One plus one equals Two?

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: $\frac{18}{100} = \frac{x}{5482}$ And then he cross-multiplied, getting: $18 \times 5482 = 100 \times x$ $98676 = 100 \times x$ and so $x = 986.76$. 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.

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.

Bad Math Books and Cantor Cardinality

A bunch of readers sent me a link to a tweet this morning from Professor Jordan Ellenberg:

The tweet links to the following image:

(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 $i$, and map it to exactly one negative integer, $0 - i$.

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 $i$ to exactly one even natural number $2i$. 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, $\pi$. Or the square root of 2. Or $e$. )

This is terrible on multiple levels.

1. It’s a math textbook written and reviewed by people who don’t understand the basic math that they’re writing about.
2. It’s teaching children something incorrect about something that’s already likely to confuse them.
3. 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.
4. 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:

1. The set of fractions is a strict subset of the set of ordered pairs of natural numbers.
2. 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.
3. On a two-d grid, put the natural numbers across, and then down.
4. 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).
5. 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?

Not a theory! Really! It’s not a theory!

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 Planck’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 $e=mc^2$, 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 $e=mc^2$ 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 can’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 can’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 world’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.