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Topic: Rigor requires rigor mortis for math objects
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Charles Wells

Posts: 42
From: Oberlin, Ohio
Registered: 7/28/06
Rigor requires rigor mortis for math objects
Posted: May 1, 2008 11:13 AM
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In a recent blog post at
http://gyregimble.blogspot.com/2008/04/representations-ii-dry-bones.html
I talked about the particular mental representation ("dry bones") of
math that we use when we are being "rigorous" ? we think of
mathematical objects as inert, not changing and affecting nothing.
There is a reason why we use this representation, and I didn't say
anything about that.

Rigor requires that we use classical logical reasoning: The logical
connectives, implication in particular, are defined by truth tables.
They have no temporal or causal connotations. That is not like
everyday reasoning about things that affect each other and change over
time. (See Note 1),

Example: "A smooth function that is increasing at x = a and
decreasing at x = b has to turn around at some point m between a and
b. Being smooth, its derivative must be 0 at m and its second
derivative must be negative near m since the slope changes from
positive to negative, so m must occur at a maximum". This is a
convincing intuitive argument that depends on our understanding of
smooth functions, but it would not be called "rigorous" by many of us.
If someone demands a complete rigorous proof we probably start
arguing with epsilons and deltas, and our arguments will be about the
function and its values and derivatives as static objects, each
thought of as an unchanging whole mathematical object just sitting
there for our inspection. That is the dry-bones representation.

In other words, we use the dry bones representation to make classical
first order logic correct, in the sense that classical reasoning about
the statements we make become sound, as they are obviously not in
everyday reasoning.

This point may have implications for mathematical education at the
level where we teach proofs. Perhaps we should be open with students
about images and metaphors, about how they suggest applications and
suggest what may be true, but they have to "go dead" when we set out
to prove something rigorously. We have been doing exactly that at the
blackboard in front of our students, but we rarely point it out
explicitly. It is not automatically the case that this explicit
approach will turn out to help very many students, but it is worth
investigating. (See Note 2).

It may also have implications for the philosophy of math.

Note 1: The statement "If you eat all your dinner you can have
dessert" does not fit the truth table for classical (material)
implication in ordinary discourse, where it means: "You can't have
dessert until you eat your dinner". Not only is there a temporal
element here, but there is a causal element which makes the statement
false if the hypothesis and conclusion are both false. Some
philosophers say that implication in English has classical implication
as its primary meaning, but idiomatic usage modifies it according to
context. I find that hard to believe. I don't believe any
translation is going on in your head when you hear that sentence: you
get its nonclassical meaning immediately and directly with no thought
of the classical vacuous-implication idea.

Note 2: I used to think that being explicit about the semiotic aspects
of various situations that take place in the classroom could only help
students, but in fact it appears to scare some of them. "I can't
listen to what you say AND keep in mind the subject matter AND keep in
mind rules about the differences in syntax and semantics in
mathematical discourse AND keep in mind that the impersonality of the
discourse may trigger alienation in my soul AND?" This needs
investigation.

Charles Wells
professional website: http://www.cwru.edu/artsci/math/wells/home.html
blog: http://www.gyregimble.blogspot.com/
abstract math website: http://www.abstractmath.org/MM//MMIntro.htm
personal website: http://www.abstractmath.org/Personal/index.html



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