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.