Searching for Lakatos

Should a textbook always give the whole truth?  Being sticklers for detail, as mathematicians generally are, the answer would seem to be an obvious “yes.”

In some cases it’s even difficult to imagine what the “partial truth” would be.

But in other cases it’s equally obvious that certain things would be easier to absorb if a few pesky details were forgotten (like a few hypotheses of a complicated theorem.)

For the purposes of discussion let me frame (probably in caricature) two opposite pedagogical philosophies:

P1:  The best way to teach a mathematical topic is to start with a concise definition of the object under discussion.  The technical definition may be bewildering at first, but after using it (and the accompanying theorem) students will become acclimatized, and feel comfortable with the concept.

P2:  The best way to teach a mathematical topic is to start with a picture or generic example of the situation under discussion.  Once the students have the basic idea, you can give them the accompanying technical jargon, which then won’t seem so bewildering. You can later point out possible exceptions to the generic case, if any exist.

I have used both of these strategies successfully as well as unsuccessfully. Usually I have to consciously employ the P1 strategy, as my natural impulse as a teacher is to show the students the “picture in my head” that’s relevant to the topic at hand.

I think it’s fair to characterize the P1 strategy as “modern” in the sense that there is a reduced emphasis on semantics and geometry, and a corresponding emphasis on formalism and deductive technique.  To frame things in terms of algebraic geometry, the formulas are what is “real,” and the corresponding geometric objects are “manifestations” of the syntax.  The geometric aspects are shadows of the equations, and may vary with context and point of view.

It is tempting to characterize the other approach, by contrast, as “romantic”: the geometric objects exist in the principal reality, and are of primary interest. They are mysterious or transcendental in character. The associated syntactical objects (i.e. equations) are a necessary but rather cold way of coming to grips with them, like a pipette or a strip of litmus paper. Moreover, the exact formal statement of what is being exhibited is not necessarily more valuable than a kind of non-linguistic geometrical insight.

A professional mathematician sees the symmetry between geometry and algebra, but we should be conscious that our students are getting a viewpoint that is heavily biased in favor of algebra.  We teach them to calculate numerically, and then to manipulate unknowns represented by letters.  A few years later, algebraic expressions suddenly become “functions”, and then these functions start emanating “graphs” like smoke from a choo choo train.

We do not, by contrast, start with a directrix and a focus, construct a parabola, and then exhort our students to find an equation that models the curve.  The parabola comes to $x^2$, not the other way around.

There is a definite efficiency to this approach.  But there is a downside:  it industrializes the production of geometric forms, and consequently cheapens them.  The hyperbola is no longer a sort of friend that steps recurrently in and out of the student’s life at different stages in their education. It is just one more of an enormous array of cheaply produced graphs, which are without inherent importance.

Be that as it may, there is another noteworthy difference between P1 and P2.  The P1 approach must capture the exact truth, whereas P2 may capture the “essence” of the truth.  I do not mean that the P2 approach is less mathematical.  I mean that when something is defined ostensively it is never completely clear what rule is being exhibited.

A natural example can be found in the introduction to Armstrong’s Basic Topology.  He presents a variety of polyhedra, and states the Descartes-Euler theorem, namely that the number of vertices (v) minus the number of edges (e) plus the number of faces (f) is always 2.  In other words, v-e+f = 2.  From the obviously large and diverse number of cases for which the formula is correct, it is immediately clear that a significant regularity has been discovered.  On the other hand, since the rule is given as an example, the ultimate class of objects to which it applies is not exactly clear.

To discover the precise truth, we must undergo a long dialectical process in which P1 style correlates of the theorem are proposed and rejected.  This in fact happened in the real history of Euler’s theorem.  This historical dialectic was translated into a dialogue in the book Proofs and Refutations, by Imre Lakatos (which appeared in print three years before Armstrong’s.)

Lakatos sees scientific (and mathematical) progress as a dialectical process of theory and counterexample.  He also advocated that this is the format in which mathematics should be taught.

We tend to “catch students up” on the dialectic to date by summarizing its history in complicated caveats.  The mean value theorem is true if $f$ is differentiable on $(a,b)$ and continuous on $[a,b]$.  How many students ever really come to understand the purpose of these hypotheses?  How long would it take to cajole the class toward a notion of a non-differentiable point on a curve?

Does the student who does not know that a curve can have problematic singularities eventually come to realize it?  I think probably so; critical points are not particularly exotic monkeywrenches.  Perhaps the material presented in class should be just as sophisticated as the students can make it, with the benefit of the instructor’s nettlesome questioning.

Which approach is a better reflection of science (including mathematics) as it actually occurs?  The P1 approach is purely retrospective, whereas P2 is used whenever a new phenomenon must be incorporated into an existing body of knowledge.  Assuming that the primary virtue of a higher mathematics class for a non-math major is to encourage the development of powers of abstract reasoning, is it not better to teach in the mode in which discovery actually takes place?

Of course some facts are important and should be clearly stated.  Future mathematicians in particular need much exposure to the P1 style, and they must be able to handle theorems. But for mathematics classes taught to scientists, it may be just as well to let ontogeny imitate phylogeny from a pedagogical viewpoint.

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6 Responses to Searching for Lakatos

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3. Pulau Sepa says:

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