Tuesday, September 29, 2009

empty promises

I promise that this blog won't always be about the empty set, but ...

Within the category of sets, Ø is what is called an initial object. That just means that for every set S there is exactly one "morphism" from Ø to S, in other words one function which associates to every member of the empty set a member of S. (The word "every" in this last phrase may take some getting used to.)

The empty set is also "initial" in a different sense. The reason why it took Russell and Whitehead hundreds of pages to get their foundations for mathematics developed to the point where they could prove that 1+1=2 was that they were defining "number", and some particular numbers, from scratch, or rather from set theory, and then defining "addition". They thought of the number 2 as, more or less, the set of all sets having exactly two elements; but in order to make sense of that idea in a non-circular fashion they had to actually define 2 to be the set of all sets of the same "size" as some particular set having just two elements. The set that they chose for this purpose was the set whose elements are the numbers 0 and 1. So they had to define 0 and 1 before defining 2. Of course, 1 was the set of sets of the same size as the set consisting of 0 alone. And of course 0 was the set of all sets having the same size as the empty set. Something like that. So the empty set was a starting point.

For me, part of the appeal of emptiness (and now I am not talking particularly about the empty set, but of empty vessels, blank minds, open-ended questions, clear skies... ) is the sense of beginning. So I'm glad that in category theory Ø is called an initial object.

On the other hand, a synonym for initial object is universally repelling object. So much for the found poetry of technical jargon.


Stuart said...

one function which associates to every member of the empty set a member of S. (The word "every" in this last phrase may take some getting used to.)

One way to avoid the mysteries of "every", at the cost of a bit more detail, might be:

A function from A to B is defined to be a subset of A X B (with certain additional properties which are not relevant here). In the case of Ø and S, Ø X S is just the empty set Ø. Ø is a subset of Ø, so Ø itself is the function from Ø to S that is meant here.

Some people can understand certain ways of explanation better than they can others. The goal of course is to move on to ways of understanding that let you push the old ones into the background. Quantors help mathematicians stop thinking about "every". Those who can't or won't stop thinking about "every" become philosophers of logic.

Stuart said...

empty, what have you changed in the settings that makes it so hard to add a comment here? Compare how easy it is to add a comment at my blog.

empty said...

As far as I know, all I have done with the settings is to select "Anyone" in answer to the question of who can comment. Is there something else I need to do, or can do, or should do?


Stuart said...

On the comment setting page, changing the value of "Comment form placement" to "pop-up window" produces the editing behavior I'm familiar with at blogspot.com.

With this setting, a new window pops up when you click on the "x comments" icon. However, you can use (in Firefox) the right mouse button to open the edit window in the same browser frame. I myself don't like too much stuff floating around on my computer desktop.

Your current value for "Comment form placement" is "embedded below post". The editor there doesn't behave properly. I can't copy text in it to work on elsewhere. I want to do this sometimes when I have started what I thought would be a brief comment which then gets longer and longer. For a new comment, I usually open a new file in a particular directory on my notebook for blog comments, and work in the file, saving it regularly. That way I don't lose anything if the browser freezes up.

Stuart said...

Your page layout is different from mine. It squeezes the text into a narrow vertical column. Perhaps you prefer that. If not, you might want to try "Minima Stretch", the template I use. You find the layout templates under "Layout" / "Pick new template".

Think of the vastness of the void. There's plenty of room there for wider blogs.

Arthur Rown said...

Empty space is the world's greatest luxury, Grumbly.

Stuart said...

Since I imagine architects pine for empty space, your remark would imply that architects pine for luxury. That would explain your insistence on equipping your Norwegian outhouse with comfy seating.

Or is it that architects fear the void, and so try to fill it with buildings?

empty said...

pop-up it is, then, and we'll try minima stretch, too

empty said...

It feels good to stretch out, but I might need to try out a few different layouts.

empty said...

Stuart: Thanks for the settings tips.

A function from A to B is defined to be a subset of A X B (with certain additional properties which are not relevant here). In the case of Ø and S, Ø X S is just the empty set Ø. Ø is a subset of Ø, so Ø itself is the function from Ø to S that is meant here.

Yeah, if I were really teaching this, I'd say it that way at some point. Certainly that's how you make a workable definition of "function" in terms of set theory. But first you have to define the product of two sets. So first you have to define ordered pairs. All of which Russell and Whitehead do ...

But your point was pedagogical, and I agree that an effort at communicating would be preferable to just saying "this 'every' takes some getting used to". However, how many people who are not used to this use of "every" will know what AxB means? And, here's the kicker: the "certain additional properties [which are not relevant here?]" are that for every element a of the (possibly empty) set A there exists one and only one element b of B such the pair (a,b) is a member, so aren't we still stuck in the same place?

Stuart said...

I might need to try out a few different layouts

On the "Layout" page, you can add widgets to the right of the page, and to the top and bottom. Then it looks like it used to - and like mine, like language hat's, Crown's, with the "Archive" list etc.

so aren't we still stuck in the same place?

I knew you were going to take me up on that. Of course you're right. My idea is to offer not only alternative explanations - so that students can see the different ways of thinking about things, and perhaps find one or two that are more congenial to them than the others. In addition, depending on the class, one wants to see which students can be weaned from natural language. Those will be the ones who may have mathematical talent, the ones who survive everyday-word deprivation.

But they all need to strengthen their ability to analyze things more stringently than usual. Some find more and more notation and concepts to be a help, others don't. A good instructor knows about all these issues, and pays attention to individual ability. I learned this leading Übungsgruppen for the first 6 semester courses at Bonn U, in the 70s.

It suddenly seems to me that I may appear to be lecturing you on teaching technique. No, I'm just rehearsing my own ideas and developing them a little.

Do you know R.L. Moore? I actually did a semester of topology with him, in the 60s when he was an old man, but sharp as a knife. In our proofs or explanations, he did not countenance any diagrams or references to theorems other than those which had been presented and proved in the course. That was a mathematical boot-camp I'll never forget, it was great.

empty said...

I must have learned something about teaching in all of these years, but it's not much.

I do believe in offering multiple points of view, multiple understandings, both because one might work better for one student, and because my own understanding of something may involve multiple points of view.

I envy you your experience of Moore. That boot-camp style of his was legendary.

Stuart said...

Take a look at Moores's Axioms 0 and 1 from the beginning of Foundations of Point-Set Theory. As I remember, there is not a word of motivation until much later, if at all - just proof after proof. It was only a few years ago that I finally found a copy of it. It's since been reprinted, apparently. It wasn't used in the course.

I've been trying to remember how I got into the course. I think someone primed me to play dumb. Also I was only 16 or 17 at the time, so perhaps Moore just assumed my mathematical virginity. In fact there was hardly anything still intacta about me then.

About Me

I am a professor of mathematics. (I began calling myself "Empty" or Ø when hanging around at blogs, because I am somewhat fixated on the empty set. Students and colleagues know that I can be a bit of an ancient mariner about it.)