What are some conceptualizations that work in mathematics but are not strictly true?

I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to iteratively learn and unlearn along the way.

I'm looking for examples in mathematics (and possibly physics) where students are commonly taught something that's not strictly true, but works, at least in some restricted manner, and is a good way to understand a concept until one gets to a more advanced stage.

Solution 1:

How about this notorious one I remember from high school?

"$f(x)$ is just a fancy name for $y$."

Solution 2:

Thinking of Dirac delta function as a function works reasonably well up to a certain point. For example, every physicist knows that $$\int_{-\infty}^{\infty}e^{i\omega x}dx=2\pi\cdot\delta(\omega),$$ but only a small part of them really studied the theory of distributions.

Solution 3:

Here are two false concepts taught in lower grades (explicitly or implicitly):

1) Every plane figure has an area. Elementary kids are not told that some figures have no area (or might not have an area, if you leave out the Axiom of Choice).

2) A set is a collection of objects satisfying any particular relation. (Some middle-school books avoid this by always discussing sets within some particular universal set, but this approach just caused me to raise questions even before I studied formal set theory.)

Solution 4:

Multiplication is repeated addition. So $\sqrt{2}\cdot \sqrt{8}=4$. Similarly division is repeated subtraction.
Solve the differential equation $$\dfrac{dy}{dx}=y$$ Easy! Multiply both sides by $dx$, so $\dfrac{dy}{y}=dx$; now, integrating both sides, $\ln{y}=x+C$. I don't know how many people understand this multiplication by $dx$ (frankly, I don't). Even university students (at my university) take this step for granted. (Relevant: Is $\frac{dy}{dx}$ not a ratio?)
The cancellation trick: $\require{cancel}x\cdot y= y\cdot z\implies x\cdot\cancel{y}=\cancel{y}\cdot z\implies x=z$. People use this 'trick' to prove $1=2$ etc.
High school teacher: Square root of negative number is not defined. So $x^2+1=0$ has no roots.

Secondary school teacher: define $\sqrt{-1}=i$ and $x^2+1=0$ has two roots $x=-i,+i$.
"$1/0$ is $\infty$" Refer to point 5 here, this is correct.
$\dfrac{\partial^2 f(x,y)}{\partial y \, \partial x}=\dfrac{\partial^2 f(x,y)}{\partial x \, \partial y}$. This can't be false in Electrodynamics.
$1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+\cdots =2$. The equality is little confusing. It should rather be considered as a limit of the sum (or limit of partial sums).
$\sqrt{mn}=\sqrt{m}\cdot\sqrt{n}$. This is only true when at least one of $m$ and $n$ is positive. This 'trick' can be used to prove $1=\sqrt{1\cdot 1}=\sqrt{-1 \cdot -1}=\sqrt{-1}\cdot\sqrt{-1}=i^2=-1$.

Solution 5:

We tell students in an introductory algebra class, say, that mathematicians invent axioms and then study the properties of the resulting object or theory. The story goes that some bright mathematician had the idea to write down the four group axioms before anyone knew anything about group theory, and then he and other mathematicians worked out all the known properties of groups from those four axioms.

In reality, it took mathematicians decades of working with concrete examples of groups -- permutation groups and groups arising from geometric symmetries -- to finally notice that all of those examples shared a few fundamental properties. These properties then became the axioms for the group, but only because the objects satisfying those axioms had already appeared naturally in other fields of mathematics.

I think it is very rare for a mathematician to define some random set of axioms and then have something interesting to say about them. Instead, axiomatizing a theory is usually one of the final steps in making that theory rigorous. Axioms are more a sign of rigor mortis than of the potential for new results. Moreover, one theory -- most notably set theory -- can have competing axiomatizations which are all studied.

And yet, I think it is pedagogically useful to introduce students to the idea of an axiom in this way. A student who has studied no formal mathematics will not yet appreciate the subtle interactions between hypotheses, conclusions, and observations that govern mathematical progress. It is easier to give him or her a concrete set of "rules" to work with, and the cleanness of the ensuing theory -- by which I mean that steady march of (possibly unmotivated) definitions, lemmas, and proofs across the pages of math textbooks -- shows him or her our ideal for formal mathematics.

Edit: I realized that what I have written so far can be interpreted to mean something with which I vehemently disagree. Even though I think expositions of mathematics at the level of a first- or second-year undergraduate can benefit from an axiomatic treatment, any expositions that are more advanced -- certainly graduate textbooks, for instance -- should complement the rigid axiomatic method with thorough motivation for all definitions. The motivation is as important, if not more important, than the definitions and theorems!