Produced by Charles Wells Revised 2017-04-08 Introduction to this website website TOC website index blog

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

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

about how to calculate it.

**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.

See also Derivatives.

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.

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. - See also Concept and computation.

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.

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

A straight line is a math object and therefore **abstract**,

although it may have physical representations

that approximate how you think about 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

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.

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.

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.

This is about how you should think about the decimal expansion. 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.

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

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.

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

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 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.

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.

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.)

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.

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