Aside

# 2. Methodology, Handwaving, and Diagrams

I will try—as much as possible—to keep the material accessible to people who have never seen, or even heard of linear algebra. At the same time, I expect that many (most?) readers will be at least somewhat acquainted with the concepts. So for those in the know, I will sometimes include information about how what they already know can be understood in the different style that we will use. But if you’re an expert and think that the formula for multiplying matrices is so basic and natural that no one in their right mind should be writing blogs on the subject, then I suspect I will have a hard time trying to win you over.

I’m an academic. In my job I teach, do research, and write papers. I love teaching and doing research. Writing scientific papers is not so great. Writing, at least for me, requires massive amounts of work. A typical 15 page conference paper is, at minimum, a solid 60+ hours of work, not including the actual research work that comes beforehand. The amount of work is not linear in the length, and so twice the length means much more than twice the work. I once spent a solid 3 months working on a 60 page paper. After all the blood, sweat and tears, what happens next is somewhat depressing: the paper goes through (anonymous) peer review, and if you’re lucky it gets accepted with some lukewarm reviews. Sometimes reviewers are really petty and mean.  It’s not pleasant.

If the internet has taught us anything, it’s that

humans + anonymity = unpleasantness.

History, moreover, has taught us time and time again that

humans + power = unpleasantness.

Anonymous peer-review combines humans, some anonymity, and a little bit of power. The results are not surprising.

The next step is for your paper to get published, and maybe presented at a scientific meeting, in front of an audience of anywhere between 10 and 100 people, 50% of whom will spend the entire presentation reading emails. Again, not a great feeling. Then, after your paper gets published, sometimes your colleagues will cite you. My most cited paper has around 150 citations. That’s considered to be not bad, by the way.

So, to get back on topic, I want to have fun writing this blog. Don’t expect anything like scientific writing. It will be totally unprofessional. I want to write about Makélélé and magic Lego. Sometimes I will present my opinions in somewhat “Australian” style. I spent my childhood in Poland, but I grew up in Australia. In this blog I will be channelling my Australian-ness. Australians, like the Scots, tend to call a spade a spade. The English aren’t usually so direct. Englishness is all about subtlety and insinuation.  Over the centuries, they have refined their public discourse and developed high-level, advanced techniques like (damning with) faint praise.

It’s funny that famous, English, public-school educated, all round posh Oxbridge professors are often regarded in their international scientific communities as the perfect gentlemen. Some of them actually are, of course. Some are often as rude as their extreme Englishness can possibly allow them, it’s just that their worst insults are misunderstood as compliments. They get to have their cake and eat it too — this makes me really jealous. Let me explain for the non-English aware: say you ask their opinion about your work and they reply “I think that your work is very interesting”. Sounds like a compliment, right? Wrong. It’s a serious put-down, they hate it. Think about it: “very interesting” is not a particularly strong compliment; it can be understood in several ways. At least we regularly thrash them in the cricket.

As you can see, things will sometimes get a bit personal here. The people that know me, know what I mean. Some of them, even some English people, still talk to me.

I promise one thing: I will try to be serious about the maths. I mean, I will try very hard not to handwave. Handwaving is when you kind of sound like you know what you’re talking about, but you’re really only speculating, using vague analogies, etc. When people do this, they naturally tend to wave their hands a lot, hence the name. So, while I will give lots of intuition, all intuition will eventually be explained. It will be hard, because I really, really love handwaving. Please let me know if you notice any, and tell me to stop. I will also try to cite related work and acknowledge sources, and not make far-fetched claims about what the maths can be used for. But that’s about as professional as you’re going to get.

In our discussions we will use many diagrams. Hence the “graphical” in graphical linear algebra.  Diagrams have got a lot of bad press in science and mathematics over the years, for being somehow less formal, less rigorous, less “mathsy” than formulas. Unfortunately, it is true that diagrams are often used by people who like to wave their hands a lot. When sceptical captains of science see a diagram, they recoil slightly. They look around the room. They shake their head knowingly at the most important person in the room. I mean, why not just write an honest-to-goodness formula? You know, like $e^{\pi i} = -1$. Formulas will sometimes appear in this blog, mainly to appease the grizzled cognoscenti.

The point is that—if used carefully—diagrams can be extremely useful, rigorous and totally 100% no-hands-waved formal. As honest as the best formulas, but much more concise, and much easier to read. Even better than rectangles of numbers, which—as I have said—are very important. We all know the cliché “a picture tells a thousand words”. We will see how, quite literally, a diagram can tell a thousand formulas.

There’s a branch of mathematics called category theory, invented by Saunders Mac Lane and Samuel Eilenberg in the 1940s, where diagrams have been used since the very beginning. Category theorists are comfortable with diagrams, and we will use category theory to do graphical linear algebra. A lot of great category theory comes from Australia, by the way.

Maybe you already know something about category theory. If not, I’ll let you in on a little sociological curiosity: there’s a lot of animosity towards category theory out there in science-land. I’ve had several colleagues discreetly tell me things like “I’ve tried category theory once, but never again” or “One of my friends used it once and it was a total disaster. It’s just all too abstract. I like to do concrete things.” I normally manage to keep a straight face. But I wonder how many carpenters have ever said “I tried using an electric drill once, but it was a complete disaster. The thing made a hole in completely the wrong place. And the hole was the wrong size. Never again.” I mean, don’t get me wrong, it’s totally fine to be a carpenter and not use electric drills.

It is true that many category theorists like to explore the deep end of the pool. Sometimes the pool is not quite deep enough and they go away and imagine something deeper. Typically we end up with several, possibly non-equivalent, infinitely deep pools. It’s cool, but it makes my head spin.

So don’t panic. I’m not really a category theorist, although I’m lucky enough to have met and learned from some truly great category theorists. You can trust me. We will mostly stay in the shallow end, and do very concrete things, like adding. In fact, adding is the subject of the very next episode.

## 7 thoughts on “2. Methodology, Handwaving, and Diagrams”

1. Very minor stylistic comment: Be consistent in your use of commas, e.g. “In my job I teach, do research, and write papers.” vs. “After all the blood, sweat and tears”. It’s hard to argue which is better, but I respectfully suggest you choose one and stick with it.

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• Thanks for the suggestion. I quite like the Oxford comma, but sometimes it doesn’t really work. For example, in the expression “blood, sweat and tears”, the way people normally say it, there isn’t a pause after the “and”.

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• It really is semantically equivalent and that comma thing is about syntax.

Meanwhile fixing things like non-negative could provide some genuine value.

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2. atucker says:

As an Englishman who has spent time in the company of Australians, I can confirm they are just as capable of misrepresentation to save others’ feelings. You do seem to be an exception, not sure it’s about honesty though, more like you take pleasure in slagging off an entire nation. Interesting topic (no faint praise intended) but I’m out.

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3. mear says:

As an “Englishman” … I’m in! This is great, and it’s beautiful to see high level category theory ramify into radically clarifying pedagogy.

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4. In the U.S., I have had people say, “Hmmm…Interesting. I never thought of it that way.” It’s a polite way of saying, “I’m going to politely walk away from you now and never think of what you just said.”

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