abstractmath.org 2.0
help with abstract math

Produced by Charles Wells     Revised 2017-04-08

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# DIAGNOSTIC EXAMPLES

This chapter provides examples of the kinds of problems students have with abstract math.

## A definition has to say how to calculate it!

#### 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.

On the contrary, IT IS A DEFINITION
because it tells you exactly what "derivative" 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

Definition and calculation are two different things you do with math. For derivatives, however, you can often figure out directly from the definition how to estimate the value of a derivative at a point. A detailed example is given in Concept and Computation. The method described there can be used to derive formulas for the derivatives of many functions, as you may know if you have taken first-semester calculus.

## 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 $\sqrt{2}$ is approximately equal to $1.414...$. (This is also an example of wanting a definition to tell you how to calculate the thing being defined.)

But consider this:

• To say that $\sqrt{2}$ is “the positive real number whose square is $2$” is an exact description of $\sqrt{2}$.
• Giving its first few decimal places is only an approximation.
• Even so, you can directly use the definition to determine (not very rapidly) as many decimal places of $\sqrt{2}$ as you want, as described in the article Definitions. The definition of a concept really is the source of all truth about the concept.

## 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.

### That doesn't make sense (1)

#### “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.

A straight line is a math object and therefore abstract,
although it may have physical representations
that approximate how you think about it.

### That doesn't make sense (2)

#### “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 the real line does not have the 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 on the real line.
There is always a point between them,
and a point between them, and a point between them,
and so on forever.

• $1.05$ is between $1$ and $1.1$.
• $1.01$ is between $1$ and $1.05$.
• $1.001$ is between $1$ and $1.01$.
• $1.00001$ is between $1$ and $1.001$.
• $1.0000001$ is between $1$ and $1.00001$.
• and so on forever

### That doesn't make sense (3)

#### “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. Many mathematicians use parenthetic assertions in the research literature. Some students understand them intuitively, and some do not.

### That doesn't make sense (4)

#### 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.

## 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.

On the contrary, 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
of the number $\frac{1}{3}$ is $3$ right now.
All the digits in "$.3333\ldots$" are already there.

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.

## 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 and chunking.

## 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 $f(a) = f(b)$. The definition of injective requires that if $f(a) = f(b)$, then $a = b$. More about that in dysfunctional.

## 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.

## This so-called proof assumes what you want to prove!

I will give a proof by induction of the theorem below. Proof by induction is explained in Discrete Mathematics, Chapter 103 starting on page 158.

For all positive integers $n$, the sum of the first $n$ odd positive integers is $n^2$.

This is plausible, because $1=1$, $1+3=4$, $1+3+5=9$, and so on.

Proof:

• This proof makes use of the fact that the $n$th odd positive integer is $2n-1$. For example, $1=2\times1-1$, $3=2\times2-1$, $5=2\times3-1$, and so on.
• The basis step: $1=1$, so the theorem is true for $n=1$.
• For the induction step, we need to show for each positive integer $n$, that if the sum of the first $n$ odd positive integers is $n^2$, then the sum of the first $n+1$ odd positive integers is $(n+1)^2$.
• The sum of the first $n+1$ odd positive integers is the sum of the first $n$ odd positive integers plus $2n+1$, because $2n+1$ is the $(n+1)$st odd positive integer.
• That sum is $n^2+2n+1$, which is $(n+1)^2$, which is what we wanted to prove.

The proof says, "if the sum of the first $n$ odd positive integers is $n^2$, then..." It is a hypothesis, not an assertion of truth.

The proof by induction proves the theorem because if there is an $n$ for which the theorem is false, there is a smallest $n$ for which it is false, and that means the theorem is true for $n-1$ and false for $n$. But we have proved that if it is true for $n-1$ then it is true for $n$, so there cannot be an $n$ for which the statement is false.

## 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 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 a 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. The Wikipedia article Four-dimensonal space discusses many aspects of the concept, including ways of visualizing such a space.

A five-dimensional space is also an abstract idea – each point has five coordinates – which can model many things, as discussed in the Wikipedia article Five-dimensional space.

Mathematicians and physicists study spaces of many dimensions, and when they do they gain (often with great difficulty) some intuition about them in spite of the fact that human beings have a built-in understanding only of spaces of dimensions one, two and three.

• In the article Interview with John Milnor, Milnor discusses how he learned to think about specific kinds of multidimensional spaces over his career.
• The mathematician Srinivasa Ramanujan was noted for having extraordinary intuition about infinite series, especially those involving integers, but he was not interested in and perhaps not capable of explaining how he thought about them.
• Chapter 5 of Timothy Gowers' book Mathematics: a very short introduction describes many aspects of higher-dimensional spaces quite clearly (but understanding them is still difficult!). I don't usually recommend books that are not free on the internet, but this books, like all the Very Short Introductions, is cheap, and I recommend it for beginners in abstract math.