## The King is Dead – Long Live the King

No, this isn’t a reference to Martin Luther King, though I could do far worse than refer you to the “Tell Him About The Dream, Martin” story about Mahalia Jackson.

Nor is it a reference to Barack Obama taking over from George W Bush, nor even a reference of Barack Obama taking over for Martin Luther King – though I could do worse than refer you to a two minute video from February 2007 where he brings a Martin Luther King quote that arch of history is long but bends towards justice forward.

Instead, I’m using the old phrase “The King is Dead – Long Live the King” to allude to a phenomenon where there is both continuity and serious discontinuity, depending on one’s perspective.  The old king is dead – and we may or may not know much about the new king.  But there is a king, and king-dom went on unabated.

What does this have to do with math?  I want to write a series – with this as the first installment – on math ideas we learn somewhere along the way, which we later unlearn.  I suggest this is normal and healthy.  The primary danger: whichever one we learn last, we think that that one is the real truth, the final truth.  The secondary danger: that we think that if we just were to teach the one we ourselves learned last, we make things better for the kids we teach.

So here’s the first installment:

“Time” is always the independent variable

This one is based on a notion that the horizontal axis in a graph is for the ‘independent variable’ – something that’s independent, hence not dependent on the other variable.  Since time progresses no matter what we do or don’t do, time is not dependent on the other variable, so is independent.

Yet later, Ms Johnson in Physics class may have the students drop objects from different heights, and measure how long it takes for the object to hit the ground.  (Ms Johnson, following Galileo’s lead, may instead use a ramp, and have objects roll down the ramp, in a way to slow down the fall so that it can be measured more easily.)  In this experiment, the height from which the object is dropped is the control variable, the time it takes to fall is the dependent variable.  Height is shown on the horizontal ais, time is shown on the vertical axis.  Gone is the relevance of the notion that time blithely proceeds on its own, and the importance is placed on how the experiment is organized.

Later on in secondary school,  you may learn about functions that have an inverse, e.g. square and square root, so that if $p = q^2$, then $q = \sqrt p$, and you can just as well regard p as depending on q as q depending on p, and either one can go on the horizontal axis of a graph.  (You get one graph from the other – in the upper right quadrant, anyway- by flipping it over the diagonal line p=q.)

Later still, professor Jablonsky may teach you the modern notion of a function as having nothing whatever to do with dependency at all.  A function, you learn, is simply a pairing of values (with some conditions placed on what pairs we allow – without those conditions, it wouldn’t be called a function, but a relation.)  So, if you write the pairs one way you get squares, if you write them the other way you get square roots:

$\begin{matrix} p & q \\\ 1 & 1 \\\ 2 & 4 \\\ 3 & 9 \\\ 4 & 16 \end{matrix} \quad \quad \quad \quad \begin{matrix} q & p \\\ 1 & 1 \\\ 4 & 2 \\\ 9 & 3 \\\ 16 & 4 \end{matrix}$

In a sense, professor Jablonsky would have you think that the table is the function and that the function is the table, taking away any notion that a function is like a piece of machinery.  Jablonsky may even have a fancy word for this approach: the extensional theory of functions.  This way of looking at functions isn’t all that old, historically speaking – just over a century.  As far as I know, the extensional theory is still the reigning king – though that may say more about me than about mathematics.

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### 3 Responses to The King is Dead – Long Live the King

1. Sam Krishna says:

Bert,

There’s actually a decent write-up of this, called “A Tour of the Calculus”. The history of functions is pretty fascinating, esp. in the context of Real Analysis and the development of the Algorithm during the rise of Computer Science.

Given how arbitrary functions can be, it’s quite fascinating how we’ve built so much machinery on them. Truly the stuff of imagination!

-S

2. Bert Speelpenning says:

Sam,