Representations III: Rigor and Rigor Mortis

In a recent blog post , 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.

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4 thoughts on “Representations III: Rigor and Rigor Mortis”

  1. As an Adjunct Professor of Math, I told my 3-year-old at the dinner table: “If you don’t eat your dessert, you shall have no broccoli.”

    My wife, a Physics professor, likewise intentionally used the language of Science and Math in unconventional ways with our son.

    Result? The boy entered university at age 13, graduated with a double B.S. in Math and Computer Science at age 18, is published as a poet and mathematical physicist, and completed his first year at USC’s Gould School of Law at age 19, intending to focus on Intellectutal Property (Patents in software, for instance).

    Let me point out that the relationship between how Mathematicians, Scientists, and Poets use language can be particularly illuminating — especially when the author is at least two or all three of those classes.

    Poetry was at one time the language of philosophy, science, and all serious thought. Major treatments of Science expressed as Poetry included the works of
    Lucretius (especially De Rerum Natura), Parmenides of Elea, Archytas (Pythagorean
    general, statesman, philanthropist, educator) and Empedocles of Acragas, plus
    the “Phaenomena” of Aratus and the Latin “Astronomica” of Manilius.

    Ancient Greek MOUSIKE of Homer, Pindar, and Anaximander preceded prose culture,
    until Pythagoras identified music with mathematics, Aristotle distinguished
    Poetry from Rhetoric, and poetry began to separate from science.

    Aristotle’s Poetics [Translated by Thomas Twining, New York: Viking, 1957]
    states:

    “For even they who compose treatises of medicine or natural philosophy in verse are denominated poets: yet Homer and Empedocles have nothing in common except their metre; the former, therefore, justly merits the name of the poet; while the other should rather be called a physiologist than a poet”
    [I:2].

    Aristotle has begun the split between “high art” and mere science or science fiction in verse.

    See this web page on the history of Science Poetry:

    http://www.magicdragon.com/UltimateSF/sfpo-2pt1.html

    — Jonathan Vos Post

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