Mathematical structures, 2

The second week of the course has been and gone. We started on the real business this week, with a chapter about sets. Although I didn’t quite finish this rather long chapter in a week (the next one will be shorter), I have posted the notes anyway.

The students continue to be very engaged with the course. In fact, they were so engaged in the first lecture that I departed from my plans and told them about Bertrand Russell and the set of all sets which are not members of themselves. This caused a certain amount of head-scratching, and several students came to see me afterwards to discuss it.

I told them in week 1 not to take anyone else’s version of things. This week I said that mathematicians did find a work-around for this horrendous paradox, but the work-around is a bit too complicated for me to explain it to them. One student wanted to know if she would meet it in the second year. Sad to say, since we cancelled our course on “Sets, Logic and Categories”, the students don’t meet axiomatic set theory anywhere in our study programmes! But do I feel hypocritical for making these two statements? Not really; this is the way mathematics (and every other subject) works, and the students have to think it out for themselves.

Indeed, when I was a student, I was told about the Jordan curve theorem on a number of occasions, and always either fobbed off with a proof of a special case, or told that I would see the proof later on. I have never seen a general proof of this theorem. Life is too short for full-blown idealism.

The tutors mostly report that the tutorials are going well and the students are engaging with the material, being prepared to explain their ideas where necessary. Inevitably, not all of my colleagues are entirely happy with this; but the majority seem to be making it work. Comments include this:

Good engagement with the group in the tutorial. But reluctance to get involved – “can’t see how to write it down for case n” when they argued it perfectly for n=4 … I had 3 of the 6 stand up and write on the board which wasn’t bad.

This raises an interesting point. Some people would say that only the general case counts as real mathematics, while n=4 is just a puzzle. (Perhaps G. H. Hardy would agree with this; in A Mathematician’s Apology, he dismisses chess problems as being not real mathematics.) Others argue that a special case which shows all the features of the general case is mathematics, and writing out a proof with n rather than a particular number like 4 is an additional skill which it is our job to teach, but which was not always deemed necessary. Thus, R. J. Gillings in Mathematics in the Time of the Pharaohs, discussing the Egyptian solution to the quadratic, says,

A non-symbolic argument or proof can be quite rigorous when given for a particular value of the variable; the conditions for rigor are that the particular value of the variable should be typical, and that a further generalization to any value should be immediate.

Two students have already submitted solutions to the first project, writing an account of what mathematics is. Another student challenged me with a problem. I decided to give it a twist and throw it back at the students. If you are really curious, you can find it on the web page; I will reveal all here when the students have had a go at it, so please hold your fire until then.

As I was entering the lecture room this morning, a student came up to me and asked if I had been on the BBC programme about infinity. I told him that I would be talking about the same stuff in that room next week. But he was an engineering student leaving the preceding lecture, and wouldn’t be there for that.

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About Peter Cameron

I count all the things that need to be counted.
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2 Responses to Mathematical structures, 2

  1. Bertrand Russell’s ‘set of all sets which are not members of themselves’ I always find a bit of a mind bender.

    One trick I use, which is conceptual rather than mathematical, is to consider a description of ‘sets which are not members of themselves’ as a set of empty surfaces or boundaries, or closed dimensions. So we have a set of attributised empty sets. A set of any type has a boundary of quality or attribute, or dimensionality, which describes the ‘contents’. So Russell’s ‘set of all sets which are not members of themselves’ can be considered more of a dimensional description of attributised empty sets, and viewed from the ‘outside’ as it were – a container of all empty containers – dimensional realms, realms of unexpressed potentiality containing no expressed potentialities, which would otherwise become the members of the set. However, as soon as the containers contain an expressed potentiality, they are ‘actualised’, and become’ a member of themselves’.

    It seems to me that there is a component of set theory which is dimensional and non-computational. In the description ‘set of all sets which are not members of themselves’, we open up the possibilities for using descriptions of dimensionality to resolve the paradoxes. And in doing so we have also extended our definition of ‘dimension’.

    Its a bit difficult to express in words, but there again I’m no mathematician!

    • This is not so different from the way I think about it. Imagine that sets are constructed in stages; at the first stage we have only the empty set, and at each stage we can gather up sets constructed at an earlier stage. The “set of all sets” cannot arise, since there is no stage at which all sets have been completed.

      But I won’t be going there in this class, I am afraid.

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