Posted by: Alexandre Borovik | June 17, 2018

## Abstraction is multiple realisability

A brilliant blogpost from   Double-entry bookkeeping and Galileo: abstraction vs idealization.  He gives a definition of abstraction, as used in computer science:

abstraction is multiple realisability.

I am currently designing a first year, first semester course, something like Introduction to Algebra. One part of it will be about integers and polynomials:

• start with division with remainder of integers and polynomials
• develop, in parallel, divisibility theory and uniqueness of  prime factorisation – at every step  stating two theorems, one for integers, another for polynomials, but proving only one of them and leaving writing up a proof of the other as an exercise for students;
• conclude this part of the course by proving the Chinese Reminder Theorem for integers and Lagrange’s Interpolation Formula for polynomials and explaining why this is one and the same theorem;
• perhaps only then introduce rings, fields, homomorphisms, and Euclidean rings;
• and give one more exercise to students: uniqueness of prime factorisation of Gaussian integers.

I think that, at start of undergraduate mathematics, an abstract concept should be introduced only after students have well familiarised themselves with its several realisations.

## Responses

1. From my point of view, the magic of mathematics is that one phenomena can be described with many different mathematical models and vice versa the same model can describe many very different phenomenas.
Good luck with your project!

2. Thank you for the kind words! I didn’t argue too strongly for the abstraction as multiple realizability in the post you linked, but kind of took it for granted. This was because I argued this two weeks prior in abstract is not the opposite of empirical.

I really like your teaching plan. This process of introducing the concrete and letting the students infer the abstract by working through another concrete example that implements the same abstraction is a really good approach. In the past, I’ve heard of it referred to as the Russian method. It is usually held in contrast to the formal Bourbaki approach that starts with abstract definitions. In fact, I think the Russian method got its name because of V.I. Arnol’d’s attacks on Bourbaki. I’ve been meaning to write about this for a while, but it has languished in the half-finished drafts pile.

• Yes, I red your previous post — but it focuses more on computer science. But Luca Pacioli is one of the unsung heroes of mathematics.

I belong to a peculiar cohort of mathematics students at the Novosibirsk University who were given, at an Year 1 course of analysis, definition of limit as limit of filter. Our lecturer, GP Akilov, was a Bourbakist. I easily survived because prior to that, at PhMSh, I took a special course based on a classical memoire by Baire, “Theory of discontinuous functions”, one of the founding works of what became the theory of functions of real variables (translated to Russian by Khinchin in 1932 from the French original of 1904). So my own education was an interesting example of interaction of concrete and abstract. I learned some lessons from that experience and apply them in my own teaching.

Luca Pacioli is one of the unsung heroes of mathematics.

Please finish your paper on the Russian method. I did something like that, I do not regret my time spent on writing: bit.ly/293orpk . And it is obvious that you have a lot say.