The poetry of definitions in maths

Coming up with an elegant mathematical definition is as much of an art as writing poetry. The analogy works even further, for like a well-written piece of poem, a definition in mathematics packs a lot of meaning in a few words. One has to unpack the meaning to appreciate either, and the process can be frustrating. But once the understanding is complete, it is easy to see why the definition/poem works, and it is infinitely preferable to use it rather than explain what one means.

Here are some examples.

Poetry : Definition of a Free group via the universal property – A subset X of a group F is said to be a free basis for F  if for every group G and every (set) function \phi: X \rightarrow G, there is a unique homomorphism \bar{\phi}: X \rightarrow G satisfying \bar{\phi} |_X = \phi. A group F is called a free group if it has some subset that is a free basis.

Prose: Homomorphisms carry the algebraic structure of an object. If for example the Cayley group of X (i.e. F) has a loop, any (non-trivial) group homomorphism must carry it into a loop. The only way you can extend any set function whatsoever to a group homomorphism is if X has no algebraic structure to be carried over. This is the property that this definition captures.  Notice that we do not require X to generate $F$. But that is exactly what will happen, because we require the homomorphism \bar{\phi} to be unique.

The second example needs some set-up.

Quasi-isometric embedding. Let (E_1,d_1) and (E_2,d_2) be two metric spaces. Let \lambda, \epsilon \in \mathbb{R} with \lambda \geq 1 and \epsilon \geq 0. A map \phi E_1 \rightarrow E_2 is called a (\lambda, \epsilon)-quasi isometric embedding if \forall x,y \in E_1

\frac{1}{\lambda}d_1(x,y) - \epsilon \leq d_2(\phi(x),\phi(y)) \leq \lambda d_1(x,y) + \epsilon.

Poetry:  A Quasi-isometric embedding is called a quasi-isometry if \exists C \geq 0 such that \forall x_2 \in E_2, \exists x_1 \in E_1 with d_2(\phi(x_1),x_2) \leq C.

Prose: We are looking for a notion that is not quite a surjective map, but close to being a surjective map. That is why we ask that if an element in E_2 is not actually in the image of \phi, it is not too far away from such an element.   So any element in E_2 is close to some element in \phi(E_1).

As they say in Latin, quod erat demonstrandum.

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Crying Wolf

You might have heard this story earlier.

There is a girl who is very naughty. She goes out to play in the woods every day and every day, she cries wolf. All the villagers come running to her rescue, only to find her laughing so hard that tears ran from her eyes. One day, when a wolf really comes and she cries for help, no one comes. None of the villagers come to help her, but for a friend, who loved her dearly. Every time the naughty girl cried wolf, he came running-never doubting for a moment that the girl needed help. Yet every time the imp laughed, he laughed with her. When she played the same prank over and over again, he never stopped playing along-not in the way you humor a child, but in the way you are caught in a friend’s joke, in spite of yourself. When the wolf came, he didn’t feel he was proved right after all. He didn’t remember that there had been a joke about it. He chased away the wolf and let the girl cry on his shoulder and took her home.

“Love is very patient and kind, never jealous or envious, never boastful or proud, never haughty or selfish or rude. Love does not demand its own way. It is not irritable or touchy. It does not hold grudges and will hardly even notice when others do it wrong. It is never glad about injustice, but rejoices whenever truth wins out. If you love someone you will be loyal to him no matter what the cost. You will always believe in him, always expect the best of him, and always stand your ground in defending him.”

Maybe the girl wasn’t playing a prank. Perhaps she heard a rustling sound in the woods. May be she really thought that there was a wolf and cried out. And when you saw her laughing, may be she was laughing with relief-the tears streaming down her cheeks only confirming how scared she had been.

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Using an image as the axis text in R plot

There were plenty of good suggestions on this page .
But the one that worked for me was in one of the comments. Use the package “tikzDevice” and you can add the awesomeness of Latex to the plotting prowess of R. All you have to do is to construct a Latex directive in R, and use it in the R plot commands, wherever you want. tikzDevice will create a tex document for you. Compile it and you are ready to go.

Example
library(tikzDevice)
library(ggplot2)
axislabels=c()
for (i in seq(1,3))
{
fname=paste(“placeholder”,i,sep=””) #img is of course present in the wd.
al=paste(“\\includegraphics[scale=0.05]{“,fname,”}”,sep=””)
axislabels=c(axislabels,al)
}
dat <- data.frame(cond = rep(c(“Good”, “Bad”), each=10),xvar = 1:10 + rnorm(10,sd=1), yvar = 1:10 + rnorm(10,sd=10))
p=qplot(1:3,1:3)+scale_x_discrete(limits=c(1,2,3),labels=axislabels)
tikz(“annotated.tex”,standAlone=T)
print(p)
dev.off()

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Teaching fractals

As a a volunteer for CSIRO’s Mathematicians in Schools Program, I was asked to give a talk on fractals to girls in grade 9 at a school in Sydney.

I am not an expert on fractals, but as a doctoral candidate in mathematics, I figured I knew more than the the girls would, so I thought, why not? And I agreed.
This school was participating in the MegaMenger project. Their teacher wanted me to motivate the study of fractals by talking about some real world applications. I did talk about the usual suspects – animation, modeling natural systems etc. But I also managed to sneak in a point that mathematics educators always make – that ” what is this good for”, is not the right question to ask in a maths class. Paul Lockhart in his famous essay says it better than I could. But I like to extend his lament to the whole of education – how will this help me get a job is not quite the question that should be uppermost on young minds. The gifts of education are rather more intangible – curiosity and the ability to learn things, that’s what is going to get you a job. I know that these are not things one could put in one’s CV, but then, these are teens who have not yet outgrown their love for Beiber. And yet, theirs are the loudest voices in that dreaded chorus – “why are we doing this? When are we ever going to use it?”.

My rant aside, I think the activity went well. Here’s an appraisal of what worked and what didn’t.

What worked – I am usually sceptical of giving presentations to young students, as there’s no surer way of putting them to sleep. But this one was a talk with just a picture on each slide, with me talking around the picture. I also put in a fractal paper cutting activity. I was afraid that the students might find it boring, but everyone eagerly participated.

What didn’t – I did ask them a bunch of questions – what other examples of fractals they could think of, where did they think fractals could be applied etc. In hindsight, I think it would have been better to tie their answers into a discussion. Instead of accepting one workd answers, I should have asked them to elaborate.

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What’s with the calculators?

Reaching out for the calculator has become a reflex when students see anything that looks vaguely mathematical. It is very discouraging to see students painstakingly type in “3*1/25″ when presented with the fraction “3*1/25″. How did things come to such a pass, I wonder?

We are always talking about “real life” uses of mathematics. We should probably be thinking about the mathematical uses of  “real life”. I am pretty confident that the same students can easily work out how much they will have to pay for 3 dozen eggs when a dozen cost $3. I am also sure they can work out the discounts in their head, choose the cheaper option. I don’t see hordes of people walking about in the supermarket punching numbers into their calculators.

I am aware of the term  “innumeracy”, I know that lottery is a thriving industry, that people scare themseleves to death about Ebola, when they are more likely to be killed on their way home. But I am not talking about that. I am talking about the urge to pull out a calculator when encountering numbers in a classroom setting – even when multiplying by 100. It is almost as if having a calculator gives them comfort, and that is deeply worrying. I don’t care if my students do not take advanced mathematics, but I do care if numbers scare them.

How can I lead them from this fearfulness to fearlessness? This occupies a lot of my thought space these days, not that I have an answer for all that. It is an evolving story that I’ll continue to transcribe faithfully.

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Nothing to be frightened of – Julian Barnes

Our lack of originality is something we usually forget as we hunch over our – to us – ever fascinating lives. My friend M., leaving his wife for a younger woman used to complain, ‘People tell me it is a cliché. But it doesn’t feel like a cliché to me.’ Yet it was, and is. As all our lives would prove, if we could see them from a greater distance – from the viewpoint, say, of that higher creature imagined by Einstein.

… there is something infinitely touching when an artist, in old age, takes on simplicity. The artist is saying: display and bravura are tricks for the young, and yes, showing off is part of ambition: but now that we are old, let us have the confidence to speak simply. For the religious, this might mean becoming as a chid again in order to enter heaven; for the artist, it means becoming wise enough and calm enough, not to hide. Do you need all those extravagances in the score, all those marks on the canvas, all those exuberant adjectives? This is not just humility in the face of eternity; it is also that it takes a lifetime to see, and say, simple things.

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The Salmon Of Doubt

By one and only Douglas Adams.

“A couple of hours later he had the answer, or at least some kind of an answer. Nothing that went so far as to make any kind of actual sense, but enough to make Dirk feel an encouraging surge of excitement: he had managed to unlock a part of the puzzle. How big a part he didn’t know. As yet he had no idea how big a puzzle he was dealing with. No idea at all.”

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