ,12), using the convention (2 ≤ n < 13) is appropriate because
For a sequence starting with zero, like (0, 1, 2, 3), the left hand condition leaks into unnatural numbers if you use “less than”: (-1 < n).
For an empty sequence, the right hand also leaks into the unnatural if you use “less than or equal to”: (n ≤ 0)
And minorly, because these are the true of another convention (1 < n ≤ 12)
Difference between bounds (13 - 2 = 11) is the length of the sequence
I know that these two sequences are adjacent: (2 ≤ n < 13) and (13 ≤ n < 24)
All that’s prep for:
When dealing with a sequence of length N, the elements of which we wish to distinguish by subscript, the next vexing question is what subscript value to assign to its starting element. Adhering to convention a) yields, when starting with subscript 1, the subscript range 1 ≤ i < N+1; starting with 0, however, gives the nicer range 0 ≤ i < N. So let us let our ordinals start at zero: an element’s ordinal (subscript) equals the number of elements preceding it in the sequence. And the moral of the story is that we had better regard – after all those centuries!2 – zero as a most natural number.
There’s also this little nugget
I think Antony Jay is right when he states: “In corporate religions as in others, the heretic must be cast out not because of the probability that he is wrong but because of the possibility that he is right.”
