It’s said that necessity is the mother of invention. Seems sensible enough – we probably are more likely to break out of conventional thinking and discover new ways to see and solve a problem when we’re faced by a powerful need, if only out of desperation.
But I think the aphorism misses the real heart of invention.
It certainly breaks down if we expand our definition of “invention” to include art; we are forced to retreat to generalities (the world needs art) or personal motivation (the artist needs to create) and to me that feels like playing word games. It similarly skirts much of our theory-building effort in science and seems to me to force out almost the whole of current human endeavor in mathematics.
On reflection, it seems a pretty poor mother to me. Plus what about the dad?
What got me thinking about this was a recent conversation with my friend Rudi Seitz. We were talking about the problem of counting all the distinct chord types in a 12-tone equal temperament musical scale. This is equivalent to determining the number of distinct bracelets made of up 12 black or white beads (where two bracelets are equivalent if they can be rotated to match up exactly). This is a solved problem … it’s easy to either generate the 4096 possible combinations and eliminate equivalents or just apply the Polya enumeration theorem to crank out the correct answer of 352. There is no need for any further invention here.
Rudi is not satisfied, however. He wants an algorithm that will not just generate and count the necklaces, but also somehow give visual insight into why the answer is what it is and ideally even into what the properties of distinct necklace (or chord) types might be. He confessed to suffering from a similar fetish throughout graduate school. A short, clear, elegant proof was fine as far as it went, but how much more wonderful it would be to find a visual proof that showed the reasoning required to get from hypothesis to conclusion. Just before parting, we talked about whether this was a silly thing to want to do.
This post is mostly an attempt to figure out what my answer to that question actually is and why.
I first went to arguments about why such an endeavor was useful. At the least there is some pedagogical value to an approach that leverages more of students’ senses. But the more I thought about it, the less this line of thought appealed to me. I have no idea whether Rudi’s effort is actually useful or whether anyone other than Rudi needs to have it done. Nevertheless my intuition is that this is very much the kind of activity that lies at the heart of innovation.
In fact it includes what I propose as the two true parents of invention: curiosity and constraint (I’ll avoid controversy by leaving gender-assignment as an exercise for the reader).
Necessity may or may not be what gets us started, but I think curiosity is probably the most powerful driver we have for trying something new. I wonder what will happen if … is a secret formula for new perspectives. At its best, curiosity is often irrational and silly – it stands things on their heads for no good reason at all. It just wants to see what will happen. That is its strength. Rationality always leads us to extend what we already know and to do things as we have always done. Curiosity takes us outside the lines to find something new.
But curiosity needs a partner and I think the real partner is constraint. Can we do it twice as fast? Can we do it with just these resources? Can we do it with fewer? Can we make it beautiful too? Can we push the tolerance down another factor two or ten or one hundred? Can we do it in 140 characters? Can we do it using this technique or without using that one? Can we do it with just primary colors or from a new point of view? Or how about this favorite of mine for wanna-be sciences like economics: can we do it so that it actually gives effective and useful predictions and prescriptions?
In many cases there are practical reasons why a constraint is important, but it can also be a simple question of elegance or preference, as in Rudi’s case or, to pick just one of the nearly infinite examples in art: haiku. If the constraint is a severe one, it forces us to look at the task differently. I cannot condense a 1000-word essay into 17 syllables by editing it down – a radical shift is required to convey the ideas in this new format. And in the process, it is highly likely that I’ll gain new insight, perhaps discover something I didn’t previously know about my own ideas. I believe the same holds true in any area, whether art or physics, commercial products or theoretical mathematics: even when a constraint is entirely voluntary or even whimsical, it has the potential to unleash interesting and powerful results.
I think mathematicians know this, although they have a strong preference for certain kinds of constraints. Take the recent proof of the bounded gaps conjecture by Yitang Zhang (a very cool story, by the way). He was able to prove that there are infinitely many pairs of primes p and q such that the gap between them is less than some number H and was able to show that H is at least as small as 70,000,000. As soon as the result was out, the math community set out to improve on it, to narrow the gap and, ideally, to find shorter and more elegant ways to achieve it. As of June 30, a collaborative project hosted by Terence Tao had pushed H down to 12,006 and provisionally as low as 6,996. They already know that the path they’re on won’t get them all the way to the ultimate H=2 twin prime conjecture, but they work on anyway. Why? I suspect because they know that such efforts are often fruitful. They may only lead to a better understanding of the current method and why it can’t take them all the way, but they may also lead to a creative leap that does. And there may be gems along the way, anything from a connection to other mathematical results to a glimpse of a whole new field to explore. Of course, this is a very practical example of the principle – Tao and collaborators can be pretty certain that their effort will have some pay-off, even if none of the more exciting possibilities arise.
Rudi’s constraint is far more whimsical and less likely to yield any results of practical interest. At the same time, I think it’s also the kind of constraint that’s much more likely to generate really interesting results if it generates any at all.
So I guess my answer to the question “Is it a silly thing to do?” is that while it’s probably a silly thing to do, I think it’s probably also a worthwhile thing to do.