One of the themes of abstractmath.org is that we should pay attention to how we think about mathematical objects. This is not the same questions as “What are mathematical objects?”. This post addresses the question: How do we think about variables? What follows are extracts from newly rewrittens sections from Variables and Substitution and Mathematical Objects.
If the author says “x is a real variable” then x plays the role of a real number in whatever expression it occurs in. It is like an actor in a play. If the producer says Dwayne will play Polonius you know that Dwayne will hide behind a curtain at a certain point in the play. When x occurs in the expression you know that if a number is substituted for x in the expression, the expression will then denote the result of cubing the number and subtracting 1 from it.
Slot or cell
The variable x is a slot into which you can put any real number. If you plug 3 into x in the expression you will get 26.
This is like a blank cell in a spreadsheet. If you define another cell with the formula “” and put 3 in the cell representing x, the other cell will contain 26.
What’s wrong with this metaphor: In Excel, a blank cell is automatically set to 0. To be a better metaphor the cell shouldn’t have a value until it is given one, and the cell with the formula “” should say “undefined!”. (I am not saying this would make Excel a better spreadsheet. Excel was not invented so that I could make a point about variables.)
Variable mathematical object
The two metaphors above refer to the name x. You can instead think of x as a variable mathematical object, meaning x is a genuine mathematical object, but with limitations about what you can say or think about it. This sort of thinking works for both the symbolic language and mathematical English, and it works for any kind of mathematical structure (“Let G be an Abelian group…”), not just numbers in a symbolic expression. There are two related points of view:
1. Some statements about the object are neither true nor false.
This means x is a genuine mathematical object and you can make assertions about it, but some of the assertions might have no truth value. From “Let x be a real number” you know these things:
- The assertion “Either x > 0 or ” is true.
- The assertion “ ” is false.
- The assertion “x > 0” is neither true nor false.
The assertion “x is a real number” is in a certain sense the most general true statement you can make about x. In other words, x is a mathematical object given by an incomplete specification, so you are limited in what you can say about it or in what conclusions you can draw about it.
If you say, “Let n be an integer divisible by 4, you cannot assume it is 8 or 12, for example. In other words, the statement “n is divisible by 4” is true, and “n = 3” is false, but the statement “n = 8” is neither true nor false, and you can’t derive any conclusions from n being 8.
2. The object is fixed but some things are not known about it.
If you say x is a real number, you know x is a real number (duh) and:
This way of looking at it involves thinking of x as a particular real number. During the process of solving the equation you are thinking of x as a specific real number, but you don’t know which one.
These points of view (1) and (2) provide genuinely different metaphors for variables. In (1) I say certain statements are neither true nor false, but (2) suggests that all statements about the object are either true or false but you don’t know which. However, note when solving the equation
that, when you are finished, you still don’t know whether x = 2 or x = 3. This factcauses me cognitive dissonance, but the point of view that some statements are neither true nor false upsets other people. I prefer (1) over (2) but I have to admit that (1) is much less familiar to most mathematicians.
View (1) is advocated by category theorists because it allows you to think of a quantity holistically as a single thing rather than as a table of values. The height of a cannonball is different at different times but the “height” is nevertheless one continuous mathematical quantity. People who know more about history than I do believe that that is the simple and uncomplicated way nineteenth-century mathematicians thought about variable quantities.
We need good tools to do math. This means good images and metaphors as well as good tools for reasoning. Having simple and uncomplicated ways to think about math objects (along with guidelines for the way you think about them, such as dropping the law of the excluded middle in some cases!) is every bit as important as making sure our reasoning follows carefully thought out rules that lead from truth only to truth.
Note: Heyting valued logic actually provides sound but non-classical reasoning for thinking about variable objects, but most mathematicians with sound intuitions nevertheless use classical reasoning and come up with correct conclusions. Some of us are now in the practice of using non-classical logic to study differentials and other things, and that is a Good Thing, but it would be a complete misunderstanding if you read this post as advocating that mathematicians change over to that way of doing things. This post is about how we think about variability.