# 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.

# Run! Hide your children! Protect them from math with letters!

Normally, I don’t write blog entries during work hours. I sometimes post stuff then, because it gets more traffic if it’s posted mid-day, but I don’t write. Except sometimes, when I come accross something that’s so ridiculous, so offensive, so patently mind-bogglingly stupid that I can’t work until I say something. Today is one of those days.

In the US, many school systems have been adopting something called the Common Core. The Common Core is an attempt to come up with one basic set of educational standards that are applied consistently in all of the states. This probably sounds like a straightforward, obvious thing. In my experience, most Europeans are actually shocked that the US doesn’t have anything like this. (In fact, at best, it’s historically been standardized state-by-state, or even school district by school district.) In the US, a high school diploma doesn’t really mean anything: the standards are so widely varied that you can’t count on much of anything!

The total mishmash of standards is obviously pretty dumb. The Common Core is an attempt to rationalize it, so that no matter where you go to school, there should be some basic commonality: when you finish 5th grade, you should be able to read at a certain level, do math at a certain level, etc.

Obviously, the common core isn’t perfect. It isn’t even necessarily particularly good. (The US being the US, it’s mostly focused on standardized tests.) But it’s better than nothing.

But again, the US being the US, there’s a lot of resistance to it. Some of it comes from the flaky left, which worries about how common standards will stifle the creativity of their perfect little flower children. Some of it comes from the loony right, which worries about how it’s a federal takeover of the education system which is going to brainwash their kiddies into perfect little socialists.

But the worst, the absolute inexcusable worst, are the pig-ignorant jackasses who hate standards because it might turn children into adults who are less pig-ignorant than their parents. The poster child for this bullshit attitude is State Senator Al Melvin of Arizona. Senator Melvin repeats the usual right-wing claptrap about the federal government, and goes on
to explain what he dislikes about the math standards.

The math standards, he says, teach “fuzzy math”. What makes it fuzzy math? Some of the problems use letters instead of numbers.

The state of Arizona should reject the Common Core math standards, because the math curicculum sometimes uses letters instead of numbers. After all, everyone knows that there’s nothing more to math than good old simple arithmetic! Letters in math problems are a liberal conspiracy to convince children to become gay!

The scary thing is that I’m not exaggerating here. An argument that I have, horrifyingly, heard several times from crazies is that letters are used in math classes to try to introduce moral relativism into math. They say that the whole reason for using letters is because with numbers, there’s one right answer. But letters don’t have a fixed value: you can change what the letters mean. And obviously, we’re introducing that into math because we want to make children think that questions don’t have a single correct answer.

No matter where in the world you go, you’ll find stupid people. I don’t think that the US is anything special when it comes to that. But it does seem like we’re more likely to take people like this, and put them into positions of power. How does a man who doesn’t know what algebra is get put into a position where he’s part of the committee that decides on educational standards for a state? What on earth is wrong with people who would elect someone like this?

Senator Melvin isn’t just some random guy who happened to get into the state legislature. He’s currently the front-runner in the election for Arizona’s next governor. Hey Arizona, don’t you think that maybe, just maybe, you should make sure that your governor knows high school algebra? I mean, really, do you think that if he can’t understand a variable in an equation, he’s going to be able to understand the state budget?!

# Everyone should program, or Programming is Hard? Both!

I saw something on twitter a couple of days ago, and I promised to write this blog post about it. As usual, I’m behind on all the stuff I want to do, so it took longer to write than I’d originally planned.

My professional specialty is understanding how people write programs. Programming languages, development environment, code management tools, code collaboration tools, etc., that’s my bread and butter.

So, naturally, this ticked me off.

The article starts off by, essentially, arguing that most of the programming tutorials on the web stink. I don’t entirely agree with that, but to me, it’s not important enough to argue about. But here’s where things go off the rails:

But that’s only half the problem. Victor thinks that programming itself is broken. It’s often said that in order to code well, you have to be able to “think like a computer.” To Victor, this is absurdly backwards– and it’s the real reason why programming is seen as fundamentally “hard.” Computers are human tools: why can’t we control them on our terms, using techniques that come naturally to all of us?

For some reason, so many people have this bizzare idea that programming is this really easy thing that programmers just make difficult out of spite or elitism or clueless or something, I’m not sure what. And as long as I’ve been in the field, there’s been a constant drumbeat from people to say that it’s all easy, that programmers just want to make it difficult by forcing you to think like a machine. That what we really need to do is just humanize programming, and it will all be easy and everyone will do it and the world will turn into a perfect computing utopia.

First, the whole “think like a machine” think is a verbal shorthand that attempts to make programming as we do it sound awful. It’s not just hard to program, but those damned engineers are claiming that you need to dehumanize yourself to do it!

To be a programmer, you don’t need to think like a machine. But you need to understand how machines work. To program successfully, you do need to understand how machines work – because what you’re really doing is building a machine!

When you’re writing a program, on a theoretical level, what you’re doing is designing a machine that performs some mechanical task. That’s really what a program is: it’s a description of a machine. And what a programming language is, at heart, is a specialized notation for describing a particular kind of machine.

No one will go to an automotive engineer, and tell him that there’s something wrong with the way transmissions are designed, because they make you understand how gears work. But that’s pretty much exactly the argument that Victor is making.

How hard is it to program? That all depends on what you’re tring to do. Here’s the thing: The complexity of the machine that you need to build is what determines the complexity of the program. If you’re trying to build a really complex machine, then a program describing it is going to be really complex.

Period. There is no way around that. That is the fundamental nature of programming.

In the usual argument, one thing that I constantly see is something along the lines of “programming isn’t plumbing: everyone should be able to do it”. And my response to that is: of course so. Just like everyone should be able to do their own plumbing.

That sounds like an amazingly stupid thing to say. Especially coming from me: the one time I tried to fix my broken kitchen sink, I did over a thousand dollars worth of damage.

But: plumbing isn’t just one thing. It’s lots of related but different things:

• There are people who design plumbing systems for managing water distribution and waste disposal for an entire city. That’s one aspect of plubing. And that’s an incredibly complicated thing to do, and I don’t care how smart you are: you’re not going to be able to do it well without learning a whole lot about how plumbing works.
• Then there are people who design the plumbing for a single house. That’s plumbing, too. That’s still hard, and requires a lot of specialized knowledge, most of which is pretty different from the city designer.
• There are people who don’t design plumbing, but are able to build the full plumbing system for a house from scratch using plans drawn by a designer. Once again, that’s still plumbing. But it’s yet another set of knowledge and skills.
• There are people who can come into a house when something isn’t working, and without ever seeing the design, and figure out what’s wrong, and fix it. (There’s a guy in my basement right now, fixing a drainage problem that left my house without hot water, again! He needed to do a lot of work to learn how to do that, and there’s no way that I could do it myself.) That’s yet another set of skills and knowledge – and it’s still plumbing.
• There are non-professional people who can fix leaky pipes, and replace damaged bits. With a bit of work, almost anyone can learn to do it. Still plumbing. But definitely: everyone really should be able to do at least some of this.

• And there are people like me who can use a plumbing snake and a plunger when the toilet clogs. That’s still plumbing, but it requires no experience and no training, and absolutely everyone should be able to do it, without question.

All of those things involve plumbing, but they require vastly different amounts and kinds of training and experience.

Programming is exactly the same. There are different kinds of programming, which require different kinds of skills and knowledge. The tools and training methods that we use are vastly different for those different kinds of programming – so different that for many of them, people don’t even realize that they are programming. Almost everyone who uses computers does do some amount of programming:

• When someone puts together a presentation in powerpoint, with things that move around, appear, and disappear on your command: that is programming.
• When someone puts formula into a spreadsheet: that is programming.
• When someone builds a website – even a simple one – and use either a set of tools, or CSS and HTML to put the site together: that is programming.
• When someone writes a macro in Word or Excel: that is programming.
• When someone sets up an autoresponder to answer their email while they’re on vacation: that is programming.

People like Victor completely disregard those things as programming, and then gripe about how all programming is supercomplexmagicalsymbolic gobbledygook. Most people do write programs without knowing about it, precisely because they’re doing it with tools that present the programming task as something that’s so natural to them that they don’t even recognize that they are programming.

But on the other hand, the idea that you should be able to program without understanding the machine you’re using or the machine that you’re building: that’s also pretty silly.

When you get beyond the surface, and start to get to doing more complex tasks, programming – like any skill – gets a lot harder. You can’t be a plumber without understanding how pipe connections work, what the properties of the different pipe materials are, and how things flow through them. You can’t be a programmer without understanding something about the machine. The more complicated the kind of programming task you want to do, the more you need to understand.

Someone who does Powerpoint presentations doesn’t need to know very much about the computer. Someone who wants to write spreadsheet macros needs to understand something about how the computer processes numbers, what happens to errors in calculations that use floating point, etc. Someone who wants to build an application like Word needs to know a whole lot about how a single computer works, including details like how the computer displays things to people. Someone who wants to build Google doesn’t need to know how computers render text clearly on the screen, but they do need to know how computers work, and also how networks and communications work.

To be clear, I don’t think that Victor is being dishonest. But the way that he presents things often does come off as dishonest, which makes it all the worse. To give one demonstration, he presents a comparison of how we teach programming to cooking. In it, he talks about how we’d teach people to make a soufflee. He shows a picture of raw ingredients on one side, and a fully baked soufflee on the other, and says, essentially: “This is how we teach people to program. We give them the raw ingredients, and say fool around with them until you get the soufflee.”

The thing is: that’s exactly how we really teach people to cook – taken far out of context. If we want them to be able to prepare exactly one recipe, then we give them complete, detailed, step-by-step instructions. But once they know the basics, we don’t do that anymore. We encourage them to start fooling around. “Yeah, that soufflee is great. But what would happen if I steeped some cardamom in the cream? What if I left out the vanilla? Would it turn out as good? Would that be better?” In fact, if you never do that experimentation, you’ll probably never learn to make a truly great soufflee! Because the ingredients are never exactly the same, and the way that it turns out is going to depend on the vagaries of your oven, the weather, the particular batch of eggs that you’re using, the amount of gluten in the flour, etc.

To write complicated programs is complicated. To write programs that manipulate symbolic data, you need to understand how the data symbolizes things. To write a computer that manipulates numbers, you need to understand how the numbers work, and how the computer represents them. To build a machine, you need to understand the machine that you’re building. It’s that simple.

A bunch of people have been mailing me links to an article from USA today
about schools and grading systems. I think that most of the people who’ve
been sending it to me want me to flame it as a silly idea; but I’m not going to do that. Instead, I’m going to focus on an issue of presentation. What they’re talking about could be a good idea, or it could be a bad idea – but because the
way that they present it leaves out crucial information, it’s not possible to meaningfully judge the soundness of the concept.

# Idiot Math Professors, Fractions, and the Fun of Math

A bunch of people have been sending me links to a USA Today article about a math professor who wants to change math education. Specifically, he wants to stop teaching fractions, and de-emphasize manual computation like multiplication and long division.

Frankly, reading about it, I’m pissed off by both sides of the argument.

# Dirty Rotten Infinite Sets and the Foundations of Math

Today we’ve got a bit of a treat. I’ve been holding off on this for a while, because I wanted to do it justice. This isn’t the typical wankish crackpottery, but rather a deep and interesting bit of crackpottery. A reader sent me a link to a website of a mathematics professor, N. J. Wildberger, at the University of New South Wales, which contains a long, elegant screed against the evils of set theory, titled “Set Theory: Should You Believe?”

It’s an interesting article – and I don’t mean that sarcastically. It’s over the top, to the point of extreme silliness in places, but the basic idea of it is not entirely unreasonable. Back at the beginnings of set theory, when Cantor was first establishing his ideas, there was a lot of opposition to it. In
my opinion, the most interesting and credible critics of set theory were the constructivists. Put briefly, constructivists believe that all valid math is based on constructing things. If something exists, you
can show a concrete instance of it. If you can describe it, but you can’t build it, then
it’s just an artifact of logic.

Some of that opposition continues to this day, and it’s not just the domain of nuts. There are
serious mathematicians who’ve questioned the meaningfulness of some of the artifacts of modern
set-theory based mathematics. Just to give one prominent example, Greg Chaitin has given lectures in which he discusses the idea that the real numbers aren’t real: they’re just logical artifacts which can never actually occur in the real world, and rationals are the only real real numbers. (I don’t think that Greg really believes that – just that he thinks it’s an interesting idea to consider. He’s far
too entranced with set theory. But he clearly considers it valid enough to be worth thinking about