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.
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 is a subset of the positive integers , then the arithmetic density of the set is defined to be
whenever the limit exists. For example, . If is the set of multiples of the prime , then ; what is more appealing is if one easily computes that for any two primes and . 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 and defined the measure of a positive integer to equal ; in this way, the measure of the set turned out to equal
Therefore, he could define a (countably additive) probability measure on the set of positive integers by setting
Riemann then proceeded to verify what he had sensed all along, namely the fundamental property
In other words, the events and that a randomly chosen integer be divisible by one of the two primes or are independent relative to the probability .
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 are the only probabilities defined on the set 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.