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help with abstract math

Produced by Charles Wells     Revised 2015-06-26      Introduction to this website       website TOC      website index   blog

DIAGNOSTIC EXAMPLES

This page provides examples of the kinds of problems students have with abstract math with links to where to read about them.

I don’t know how to think about …

This may mean:

You don’t have a useful mental representation of the concept – metaphors or images that allow you to think about it.
You don’t have a lot of experience with the concept, so you don’t have any intuition about it. (Intuition is in fact a kind of obscure mental representation learned from thinking a lot about the concept.)

I can’t imagine five-dimensional space. I know the fourth dimension is time, but what could the fifth dimension be?

Your statement “I know the fourth dimension is time” shows a misunderstanding. An $n$-dimensional space, roughly speaking, is a space in which you can locate every point by giving $n$ coordinates. The model of space-time given by general relativity does indeed have four coordinates, one of which models time. But 4-dimensional space as an abstract math object doesn’t have to model space-time. It could be a phase space, for example, or something completely abstract. A five-dimensional space is also an abstract idea – each point has five coordinates – which can model many things.

You don’t have useful metaphors or images for five-dimensional space or any intuition about it either, so you don’t have “any way to think about it”. However, each type of five-dimensional space has a precise mathematical definition. By working example and proving theorems based on that definition you can gain some intuition about how things behave in that space.

But you didn’t say what it is!

Beginners tend to be stuck with one representation of a type of math object, and tend to think that the math object is that representation. See Representations and models.

You defined $\sqrt{2}$ to be “the positive real number whose square is 2” but you didn’t say WHAT IT IS.

That may be because you think that a real number is the same thing as its decimal expansion and you want to be told that $\sqrt{2}=1.414...$. In fact to say it is “the positive real number whose square is 2” is an exact description of $\sqrt{2}$, but giving its first few decimal places is only an approximation. This example is covered in more detail in Definitions. See also exact number and Concept and computation.

You wrote: "Definition: The derivative of a function $f(x)$ at a point $x=a$ is the slope of the tangent line at $(a,f(a))$." But that's not a definition because you didn't say how to calculate it.

It is a definition because it tells you exactly what the derivative at a point means. You very likely want to find an easy way to calculate it, but a definition of a math object does not have to tell you anything about how to calculate it. Definition and calculation are two different things you do with math. In this case, however, you can figure out from the definition how to calculate the value of a derivative at a point. See Concept and Computation and Derivatives.

You can never get all the way to infinity

You said $\frac{1}{3}$ is exactly equal to the infinite decimal expansion $0.333…$, but the decimal expansion never gets to infinity so it is only an approximation.

The infinite decimal expansion denoted by "$0.333…$" is an exact description of the whole decimal expansion. For every integer $n$, the $n$th decimal digit is already $3$. All the digits are already there. It is a completed infinity. (This is about how you should think about it. It is not a claim about physical or metaphysical existence.) More about this topic in Real decimal representations.

You are in good company if you find the idea hard to take. The idea of completed infinity was the subject of a monstrous argument in the early twentieth century. But almost all mathematicians accept it as a powerful and useful idea, while not claiming that it says anything about "reality". See constructivism.

That doesn’t make any sense

The feeling of something not making sense may coming from having mental representations of a type of object that don’t fit well the way they actually are (which means: the mental representations don’t fit what can be proved about them.) Mental representation are described in more detail in the chapter on Images and Metaphors. In the topic articles about particular parts of math, the images and metaphors that mathematicians commonly have about each topic are discussed – with specific attention to aspects that mislead.

“A straight line segment has length but its width is zero.” How could its width be zero? If its width is zero I wouldn’t be able to see the line! If it is zero, the line wouldn’t be there.

You may be thinking of a straight line segment as like a straight mark on a page or like a stick. The line segment is like a stick in some respects, but it is not a physical object and does not have to have thickness. Indeed, you can’t see the line in the sense of physical seeing; what you see on the paper or on the screen is only a visual representation of the line segment.

“The real line is dense, in the sense that between any two real numbers there are an infinite number of others. This means that for a given number $r$, there is no number just to the right of $r$.” I can’t imagine that! Surely, if you are looking at the point, there has to be another point right next to it!

You are thinking of the real line as a row of points, but it does not have that particular property that a row has, namely that the points are arranged one after another. Your image doesn’t fit the facts: There is never another point just next to a point. There is always a point between them, and a point between them, and always a point beween them, and $\dots$

“This infinite series converges to $\zeta (2)=\frac{{{\pi }^{2}}}{6}\approx 1.65$”. This sentence is incomprehensible!

It means “This infinite series converges to $\zeta (2)$, which is $\frac{{{\pi }^{2}}}{6}$, which is approximately 1.65.” This is an example of a Parenthetic Assertion.

Why do you say $i$ is an imaginary number? If it doesn’t exist, why talk about it?

This is an example of cognitive dissonance caused by a technical term (“imaginary”) that is also a word in ordinary English. Imaginary numbers have the same cognitive status as real numbers: they are mathematical objects.

How did you know that so fast?

How did you know that ${{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}$is never negative?

This is an example of pattern recognition. It is discussed in more detail under ratchet effect.

They say the same thing!

The definition of function requires that there be a unique output for every input. The definition of injective says the same thing!

You have reworded the definitions that your teacher or book gave. The word “unique” is ambiguous! The definition of function requires that if $a = b$, then $f(a) = f($). The definition of injective requires that if $f(a) = f(b)$, then $a = b. More about that here.

This proof assumes the opposite!

Theorem

If ${{n}^{2}}$is even, then $n$ is even.

Proof

Suppose $n$ is odd. Then for some integer k, n = 2k + 1. Then ${{n}^{2}}=4{{k}^{2}}+4k+1=2(2{{k}^{2}}+2k)+1$. Thus ${{n}^{2}}=2h+1$ for some integer h, so ${{n}^{2}}$ is odd. QED.

The theorem says something about when $n^2$ is even, but the proof assumes $n$ is odd! What does the proof have to do with the theorem??

The author is proving the theorem by proving the contrapositive. The contrapositive of a statement is equivalent to the statement, so when you prove one you prove the other. Authors use this all the time but rarely say what they are doing by name.

See also proof by contradiction, which is also commonly used without mentioning it.

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