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Last edited 4/10/2008 11:10:00 AM
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
Argument by analogy
An argument by analogy is the claim that because of
the similarity between certain parts there must also be a similarity between some other parts. Analogy is a powerful tool that suggests further similarities; to use it to argue for further
similarities is a fallacy. People new to
abstract math may fall into this trap, usually subconsciously.
Example
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.
If you are new to matrix multiplication, you
might fall into this trap. The product of
two matrices M and N is written MN, just like multiplication of numbers. However, the matrix product need not be commutative (MN might not equal NM), although it is
associative. This means you can’t
blindly manipulate MNM to become . More generally, a law such as
is no longer correct.
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 nearly all the others don’t know it either.) Need reference
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:
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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
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to the definition to get around this, or keep better track
of the fence by writing
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(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.
Literalism is a big bad MISTAKE
that 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}}.