reperiendi

Posted in Category theory, Math, Programming by Mike Stay on 2012 August 28

With the new ECMAScript 6 Proxy object that Firefox has implemented, you can make dot do pretty much anything you want. I made the dot operator in JavaScript behave like Haskell’s bind:

```// I'll give a link to the code for lift() later,
// but one thing it does is wrap its input in brackets.
lift(6); // [6]
lift(6)[0]; // 6
lift(6).length; // 1

// lift(6) has no "upto" property
lift(6).upto; // undefined

// But when I define this global function, ...

// Takes an int n, returns an array of ints [0, ..., n-1].
var upto = function (x) {
var r = [], i;
for (i = 0; i < x; ++i) {
r.push(i);
}
return r;
};

// ... now the object lift(6) suddenly has this property
lift(6).upto; // [0,1,2,3,4,5]
// and it automagically maps and flattens!
lift(6).upto.upto; // [0,0,1,0,1,2,0,1,2,3,0,1,2,3,4]
lift(6).upto.upto.length; // 15
```

To be clear, ECMAScript 6 has changed the API for Proxy since Firefox adopted it, but you can implement the new one on top of the old one. Tom van Cutsem has code for that.

I figured this out while working on a contracts library for JavaScript. Using the standard monadic style (e.g. jQuery), I wrote an implementation that doesn’t use proxies; it looked like this:

```lift(6)._(upto)._(upto).value; // [0,0,1,0,1,2,0,1,2,3,0,1,2,3,4]
```

The `lift` function takes an input, wraps it in brackets, and stores it in the `value` property of an object. The other property of the object, the underscore method, takes a function as input, maps that over the current value and flattens it, then returns a new object of the same kind with that flattened array as the new value.

The direct proxy API lets us create a “handler” for a target object. The handler contains optional functions to call for all the different things you can do with an object: get or set properties, enumerate keys, freeze the object, and more. If the target is a function, we can also trap when it’s used as a constructor (i.e. `new F()`) or when it’s invoked.

In the proxy-based implementation, rather than create a wrapper object and set the `value` property to the target, I created a handler that intercepted only get requests for the target’s properties. If the target has the property already, it returns that; you can see in the example that the length property still works and you can look up individual elements of the array. If the target lacks the property, the handler looks up that property on the window object and does the appropriate map-and-flatten logic.

I’ve explained this in terms of the list monad, but it’s completely general. In the code below, `mon` is a monad object defined in the category theorist’s style, a monoid object in an endofunctor category, with multiplication and unit. On line 2, it asks for a type to specialize to. On line 9, it maps the named function over the current state, then applies the monad multiplication. On line 15, it applies the monad unit.

```var kleisliProxy = function (mon) {
return function (t) {
var mont = mon(t);
function M(mx) {
return Proxy(mx, {
get: function (target, name, receiver) {
if (!(name in mx)) {
if (!(name in window)) { return undefined; }
return M(mont['*'](mon(window[name]).t(mx)));
}
return mx[name];
}
});
}
return function (x) { return M(mont[1](x)); };
};
};

var lift = kleisliProxy(listMon)(int32);
lift(6).upto.upto; // === [0,0,1,0,1,2,0,1,2,3,0,1,2,3,4]
```

Cantor’s diagonal proof

Posted in Math by Mike Stay on 2012 August 20

There’s a riddle about a library where the index had two parts, each in its own book. The first index book listed every book whose title appeared between its own covers; nearly every book in the library was listed in this index, since the title is typically on the title page. The other index book, a mere pamphlet really, was for those rare cases where the title was not between the covers of the book. Neither index had a title page, just a list of book titles and what shelf each book was on.

In which book was the first book listed? It could have been listed in itself, which is consistent with the claim that it lists all the books that contain their own titles. Or it could have been listed in the second book and not in itself, which would also be consistent.

The riddle is this: in which index was the second index book listed? If we suppose that the second book did not list itself, then it would be incomplete, since it is a book in the library that does not contain its title between the covers. If we suppose that it did list itself, then it would be inconsistent, since it is supposed to list only those books that do not contain their own titles. There is no way for the second index book to be both consistent and complete.

Georg Cantor was a mathematician who used this idea to show that there are different sizes of infinity! But before I go there, consider the Munduruku tribe of the Amazon, which has no words for specific numbers larger than five; how would a man with eight children tell if he has enough fruit to give one to each child? He would probably name each child and set aside a piece of fruit for them—that is, he would set up an isomorphism between the children and the fruit. Even though he cannot count either set, he knows that they are the same size.

We can do the same thing to compare infinite sets: even though we can’t count them, if we can show that there is an isomorphism between two infinite sets, we know they are the same size. Likewise, if we can prove that there is no possible isomorphism between the sets, then they must be different sizes.

So now suppose that we take the set of natural numbers {0, 1, 2, 3, …} and the set of positive even numbers {0, 2, 4, 6, …}. These sets both have infinitely many elements. Are they the same size? We can double every natural number and get an even one, and halve every positive even number and get a natural number. That’s an isomorphism, so they’re the same size.

How about natural numbers and pairs of natural numbers? Given two numbers like 32768 and 137, we can pad them to the same length and then interleave them: 3020716387; they come apart just as easily.

Since we can take the first number to be the denominator of a nonnegative rational number and the second number to be the numerator, we also find that the size of the set of rationals is the same as the size of the set of natural numbers.

Cantor came up with isomorphisms for each of these sets and thought that perhaps all infinite sets were the same size. But then he tried to come up with a way to match up the natural numbers and infinite sequences of bits, like the sequence of all zeros:

`    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...`

or the sequence that’s all zeros except for the third one:

`    0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...`

or the sequence whose nth bit is 1 if n is prime:

`    0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, ...`

or sequences with no pattern at all:

`    0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, ...`

He tried and tried, but couldn’t figure out a way to make them match up. Then he thought of an ingenious proof that it’s impossible! It uses the same insight as the library riddle above.

Suppose we try to match them up. We write in the left column a natural number and in the right column a binary sequence:

```    1: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
2: 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
3: 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, ...
4: 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, ...
5: 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, ...
6: 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, ...
...```

Consider the diagonal of the table above: 0, 0, 1, 1, 0, 1, … This sequence is number 6 above, and is like the first index book: the sixth bit in the sequence could be either 0 or 1, and it would be consistent.

Now consider the opposite of the diagonal, its “photographic negative” sequence: 1, 1, 0, 0, 1, 0, … This sequence is like the second index book: it cannot consistently appear anywhere in the table. To see that, imagine if we had already assigned this sequence to number 7. (The number 7 is arbitrary; any other number will work just as well.)  Then the table would look like this:

```    1: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
2: 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
3: 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, ...
4: 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, ...
5: 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, ...
6: 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, ...
7: 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, ...```

But look at the seventh bit of sequence 6! It’s supposed to be the seventh bit along the diagonal, and it’s wrong. If we correct it, then the 7th bit of sequence 7 is wrong, since it’s supposed to be the opposite of the 7th bit of sequence 6.

This proof shows that there are too many sequences of bits to match up to natural numbers. No matter how we try match them up, we can always find a contradiction by looking at the sequence defined by the opposite of the diagonal sequence.  And as if two different sizes of infinity weren’t enough, Cantor was able to repeat this process and show that there are infinitely many sizes of infinity, each bigger than the last!

Many influential mathematicians didn’t like this idea, and attacked Cantor personally rather than trying to find fault with his proof.  Religious philosophers misinterpreted his results and equated the idea of multiple infinities with pantheism!  Confronted with so much antagonism, Cantor often fought with depression.

He persevered, however, and later in life was greatly praised for his work. The Royal Society gave him their highest honor, the Sylvester Medal; and no less a mathematician than David Hilbert exclaimed, “No one shall expel us from the paradise that Cantor has created.”