Colm Bhandal commented on my article on sets in abstractmath.org.

Let me first of all say that I am impressed with your website. It gave
me a few very good insights into set notation. Now, I’ll get straight
to the point. While reading your page, I came across a section
claiming that:

“Sets do not have to be homogeneous in any sense”

This confused me for a while, as I was of the opinion that all objects in a set were of the same type. After thinking about it for a while, I came to a conclusion:

A set defines a level of abstraction at which all objects are homogeneous, though they may not be so at other levels of abstraction.

Taking the example on your page, the set {PI^2, M, f, 42, -1/e^2} contains two irrational numbers, a matrix, a function, and a whole number. Thus, the elements are not homogeneous from one perspective (level of abstraction as I call it) in that they are spread across four known sets. However, in another sense they are homogeneous, in that they are all mathematical objects. Sure, this is a very high level of abstraction: A mathematical object could be a lot of things,
but it still allows every object in the set to be treated homogeneously i.e. as mathematical objects.

You are right.  I think I had better say “The elements of a set do not have to be ‘all of the same kind’ in the sense of that phrase in everyday speech.”  Of course, a mathematician would say the elements of a set S are “all of the same kind”, the “kind” being elements of S.
 
Apparently, according to the way our brains work, there are natural kinds and artificial kinds.  There is something going on in my students’ minds that cause them to be bothered by sets like that given about or even sets such as {1,3,5,6,7,9,11} (see the Handbook, page 279).   Philosophers talk about “natural kinds” but they seem to be referring to whether they exist in the world.  What I am talking about is a construct in our brain that makes “cat” a natural kind and “blue-eyed OR calico cat” an artificial kind.  Any teacher of abstract math knows that this construct exists and has to be overcome by talking about how sets can be arbitrary, functions can be arbitrary, and so on, and that’s OK.

 This distinction seems to be built into our brains.  A large part of abstractmath.org is devoted to pointing out the clashes between mathematical thinking and everyday thinking. 

Disclaimer:  When I say the distinction is “built into our brains” I am not claiming that it is or is not inborn; it may be a result of cultural conditioning. What seems most likely to me is that our brains are wired to think in terms of natural kinds, but culture may affect which kinds they learn.  Congnitive theorists have studied this; they call them “natural categories” and the study is part of prototype theory.  I seem to remember reading that they have some evidence that babies are born with the tendency to learn natural categories, but I don’t have a reference.

Send to Kindle