Archive for October, 2011

How might we get to a new model of mathematical publishing?

October 31, 2011

This is a post I’ve been intending to write for several months, but now seems to be quite a good moment, since the issue is in the air somewhat. For example, I’ve just read a post by Michael Nielsen on a similar topic, which itself was responding to things that other people have written. However, he is addressing a different issue: that of the restriction of access to journal articles once they are published. I am more interested in whether mathematicians really need journal articles at all, now that we have the internet. (Just in case anyone hasn’t noticed, this post is not part of the series about first-year mathematics at Cambridge …)

Before I go any further, let me make clear right from the outset that I’m not merely saying, “We can stick our papers on the internet, so let’s forget about journals.” I think that journals still have a vital role to play, even though the internet exists. However, like many people, I do not think it is at all obvious that they will continue to have a vital role to play, so I’d like to discuss two questions.

1. If we didn’t have journals, then what might we have instead?

2. How could the change from journals to whatever replaces them actually take place?
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Equivalence relations

October 30, 2011

Equivalence relations are in a way a fairly simple mathematical concept. After all, it’s not that hard to learn what reflexive, symmetric and transitive mean and to remember that if you’ve got all three properties then you’ve got an equivalence relation. However, equivalence relations do still cause one or two difficulties. One, which I’ll discuss first, is that many people find that the proof that equivalence classes always form a partition is rather complicated and hard to remember. The other is that equivalence relations are closely connected to the notion of quotients, which appear in many places in mathematics, and which many people find quite hard to grasp.
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Alternative definitions

October 25, 2011

Something that happens very often in lecture courses is that you are presented with a definition, and soon after it you are told that a certain property is equivalent to that definition. This equivalence means that in principle one could have chosen the property as the “definition” and the definition as an equivalent property. To put that differently, suppose you are developing a piece of theory and have some word you want to define. To pick an imaginary example, suppose you have a notion of a set being “abundant”. Suppose that a set is defined to be abundant if it has property P, and that property P is equivalent to property Q. There may well not be much to choose between the following pair of alternatives. On the one hand you can say, “Definition: A set is abundant if it has property P,” and follow that with, “Proposition: A set is abundant if and only if it has property Q,” while on the other you can say, “Definition: A set is abundant if it has property Q,” and follow that with, “Proposition: A set is abundant if and only if it has property P.”
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Definitions

October 23, 2011

By now you will have seen several definitions in lectures. Many of them will be written in the form

Definition. A blah is …

That is, the definition is displayed and the word being defined is in italics (or underlined if somebody is writing by hand). Sometimes, one doesn’t bother with the display, and simply says, during a discussion, “We define a blah to be …”

What is likely to have been emphasized less is that there are several different kinds of definition. In this post I’d like to enumerate some of them and give examples. It’s very much worth being aware, each time you meet a definition, what kind it is.
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Permutations

October 16, 2011

I don’t have too much to say about permutations, but there are two points that I have often found myself needing to get straight in supervisions. In fact, make that three. Here they are. [Added later: I have just finished the post, and it ended up being longer than I expected.]

1. The first is a confusion that some people have about what a permutation of $\{1,2,\dots,n\}$ actually is. What could possibly be the trouble, you might ask? Well, let's take the permutation that in cycle notation is written $(124)$. My guess is that a non-negligible percentage of people reading this have worried about whether this permutation means that you cycle round the elements 1, 2 and 4 of the set $\{1,2,\dots,n\}$ or the elements in the places 1, 2 and 4.
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Domains, codomains, ranges, images, preimages, inverse images

October 13, 2011

If I were writing a textbook, I would have discussed the basics of functions before talking about injections and surjections, but this is not a textbook — it is a series of blog posts that provide a kind of commentary on some of the lecture courses. However, now that I have got on to the subject of functions, it probably makes sense to discuss them a bit more, especially as I hear that they have made an appearance in Numbers and Sets.

Let me start with the most basic question of all: what is a function? This is one of the first examples (of many, unfortunately) of a concept that you were probably reasonably happy with until your lecturer explained it to you. This is absolutely not a criticism of your lecturer (who is excellent, by the way). It’s more like a criticism of an entire mathematical tradition that goes back to the days when the foundations were laid for our subject in terms of set theory.
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Injections, surjections and all that

October 11, 2011

My spies tell me that the second lecture of the Group Theory course contained a discussion of functions, and in particular bijections — for which it was necessary to prove a few results about injections and surjections. Since a good understanding of functions is essential throughout mathematics, perhaps a post on the topic would be in order. My aim in these posts is not to cover the material all over again, but rather to stress the points that you need to understand in order to be able to write proofs. So let me give a few thoughts about functions, in no particular order.
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Basic logic — summary

October 9, 2011

Here is the promised post that I hope will be easier to refer back to than the much longer posts I’ve written on individual aspects of basic logic. What I imagine people doing is reading the longer posts and using this one to jog their memories later. If you can think of any important points that I made in earlier posts and have forgotten to mention here, I’d be grateful to know of them.

Once again, the main topics dealt with were these.

Logical connectives. AND, OR, NOT, IMPLIES (or in symbols, $\wedge,\vee,\neg,\implies$).

Quantifiers. “for every” and “there exists” (or in symbols, $\forall$ and $\exists$).

Relationships between statements. Negation, converse, contrapositive.
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Basic logic — tips for handling variables

October 7, 2011

Roughly speaking, a variable is any letter you use to stand for an unknown object of a certain type. For example, if you write $x+y=20,$ then $x$ and $y$ are variables. If you write, “Let $A$ be a subset of $\mathbb{N},$” then $A$ is a variable (it is an unknown set of a certain kind) whereas $\mathbb{N}$ isn’t (it’s the name we give to the set of all positive integers). I suppose the definition I’ve just given isn’t quite perfect, since if I asked you to solve the simultaneous equations $x+y=8$ and $x+3y=12,$ then one would normally call $x$ and $y$ variables even though their values are completely determined by the equations. Though even then one could say that they started out as “unknown”.

Just in case I’ve gone and confused the issue, let me try to clear it up instantly. It would be quite normal to say something like this: “Let $x$ and $y$ be two real numbers. Suppose that they satisfy the equations $x+y=8$ and $x+3y=12.$ Determine the values of $x$ and $y.$” It is then reasonable to call them variables, because when I started discussing them I gave no information about them whatever. I then went on to specify some relationships between $x$ and $y,$ and it so happened that from those relationships it was possible to deduce the exact values of $x$ and $y.$
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Basic logic — relationships between statements — converses and contrapositives

October 5, 2011

Converses.

What is the relationship between the following two statements?

1. If $n$ is 1 or a prime number, then $(n-1)!+1$ is divisible by $n$.

2. If $(n-1)!+1$ is divisible by $n,$ then $n$ is 1 or a prime number.

At first sight, this doesn’t look a very difficult question: the first statement is of the form $P\implies Q$ and the second is of the form $Q\implies P.$ We say that the second statement is the converse of the first. (Note that the first statement is also the converse of the second.)
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