Saw this tweet from Steven Strogatz a few days ago:

I watched it and mostly liked it. Actually loved almost all if it, but something was nagging at the back of my mind. Watched it again last night and then a few more times today and couldn’t resolve the conflict.

Part of what feels awkward to me is the idea that this is the use of the word “natural.” I understand why they are using the word, but it reminds me a little bit of two passages from Jordan Ellenberg’s *How not to be Wrong*

First, on page 78 in the section “More Pie than Plate” discussing the danger of using percentages when some numbers in your data might be negative:

“This may seem overly cautious. Negative numbers are numbers, and as such they can be multiplied and divided like any others. But even this is not as trivial as it first appears. To our mathematical predecessors, it wasn’t even clear that negative numbers were numbers at all – they do not, after all, represent quantities in exactly the same way that positive numbers do. I can have seven apples in my hand, but not negative seven. The great sixteenth-century algebraists, like Cardano and François Viète, argued furiously about whether a negative times a negative equaled a positive; or rather, they understood that consistency seemed to demand that this be so, but there was real division about whether this had been proved factual or was only a notational expedient. Cardano, when an equation he was studying had a negative number among its solutions, had the habit of calling the offending solution *ficta*, or fake. ”

Second, on page 200 in the section “Eet Ees Obvious” in discussing the idea that younger people should get charged more than older people for annuities is obvious:

“It is not obvious. Rather, it is obvious if you already know it, as modern people do. But the fact that people who administered annuities failed to make this observation, again and again, is proof that it’s not *actually* obvious. Mathematics is filled with ideas that seem obvious now – that negative quantities can be added and subtracted, that you can usefully represent points in a plan by a pair of numbers, that probabilities of uncertain events can be mathematically described and manipulated – but are in fact not obvious at all. If they were, they would not have arrived so late in the history of human thought.”

The confusion I had somehow also reminded me a little of the public lecture that Jacob Lurie gave when he won the Breakthrough Prize:

In the first 3 or so minutes of the lecture he gives some quotes from Descartes, Gauss, and Kronecker about complex numbers:

Descartes: Some numbers are “real,” but others (like ) are “imaginary”

Gauss: The true metaphysics of the square root of -1 is elusive.

Kronecker: God made the integers, all else is the work of man.

These ideas from Ellenberg and Lurie left me wondering if someone learning math would feel comfortable with the idea that the properties of the exponential function described in the first video are particularly natural.

Still, though, I really liked the video and tried to through it with my kids tonight. I enjoyed going through the ideas of “adders” and “multipliers” with the kids, and that felt like some really productive time, actually. Just for this part alone I’m happy we went through this exercise.

Trying to talk through the idea of the exponential function transforming adders into multipliers, though, proved to be difficult. Since this was all done on the fly I doubt that the explanation I provided the kids was even remotely the best possible one. But, even with that said, I still worry that part of the problem that novices will have with this video is that the ideas here aren’t quite as natural as the video makes them out to be. Similar to the examples that Ellenberg was talking about in his book, I think that the ideas only seem natural well after you learn them.

Here are the five parts of our conversation tonight – the first four are the ones about adders and multipliers:

Great discussion. I wonder if they made the connection of what happens to the complex plane when you multiply by -1.

I wonder, is the annuity point so obvious that it is actually false? Or, to put it differently, was there a point in which it made sound actuarial sense to charge more for an older person’s annuity than for a younger person? I can imagine certain times and places where the life expectancy curve has a very odd shape.

While the original video shows three interpretations of a number (point on the line, additive operator, multiplicative operator), there are actually two versions of the point on the line:

– in the domain of one of the operators

– in the range of one of the operators

I think this relates to the confusion your sons’ occasionally expressed around how the operators act. For example, does the x2 operator take 1 to 2 or 2 to 1? Well, it takes 1 in the domain to 2 in the range, but can easily be seen as taking 2 in the range to 1 in the domain.

I like the connection you make with Jordan’s chapter Eet Ees Obvious. Conflating 4 different interpretations like this probably seems very confusing to a beginner, then increasingly comfortable to the point of being obvious for a more experienced student. However, it can start to feel increasingly unclear again. In particular, how are the 4 different interpretations related? Are they still, in some sense, the same thing?

In any case, this is a powerful idea in modern algebra: taking an object in our field of study and using it to define an associated function.