From Cakes, Custards and Category Theory by Eugenia Cheng:
The idea of maths is to look for similarities between things so that you only need one ‘recipe’ for many different situations. The key is that when you ignore some details, the situations become easier to understand, and you can fill in the variables later…
…once you’ve made the abstract ‘recipe’ you will find that you won’t be able to apply it to everything. But you are at least in a position to try, and sometimes surprising things turn out to work in the same recipe.
This connects with my earlier post on what the domain of the ‘for all’ is in the closure conditions – we are taking a rather structuralist view of causal theories (or model closure schema). That is, we are saying what the structure, expressed in terms of relationships between a collection of objects, of an idealised causal theory looks like without worrying too much (for now) about the nature of objects to be ‘filled in’.
Obviously more needs to be said on the crucial ideas of idealisation and approximation (though I’ve touched on these somewhat) and hence the process of slotting objects in. This is what I’d like to focus on next, hopefully, before further linking to some of the other causal literature.
Postscript
This idea of focusing on the essence of the recipe rather than the details of the objects is of course quite generally applicable (get it!) and, I feel, has a lot of pedagogical value. For example I recently read a nice article on improving the teaching of simple significance testing here. The author takes a quite similar ‘structuralist’ (in my view) and ‘abstract recipe’ perspective. Which is somewhat ironic since, without meaning to nitpick a nice article, claims
When statistics is taught by mathematicians, I can see the temptation. In mathematical terms, the differences between tests are the interesting part. This is where mathematicians show their chops, and it’s where they do the difficult and important job of inventing new recipes to cook reliable results from new ingredients in new situations. Users of statistics, though, would be happy to stipulate that mathematicians have been clever, and that we’re all grateful to them, so we can get onto the job of doing the statistics we need to do
Ironically, as argued above, a mathematician (or at least one who likes the ‘abstract nonsense’ of category theory) would probably prefer the view expressed earlier in the same article:
Every significance test works exactly the same way. We should teach this first, teach it often, and teach it loudly; but we don’t. Instead, we make a huge mistake: we whiz by it and begin teaching test after test, bombarding students with derivations of test statistics and distributions and paying more attention to differences among tests than to their crucial, underlying identity. No wonder students resent statistics.