17 Sep 2013

Geometry Is Definitely Analytic in *This* Anecdote

I was trying to show my 8-year-old some stuff in geometry over the weekend. First I drew a generic rectangle and asked him what it was. He of course identified it as a rectangle. Then I asked him to tell me why it was a rectangle, and we began making a list that I titled, “Rules for Being a Rectangle.” We did the same procedure for a square.

Then we had the following exchange, which I thought was pretty funny:

BOB: Okay, now that we’ve figured out the rules for rectangles and squares, let me ask: Is a rectangle a square?

SON: (Pauses.) Not usually.

At first this disappointed me, since he’s very bright and we had clearly laid out the rules; I wanted him to give me the “correct” answer of no. But then I thought that maybe he meant only a special subset of rectangles were squares, and so if you randomly sampled from the set of all rectangles, it would usually not be a square, but hey once in a while you’d get a square. So, I gave him the benefit of the doubt, clarified what the precise situation was, and then moved on. This is when it got interesting.

BOB: Okay, let’s try this question: Is a square a rectangle?

SON: (Pauses.) Not usually.

Now I was really flummoxed. What the heck was this kid thinking? Let’s find out.

BOB: “Not usually”? Why do you say that?

SON: Because most people would call it a square!

I love this kind of stuff. This happened to me all the time when I was teaching at Hillsdale College. I’d write down what I thought were perfectly straightforward exam questions, and then bright students would miss tons of them. After a while I realized that their “problem” was that they were giving decent answers to their interpretation of what the question meant, it’s just that they weren’t interpreting the questions the way trained economists do.

35 Responses to “Geometry Is Definitely Analytic in *This* Anecdote”

1. Ken P says:

After the first question, I thought “brilliant”. I was disappointed after his second response until I read his reasoning. He’s clearly familiar with the subjective nature of value. Actually, I think around that age they start taking other people’s perspective into account.

The more specialized we become the more we tend to talk past each other. We share different premises, different biases and different approaches to very analogous problems. There s a sort of path dependence to our next statement.

2. terrymac says:

all squares are rectangles, all the time.

some rectangles – those with equal sides – are squares.

• Tel says:

My vote goes with the kid, if a rectangle has equal sides, most people would call it a square.

3. Daniil Gorbatenko says:

I think, in a funny way, this anecdote actually suggests that geometry is not analytic. Because the very reason for your son’s mistake lies in his (over-)reliance on convention.

In other words, I mean that the only way you can initially get an idea what a square and a rectangle are by looking at their actual external depictions. That those aren’t ideal rectangles and squares is beside the point because even if you draw a rectangle or a square in your mind it will also be imperfect. But nonetheless, you’ll “see” from those imperfect depictions the conceptual truth that a square is necessarily a rectangle but not vice versa. The Aristotelian interpretation of this (and the only one I can make sense of) is that the concepts of square and rectangle are contained somehow in their imperfect instances and our minds are capable of abstracting those concepts from those instances.

• Matt Tanous says:

“even if you draw a rectangle or a square in your mind it will also be imperfect”

Er… what? I can imagine a square in my mind that has perfectly exact side lengths and perfectly right angles. I can’t reproduce this to anyone else, but that doesn’t stop me from conceptualizing the ideal in my head…

• Daniil Gorbatenko says:

You confuse imagination and conceptualization. The picture of a triangle in your mind is as imperfect as a drawing. The concept of a triangle is not a picture of it.

• Matt Tanous says:

The picture in my mind is the ideal – the visual representation of the concept. The fact that I know my drawing is imperfect simply by looking at it – without measuring angles, etc. – requires such a visual ideal to be formulated in my mind.

• Daniil Gorbatenko says:

I don’t know, maybe you are a superhuman but when I mentally draw a triangle, it’s nothing ideal. It’s impossible for a human to imagine infinitesimally thin sides for example. Imagine in the sense of mentally draw them. IOW, concepts aren’t imaginary pictures.

• Bob Murphy says:

Daniil wrote:

I think, in a funny way, this anecdote actually suggests that geometry is not analytic. Because the very reason for your son’s mistake lies in his (over-)reliance on convention.

But the point of my post was that my son didn’t make a mistake. Given what he meant by the question, his answer was right. So I’m saying in this case, the answer to a geometrical question depended not on the “objective nature of shapes” but rather on somewhat arbitrary social conventions on what words mean.

• Daniil Gorbatenko says:

Bob, fair enough. You caught me into a trap 🙂

• Ken P says:

Like the word inflation. Sorry. Had to say that. But the thing is that it is good to understand both tbe logical aspect to meaning and the fluid/subjective aspect.

4. Yancey Ward says:

In other words, you and Krugman agree on everything, you just think you are disagreeing.

5. Innocent says:

The problem with communication is semantics.

• Z says:

The problem with squirrels is that they eat too many acorns.

6. Silas Barta says:

You actually touched on a very important phenomenon in human cognition: the significance of good (“perspicuous”, “central”) examples. When assigning labels, people don’t normally go by some hard and fast rule, but by whether it means “most” of the criteria, i.e. is a central example. When your son says that “most people wouldn’t call a square a rectangle”, that could be rephrased as “a recognizable square is not a *characteristic* example of a rectangle”.

If you want to study a field where definitions *are* hard and fast rules, you have to train out of this habit.

Once I had a dream where I received the message, “Humans learn from good examples. If you want to explain to me what a bird is, show me a robin, or a sparrow, or a bluejay. Don’t show me a penguin.”

7. Ken B says:

Cute kid.

8. Ken B says:

FWIW this square/rectangle question is a standard one in Object-Oriented programming. Where a square is usually not a rectangle … (for geeks: fails substitutibility).

• Bob Murphy says:

Ken I am delighted that my son’s answer is right in OOP. But what do you mean that it “fails substitutability”?

• Ken B says:

Hmm. Hard to explain briefly.
in OO programming we have the notion of a type. Every ob ject has a type, which is more less a defined suite of requests it answer to in sensible ways. “Fruit” is a standard example. And you might be able to ask a an object of type Fruit to tell you its weight. “Apple” is a sub type of Fruit. What that means is that any request you can make of a Fruit you can make of an apple. So an apple can tell you its weight, or what plant it comes from.
Substitutibility means I can use an apple wherever I can use a fruit.

Can I use a Square anyplace I can use a Rectangle? Maybe not. It depends on the details of the Rectangle’s type. Let’s say the Rectangle has a defined request to change it’s area while keeping its length constant. Any Rectangle must be able to do that, and a Square cannot. If I write code that asks a Rectangle to do that my code will break if I pass in an Apple. So if that request is part of the Rectangle’s type then a Square is not a Rectangle.

• Ken B says:

Technical clarification since the aspect ratio example looks forced: A more realistic method that might exist on a Rectangle is setDimensions(length, width) which for a Rect I can call with parameters 3,5.

• John Hall says:

Just create square as a subclass of rectangle and leave it at that.

• martinK says:

Just create square as a subclass of rectangle and leave it at that.

But then Square will inherit setDimensions(length, width). How is it going to handle setDimensions(3,5)?

• Matt Tanous says:

Override it to behave differently, such as throw an exception or simply use the first parameter with a warning notice.

• Tel says:

You seem to be working in a paradigm where your objects cannot change type as a result of a method call… that does seem rather limiting.

• Ken B says:

I’m not debating good design. I pointed out this is a standard example of a debate in OO.
Basically when you select an interface and a contract you can sometimes get mild surprises.
Whatever you do the Square fails substitutibility on setDimensions and possibly on other methods if they exist.
Throwing an exception or modifying some other value could violate the contract.

• Silas Barta says:

If you love your son, you won’t indoctrinate him in object-oriented design patterns.

• Ken B says:

Watch out for Adaptor; they’ll tell you it’s innocuous, and much safer then Flyweight, but don’t you believe them:it’s a gateway pattern.

• Matt Tanous says:

Yes, he should stick to a more functional methodology. Much preferable to not have to track object states.

• Matt G says:

On this, Matt Tanous, we can agree! Avoiding mutation eliminates a whole dimension of complexity and potential bugs.

• Matt Tanous says:

We agree until you have to do my job. When you are actual keeping track of the physical and logical configuration of a device in order to perform various tests upon it, OOP is pretty necessary. I generally stick to structs, though – classes are too complex for the task.

• Matt Tanous says:

I suppose technically structs aren’t actually object-oriented, but still… what is functionally the difference between that and the “get” and “set” functions of a class? Struct + specific functions relating to it… that’s essentially a class without the complexity of multiple objects.

• Tel says:

Does make Input/Output difficult, those immutable objects don’t relate to real world objects particularly well.

9. Robert Hill says:

……….When I studied the gifted back in the day (’60s), they said that the reason IQs only go up to 200 is because beyond that the test-taker gives plausible answers not thought of by the test writers. Above that,they used to call it “untestibly high”

10. John Hall says:

Usually the problem is because the teacher writes tests that are too vague. Part of my high school geometry exams always included some always, sometimes, never questions. I was always able to talk myself out of the wrong answer by thinking things like “well what if we had different postulates” or some other non-sense. Always got the extra credit proofs right, but never could get the easy questions right.

11. Greg Ransom says:

Your son is teaching you Wittgenstein.

It’s is no accident that Wittgenstein taught 8 year olds after mastering Frege & Russell …