I’ve been terrible about updating the blog lately. Personal life interferes sometimes, but I’m trying to build up a backlog of posts so that I can get the blog moving again without too much pressure.

In the meantime, I’ve got a request for you folks.

Through this blog, I met a really great SFF writer named Catherine Asaro. Catherine both writes, and also teaches math. She’s running a GoFundMe for her students, who participate in extracurricular math activities – clubs, classes and competitions.

They need money to cover travel expenses for going to math league competitions.

These kids are us: they’re the math geeks of the future. They need help from the math geeks of today. So go, give them a couple of bucks, help them out!

An extra little bit of incentive: the first five people who donate more than $25 can pick a topic, from either math or computer science, and I’ll write a blog post about it. Send me a copy of the thank-you letter you get from GoFundMe to show that you contributed, along with the topic you’d like me to write about.

These thank-you posts for contributing will skip to the top of my priority queue, so you’ll see them soon!

This is a traditional chinese dish. It’s affectionately known as chinese spaghetti. My wife has been telling me about it for years, and I’ve been trying to reproduce it for years. I’ve finally gotten it to the point where she’s happy with it.

It’s a fun dish: it’s a make-your-own. You prepare the noodles and sauce and a bunch of toppings, and then each guest adds sauce and toppings to their own taste. I’ve seen a couple of variations on the set of toppings you can serve with it: cucumbers, carrots, raw onions, shredded ginger, sliced garlic, and bean sprouts all work. I’ve also seen one version that served it with black chinese vinegar, which was really strange.

Ingredients

4 chicken thighs, ground coarsely.

2 tablespoons tobanjiang (chinese spicy bean sauce, available at asian groceries.)

1/4 plus a couple of tablespoons cup soy sauce.

1 cup chicken stock.

1/2 cup sake or shaoshing wine.

1 large onion, minced.

2 cloves garlic, minced.

1 large slice ginger, crushed and minced.

1 cucumber

1 carrot

2 eggs

2 scallions, green sliced, and whites finely minced.

1 tablespoon sugar.

1 tablespoon cornstarch mixed with some water.

1/2 teaspoon sesame oil.

Prepare the toppings

Peel the cucumber, cut it in half, scrape out the seeds, and then cut it into thin sticks. Toss with salt, and set aside for 30 minutes, and then pour off the water that comes out.

Peel the cucumber, and cut into thin sticks. Toss it with salt, and set aside for 30 minutes. Pour off the water that comes out.

Finely slice the greens of the scallions, and set aside.

Beat the two eggs with a bit of soy sauce. Put some oil into a hot non-stick pan, and pour in the eggs. Spread then into a thin layer that covers the pan. Cook until it’s mostly solid, then flip it. When it’s all cooked, move it to a cutting board, and cut into thin ribbons. Set aside.

Make the sauce:

Mix the ground chicken thighs with a tablespoon of soy sauce.

Put 1 tablespoon of oil into a hot wok. Add the chicken, and stir fry until it starts to brown. Then remove it.

Put 1 tablespoon of oil into a hot wok. Add the garlic and ginger, and stir fry until fragrant. Then add the onions and scallion whites, and stir until they soften and just barely start to brown.

Scrape the onions to the sides of the pan, then add the tobanjiang to the oil that collects of the bottom of the pan. Stir it around for 30 seconds.

Add the chicken back in, and then add the sake.

When most of the sake has cooked off, add the chicken stock, soy sauce, sugar, and a cup of water.

Reduce the heat to a simmer, and then let it cook for about 20 minutes.

Thicken with some cornstarch, and add the sesame oil. Remove from heat, put into a bowl, and set aside.

Cook some noodles. This will work with a variety of different types of asian noodles: fresh ramen, lo mien, shanghai noodle. Cook according to the instructions on the package. It’s best if it’s a bit undercooked. Drain them, and then toss with a bit of oil to stop them from sticking together.

To serve, you put the sauce into a bowl, and put it and all of the toppings onto the table. Then put a pile of noodles onto each persons plate. Then have each guest add as much sauce and toppings as they want, and dig in.

My friend Dr24hours sent me a link via twitter to a new piece of mathematical crackpottery. It’s the sort of thing that’s so trivial that I might lust ignore it – but it’s also a good example of something that someone commented on in my previous post.

This comes from, of all places, Rolling Stone magazine, in a puff-piece about an actor named Terrence Howard. When he’s not acting, Mr. Howard believes that he’s a mathematical genius who’s caught on to the greatest mathematical error of all time. According to Mr. Howard, the product of one times one is not one, it’s two.

After high school, he attended Pratt Institute in Brooklyn, studying chemical engineering, until he got into an argument with a professor about what one times one equals. “How can it equal one?” he said. “If one times one equals one that means that two is of no value because one times itself has no effect. One times one equals two because the square root of four is two, so what’s the square root of two? Should be one, but we’re told it’s two, and that cannot be.” This did not go over well, he says, and he soon left school. “I mean, you can’t conform when you know innately that something is wrong.”

I don’t want to harp on Mr. Howard too much. He’s clueless, but sadly, he’s a not too atypical student of american schools. I’ll take a couple of minutes to talk about what’s wrong with his stuff, but in context of a discussion of where I think this kind of stuff comes from.

In American schools, math is taught largely by rote. When I was a kid, set theory came into vogue, but by and large math teachers didn’t understand it – so they’d draw a few meaningless Venn diagrams, and then switch into pure procedure.

An example of this from my own life involves my older brother. My brother is not a dummy – he’s a very smart guy. He’s at least as smart as I am, but he’s interested in very different things, and math was never one of his interests.

I barely ever learned math in school. My father noticed pretty early on that I really enjoyed math, and so he did math with me for fun. He taught me stuff – not as any kind of “they’re not going to teach it right in school”, but just purely as something fun to do with a kid who was interested. So I learned a lot of math – almost everything up through calculus – from him, not from school. My brother didn’t – because he didn’t enjoy math, and so my dad did other things with him.

When we were in high school, my brother got a job at a local fast food joint. At the end of the year, he had to do his taxes, and my dad insisted that he do it himself. When he needed to figure out how much tax he owed on his income, he needed to compute a percentage. I don’t know the numbers, but for the sake of the discussion, let’s say that he made $5482 that summer, and the tax rate on that was 18%. He wrote down a pair of ratios:

And then he cross-multiplied, getting:

and so .

My dad was shocked by this – it’s such a laborious way of doing it. So he started pressing at my brother. He asked him, if you went to a store, and they told you there was a 20% off sale on a pair of jeans that cost $18, how much of a discount would you get? He didn’t know. The only way he knew to figure it out was to do the whole ratios, cross-multiply, and solve. If you told him that 20% off of $18 was $5, he would have believed you. Because percentages just didn’t mean anything to him.

Now, as I said: my brother isn’t a dummy. But none of his math teachers had every taught him what percentages meant. He had no concept of their meaning: he knew a procedure for getting the value, but it was a completely blind procedure, devoid of meaning. And that’s what everything he’d learned about math was like: meaningless procedures performed by rote, without any comprehension.

That’s where nonsense like Terence Howard’s stuff comes from: math education that never bothered to teach students what anything means. If anyone had attempted to teach any form of meaning for arithmetic, the ridiculous of Mr. Howard’s supposed mathematics would be obvious.

For understanding basic arithmetic, I like to look at a geometric model of numbers.

Put a dot on a piece of paper. Label it “0”. Draw a line starting at zero, and put tick-marks on the line separated by equal distances. Starting at the first mark after 0, label the tick-marks 1, 2, 3, 4, 5, ….

In this model, the number one is the distance from 0 (the start of the line) to 1. The number two is the distance from 0 to 2. And so on.

What does addition mean?

Addition is just stacking lines, one after the other. Suppose you wanted to add 3 + 2. You draw a line that’s 3 tick-marks long. Then, starting from the end of that line, you draw a second line that’s 2 tick-marks long. 3 + 2 is the length of the resulting line: by putting it next to the original number-line, we can see that it’s five tick-marks long, so 3 + 2 = 5.

Multiplication is a different process. In multiplication, you’re not putting lines tip-to-tail: you’re building rectangles. If you want to multiply 3 * 2, what you do is draw a rectangle who’s width is 3 tick-marks long, and whose height is 2 tick-marks long. Now divide that into squares that are 1 tick-mark by one tick-mark. How many squares can you fit into that rectangle? 6. So 3*2 = 6.

Why does 1 times 1 equal 1? Because if you draw a rectangle that’s one hash-mark wide, and one hash-mark high, it forms exactly one 1×1 square. 1 times 1 can’t be two: it forms one square, not two.

If you think about the repercussions of the idea that 1*1=2, as long as you’re clear about meanings, it’s pretty obvious that 1*1=2 has a disastrously dramatic impact on math: it turns all of math into a pile of gibberish.

What’s 1*2? 2. 1*1=2 and 1*2=2, therefore 1=2. If 1=2, then 2=3, 3=4, 4=5: all integers are equal. If that’s true, then… well, numbers are, quite literally, meaningless. Which is quite a serious problem, unless you already believe that numbers are meaningless anyway.

In my last post, someone asked why I was so upset about the error in a math textbook. This is a good example of why. The new common core math curriculum, for all its flaws, does a better job of teaching understanding of math. But when the book teaches “facts” that are wrong, what they’re doing becomes the opposite. It doesn’t make sense – if you actually try to understand it, you just get more confused.

That teaches you one of two things. Either it teaches you that understanding this stuff is futile: that all you can do is just learn to blindly reproduce the procedures that you were taught, without understanding why. Or it teaches you that no one really understands any of it, and that therefore nothing that anyone tells you can possibly be trusted.

(And yes, this is real. You can see it in context here.)

This is absolutely infuriating.

This is a photo of a problem assignment in a math textbook published by an imprint of McGraw-Hill. And it’s absolutely, unquestionably, trivially wrong. No one who knew anything about math looked at this before it was published.

The basic concept underneath this is fundamental: it’s the cardinality of sets from Cantor’s set theory. It’s an extremely important concept. And it’s a concept that’s at the root of a huge amount of misunderstandings, confusion, and frustration among math students.

Cardinality, and the notion of cardinality relations between infinite sets, are difficult concepts, and they lead to some very un-intuitive results. Infinity isn’t one thing: there are different sizes of infinities. That’s a rough concept to grasp!

Here on this blog, I’ve spent more time dealing with people who believe that it must be wrong – a subject that I call Cantor crackpottery – than with any other bad math topic. This error teaches students something deeply wrong, and it encourages Cantor crackpottery!

Let’s review.

Cantor said that two collections of things are the same size if it’s possible to create a one-to-one mapping between the two. Imagine you’ve got a set of 3 apples and a set of 3 oranges. They’re the same size. We know that because they both have 3 elements; but we can also show it by setting aside pairs of one apple and one orange – you’ll get three pairs.

The same idea applies when you look at infinitely large sets. The set of positive integers and the set of negative integers are the same size. They’re both infinite – but we can show how you can create a one-to-one relation between them: you can take any positive integer , and map it to exactly one negative integer, .

That leads to some unintuitive results. For example, the set of all natural numbers and the set of all even natural numbers are the same size. That seems crazy, because the set of all even natural numbers is a strict subset of the set of natural numbers: how can they be the same size?

But they are. We can map each natural number to exactly one even natural number . That’s a perfect one-to-one map between natural numbers and even natural numbers.

Where it gets uncomfortable for a lot of people is when we start thinking about real numbers. The set of real numbers is infinite. Even the set of real numbers between 0 and 1 is infinite! But it’s also larger than the set of natural numbers, which is also infinite. How can that be?

The answer is that Cantor showed that for any possible one-to-one mapping between the natural numbers and the real numbers between 0 and 1, there’s at least one real number that the mapping omitted. No matter how you do it, all of the natural numbers are mapped to one value in the reals, but there’s at least one real number which is not in the mapping!

In Cantor set theory, that means that the size of the set of real numbers between 0 and 1 is strictly larger than the set of all natural numbers. There’s an infinity bigger than infinity.

I think that this is what the math book in question meant to say: that there’s no possible mapping between the natural numbers and the real numbers. But it’s not what they did say: what they said is that there’s no possible map between the integers and the fractions. And that is not true.

Here’s how you generate the mapping between the integers and the rational numbers (fractions) between 0 and 1, written as a pseudo-Python program:

i = 0
for denom in Natural:
for num in 1 .. denom:
if num is relatively prime with denom:
print("%d => %d/%d" % (i, num, denom))
i += 1

It produces a mapping (0 => 0, 1 => 1, 2 => 1/2, 3 => 1/3, 4 => 2/3, 5 => 1/4, 6 => 3/4, …). It’ll never finish running – but you can easily show that for any possible fraction, there’ll be exactly one integer that maps to it.

That means that the set of all rational numbers between 0 and 1 is the same size as the set of all natural numbers. There’s a similar way of producing a mapping between the set of all fractions and the set of natural numbers – so the set of all fractions is the same size as the set of natural numbers. But both are smaller than the set of all real numbers, because there are many, many real numbers that cannot be written as fractions. (For example, . Or the square root of 2. Or . )

This is terrible on multiple levels.

It’s a math textbook written and reviewed by people who don’t understand the basic math that they’re writing about.

It’s teaching children something incorrect about something that’s already likely to confuse them.

It’s teaching something incorrect about a topic that doesn’t need to be covered at all in the textbook. This is an algebra-2 textbook. You don’t need to cover Cantor’s infinite cardinalities in Algebra-2. It’s not wrong to cover it – but it’s not necessary. If the authors didn’t understand cardinality, they could have just left it out.

It’s obviously wrong. Plenty of bright students are going to come up with the the mapping between the fractions and the natural numbers. They’re going to come away believing that they’ve disproved Cantor.

I’m sure some people will argue with that last point. My evidence in support of it? I came up with a proof of that in high school. Fortunately, my math teacher was able to explain why it was wrong. (Thanks Mrs. Stevens!) Since I write this blog, people assume I’m a mathematician. I’m not. I’m just an engineer who really loves math. I was a good math student, but far from a great one. I’d guess that every medium-sized high school has at least one math student every year who’s better than I was.

The proof I came up with is absolutely trivial, and I’d expect tons of bright math-geek kids to come up with something like it. Here goes:

The set of fractions is a strict subset of the set of ordered pairs of natural numbers.

So: if there’s a one-to-one mapping between the set of ordered pairs and the naturals, then there must be a one-to-one mapping between the fractions and the naturals.

On a two-d grid, put the natural numbers across, and then down.

Zigzag diagonally through the grid, forming pairs of the horizontal position and the vertical position: (0,0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0), (2, 1), (1, 2), (0, 3).

This will produce every possible ordered pair of natural numbers. For each number in the list, produce a mapping between the position in the list, and the pair. So (0, 0) is 0, (2, 0) is 3, etc.

As a proof, it’s sloppy – but it’s correct. And plenty of high school students will come up with something like it. How many of them will walk away believing that they just disproved Cantor?

I know I’ve been terrible about updating my blog lately. I’ve got some good excuses. (The usual: very busy with work. The less usual: new glasses that I’m having a very hard time adapting to. Getting old sucks. My eyes have deteriorated to the point where my near vision is shot, and the near-vision correction in my lenses needed to get jumped pretty significantly, which takes some serious getting used to.) And getting discussions of type theory right is a lot of work. Type theory in particular takes a lot of work, because it’s a subject that I really want to get right, because it’s so important in my profession, and because so few people have actually written about it in a way that’s accessible to non-mathematicians.

Anyway: rest assured that I’m not dropping the subject, and I hope to be getting back to writing more very soon. In the meantime, I decided to bring you some humorous bad math.

Outside the scientific community, one of the common criticisms of science is that scientific explanations are “just a theory”. You hear this all the time from ignorant religious folks trying to criticize evolution or the big bang (among numerous other things). When they say that something is just a theory, what they mean is that it’s not a fact, it’s just speculation. They don’t understand what the word “theory” really means: they think that a theory and a fact are the same class of things – that an idea starts as a theory, and becomes a fact if you can prove it.

In science, we draw a distinction between facts and theories, but it’s got nothing to do with how “true” something is. A fact is an observation of something that happens, but doesn’t say why it happens. The sun produces light. That’s a fact. The fact doesn’t say why that happens. It doesn’t have to say how it happens. But it does. That’s an observation of a fact. A theory is an explanation of a set of facts. The combined gravitational force of all of the particles in the sun compress the ones in the center until quantum tunnelling allows hydrogen atoms to combine and fuse producing energy, which eventually radiates as the heat and light that we observe. The theory of solar hydrogen fusion is much more than the words in the previous sentence: it’s an extensive collection of evidence and mathematics that explains the process in great detail. Solar hydrogen fusion – mathematical equations and all – is a theory that explains the heat and light that we observe. We’re pretty sure that it’s true – but the fact that it is true doesn’t mean that it’s not a theory.

Within the scientific community, we criticize crackpot ideas by saying that they’re not a theory. In science, a theory means a well-detailed and tested hypothesis that explains all of the known facts about something, and makes testable predictions. When we say that something isn’t a theory, we mean that it doesn’t have the supporting evidence, testability, or foundation in fact that would be needed to make something into a proper theory.

For example, intelligent design doesn’t qualify as a scientific theory. It basically says “there’s stuff in the world that couldn’t happen unless god did it”. But it never actually says how, precisely, to identify any of those things that couldn’t happen without god. Note that this doesn’t mean that it’s not true. I happen to believe that it’s not – but whether it’s true or not has nothing to do with whether, scientifically, in qualifies as a theory.

That’s a very long, almost Oracian introduction to today’s nonsense. This bit of crackpottery, known as “the Principle of Circlon Synchronicity”, written by one James Carter, has one really interesting property: I agree, 100%, with the very first thing that Mr. Carter says about his little idea.

The Principle of Circlon Synchronicity is not a Theory

They’re absolutely correct. It’s not a theory. It’s a bundle of vague assumptions, tied together by a shallow pretense at mathematics.

The “introduction” to this “principal” basically consists of the author blindly asserting that a bunch of things aren’t theories. For example, his explanation for why the principal of circlon synchronicity is not a theory begins with:

There are many different theories that have been used to explain the nature of reality. Today, the most popular of these are quantum mechanics, special relativity, string theories, general relativity and the Big Bang. Such theories all begin with unmeasured metaphysical assumptions such as fields and forces to explain the measurements of various phenomena. Circlon synchronicity is a purely mechanical system that explains local physical measurements. You only need a theory to explain physical measurements in terms of non-local fields, forces and dimensions.

This is a novel definition of “theory”. It has absolutely nothing to do with what the rest of us mean by the word “theory”. Basically, he thinks that his explanations, because they are allegedly simple, mechanical, and free of non-local effects, aren’t theories. They’re principals.

The list of things that don’t need a theory, according to Mr. Carter, is extensive.
For example:

The photon is not a theory. The photon is a mechanical measurement of mass. The photon is a conjoined matter-antimatter pair that is the basic form of mass and energy in the Living Universe. All photons move at exactly the speed of light relative to one another within the same absolute space. Photons are produced when a proton and electron are joined together to form a hydrogen atom. The emission of a photon is a mini annihilation with part of the electron and part of the proton being carried away by the photon. A photon with mass and size eliminates the need for both Planckâ€™s constant and the Heisenberg uncertainty principle and also completely changes the meaning of the equation E=MC2. This is not a theory of a photon. It is the measurements describing the nature of the photon.

This is where we start on the bad math.

A photon is a quantum of light, or some other form of electromagnetic radiation. It doesn’t have any mass. But even if it didn’t: a photon isn’t a measurement. A photon is a particle (or a wave, depending on how you deal with it.) A measurement is a fundamentally different thing. If you want to do math that describes the physical universe, you’ve got to be damned careful about your units. If you’re measuring mass, you your units need to be mass units. If you’re describing mass, then the equations that derive your measurements of mass need to have mass units. If a photon is a measurement of mass, then what’s its unit?

Further, you can’t take an equation like , and rip it out of context, while asserting that it has exactly the same meaning that it did in its original context. Everyone has seen that old equation, but very few people really understand just what it means. Mr. Carter is not part of that group of people. To him, it’s just something he’s seen, which he knows is sciency, and so he grabs on to it and shouts about it in nonsensical ways.

But note, importantly, that even here, what Mr. Carter is doing isn’t science. He’s absolutely right when he says it’s not a theory. He asserts that the whole meaning of changes because of his new understanding of what light is; but he doesn’t ever bother to explain just what that new understanding is, or how it differs from the old one.

He makes some hand-waves about how you don’t need the uncertainty principle. If his principles had a snowballs chance in hell of being correct, that might be true. The problem with that assertion is that the uncertainty principle isn’t just a theory. It’s a theory based on observations of facts that absolutely require explanations. There’s a great big fiery-looking ball up in the sky that couldn’t exist without uncertainty. Uncertainty isn’t just a pile of equations that someone dreamed up because it seemed like fun. It’s a pile of equations that were designed to try to explain the phenomena that we observe. There are a lot of observations that demonstrate the uncertainty principle. It doesn’t disappear just because Mr. Carter says it should. He needs to explain how his principles can account for the actual phenomena we observe – not just the phenomena that he wants to explain.

Similarly, he doesn’t like the theory of gravity.

We do not need a theory of gravity to explain exactly how it works. Gravity is a simple measurement that plainly shows exactly what gravity does. We use accelerometers to measure force and they exactly show that gravity is just an upwardly pointing force caused by the physical expansion of the Earth. The gravitational expansion of matter does not require a theory. It is just the physical measurement of gravity that shows exactly how it works in a completely mechanical way without any fields or non-local interactions. You only need a theory to explain a non-local and even infinite idea of how gravity works in such a way that it canâ€™t be directly measured. Gravity not a theory.

Once again, we see that he really doesn’t understand what theory means. According to him, gravity can be measured, and therefore, it’s not a theory. Anything that can be measured, according to Mr. Carter, can’t be a theory: if it’s a fact, it can’t be a theory; even more, if it’s a fact, it doesn’t need to be explained at all. It’s sort-of like the fundamentalists idea of a theory, only slightly more broken.

This is where you can really see what’s wrong with his entire chain of reasoning. He asserts that gravity isn’t a theory – and then he moves in to an “explanation” of how gravity works which simply doesn’t fit.

We do not need a theory of gravity to explain exactly how it works. Gravity is a simple measurement that plainly shows exactly what gravity does. We use accelerometers to measure force and they exactly show that gravity is just an upwardly pointing force caused by the physical expansion of the Earth. The gravitational expansion of matter does not require a theory. It is just the physical measurement of gravity that shows exactly how it works in a completely mechanical way without any fields or non-local interactions. You only need a theory to explain a non-local and even infinite idea of how gravity works in such a way that it canâ€™t be directly measured. Gravity not a theory.

The parade of redefinitions marches on! “Exactly” now means “hand-wavy”.

We’re finally getting to the meat of Mr. Carter’s principle. He’s a proponent of the same kind of expanding earth rubbish as Neal Adams. Gravity has nothing to do with non-local forces. It’s all just the earth expanding under us. Of course, this is left nice and vague: he mocks the math behind the actual theory of gravity, but he can’t actually show that his principal works. He just asserts that he’s defined exactly how it works by waving his hands really fast.

I can disprove his principle of gravity quite easily, by taking my phone out of my pocket, and opening Google maps.

In 5 seconds flat (which is longer than it should take!), Google maps shows me my exact position on the map. It does that by talking to a collection of satellites that are revolving around the earth. The positions of those satellites are known with great accuracy. They circle the earth without the use of any sort of propellant. If Mr. Carter (or Mr. Adams, who has a roughly equivalent model) were correct – if gravity was not, in fact, a force attracting mass to other masses, but instead was an artifact of an expanding earth – then the “satellites” that my phone receives data from would not be following an elliptical path around the earth. They’d be shooting off into the distance, moving in a perfectly straight line. But they don’t move in a straight line. They continue to arc around the earth, circling around and around, without any propulsion.

In any reasonable interpretation of the expanding earth? That doesn’t make sense. There’s no way for them to orbit. Satellites simply can’t work according to his theory. And yet, they do.

Of course, I’m sure that Mr. Carter has some hand-wavy explanation of just why satellites work. The problem is, whatever explanation he has isn’t a theory. He can’t actually make predictions about how things will behave, because his principles aren’t predictive.

In fact, he even admits this. His whole screed turns out to be a long-winded advertisement for a book that he’ll happily sell you. As part of the FAQ for his book, he explains why (a) he can’t do the math, and (b) it doesn’t matter anyway:

The idea that ultimate truth can be represented with simple mathematical equations is probably totally false. A simple example of this is the familiar series of circular waves that move away from the point where a pebble is dropped into a quiet pool of water. While these waves can be described in a general way with a simple set of mathematical equations, any true and precise mathematical description of this event would have to include the individual motion of each molecule within this body of water. Such an equation would require more than the worldâ€™s supply of paper to print and its complexity would make it virtually meaningless.

The idea of the circlon is easy to describe and illustrate. However, any kind of mathematical description of its complex internal dynamics is presently beyond my abilities. This deficiency does not mean that circlon theory cannot compete with the mathematically simplistic point-particle and field theories of matter. It simply means that perhaps ultimate truth is not as easily accessible to a mathematical format as was once hoped.

It’s particularly interesting to consider this “explanation” in light of some recent experiments in computational fluid dynamics. Weather prediction has become dramatically better in the last few years. When my father was a child, the only way to predict when a hurricane would reach land was to have people watching the horizon. No one could make accurate weather predictions at all, not even for something as huge as a storm system spanning hundreds of miles! When I was a child, weathermen rarely attempted to predict more than 2 days in advance. Nowadays, we’ve got 7-day forecasts that are accurate more often than the 2-day forecasts were a couple of decades ago. Why is that?

The answer is something called the Navier Stokes equations. The Navier-Stokes equations are a set of equations that describe how fluids behave. We don’t have the computational power or measurement abilities to compute N-S equations to the level of single molecules – but in principle, we absolutely could. The N-S equations – which demonstrably work remarkably well even when you’re just computing approximations – also describe exactly the phenomenon that Mr. Carter asserts can’t be represented with mathematical equations.

The problem is: he doesn’t understand how math or science work. He has no clue of how equations describe physical phenomena in actual scientific theories. The whole point of math is that it gives you a simple but precise way of describing complex phenomena. A wave in a pool of water involves the motion of an almost unimaginable number of particles, with a variety of forces and interactions between those particles. But all of them can be defined by reasonably simple equations.

Mr. Carter’s explanations are, intuitively, more attractive. If you really want to understand relativity, you’re going to need to spend years studying math and physics to get to the point where its equations make sense to you. But once you do, they don’t just explain things in a vague, hand-wavy way – they tell you exactly how things work. They make specific, powerful, precise predictions about how things will behave in a range of situations that match reality to the absolute limits of our ability to measure. Mr. Carter’s explanations don’t require years of study; they don’t require to study esoteric disciplines like group theory or tensor theory. But they also can’t tell you much of anything. Relativity can tell you exactly what adjustment you need to make to a satellite’s clock in order to make precise measurements of the location of a radio receiver on the ground. Mr. Carter’s explanations can’t even tell you how the satellite got there.

There’s an image going around that purports to explain the origin of the arabic numerals. It’s cute. It claims to show why the numerals that we use look the way that they do. Here it is:

According to this, the shapes of the numbers was derived from a notation where for each numeral contains its own number of angles. It’s a really interesting idea, and it would be really interesting if it were true. The problem is, it isn’t.

Look at the numerals in that figure. Just by looking at them, you can see quite a number of problems with them.

For a couple of obvious examples:

Look at the 7. The crossed seven is a recent invention made up to compensate for the fact that in cursive roman lettering, it can be difficult to distinguish ones from sevens, the mark was added to clarify. The serifed foot on the 7 is even worse: there’s absolutely no tradition of writing a serifed foot on the 7; it’s just a font decoration. The 7’s serifed foot is no more a part of the number than serifed foot on the lowercase letter l is an basic feature of the letter ls.

Worse is the curlique on the 9: the only time that curly figures like that appear in writing is in calligraphic documents, where they’re an aesthetic flourish. That curly thing has never been a part of the number 9. But if you want to claim this angle-counting nonsense, you’ve got to add angles to a 9 somewhere. It’s not enough to just add a serifed foot – that won’t get you enough angles. So you need the curlique, no matter how obviously ridiculous it is.

You don’t even need to notice stuff like that to see that this is rubbish. We actually know quite a lot about the history of arabic numeral notation. We know what the “original” arabic numerals looked like. For example, this wikipedia image shows the standard arabic numerals (this variant is properly called the Bakshali numerals) from around the second century BC:

It’s quite fascinating to study the origins of our numeric notation. It’s true that we – “we” meaning the scholarly tradition that grew out of Europe – learned the basic numeric notation from the Arabs. But they didn’t invent it – it predates them by a fair bit. The notation originally came from India, where Hindu scholars, who wrote in an alphabet derived from Sanskrit, used a sanskrit-based numeric notation called Brahmi numerals (which, in turn, were derived from an earlier notation, Karosthi numerals, which weren’t used quite like the modern numbers, so the Brahmi numerals are considered the earliest “true” arabic numeral.) That notation moved westward, and was adopted by the Persians, who spread it to the Arabs. As the arabs adopted it, they changed the shapes to work with their calligraphic notations, producing the Bakshali form.

In the Brahmi numerals, the numbers 1 through 4 are written in counting-based forms: one is written as one horizontal line; 2 as two lines; 3 as three lines. Four is written as a pair of crossed lines, giving four quadrants. 5 through 9 are written using sanskrit characters: their “original” form had nothing to do with counting angles or lines.

The real history of numerical notations is really interesting. It crosses through many different cultures, and the notations reform each time it migrates, keeping the same essential semantics, but making dramatic changes in the written forms of individual numerals. It’s so much more interesting – and the actual numeral forms are so much more beautiful – than you’d ever suspect from the nonsense of angle-counting.

A few weeks ago, my wife took me out to dinner at an Izakaya that she found. If you don’t know about Izakayas, you really need to find one and try it. An Izakaya is basically a Japanese sake bar – it’s sort-of the Japanese equivalent of a Tapas bar – you order a variety of small dishes, and eat them with your sake. Izakayas are really terrific – they’re a nice informal place to have dinner with good sake, and a variety of delicious dishes.

One of the dishes we had was Tsukune, which is a standard at Izakayas. It’s basically a meatball made from ground chicken, and served drizzled with a sauce that’s a lot like a thick, rich teriyaki. These suckers were the tenderest, most succulent meatballs I’ve ever eaten. I was strongly motivated to try to reproduce them.

So I did some research. And it turned out that there were two tricks to how really good tsukune come out that way.

One part is what goes in: pretty much any part of the bird that can get finely minced. Breast, thigh, neck, legs, wings. If the cartilage is soft enough to be minced, it goes right in with the meat.

The other part is how it’s treated. The meat has the living hell beaten out of it. It’s finely minced, and then the ground meat is kneaded and pounded until it’s all soft and firmly stuck together. That process both binds the meat without adding any extra binding agents like eggs, and tenderizes it.

I really don’t like working with cartilage. I’ve tried it in some recipes in the past, and I find it really unpleasant. It’s slimy, and it tends to sometimes form sharp edges that can cut you. Getting cut by raw meat is not something I like doing.

So I decided to try to reproduce as much of the flavor as I could, using ingredients that I had on hand. And I tried couple of tricks to get the tender texture. It’s not as tender as a true Izakaya tsukune, but it was awfully good.

I relied on three things to get the tenderness I wanted.

Force: I beat the living hell out of the meat. I didn’t do it by kneading; instead, I took chicken thighs, cut the meat off the bones, and then beat the crap out of it with the flat of my meat tenderizer. By the time I was done, I wouldn’t describe the result as a particular thickness; I’d say that it had been reduced to more of a chicken paste smeared across my cutting board.

Chemistry: I took that, and put it into a flat bottomed baking dish, and poured a cup of apple juice over it. There’s an enzyme in fruits from the apple family that tenderizes meat, so I let that do some work.

Slow heat: Finally, I browned the meatball quickly in a pan, and then transferred them to a baking dish with the sauce, and put them into a low oven for a long time.

The result was, honestly, not very much like the tsukune from the Izakaya. But it was really, really delicious. I mean really, seriously delicious – like some of the best meatballs I’ve ever eaten. Not quite as good as the Izakaya ones, and quite different – but so, so good.

It’s not a difficult dish to make, but it is time consuming. Not really that different from other meatball recipes I’ve made: meatballs are often the kind of dish that cooks all day.

The sauce cooks for a while on its own, and then cooks more with the meatballs in it. But the chicken needs to marinate for an hour or two – so you really need to start both around the same time.

The Sauce

The sauce is basically just a slightly amped up teriyaki sauce – but homemade is so much better than the crap you get out of a bottle, and it’s really easy to make.

Ingredients

White part of one scallion, minced.

3 cloves finely minced garlic.

1 inch section of fresh ginger, finely minced.

1/3 cup soy sauce (ideally good quality shoyu!).

1/2 cup sake

1/4 cup mirin

1/2 cup apple juice

1 cup chicken stock.

2 tablespoons brown sugar.

Instructions

Heat a bit of oil in a pot. When it’s hot, add the garlic, ginger, and scallions, and cook until it’s nice and fragrant.

Add the sake, and then let it reduce by half.

Reduce the heat to medium-low, and add the rest of the liquids and the brown sugar.

Simmer for about an hour.

Remove from heat, and set aside.

The meatballs

Ingredients

6 boneless chicken thighs

1/4 onion, finely minced

1/4 cup apple juice

1/2 cup bread crumbs.

sliced scallion greens (the leftovers from the whites used in the sauce.)

flour

Instructions

With a meat tenderizer, pound the heck out of the chicken thighs until they’re severely mangled and very thin.

Put them into a dish with the apple juice, and put that into the refrigerator for an hour or two.

Put the marinated thighs into the food processor with 1/4 cup of the sauce, and pulse a couple of times until they’re minced.

Fold in the onion and the bread crumbs.

Chill the ground meat for an hour until it firms up.

Separate into about 30 meatballs – it should be around a heaped teaspoon per meatball.

Roll each meatball in flour; flatten slightly, and then brown the two flat sides in a hot pan with a bit of oil. The meatballs will not be cooked through by this – you’re just browning the meatballs, not fully cooking them.

Put the meatballs into a baking dish, cover with the sauce, and bake for 1 1/2 hours.

Remove the meatballs from the sauce, and reduce the sauce until there’s just a cup or so left. Thicken with a bit of cornstarch.

Pour the sauce back over the meatballs, and sprinkle with scallion greens.

Serve this on a bed of simple steamed rice, and some stir fried veggies on the side. Home cooking doesn’t get better than this.

Over time, I’ve come to really, really hate the number φ.

φ is the so-called golden ratio. It’s the number that is a solution for the equation (a+b)/a = (a/b). The reason that that’s interesting at all is because it’s got an interesting property when you draw it out: if you take a rectangle where the ratio of the length of the sides is 1:φ, then if you remove the largest possible square from it, youâ€™ll get another rectangle whose sides have the ratio φ:1. If you take the largest square from that, youâ€™ll get a rectangle whose sides have the ratio 1:φ. And so on.

The numeric value of it is (1+sqrt(5))/2, or about 1.618033988749895.

The problem with φ is that people are convinced that it’s some kind of incredibly profound thing, and find it all over the place. The problem is, virtually all of the places where people claim to find it are total rubbish. A number that’s just a tiny bit more that 1 1/2 is really easy to find if you go looking for it, and people go looking for it all over the place.

People claim it’s in all sorts of artwork. You can certainly find a ton of things in paintings whose size ratio is about 1 1/2, and people find it and insist that it was deliberately done to make it φ. People find it in musical scales, the diatonic and pentatonic scales, and the indian scales.

People claim it comes up all over the place in nature: in beehives, ant colonies, flowers, tree sizes, tree-limb positions, size of herds of animals, litters of young, body shapes, face shapes.

People claim it’s key to architecture.

And yet… it seems like if you actually take any of those and actually start to look at it in detail? The φ isn’t there. It’s just a number that’s kinda-sorta in the 1 1/2 range.

One example of that: there’s a common claim that human faces have proportions based on &phi. You can see a bunch of that nonsense here. The thing is, the “evidence” for the claim consists of rectangles drawn around photographs of faces – and if you look closely at those rectangles, what you find is that the placement of the corners isn’t consistent. When you define, say, “the distance between the eyes”, you can measure that as distances between inner-edges, or between pupils, or between outer edges. Most of these claims use outer edges. But where’s the outer edge of an eye? It’s not actually a well-defined point. You can pick a couple of different places in a photo as “the” edge. They’re all close together, so there’s not a huge amount of variation. But if you can fudge the width a little bit, and you can fudge other facial measurements just a little bit, you’ve got enough variation that if you’re looking for two measurements with a ratio close to φ, you’ll always find one.

Most of the φ nonsense is ultimately aesthetic: people claiming that the golden ratio has a fundamental beauty to it. They claim that facial features match it because it’s intrinsically beautiful, and so people whose faces have φ ratios are more beautiful, and that that led to sexual-selection which caused our faces to embody the ratio. I think that’s bunk, but it’s hard to make a mathematical argument against aesthetics.

But then, you get the real crackpots. There are people who think φ has amazing scientific properties. In the words of the crank I’m writing about today, understanding φ (and the “correct” value of π derived from it) will lead humanity to “enter into a veritable Space Age”.

I’m talking about a guy who calls himself “Jain 108”. I’m not quite sure what to call him. Mr. Jain? Mr. 108? Dr 108? Most of the time on his website, he just refers to himself as “Jain” (or sometimes “Jain of Oz”) so I’ll go with “Jain”).

Jain believes that φ is the key to mathematics, science, art, and human enlightenment. He’s a bit hard to pin down, because most of his website is an advertisement for his books and seminars: if you want to know “the truth”, you’ve got to throw Jain some cash. I’m not willing to give money to crackpots, so I’m stuck with just looking at what he’s willing to share for free. (But I do recommend browsing around his site. It’s an impressive combination of newage scammery, pomposity, and cluelessness.)

What you can read for free is more than enough to conclude that he’s a total idiot.

I’m going to focus my mockery on one page: “Is Pi a Lie?”.

On this page, Jain claims to be able to prove that the well-known value of π (3.14159265….) is wrong. In fact, that value is wrong, and the correct value of π is derived from φ! The correct value of π is , or about 3.144605511029693.

For reasons that will be soon explained, traditional Pi is deficient because historically it has awkwardly used logical straight lines to measure illogical curvature. Thus, by using the highest level of mathematics known as Intuitive Maths, the True Value of Pi must be a bit more than anticipated to compensate for the mysterious “Area Under The Curve”. When this is done, the value, currently known as JainPi, = 3.144… can be derived, by knowing the precise Height of the Cheops Pyramid which is based on the Divine Phi Proportion (1.618…). Instead of setting our diameter at 1 unit or 1 square, something magical happens when we set the diameter at the diagonal length of a Double Square = 2.236… which is the Square Root of 5 (meaning 2.236… x 2.236… = 5). This is the critical part of the formula that derives Phi , and was used by ancient vedic seers as their starting point to construct their most important diagram or â€˜Yantraâ€™ or power-art called the Sri Yantra. With a Root 5 diameter, the translation of the Phiâ€™s formula into a geometric construct derives the royal Maltese Cross symbol, concluding that Phi is Pi, that Phi generates Pi, and that Pi must be derived with a knowledge of the Harmonics of Phi. When this is understood and utilized, we will collectively enter into a veritable Space Age.

How did we get the wrong value? It’s based on the “fact” that the computation of π is based on the use of “logical” straight lines to measure “illogical” curvurature. (From just that one sentence, we can already conclude that Jain knows nothing about logic, except what he learned from Mr. Spock on Star Trek.) More precisely, according to Jain:

In all due good respects, we must first honour Archimedes of Syracuse 2,225 years ago, who gave the world his system on how to calculate Pi, approximated to 22Ã·7, by cutting the circle into say 16 slices of a pizza, and measuring the 16 edge lengths of these 16 triangular polygons (fig 3), to get a good estimate for the circumference of a circle. The idea was that if we kept making the slices of pizza smaller and smaller, by subsequently cutting the circle into 32 slices, then 64, then 128 then 256 slices, we would get a better and more accurate representation for the circumference. The Fundamental Flawed Logic or Error with Archimedeâ€™s Increasing Polygon Method was that he failed to measure The Area Under The Curve. In fact, he assumed that The Area Under The Curve, just magically disappeared. Even in his time, Archimedes admitted that his value was a mere estimate!

This explanation does a beautiful job of demonstrating how utterly ignorant Jain is of math. Archimedes may have been the first person from the western tradition to have worked out a mechanism to compute a value for π – and his mechanism was a good one. But it’s far from the only one. But let’s ignore that for a moment. Jain’s supposed critique, if true, would mean that modern calculus doesn’t work. The wedge-based computation of π is a forerunner of the common methods of calculus. In reality, when we compute the value of almost any integral using calculus, our methods are based on the concept of drawing rectangles under the curve, and narrowing those rectangles until they’re infinitely small, at which point the “area under the curve” missed by the rectangles becomes zero. If the wedge computation of π is wrong because it misses are under the curve, then so will every computation using integral calculus.

Gosh, think we would have noticed that by now?

Let’s skip past that for a moment, and come back to the many ways that π comes into reality. π is the ratio of the diameter of a circle to its radius. Because circles are such a basic thing, there are many ways of deriving the value of π that come from its fundamental nature. Many of these have no relation to the wedge-method that Jain attributes to Archimedes.

For example, there is Viete’s product:

Or there’s the Gregory-Leibniz series:

These have no relation to the wedge-method – they’re derived from the fundamental nature of π. And all of them produce the same value – and it’s got no connection at all to φ.

As supportive evidence for the incorrectness of π, Jain gives to apocryphal stories about NASA and the moon landings. First, he claims that the first moon landing was off by 20 kilometers, and that the cause of this was an incorrect value of π: that the value of π used in computing trajectories was off by 0.003:

NASA admitted that when the original Mooncraft landing occurred, the targeted spot was missed by about 20km?
What could have been wrong with the Calculations?
NASA subsequently adjusted their traditional mathematical value for Pi (3.141592…) by increasing it in the 3rd decimal by .003!

Let’s take just a moment, and consider that.

It’s a bit difficult to figure out how to address that, because he’s not mentioning what part of the trajectory was messed up. Was it the earth-to-moon transit of the full apollo system? Or was it the orbit-to-ground flight of the lunar lander? Since he doesn’t bother to tell us, we’ll look at both.

π does matter when computing the trajectory of the earth-to-moon trip – because it involves the intersection of two approximate circles – the orbit of the earth around the sun, and the orbit of the moon around the earth. (Both of these are approximations, but they’re quite useful ones; the apollo trajectory computations did rely on a value for π.

Let’s look at earth-to-moon. I’m going to oversimplify ridiculously – but I’m just trying to give us a ballpark order-of-magnitude guess as just how much of a difference Mr. Jain’s supposed error would cause. THe distance from the earth to the moon is about 384,000 kilometers. If we assume that π is a linear factor in the computation, then a difference in the value of pi of around 1 part in 1000 would cause a difference in distance computations of around 384 kilometers. Mr. Jain is alleging that the error only caused a difference of 20 kilometers. He’s off by a factor of 15. We can hand-wave this away, and say that the error that caused the lander to land in the “wrong” place wasn’t in the earth-moon trajectory computation – but we’re still talking about the apollo unit being in the wrong place by hundreds of kilometers – and no one noticing.

What if the problem was in the computation of the trajectory the lander took from the capsule to the surface of the moon? The orbit was a nearly circular one at about 110 kilometers above the lunar surface. How much of an error would the alleged π difference cause? About 0.1 kilometer – that is, about 100 meters. Less than what Jain claims by a factor of 200.

The numbers don’t work. These aren’t precise calculations by any stretch, but they’re ballpark. Without Jain providing more information about the alleged error, they’re the best we can do, and they don’t make sense.

Jain claims that in space work, scientists now use an adjusted value of π to cover the error. This piece I can refute by direct knowledge. My father was a physicist who worked on missiles, satellites, and space probes. (He was part of the Galileo team.) They used good old standard 3.14159 π. In fact, he explained how the value of π actually didn’t need to be that precise. In satellite work, you’re stuck with the measurement problems of reality. In even the highest precision satellite work, they didn’t use more that 4 significant digits of precision, because the manufacturing and measurement of components was only precise to that scale. Beyond that, it was always a matter of measure and adjust. Knowing that π was 3.14159265356979323 was irrelevant in practice, because anything beyond “about 3.1416” was smaller that the errors in measurement.

Mr. Jain’s next claim is far worse.

Also, an ex-Engineer from NASA, “Smokey” admitted (via email) that when he was making metal cylinders for this same Mooncraft, finished parts just did not fit perfectly, so an adjusted value for Pi was also implemented. At the time, he thought nothing about it, but after reading an internet article called The True Value of Pi, by Jain 108, he made contact.

This is very, very simple to refute by direct experience. This morning, I got up, shaved with an electric razor (3 metal rotors), made myself iced coffee using a moka pot (three round parts, tight fitted, with circular-spiral threading). After breakfast, I packed my backpack and got in my car to drive to the train. (4 metal cylinders with 4 precisely-fitted pistons in the engine, running on four wheels with metal rims, precisely fitted to circular tires, and brakes clamping on circular disks.) I drove to the train station, and got on an electric train (around 200 electric motors on the full train, with circular turbines, driving circular wheels).

All those circles. According to Jain, every one of those circles isn’t the size we think it is. And yet they all fit together perfectly. According to Jain, every one of those circular parts is larger that we think it should be. To focus on one thing, every car engine’s pistons – every one of the millions of pistons created every year by companies around the world – requires more metal to produce than we’d expect. And somehow, in all that time, no one has ever noticed. Or if they’ve noticed, every single person who ever noticed it has never mentioned it!

It’s ludicrous.

Jain also claims that the value of e is wrong, and comes up with a cranky new formula for computing it. Of course, the problem with e is the same as the problem wiht π: in Jain’s world, it’s really based on φ.

In Jain’s world, everything is based on φ. And there’s a huge, elaborate conspiracy to keep it secret. Any Jain will share the secret with you, showing you how everything you think you know is wrong. You just need to buy his books ($77 for a hard-copy, or $44 for an ebook.) Or you could pay for him to travel to you and give you a seminar. But he doesn’t list a price for that – you need to send him mail to inquire.

I made an amazing dinner tonight, out of a total ad-lib. So I want to write it down so I’ll remember it. This turned out to be the best steak I’ve ever made.

The Steak

1 pound skirt steak

Kombu leaves

1 teaspoon finely minced garlic

1 teaspoon finely minced ginger

1/4 cup soy sauce

1/4 cup mirin

1/4 cup sake

Mix together the liquid ingredents in a baking dish. dip the steak into it, to coat both sides.

Press kombu leaves into both sides of the steak, and then put it in the dish, and leave it in the fridge for a couple of hours.

Take the steak out, pat dry.

Toss the steak on a very hot grill, and cook for 2 minutes on one side, and about one minute on the other.

Let rest for a few minutes, then slice on the bias. (Save any juices that drip off while it’s resting, and add them to the sauce!)

The Sauce

1/4 cup sugar

1/4 cup water

1/4 cup apple cider vinegar

1/4 cup sake

1 tablespoon shoyu

Mix together the sugar and water, and cook on high heat, whisking until the sugar is dissolved.

Keep cooking on high heat until all the water evaporates, the sugar starts to caramelize, and turns amber.

Add the cider and sake, and reduce the heat. Then whisk vigorously until everything is dissolved. (When you add the liquids to the caramel, it will seize up; you need to whisk until it’s redissolved.

Turn off the heat, then add the shoyu and any juices that dripped off the steak.

Now, we’re getting to the heart of type theory: judgements. Judgements are the basic logical statements that we want to be able to prove about computations in type theory. There’s a basic set of judgements that we want to be able to make.

I keep harping on this, but it’s the heart of type theory: type theory is all about understanding logical statements as specifications of computations. Or, more precisely, in computer science terms, they’re about understanding true logical statements as halting computations.

In this post, we’ll see the ultimate definition of truth in type theory: every logical proposition is a set, and the proposition is true if the set has any members. A non-halting computation is a false statement – because you can never get it to resolve an expression to a canonical value.

So remember as we go through this: judgements are based on the idea of logical statements as specifications of computations. So when we talk about a predicate P, we’re using its interpretation as a specification of a computation. When we look at an expression 3+5, we understand it not as a piece of notation that describes the number 8, but as a description of a computation that adds 3 to 5. “3+5” not the same computation as “2*4” or “2+2+2+2”, but as we’ll see, they’re equal because they evaluate to the same thing – that is, they each perform a computation that results in the same canonical value – the number 8.

In this post, we’re going to focus on a collection of canonical atomic judgement types:

This judgement says that A is a set.

A and Bare equal sets.

a is an element of the set A

and are equal members of the set .

is a proposition.

…

The proposition is true.

…

The definition of the meanings of judgements is, necessarily, mutually recursive, so we’ll have to go through all of them before any of them is complete.

An object is a Set

When I say that is a set in type theory, that means that:

I know the rules for how to form the canonical expressions for the set;

I’ve got an equivalence relation that says when two canonical members of the set are equal.

Two Sets are Equal

When I say that and are equal sets, that means:

and are both sets.

If is a canonical member of , then is also a canonical member of , and vice versa.

If and are canonical members of , and they’re also equal in , then and are also equal canonical members of (and vice versa).

The only tricky thing about the definition of this judgement is the fact that it defines equality is a property of a set, not of the elements of the set. By this definition, tt’s possible for two expressions to be members of two different sets, and to be equal in one, but not equal in the other. (We’ll see in a bit that this possibility gets eliminated, but this stage of the definition leaves it open!)

An object is a member of a set

When I say , that means that if I evaluate , the result will be a canonical member of .

Two members of a set are equal

If and and , that means when if you evaluate , you’ll get a canonical expression ; and when you evaluate , you’ll get a canonical expression . For to be true, then , that is, the canonical expressions resulting from their evaluation must also be equal.

This nails down the problem back in set equivalence: since membership equivalence is defined in terms of evaluation as canonical values, and every expression evaluates to exactly one canonical expression (that’s the definition of canonical!), then if two objects are equal in a set, they’re equal in all sets that they’re members of.

An object is a proposition

Here’s where type theory really starts to depart from the kind of math that we’re used to. In type theory, a proposition is a set. That’s it: to be a proposition, has to be a set.

The proposition is true

And the real meat of everything so far: if we have a proposition , and is true, what that means is that has at least one element. If a proposition is a non-empty set, then it’s true. If it’s empty, it’s false.

Truth in type theory really comes down to membership in a set. This is, subtly, different from the predicate logic that we’re familiar with. In predicate logic, a quantified proposition can, ultimately, be reduced to a set of values, but it’s entirely reasonable for that set to be empty. I can, for example, write a logical statement that “All dogs with IQs greater that 180 will turn themselves into cats.” A set-based intepretation of that is the collection of objects for which it’s true. There aren’t any, because there aren’t any dogs with IQs greater than 180. It’s empty. But logically, in terms of predicate logic, I can still say that it’s “true”. In type theory, I can’t: to be true, the set has to have at least one value, and it doesn’t.

In the next post, we’ll take these atomic judgements, and start to expand them into the idea of hypothetical judgements. In the type-theory sense, that means statements require some other set of prior judgements before they can be judged. In perhaps more familiar terms, we’ll be looking at type theory’s equivalent of the contexts in classical sequent calculus – those funny little s that show up in all of the sequent rules.