Things that might be as wrong as racism

In The Descent of Man, Charles Darwin wrote:

"At some future period, not very distant as measured by centuries, the civilized races of man will almost certainly exterminate, and replace, the savage races throughout the world. At the same time the anthropomorphous apes, as Professor Schaaffhausen has remarked, will no doubt be exterminated. The break between man and his nearest allies will then be wider, for it will intervene between man in a more civilized state, as we may hope, even than the Caucasian, and some ape as low as a baboon, instead of as now between the Negro or Australian and the gorilla."

We now consider this to be not merely morally repugnant, but factually incorrect (and, yes, I did mean to put those two clauses in that order). However, when Darwin wrote this passage, such racist statements were commonplace. There was nothing even vaguely controversial about the idea that white men were better than Negroes. This poses an interesting question: what moral or political positions do we hold today that will, or might, be as utterly repugnant to future generations as Darwin's casual racism is to us? I can think of a few:

Patriotism

It is currently considered perfectly acceptable for Walmart to declare that they try to "Buy American", and for British politicians to claim that products made in this country should "proudly display the Union Flag". This is exactly as immoral as racism, for exactly the same reason. (Steve Landsburg has been pointing this out for a while - see, eg, this video). I can only hope that there will come a time in the future when such blatant tribalism: the utterly deplorable idea that I should care more about a complete stranger who happens to have been born on the right side of some arbitrary line, is considered every bit as heinous as the utterly deplorable idea that I should care more about some complete stranger who happens to have been born with the right colour skin.

Drugs

It is currently considered immoral to ingest certain substances. So much so that we have passed laws banning the ingestion of said substances. This strikes me as bizarre. It seems eminently likely that, at the very least, the substances which we consider it immoral to ingest will change over time.

Transplants

There are people who seriously suggest that it would be immoral to switch the law regarding transplantation from a default "no" to a default "yes". This is plainly ridiculous, as is the idea that a market in organs is immoral. However, I can actually imagine future generations going one step further, and being utterly horrified at the idea that we let people die from lack of transplants when there were perfectly good organs rotting away inside human carcasses: as if the dead people had some use for them! Compulsory post-mortem organ donation is certainly a plausible moral imperative.

Meat

We bring sentient beings into existence, feed them, house them, and kill them just in order to eat their dead bodies. I can easily see how a future society in which either meat has just dropped rapidly out of fashion, or in which meat production no longer requires the participation of sentient beings, might hold us to account.

Children

I am not the first person to have noticed that children are essentially treated as non-people in our society. I can easily imagine the way we allow parents to completely dictate their children's lives could be considered morally repugnant. Also relating to children: it seems quite plausible that our current attitude to Child Sexual Activity is somewhat misguided (see, eg Rind Et Al) - the very fact that I'm almost afraid to write that statement in public without a string of caveats should tell us, at least, that society's current attitudes to CSA are too emotional to be completely rational.

Other things

Almost all of the examples I've managed to come up with are examples in which I'm fairly sure that the current mainstream moral opinion is actually wrong, or at least in which I'm fairly sure it could be wrong. Clearly this is a failure of imagination on my part - there is almost certainly at least one thing that wouldn't even occur to me that will be seriously considered to be a moral issue in, say, 250 years' time - anyone have any ideas what it might be?

In Defence of Impossible Precision

John Allen Paulos has quite a good column on innumeracy, first he asks readers to assess the following headlines:

1. After the Packers' Super Bowl victory, an exuberant Aaron Rogers Shook Hands with Everyone in the Stadium.

2. Experts Fear Total US Housing Costs (Rents plus Mortgage Payments) Will Top $2 Billion in 2011.

3. Only by Completely Eliminating Foreign Aid Can We Eliminate the Deficit.

What is wrong with them? Well, I don't expect anyone reading this not to have noticed that shaking 40,000 hands would take at least 10 hours, that $2 billion comes to about $10 per person or that Foreign Aid is such a tiny portion of the US deficit that even eliminating it entirely wouldn't make a big dent (this doesn't, of course, mean that it shouldn't be eliminated, only that if you're obsessed with the deficit, you have bigger fish to fry).

He then moves onto the following headline:

4. Number of Americans with Alzheimer's Believed to Be 5,451,213.

The supposed problem?

4. The problem here is that the number is ridiculously precise. Definitions of Alzheimer's vary and it's difficult to determine whether a single individual is suffering from it, much less whether five million plus are. Such impossible precision is common.

Well, yes, the number is ridiculously precise. No, no-one does think that we can measure the number of Americans with Alzheimer's to that degree of accuracy, but so what? If you do a survey of Americans, do some calculations, and your best estimate of the number of Americans with Alzheimer's comes out as 5,451,213, what number, exactly, does Paulos want you to report?

Assuming that you've done your sums correctly, 5,451,213 is an unbiased estimator of the number of Americans with Alzheimer's. Rounding your guess to 5.5 million does systematically worse than just reporting the estimator you got out of your calculations, so what exactly is the rationale behind it?

Yes, numbers like this should probably be reported along with some estimate of variance, and maybe it's a convention that we assume the number of signficant figures of a number to be a proxy for the size of its error bars, but it doesn't have to be that way: I look forward to a day when numbers like "5.5 million" get scoffed at by popular mathematics writers for being "overly round" or "not accurate enough".

Why aren't all journals open access?

Here is the way the current system of academic publishing works, as far as I can tell: universities employ researchers who do original research, and produce journal papers; universities employ researchers who do peer-review, and make sure journal papers are up to standard; journals employ editors, who put the content together, and organise the referees; universities pay large amounts of money to journals in order to be allowed to read the articles.

Now, as you can see all of the money in the system comes from the universities. Universities pay the wages of the researchers and the reviewers directly, and they pay the wages of the editors indirectly (through journal subscriptions). So, here's an idea; why don't the universities club together to buy the journals, employ the editors directly, and publish all the content for free?

Note that buying the journals doesn't cost the universities (as a group) anything in the long-run, as the entire current value of the journal companies comes from the amount of money they expect to be paid in journal subscriptions by universities in the future. And there's no need for the journals to charge "submission fees", as those were all being paid by the universities in the first place: they can just come out of the communal pot.

So far as I can see, there is literally no downside to this - assuming coordination can be achieved, you have the same universities paying the same amount of money to the same people to produce the same articles, but the articles are now all available open-access. I admit that "assuming coordination can be achieved" is a fairly hefty assumption, but given the massive upsides, why isn't anyone at least suggesting this sort of approach?

There seems to be a general trend towards open-access publishing anyway, which is a Good Thing, but I don't undestand why this model isn't a strict Pareto improvement on the current system.

What should be in a maths class?

I have recently (well, in the past two years) read two very interesting essays on the teaching of mathematics. At first glance, they seem to be almost diametrically opposed, but I tend to find myself agreeing, overall, with the thrust of both. The essays are Paul Lockhart's A Mathematician's Lament and Conrad Wolfram's TED talk on mathematical education. A couple of quotes from each which provide a brief summary.

From Wolfram's talk (trancript available here):

I want to see a completely renewed, changed math curriculum built from the ground up, based on computers being being there, computers that are now ubiquitous almost. calculating machines are everywhere and will be completely everywhere in a small number of years. Now I'm not even sure if we should brand the subject as math, but what I am sure is it's the mainstream subject of the future.

From Lockhart's Lament:

The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

So, Lockhart thinks we should be teaching mathematics as an art form, and Wolfram thinks we should be introducing more computers into mathematics lessons so that people can concentrate on doing the bits that are actually useful. Which of them is right? Well... both, but mostly Wolfram.

The question we have to ask ourselves before we can even begin to compare the two essays is why do we teach mathematics at all? So far as I can see, the only sensible justification for having mathematics as a subject that everyone in the world should be taught up to a relatively high level is because it's useful. You can't do physics, or engineering, or any sort of science, or do derivatives trading, or even decide which mortgage to get, without knowing quite a lot of mathematics. For this reason, everyone should be taught the basic mathematics that they need to know in order to do these things (or the basics they need to know in order to learn the specific maths they wnat to use).

Note that one corollary of this mode of thinking is that most of the mathematics you learn probably shouldn't be taught in a maths class. It's much easier to learn how to get from acceleration to speed than it is to learn how to differentiate a function. Yes, it is useful to then point out the possible generalisations (getting from acceleration to speed is the same as getting from jerk to acceleration) but I don't see any reason why these topics can't be introduced concretely. I personally have serious trouble doing any calculus that I can't do using physical intuition, and I think it's fair to say that I am an above-average student when it comes to learning maths. Calculus should be taught when you're doing engineering, statistics should be taught when you're analysing the results of experiments, graph theory should be taught when you're trying to solve scheduling problems.

Lockhart, on the other hand, seems to think that we should be learning mathematics because, essentially, mathematics is awesome. I happen to agree with him that mathematics is an exceptionally beautiful art form. I'm happy to sit back and bask in the glory of Cantor's diagonalisation argument, or the ingenuity of Karp's reductions between NP problems, but I'm not sure that I'm willing to contend that everyone should be forced to. Yes, if you want to be a mathematician you have to learn that mathematics is actually an art, but most people who study mathematics don't want to be mathematicians, and most people who study mathematics *shouldn't* want to be mathematicians. For these people, learning about the art of mathematics is little more than an intellectual curiosity, on a par with learning about Titian or Shakespeare.

In other words, Lockhart is right, inasmuch as we want people to study mathematics for it's own sake. Wolfram is right, inasmuch as we want people to study mathematics because it's useful.

Now, I happen to tend strongly towards the idea that the only things we should be teaching in schools are things which are potentially useful, but that obviously isn't the prevailing wisdom - everyone in this country is still forced to do an English Literature GCSE. Lockhart-style mathematics is a perfectly good substitute for art class, or critical theory. Wolfram's mathematics is a necessary prerequisite for doing just about anything else.

The Story of a Proof

I started writing this blogpost about 6 months ago after reading an interesting post by RJ Lipton on why mathematicians prove things. I can't find that post any more (maybe it wasn't by RJ Lipton...?), but it was general speculation on why mathematicians are actually interested in formal proofs of their theorems.

A recent experience of mine in my own research seems particularly relevant here, and I'm going to try and explain what happened (this will, at the very least, satisfy my promise to Michael Brough at about the same time of getting some more serious mathematics into my blog). This is pretty much the first time I've ever tried to explain something technical in a non-technical setting, so let's see how it goes...

Click here to read the rest

Writing Mathematics: Formalism obscures intuition?

I have read quite a large percentage of the books that are currently widely available on the topic of writing mathematics, and several that aren't. I've also been involved in teaching the Mathematical Writing course run at the QMUL. There is one serious issue that I've encountered: the main problem seems to be that most people start with the assumption that you already know everything that you're going to write, and the basic structure of your proof, but don't know the best order to put the words on the paper. In my experience, this is far from true.

I'm currently writing up my thesis, and have several new ideas which I've never written down in complete formal detail before. This process is difficult. The ideas in are generally simple enough that I think I could explain them to reasonably bright 15 year old in about 10 minutes given a piece of paper, but sufficiently abstract that to write them down formally has involved pages of writing that even I barely understand. These are proofs that I can (and have) run through in the pub with non-experts on the back of a napkin, and they translate into 10 pages of symbols that even I can barely follow. (I'm working on a post containing an example, but it will require drawing some pictures, so I'll probably get round to it some time over the weekend).

It is all very well saying that one should write an informal verbal summary of what you are going to do before you do it, but when the formalism is so far removed from the key idea, this becomes difficult. Also, when you are writing a maths paper, you have to make damn sure that the "summary" is still technically accurate. There can be no hand-wavey 'look, it just works' and (importantly) no interaction with the audience - you dont' know if they 'get it', so you have to put down everything it might take for them to get it. In my experience, the heavy formalism has often meant that the proofs I write down end up so complicated that I'm not even sure I would 'get' them if I had to pick up the ideas directly from reading the papers. The formalism masks the intuition, and the intuition isn't quite formal enough to be appropriate for the paper.

This is an interesting topic. It's one that several people have no doubt spent a lot of time thinking about, but it's not one that seems to be discussed. Even in a mathematical writing course, the tendency is to focus on technicalities. To be fair, this is usually adequate for undergraduates, and you do need to get the technical stuff right even when writing down simple ideas, but this is something that needs addressing: exposition of mathematics is a difficult skill, and probably one that there's not enough focus on in training academics.

Maybe not everyone has this problem - maybe not everyone has elementary ideas which don't translate nicely into formalisms - maybe their ideas are inherently more technical, or maybe they're just better at finding formalisms than I am - but I do find it interesting that it's happened to me almost every time I've tried to write out a formal version of a proof that I've created myself.

PS - the vast majority of this post was written over a year ago - the posting of it was sparked by having a discussion with Andy in which he mentioned that he is currently having exactly the same problem, and by me finally figuring out how to access the drafts of my old posts...

Philosophy of Education

I just came across this on Rosemary Bailey's website. While I think the ideals are nice, I think in practice I disagree with just about all of it:

My vision of a university was succintly described by David L. King, writing in The Times Higher on 9 April 2004. It is:

• “the belief in a community of scholars and not a confederacy of self-seekers;
• the idea of openness and not ownership;
• the professor as a pursuer of truth and not an entrepreneur;
• the student as an acolyte whose preferences are to be formed, not a consumer whose preferences are to be satisfied.”

I believe that university students should be able to be confident that they are being taught by people who are immersed in the subject in other ways than teaching. I collaborate with a range of scientists on the design of their experiments and the analysis of their data, so I teach Statistics. I still prove theorems in Combinatorics and Algebra, so I also teach those subjects.

I've no idea what a "confederacy of self-seekers" is, so I'm not going to address that point. I don't think anyone believes in a professor as an entrepreneur, one thing that I think people can sensibly expect professors to be, at least in the current system, is teachers. Incidentally, I happen to think that Rosemary is a very good teacher, and one who thinks a lot about doing the best for her students, so I hope she won't be offended by any of this.

The problem I have mostly is with the last point in the bulleted list: "the student as an acolyte whose preferences are to be formed, not a consumer whose preferences are to be satisfied.". Now, that might be a nice model for a university if the point of going to university was to learn about the subject you were studying. The problem is that for the vast majority of people who go to university currently this just isn't the case. They go because in order to get the better paid jobs you have to have a little piece of paper which said that you went to university. They go to university to get this piece of paper and, and I think this is the important part. This system is not the students' fault.

The students in our department are not there because they are particularly interested in mathematics and (and I think this is something that academics sometime struggle with, and I will almost certainly get round to making a separate post about this one day) the vast majority of them are not interested in pursuing a career in academia. The overwhelming majority of students are at university to get a degree because it will help them get a better job. And they are right. In the world we happen to live in people do need degrees to get good jobs, and universities are there to supply these degrees, and to make sure that the degrees are a useful signal of ability for employers.

It might be that the vision of the university described by David King is a good way to train people who want to obtain the sort of detailed knowledge of an academic subject one needs to do research, but it absolutely is not the case that 50% of the population want, or need to obtain that sort of detailed knowledge of any subject. So, while we have a system where something pushing 50% of young people go to university (and, remember, pay the fees that pay the salaries of the people who work at the university).

Most people are there to get a degree because getting a degree is necessary prerequisite to doing whatever it is they actually want to do. Like it or not, the job of the teaching staff in a university is to help them get that degree. If you don't like it, by all means do your best to change it, but while you're accepting money from the university as it exists now, I'm afraid I think you are morally obligated to teach in the way that is appropriate for the majority of students who are there now. More to the point, if you don't like the current system, you're going to have to figure out another way to pay for researchers - at the moment research is cross-subsidised by an awful lot of socially innefficient teaching.

I personally would love to live in a world where universities could all be like Cambridge and Oxford, and where the vast signalling game that is university education as it exists now didn't exist. I admit that this world would probably be a world in which less "blue skies" research got done, and I find it hard to believe that would be a particularly bad thing.

This post is getting long, and I'm not sure my thoughts on the last paragraph of Rosemary's statement are fully-formed, so I won't write much about that for now. I will however finish with one thought that I have on the issue: it would be a remarkable coincidence if the best teachers in the world also happened to be the people who were best at doing original research. I happen to think that I personally am quite a lot better at the former than the latter. I'm not going to give an example of someone whose abilities are skewed in the other direction, but I'm almost certain everyone who has studied or worked at a university already has one in mind.