Why Is It Lovely

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Recently on one of my "social network feeds" I saw a post saying:

  • X% of Y equals Y% of X.

There were already several comments with people expressing various reactions, so I idly commented as follows:

1% of 100 is 1, by definition. So 2% of 100 is 2, 3% of 100 is 3, and 4% of 100 is 4. Halve things, so we have 4% of 50 is 2.

The general formula can be derived, that X% of Y is (X*Y)/100. But that's equal to (Y*X)/100, which is Y% of X, and there's your result.

For those of us who did maths this is obvious, but for those for whom maths was just a list of formulas and processes, applied without any real understanding, I can see that this can come as a surprise.

It's genuinely a lovely thing, and occasionally useful.

Just to be clear - this is a heavily edited and anonymised account of a discussion that happened in a closed group.
Someone engaged with my response, and after some considerable back and forth I ended up writing a few rather longer comments that I'd hoped would explain my thoughts and feelings about some things. Instead of leaving them in that walled garden, I thought I'd extract them and post them here.

So someone asked:

I was trying to get to the 'understand' part.

What does it mean to 'understand' a formula?

How do you enable people to 'understand' such that they can appreciate the beauty you see?

Those are good questions, although it turned out that it wasn't what they were really asking. Even so, I replied:

This is very much a two-stage process, but iterated multiple times. You need to do a bunch of stuff so you become familiar with a thing and gain competence, then you need to reflect on it to allow connections to be made. Then lather, rinse, and repeat. Sadly, most people's experiences of "maths" stops at the "learn how to recognise this magic word and apply this mysterious process."

To get beyond that requires more time than people are usually given, and processes with which the students (of all ages) will engage. That doesn't necessarily mean "relevant", but it does mean engaging - something that the students, each of them, will find intriguing for whatever reason is relevant to them.

For some, that's a practical application, for others it's the thrill of the chase, simply to find out "why is it so?"

Either way, it takes time.

Well, as I said, that wasn't actually the question intended. There was a follow-up:

What triggered me was this part: "It's genuinely a lovely thing,"

What interests me is how that loveliness is being, or can be, conveyed, and to whom, and why.

The reason it triggered me is I think the irritation I often feel when specialists write to each other in a public forum, showing their delight at something but failing to convey it. The public are left on the sidelines, closed out of the delight.

There is an idea much used in writing: don't tell, show. Don't tell me this is beautiful, show me its beauty.

My challenge to you, and to any specialist writing on a public forum, is to help the rest of us share in your delight at the things that please you.

That's a real challenge. How do you explain why something is lovely? Why something is beautiful? Beauty is in the eye of the beholder, and depends critically on one's background and knowledge, one's skills and abilities.

But I replied:

That's a good question, and it's a hard question, one that I spend my life grappling with, because I am a professional science communicator. But I'm not trying to get people sucked into things from the outside, I'm trying to advance them once they've already got a foot, or a toe, in the water.

In this case I wasn't trying to educate, I was expressing my pleasure. Anyone who is already capable enough then knows that we are kindred spirits. For someone in (I assume) your position, I'm making it clear that there is pleasure to be had/found in this particular instance.

But just as late Beethoven quartets are completely opaque to one who doesn't have the experience of early Beethoven, late Haydn, early Haydn, and some "pop" music, so things like this are (or certainly can be) opaque to those without the experience.

Do you know how it feels to hit a spectacularly good shot in tennis, where it hits the sweet spot, goes exactly where you intend, almost without effort? Without putting in time to play the game, you can't appreciate that experience. So it is here. There is no royal road to geometry.

But it's not made deliberately inaccessible, there are plenty of ways to "come up to speed", it's just that reading about maths won't give you the necessary background or understanding. Maths isn't a spectator sport, it's a full-contact sport. You need to get involved to start to appreciate these things.

My direct personal experience is that people tend to start out enthusiastically, wanting to understand something, wanting to appreciate how it works and why mathematicians might call it beautiful, or lovely, or even just "nice", and then give up when they realise how long it will take to get the necessary degree of understanding and ability.

And I get that - people have other things to do with their lives.

But there's no shortcut. I can't immediately explain to you why a particular Henry Moore statue is moving, or why I'm affected by a particular piece of abstract art - you need the experience, not the "understanding".

Perhaps that's overly pessimistic, but actually, I'm full of optimism, because at least some of the thousands of people I come into contact with each year get enthused about things they hadn't previously realised were genuinely exciting.

But it takes time, effort, time, experience, time, work, and time.

The other commenter went on:

  • I do understand that sometimes there is a need for foundations, but the art of the science writer is surely to find the best way do without.

    Beauty is often about cutting through complexity to reach meaning - that's the delight - you've bypassed the hard work; you've found a way "home" by locating a simple sign, that you recognise by the pleasure it gives you.

    Perhaps it's the communication that takes time, work, effort - or just intent.

Let me reply to these three statements in a different order.

  • Perhaps it's the communication that takes time, work, effort - or just intent.

Certainly effective communication takes time, work, effort, as well as skill, practice, and intent. That seems indisputable.

  • Beauty is often about cutting through complexity to reach meaning - that's the delight - you've bypassed the hard work; you've found a way "home" by locating a simple sign, that you recognise by the pleasure it gives you.

I don't understand this - I certainly don't recognise this attempt to characterise "beauty". To me, certainly in mathematics, the pleasure is in seeing things come together, but without the work there is nothing there to do the coming together. So while there are perhaps echoes of what I think "beauty" and "pleasure" in the field is about, that's not really it.

And then:

  • I do understand that sometimes there is a need for foundations, but the art of the science writer is surely to find the best way to do without.

This hits what I think is an important point. I replied thus:

There's a difference between writing to give a sense of an experience, and writing to give someone actual understanding. It's the difference between experiencing the smell of food to appreciate how wonderful it is, and actually eating it to get the nourishment.

There's a lot of "Science Communication" that's about the scent, not the sustenance.

This is something I think about a lot. There are a lot of "Popular Science" books and "Popular Maths" books that talk about the science or about the maths. They're engaging, entertaining, and can provide insights as to what maths is about, but at the end, when the reader puts the book down, what can they do that they couldn't do before?

And does that matter?

I don't have answers to either of those questions, but my worry is that the answer to the first is "nothing" and the answer to the second is "not as much as I think it should."

Being a popular speaker, or a popular writer, is about being entertaining. People would be happy to have been educated, just as I want to have written a book, but people really don't want to be educated, they want to be entertained. In all likelihood nothing we say or do will ever change that, so the question becomes this:

Given the unalterable underlying
ground truth that people don't
want to be educated, they want
to be entertained - what should
we be doing?

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