Equivalence Classes and Equivalence Relations

12 December 2009

In a recent post, I mentioned that we often don’t directly define the objects we care about.  We define ways to represent the objects we care about, and then we define what it means for two representations to represent the same object.  This may seem like a roundabout way to explain something, but in fact it is very common.  The example I gave there was fractions.

I thought it was worth a little more attention to see the machinery that makes this go.  The key ideas are “equivalence relation” and “equivalence class”.

Read the rest of this entry »


“Well-Defined”: More Examples

10 December 2009

In this recent post, I explained the metamathematical concept of well-definition, and I made the claim that this notion is pervasive in the organization of mathematical and nonmathematical ideas.  I gave a few examples, but barely scratched the surface.  So a few more won’t hurt. Read the rest of this entry »


Math and Poetry

5 December 2009

In my class, I have a quote of the day; today I had a double feature.  A matched set, if you like.

A man should be learned in several sciences, and should have . . . , in some measure, a mathematical mind, to be a complete poet. —Dryden

alongside

A mathematician who is not also something of a poet will never be a complete mathematician. —Weierstraß

Read the rest of this entry »


Mathematician-Speak: Well-Defined

4 December 2009

I often find, when I am thinking and speaking about things not directly related to traditional mathematics, that my mathematical vocabulary gives me a way of expressing an idea that doesn’t have any obvious analogue in common English.  (I really shouldn’t be surprised by this, since mathematical thinking in some sense attempts to work with and manipulate ideas at their most elemental, but nonetheless I often am.)  One of the words I very often want to say in ordinary conversation is well-defined.

In the most general context, something is well-defined if it has a clear, unambiguous definition.  It should be not be defined in terms of other terms requiring a definition, etc.  But the term is usually reserved for a more specific sort of potential ambiguity in a definition — dependence on a “choice”.  It’s not unusual for a definition to depend, or at least to appear to depend, on one or more choices, but if the definition is going to make sense then these choices  should not matter.

Now, it is always possible to formulate a definition without apparent ambiguities, and if you’re interested in pure logic or foundations of mathematics, or if you like all your math to be written in strings of logical symbols for use, say, by a computer, you probably always want to.  But if you want what you write to be readable, and if you want the literal meaning of what you write to mirror as closely as possible the motivating ideas, it can often be clearer to write an aesthetically pleasing but a priori sketchy definition and then prove it makes sense.

Examples

  1. What is the numerator of a rational number?  We could say that the numerator of r is p if r=\frac{p}{q}, provided p,q are integers, but that’s not well-defined.  A rational number has lots of expressions as a fraction, and they all have different “top numbers”.  So this definition has an implicit choice (the choice of fraction representation), and it does depend on the choice.  We need to amend the definition.  Better is that the numerator of r is p if r=\frac{p}{q}, provided p,q are integers with no common factors, and q>0.  In this case the ambiguity is removed by specifying which choice we have to make.
    This is not just frivolous.  The point is that, if we want the numerator to mean numerator of a number, then we have to accept that if m,n are positive integers, the numerator of \frac{m}{n} is not necessarily m.  (On the flip side, if you really want the numerator of  \frac{m}{n} to be m, then you have to accept that you are no longer considering \frac{2}{3} and \frac{14}{21} to be the same thing, so you aren’t talking about numbers anymore.)
  2. How do we add and multiply numbers in modular arithmetic?  Divide the numbers into 12 families according to their remainder modulo 12 (that is, two numbers are in the same family if their remainders mod 12 are the same, i.e. if 12 divides their difference).  I claim that we can add and multiply the families. I define the sum of two families as follows: pick a member of each family, add those numbers, and see what family that number belongs to.  Likewise for products.  Again, there is an implicit choice in the definition (the choice of the representative for each family), but in this case you can prove that the sum/product winds up in the same family regardless.  So the arithmetic of number familes, beautifully, is well-defined.
  3. I said this concept is not limited to “numerical” objects.  In many languages, such as Spanish, French, and Portuguese, a grammatical feature of a noun is its gender, masculine or feminine.  Suppose I hold up an postage stamp and ask whether this it is masculine or feminine in Spanish.  Well, it depends what you call it; did you first think el sello or la estampilla? Both words mean “stamp”.  Or maybe you just though la cosa (thing) or el objeto (object).  Certainly all four nouns apply, as well as many others.  So is a stamp masculine or feminine?  It’s a bad question.  Gender is a property words have, not things, and it does depend on the word choice.  The grammatical gender of an actual thing is not well-defined (some people use ill-defined as the negation).

The first example deserves a little extra attention to a fine point: the difference between rational numbers and the fractions which represent them.  From a starting point of the integers, it’s not so easy to directly define “rational number”.  Instead we define “fraction”, which is a pair of integers, the latter not allowed to 0, and then we define what it means for two fractions to represent the same number.  This two step process is very common in the way ideas (not just mathematical) are organized.  It is too hard to define exactly the objects we care about, so we define another kind of object, then define what it means for two of that object to be the “same”.  (For example, what is a hand in poker?  Well, it’s a list of 5 cards out of the deck, but it isn’t literally that, it’s a list of 5 cards out of the deck with the understanding that we don’t care about the order.)

It is precisely because this sort of construction is so utterly pervasive that well-defined is a good word to know.  What we care about are rational numbers (or poker hands, etc.) but what we can conveniently write down are fractions (or lists of 5 cards, etc.), so it’s convenient to have our definitions and operations be about fractions.  And when we build up our vocabulary about fractions, we hope that what we do is actually informing us about rational numbers — and it is, precisely to the extent that our concepts are well-defined for rational numbers.

Well-definition is a notion that is key to the way we process and manipulate ideas.  It is very common to have multiple concepts which overlap or are otherwise deeply related but not quite the same: objects and nouns, people and names, rational numbers and fractions; in these situations one wants to know when it does and doesn’t make sense to blur the distinctions, to know which abuses of language make sense and which don’t.


Follow

Get every new post delivered to your Inbox.