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DYSFUNCTIONAL ATTITUDES AND BEHAVIORS
Contents
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Boundary values of definitions
I never would have thought of that
Ooh, they used a word I don't know
Reading variable names as labels
The representation is the object
Variable clash
Argument by analogy
When William Rowan Hamilton was trying to understand the new type of number called quaternions (MW, Wik) that he invented, he assumed by analogy that like other numbers, quaternion multiplication was commutative. It was a major revelation to him that they were not commutative.
Analogy
may suggest new theorems or ways of doing things. But it is fallible. What happens particularly often in abstract
math is applying a rule to a situation where it is not appropriate. This is an easy trap to fall into when the notation
in two different cases has
the same form that is an example of formal analogy.
Matrix multiplication
Matrix multiplication is not commutative
If r and s are real numbers then the products rs and sr are always the
same number; multiplication of real numbers is commutative: rs = sr. However, subtraction
is not
commutative. In general, .
The product of two matrices M and N is written MN, just as for numbers. But matrix multiplication is not commutative. For example,
Because rs = sr for numbers, the formal similarity of the notation suggests MN = NM, which is wrong.
This means you can’t blindly manipulate MNM to become 
. More generally, a
law such as 
is no longer correct.
The product of two nonzero matrices can be 0
If the product of two numbers is 0, then one or both of the numbers is zero. But that is not true for matrix multiplication:
Canceling sine
Beginning calculus students have already learned algebra.
¨ They have learned that an expression such as xy means x times y.
¨ They have learned to cancel like terms in a quotient, so that for example
¨ They have learned to write the value of a function f at the input x by f(x).
¨ They have seen people write sin x instead of sin(x) but never really thought about it.
¨ So they write 
(This happens fairly often in freshman calculus classes. But you wouldn’t do that, would you?)
Boundary values of definitions
Definitions of math concepts usually include the special cases they generalize.
Examples
¨ A square is a special case of rectangle, and as far as I know texts that define “rectangle” include squares in the definition. Thus a square is a rectangle.
¨ A straight line is a curve.
¨
An integer is a real number. (But not always in computing languages see here.)
But not always…
¨ The axioms of a field include a bunch of axioms that a one-element set satisfies, plus a special axiom that does nothing but exclude the one-element set. So a field has to have at least two elements, and that fact does not follow from the other axioms.
¨ Boolean algebras are often defined that way, too. (MathWorld gives definitions of Boolean algebras that differ with each other on this point. See the discussion about the Wikipedia entry here.)
Blunders
A definition that includes such special cases may be called inclusive; otherwise it is exclusive. People new to abstract math very commonly use words defined inclusively as if their definition was exclusive.
¨ They say things such as “That’s not a rectangle, it is a square!” and “Is that a group or a semigroup?”
¨
They object if you say “Consider
the complex number 
.”
This appears to be natural linguistic behavior. (find out what linguists and cognitive scientists know about this) Even so, such usage is technically incorrect.
Boundary values
These special cases may be regarded informally (and in many cases formally) as boundary values or extreme values. Definitions may or may not include other types of boundary values.
Examples
¨ If S is a set, it is a subset of itself. The empty set is also a subset of S.
¨ Similarly the divisors of 6 are 1, 2, 3 and 6, not just 2 and 3.
But…
¨ The positive real numbers include everything bigger than 0, but not 0. (Note).
Co-intimidator
¨ You attend a math lecture and the speaker starts talking about things you never heard of.
¨ Your fellow students babble at you about manifolds and tensors and you thought they were car parts and lamps.
¨ You suspect your professor is deliberately talking over your head to put you down.
¨ You suspect your friends are trying to make you believe they are much smarter than you are.
¨ You suspect your friends are smarter than you are.
¨ In short, you are intimidated.
There are two possibilities:
® They are not trying to intimidate you (most common).
® They are deliberately setting out to intimidate you with their arcane knowledge so you will know what a worm you are. (There are people like that.)
Another possibility, which can overlap with the two above, is:
® You expect to be intimidated. You may be what might be called a co-intimidator. (Like someone who is codependent wants other people to be dependent on them.)
There are many ways to get around being intimidated.
¨ Ask “What the heck is a manifold?”
¨ (In a lecture where it might be imprudent or impractical to ask) Write down what they say then later ask a friend or look it up.
¨ Most teachers like to be asked to explain something. Yes, I know some professors repeatedly put down people. Change sections! If you can’t, live with it! Don’t think it says anything about you.
¨ Remember:
If you don’t know something
probably many other students don’t know it either.
(Surveys show that remarkably often when one student doesn’t know something most of the others don’t know it either.)
Element is a property
THIS IS A MYTH: There are two kinds of mathematical objects: "sets" and "elements".
This is the truth: Being an element is not a property that some math objects have and others don’t.
Any mathematical object can be an element of a set
In particular, any set can be the element of another set. More about that here.
Enthymeme
Existential bigamy
Beginning abstract math students sometimes make a particular type of mistake that occurs in connection with a property P of an mathematical object x that is defined by requiring the existence of an item y with a certain relationship to x. When students have a proof that assumes that there are two items x and x' with property P, they sometimes assume that the same y serves for both of them. This mistake is called existential bigamy: The fact that Muriel and Bertha are both married (there is a person to whom Muriel is married and there is a person to whom Bertha is married) doesn't mean they are married to the same person.
Example
Let
m and n be integers. By definition, m divides n if there
is an integer q such that 
. Suppose you are
asked to prove that if m divides both
n and p, then m divides 
. If you begin the proof by saying, "Let n = qm and p = qm... " then you are committing existential
bigamy. You need to
begin the proof this way: “Let n = qm and p = q’m…”
Extrapolate
Every number is an integer
False symmetry
Formal analogy
I never would have thought of that
Jump the fence
If you
are working with an expression whose variables are constrained to certain values, and you substitute a
value in the expression that violates the constraint, you jump the fence
(also called a fencepost error).
Example
The Fibonacci numbers (MW, Wi) are usually defined inductively like this:
In
calculating a sum of Fibonacci numbers, you might write
This
contains errors: the sums on the right involve 
and 
,which are not defined by the definition above. You could add
to the definition to get around
this, or keep better track of the fence by writing
(For
the “(n > 1)” notation see here.)
Literalism
Definitional literalism
Every type of math object has to have a definition. In giving a definition, a few of the many
ingredients that are involved in that type of object are selected as a basis
for the definition. They are not necessarily the
most important parts. People who make definitions try to use as
little as possible in the definition so that it is easier to verify that
something is an example of the thing being defined.
A
literalist is
someone who insists on thinking about a type of math object primarily in terms of
what the definition says it is.
Definitional literalism inhibits your understanding of abstract
math.
Example
One of the major tools in the study of the foundations of mathematics is to try to define all mathematical objects in terms of as few as possible objects. The most common form this takes is to define everything in terms of sets. For example, the ordered pair (a, b) can be defined to be the set {{a}, {a, b}}. (See Wi).
A literalist will conclude that the ordered pair (a, b) is the set {{a}, {a, b}}.
That is not how we should think about ordered pairs. What is important about an ordered pair is that it has
a first coordinate and a second coordinate and what those two coordinates are
completely determine the ordered pair.
It is ludicrous to say something like 
. The definition that
(a, b) is the set {{a}, {a, b}} is done purely for the purpose of
showing that the study of ordered pairs can be reduced to the study of sets. It is not a fact about ordered pairs that we can use.
A
similar example is the definition of the number 2 as 
. Would you ever want
to know that 
?
Example
An
equivalence relation on a set S is a relation on S with certain
properties. A partition on S is a set of
subsets with certain properties. The two definitions can be
proven to give the same structure (that is done here). I have heard
literalists say, “How can they give the same structure? One is a relation and one is a
partition.” This is definitional
literalism. It skips the important part and
keeps the details.
Example
The
(less strict) definition
of function says that a function is a set of ordered pairs with
the functional
property.
This
does not mean that if your function is F(x) = 2 x + 1, then you
would say 
. Common practice is to say F(3) = 7 or “the
value of F at 3 is 7” or something of the sort. However, I do know mathematicians who tell me
that they really
do think of a
function as a set of ordered pairs and would indeed say . That is why I referred to “common practice” at the
beginning of this paragraph instead of “normal practice”!
Antimetaphorism
Many years ago I had a math professor who hated it with a
purple passion if anyone said a function vanishes at some number a. He would say things like, “The function 
‘vanishes at 1’ Pah!
The function is still
there isn’t
it?”
He was an antimetaphorist, which is
a kind of
literalist. Metaphors
are one of our primary ways of thinking about things, especially abstract
things. There is really nothing wrong
with using metaphors to describe math objects as long as you remember that
metaphors fit in
some ways and don’t fit in others, and they don’t belong in proofs. See Images
and Metaphors.
Method addiction
Beginners at abstract math
sometimes have the attitudes that a problem must be solved or a proof constructed by a specific procedure. They
become quite uncomfortable when faced with problem solutions that involve guessing or conceptual proofs that involve little or no calculation.
Example
Once I gave a problem in my Theoretical Computer Science class
that in order to solve it required finding the largest integer n for which n! <
Trial
and error is a perfectly valid method.
Example
Students at a more advanced level may feel insecure in the case
where they are faced with solving a problem for which they know there is no
known feasible algorithm, a situation that occurs mostly in senior and graduate level
classes. For example, there are no known feasible general algorithms for determining if two finite groups given by their multiplication tables are isomorphic, and there is
no algorithm at
all to determine if two
presentations (generators and relations) give the same group. Even so, the
question, "Are the dihedral group of order
See also look ahead and conceptual.
Myths
Ooh, they used a word I don't know
Proof by Example
Definition:
An integer is even if
it is divisible by
Theorem: Prove that if n is an even integer then so is
.
This is proved by universal generalization.
One type of mistake made by beginniers for
proofs like this would be the following:
“Proof: Let n = 8.
Then 
and
This violates the requirement of universal
generalization that you have “made no restrictions on c” you have restricted it to being a particular even integer!
But I doubt
that people who make this kind of mistake don’t understand universal generalization. Instead, I believe the mistake is caused by
misreading the phrase “An integer is even if…” to read that you can prove the
statement by picking an integer and showing that it is true for that integer. But in fact, “an”
in a statement like this means “any”. See
indefinite
article.
Reading variable names as labels
Say it in your own words 
See also unique.
The representation is the object
Unique
By definition, a set R of ordered
pairs has the functional
property if
two pairs in R with the same first coordinate have to have the same second
coordinate
It
is wrong to rephrase the definition this way: “The first
coordinate determines a unique second coordinate.”
That use of “unique” is
ambiguous. It could mean the set {(1,2), (2,4), (3,2),
(5,8)} does not have the functional property because the first
coordinate in (1,2) determines 2 and the first coordinate in (3,2) determines
2, so it is “not unique”. This statement is wrong. The set does
have the functional property. A related
error is to reword the definition of injective
by saying” “For each input there is a unique output.” It is easy to read this and think injectivity
is merely the functional property.
It
seemed to me that during the 35 years I taught calculus and discrete math,
students fell into this trap about
100,000 times. Of course, this could be
a slight exaggeration.
Avoid
rewording any definition that does not use the word unique
so
that it does use the word unique.
Such activity fries your brain and turns A’s into B’s.
Unnecessarily weak assertion
Examples
¨
The statement "Either x > 0 or x < 2"
is true (for real numbers). Yes, you
could make a stronger statement, for example “Either 
or x > 0” . But the statement "Either x > 0 or x < 2" is still
true.
¨
Some students have problems with the true
statements "
" and with "
" for a similar reason, since in fact 2 = 2 and 2 <
3.
¨ You may get a twinge if someone says “Many primes are odd”, since in fact there is only one that is not odd. But it is still true that many primes are odd.
An unnecessarily weak assertion may occur in math texts because it is the form your proof gives you, or it is the form you need for a proof. In the latter case you may feel you are walking blindfolded (below).
There is another example here.
It is not wrong for an author to make an unnecessarily weak assertion.
Read ahead maybe you will find out why it is in that
form.
(And maybe you won’t.)
Variable clash
Walking blindfolded
Sometimes when you
are reading or listening to a proof you will find yourself following each step
but with no idea why
these steps are
going to give a proof. This can happen
with the whole structure of the proof or with the sudden appearance of a step
that seems like the prover pulled a rabbit out of
a hat. You feel as if you are walking
blindfolded.
Example (mysterious proof structure)
The lecturer
says he will prove that for an integer n, if is even then n is even. He begins the proof: Let n be
odd” and then continues to the conclusion, “Therefore n is odd.”
WHY did he begin a proof about 
being even with the
assumption that n is odd??
The answer is that in this case he is doing
a proof by contrapositive. If you don’t recognize the pattern of the
proof you may be totally lost. This can
happen if you don’t recognize other forms, for example contradiction and induction.
Example (rabbit out of hat)
You are reading a proof that 
. It is an 
proof, so what must be proved is:
(*) For any positive real number 
,
there is a positive real number 
for which:
If 
then 
.
Proof
Here is the proof, with what I imagine might be your agitated reaction to certain
steps. Below is a proof with detailed explanations.
1)
Suppose 
is given.
2)
Let 
(the minimum of the
two numbers 1 and 
). Where the *!#@! did THAT come from? They pulled it out of thin air! I can’t see where we are going with this
proof! I feel like I’m walking
blindfolded!
3)
Suppose .
4)
Then 
by (2) and (3).
5) So 
by algebra. Well, so what?
We know that 
and lots of other
things, too? Why did they do THIS?? How do I know I am not about to fall off a
cliff??
6)
Also 
by (2).
7)
Then 
by (5) and (6).
This proof is typical of proofs in texts, except that it is unusually
detailed. Steps 2) and 5) look like they
were rabbits
pulled out of a hat; the author gives no explanation
of where they came from.
Nevertheless, each step of the proof follows from previous steps, so the proof
is correct.
In order to understand a proof,
you do not have to know where the rabbits came from!
Proof with detailed explanations
1)
Suppose is given. We
are starting a proof by universal generalization.
2)
Let (the minimum of the
two numbers 1 and ). Rabbit out of the hat. You can let any symbol mean anything you
want, so this is a legitimate thing to do even if you don’t see where this is
all going.
3) Suppose . We are about to prove the conditional
statement “If 
then 
” and we are proceeding by the direct method.
4)
Then by (2) and (3). The
fact that means that 
and that 
. Since , the statement follows by
transitivity of “<”.
5) So by algebra. means that 
. Add 4 to each term
in this equation to get 
. This is another
rabbit, but it is a correct statement!
6)
Also by (2). Same reasoning as in (4).
7) Then by (5) and (6).
Reading
In general, the author did not think up the
proof steps in the order they occur in the proof. (See this
remark in the section on Forms of Proofs.)
When you are reading a proof and some step is surprising,
remember:
Whether you are surprised or not has nothing to do with whether
it is correct.
If you continue to plow through the proof you may
eventually discover why the surprising steps are the way they are.
In the proof
above the author looked
ahead at the goal of proving that and thought of factoring the left side. Now she must prove that 
. But if x 2 is small then x has to be close
to 2, so that x + 2 can’t be too big. Since the
only restriction on
is that it has to be positive,
let’s restrict it to being smaller than 1.
(The choice of 1 is purely arbitrary.
Any positive real number would do.)
In that case step (5) shows that . So how
small do you have to make to make ? In other words, how small do you
have to make to make 
(remembering that ). Well, clearly will do!
You can check that if she had chosen to restrict to being less than 42, then we would need 
.
See also look
ahead.
Acknowledgments
Thanks to Robert Burns for corrections and suggestions.