Mathematical historical fiction

Bill Gasarch is right – writing technical posts is tiring! (I’ve been trying to finish the next GILA post for days.) So I’ll share some more thoughts instead. Today’s thought was triggered by David Corfield:

In the first of the above posts I mention Leo Corry’s idea that professional historians of mathematics now write a style of history very different from older styles, and those employed by mathematicians themselves. …

To my mind a key difference is the historians’ emphasis in their histories that things could have turned out very differently [emphasis mine], while the mathematicians tend to tell a story where we learn how the present has emerged out of the past, giving the impression that things were always going to turn out not very dissimilarly to the way they have, even if in retrospect the course was quite tortuous.

This in turn reminded me of something else Rota wrote about his Walter Mitty fantasies:

Let us begin with a piece of history-fiction, and imagine how Riemann might have discovered the Riemann zeta function.

Professor Riemann was aware that arithmetic density is of fundamental importance in number theory. If Image may be NSFW.
Clik here to view. is a subset of the positive integers Image may be NSFW.
Clik here to view., then the arithmetic density of the set Image may be NSFW.
Clik here to view. is defined to be

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Clik here to view.

whenever the limit exists. For example, Image may be NSFW.
Clik here to view.. If Image may be NSFW.
Clik here to view. is the set of multiples of the prime Image may be NSFW.
Clik here to view., then Image may be NSFW.
Clik here to view.; what is more appealing is if one easily computes that Image may be NSFW.
Clik here to view. for any two primes Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view.. If density were a (countably additive) probability measure, we would infer that the events that a randomly chosen number is divisible by either of two primes are independent [emphasis mine]. Unfortunately, arithmetic density shares some but not all properties of a probability measure. It is most emphatically not countably additive.

After a period of soul-searching, Professor Riemann was able to find a remedy to some deficiencies of arithmetic density by a brilliant leap of imagination. He chose a real number Image may be NSFW.
Clik here to view. and defined the measure of a positive integer Image may be NSFW.
Clik here to view. to equal Image may be NSFW.
Clik here to view.; in this way, the measure of the set Image may be NSFW.
Clik here to view. turned out to equal

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Clik here to view..

Therefore, he could define a (countably additive) probability measure Image may be NSFW.
Clik here to view. on the set Image may be NSFW.
Clik here to view. of positive integers by setting

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Clik here to view..

Riemann then proceeded to verify what he had sensed all along, namely the fundamental property

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Clik here to view..

In other words, the events Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. that a randomly chosen integer Image may be NSFW.
Clik here to view. be divisible by one of the two primes Image may be NSFW.
Clik here to view. or Image may be NSFW.
Clik here to view. are independent relative to the probability Image may be NSFW.
Clik here to view..

The Riemann zeta function was good for something after all.

…

Long after Riemann was gone, it was shown, again subject to technical assumptions, that the probabilities Image may be NSFW.
Clik here to view. are the only probabilities defined on the set Image may be NSFW.
Clik here to view. of natural integers for which the events of divisibility by different primes are independent. This fact seems to lend support to the program of proving results of number theory by probabilistic methods based on the Riemann zeta function.

Rota doesn’t quote any sources, but this is a brilliant piece of insight which Rota goes on to dwarf with an even better one. But I want to use this example because I want to see more writing like this, and I’ll tell you why.

I don’t believe that any particular development in mathematical history was inevitable. I don’t believe that either the mathematical history or the axiomatic presentation of a subject are the only things to say about its origins. As the mathematical community takes a new idea, such as category theory, and begins to apply it to fields other than the field in which it originated, we gain hindsight. The problem with focusing on axioms, as I have said, is that an axiomatic treatment of a subject tends to ignore that this process of gaining hindsight ever happened. It also lends to a view of mathematics as “God-given,” as opposed to being driven strongly by human concerns and human modes of understanding. On the other hand, abstraction and generality are our best tools for compressing mathematical insight, and often it’s not clear what the clearest or most general way to study a particular phenomenon is until decades after the first attempt.

With these concerns in mind, I want to propose a way of introducing certain concepts or subjects based on historical fiction. Instead of axioms or the actual historical progression of a concept, perhaps it would be instructive to invent a fictional history fueled by modern hindsight. For example, identify a mathematician who was in a position to discover a concept but lacked a key insight, and imagine, as Rota did, that the insight was available. For simpler concepts, it might suffice to tell a story with the reader as protagonist about how one might reasonably go about writing down the axioms of, say, a group.

Perhaps I’m just asking for people to motivate things better in general. But does anyone know of any textbooks or expository articles employing primarily this strategy? I would love to see a textbook begin every chapter with a story.