Math Definitions

It was sometime before noon, when my wife and I heard a boink, boink somewhere in the house. What the…?

A short investigation revealed that a female cardinal and her mate had perched upon a branch just beyond our second floor porch *. We think they may have had a baby or babies with them.

The female appeared to be attacking her reflection in one of the large windows on the south side of our house. Every few seconds she would furiously fly into the window with a thud. This went on for quite some time.

Mimi, my wife, told me that ,down the hill at her sister’s house, they had a cardinal that did that for a long time and they couldn’t figure out a way to stop it.

After a while we decided to take action. I got a long stick and tried to shoo the bird from the branch. To no avail. Every time I approached the tree from the porch the bird would see me and fly away. As soon as I would leave, the bird would come back.

Then I went and got a bucket of water. I thought I might douse the bird. The bird was having none of that, though. Just as before, as soon as I’d show up with the bucket, the bird would hightail it, and as soon as I left, she’d return.

Mimi suggested we hang something from a branch that would make noise when the cardinal landed. It sounded like a good idea to me, so we went and got a set of measuring spoons and tied them to the end of a string. Then, from the porch, I tried to throw them over the branch that the bird had perched on. After several attempts, it seemed like it worked. So Mimi went back to her reading, and I went back to my writing.

Minutes later, the little birdbrain was back at it. By this time she had flown into the window at least 100 times in the last half hour or so. No wonder they call them birdbrains. What is it in her little noggin that kept her at this unproductive activity?

We then tried another tactic. We have a broom with a telescopic handle that extends to about 20 feet. We stretched it to reach meanfrom the inside up to the top window, which the bird had been attacking. We thought that maybe the sight of something on the other side of the window would scare the bird.

I must admit that by now I had started feeling like Wiley Coyote versus the Road Runner.

Have you ever heard the expression, “The definition of insanity is doing the same thing over and over again and expecting different results?” At first I was wondering why this bird would keep repeating her actions and expecting different results. Now I was wondering the same thing about me!

It’s been a week since I first started this post. Who do you think has won? Yes, that little feathered perpetual motion machine is still at it. Furthermore, she must have phoned her family and told them about her success, because as of today we’ve had another cardinal smashing herself into one of the windows on the western side of our house. I don’t know what they get out of this besides blunted beaks. But my wife and I have learned acceptance and patience with our fluttering friends. At least we had for a while, until the groundhog showed up in our bedroom. But more about that in the next post.

* (This sentence just reminded me of a line from Edgar Allan Poe’s “The Raven”… “perched upon a bust of Pallas, just above my chamber door…”)

Regarding the the expression, “The definition of insanity is doing the same thing over and over again and expecting different results” …

I’ve seen it attributed to Albert Einstein, but I don’t believe that one bit. It seems like too much of a trite truism to be from Einstein. Whenever someone considers something slightly pithy, they tend to want to attribute it to either Einstein, Mark Twain, Churchill, or George Carlin.

The quote has some merit, but not the depth that is generally ascribed to it. It may concern an attribute of insanity, but it certainly isn’t the definition.

Definitions are funny things. So often, people are content to settle for generalizations that have some truth to them as a full definition. If you are following the idiotic rhetoric from some American “news personalities,” then you’ve had lots of chances to see how minor, occasional attributes can be twisted into seeming like full-blown definitions.

People like simple solutions. And the simpler the person, the simpler they like their solutions, even if it concerns complex questions.

Somehow, in benighted corners of society, “simple” has become a badge of honor. “I’m just a simple man, and I just want simple answers…”

Yeah, and I want a flying pony.

Simple answers for complex questions are for simpletons, or “simps.” One thing that Einstein did say, is something like, “Everything should be made as simple as possible, but no simpler.”

It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.

wikiquote.org: “On the Method of Theoretical Physics” The Herbert Spencer Lecture, delivered at Oxford (10 June 1933); also published in Philosophy of Science, Vol. 1, No. 2 (April 1934), pp. 163-169. [thanks to Dr. Techie @ www.wordorigins.org and JSTOR]

(That quotation is very similar to Occam’s Razor. You can read more about Occam’s Razor at a previous Math Mojo Chronicles post.)

Mathematical Definitions – Why are they so elusive?

Usually, as you learn math in school, you are given certain definitions for concepts. For example, you are told what a digit is. You are told what a whole number is, what a fraction is, etc. Then you are told what addition is, what subtraction is, etc.

From there, you are taught how to manipulate numbers, based on those concepts.

But are those definitions correct, complete, useful, or perhaps even misleading and harmful? Many problems people have with math stem from the poor explanations they were given. This leads to frustration, and often to the wrong conclusion that, “Oh, I’m just not good at math.”

Maybe it’s not that the person is not good at math – maybe it’s that what they were taught was not good math. Had they been properly introduced to math in a better way, they would surely have had less problems with it, and maybe even come to appreciate the joy and romance of it.

Let me give you an example. Often, we are taught that “multiplication is just repeated addition.”

Oh, really? How does that work for fractions? It doesn’t. The definition of multiplication as repeated addition is what drove you crazy when you first learned to multiply fractions.

There are plenty of definitions in math that aren’t definitions at all, just descriptions of some attributes. How about this one, “A fraction is a number that can be put into the form of m/n.”

That is a description of an attribute. It is not incorrect, it’s just not a good definition. It’s sometimes useful, and sometimes not.

Calling things “definitions” is one of the problems in math education. We are a society that is obsessed with easy answers for complex questions. Ah, if only life worked that way! But of course, it doesn’t. But we like to pretend it does, so we give expedient and incomplete definitions, to get the matter out of the way.

The problem is that later, it comes back to bite us in the butt. A child is faced with multiplying fractions, and is confused that you cannot just repeatedly add to get the answer. Now he’s mad that you told him one thing, then, but it doesn’t work, now. You pulled the rug out from under him. That’s probably what makes math so frustrating for most people.

Or in another case, we may teach children that the “natural numbers” are “1,2,3…” Then some day they read that the natural numbers are “0,1,2,3…” (This time including zero). What’s up with that?

Both are correct. The former is used for number theory, and the latter is used for set theory and other branches of math. The former is probably what you want to be using when you are talking about arithmetic.

Do we even hint that there are distinctions? Of course not. Most teachers don’t even know that there are distinctions. No matter – the kids are to be tested on it anyway, and if they don’t give the answer we want, why, then we fail the little suckers.

Maybe you, like I, remember sitting in math classes throughout your school years, listening to teachers drone on, and wondering “What the hell are they talking about?” Among the many reasons that many children have this experience, is because we expect them to accept definitions that only work in certain cases, as black-and-white all-encompassing final definitions. Then we reinforce this by testing them on it, and hammering in the notion that simple explanations are sufficient to explore complex subjects.

What’s the solution? At some point we, as a society, have to mature to the point that we understand that not everything is simple. There is always more to be learned. Take everything with a grain of salt. It’s best to teach that we cannot pin everything down at the level of the learner. There are things about math that you have to try now, and try later, when you have more experience with the subject, and that later you will grasp it better.

We also have to stop the obsessive testing. Things take time to sink in. I dare you to define the word “number” right now, or even “math.” Of course your answer will be more or less incomplete. So if you and I can’t define these things adequately, how fair is it to be testing children on concepts we cannot adequately define?

It turns out that what we are testing for is not what is important. Even the way we are testing makes it harder to teach and learn. We’re doing exactly the wrong things, and we’re doing them systematically.

Don’t take my word for it…

One of the greatest mathematician/philosophers of all time, Bertrand Russell, in Introduction to Mathematical Philosophy (1919) concerning definitions of basic concepts, like “number”, wrote on pages 3-5,

But though familiar, they are not understood. Very few people are prepared with a definition of what is meant by ” number,” or ” 0,” or “1.” It is not very difficult to see that, starting from 0, any other of the natural numbers can be reached by repeated additions of 1, but we shall have to define what we mean by “adding 1,” and what we mean by “repeated.” These questions are by no means easy. It was believed until recently that some, at least, of these first notions of arithmetic must be accepted as too simple and primitive to be defined. Since all terms that are defined are defined by means of other terms, it is clear that human knowledge must always be content to accept some terms as intelligible without definition, in order to have a starting-point for its definitions.

It is not clear that there must be terms which are incapable of definition: it is possible that, however far back we go in defining, we always might go further still. On the other hand, it is also possible that, when analysis has been pushed far enough, we can reach terms that really are simple, and therefore logically incapable of the sort of definition that consists in analysing. This is a question which it is not necessary for us to decide; for our purposes it is sufficient to observe that, since human powers are finite, the definitions known to us must always begin somewhere, with terms undefined for the moment, though perhaps not permanently.

In Introduction to Mathematical Thinking, by Friedrich Waismann, 1951 (Originally Einfuehrung in das mathematische Denken ) Karl Menger wrote in the foreword on page VII, concerning a particular definition:

In regard to the definition of numerical equivalence, which goes back to Cantor and Frege, one will willingly grant that it represents only one of many possible attempts at precise formulation of the vague and ambiguous use of this word in colloquial language-a one-sidedness that, to be sure, it shares with every attempt at precise definition of any part of colloquial speech. Furthermore, one will perhaps agree that the criteria given by this definition of numerical equivalence for its application to experience, are none the more precise if compared with those of the usual definitions of fundamental physical concepts, such as equality of lengths, simultaneity, etc. Neither does it seem to me, however, that they are less precise. Indeed, the great mathematical significance of this definition rests preponderantly on its fruitfulness, i.e., on the fact that so many conclusions can be deducted from it. However, just because this definition has proved to be so very fruitful that no other definition up to now has been able to compete, it certainly is of value to point out other similar possibilities (especially to prevent the erroneous view that it is the only conceivable definition).
(Emphasis mine)

Paul Lockhart, in his absolutely must-read, A Mathematician’s Lament, puts it like this,

The point is you don’t start with definitions, you start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction. Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion.

So, in essence, you have to start somewhere, but you should not get locked into the Idea that your working definition is complete, nor the only possible definition. There is always more to learn and refine. Keeping an open mind is essential to understanding.

By the way, I can see no reason why these three resources should not be read and digested by everyone who teaches mathematics, especially elementary school teachers. They contain the seeds of the answers to almost all the questions that curious students will ask when taught about numbers and math, as does Karl Menger’s The Basic Concepts of Mathematics (1957) (Out of print – Okay, I can see why this one might not be read by all teachers).

All too often teachers are expected to teach “a mile wide and an inch deep.” Fortunately, many pupils’ curiousity extends a lot deeper than that. It’s books like these, which delve deep into the “why” of math, that help teachers, students and the general public get a grip on what is a lot more important than simply teaching the mechanics of passing less-than-meaningful tests.