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The mathematics of discovering new things

A visualization of the new model for how one new idea allows others. (Tria et al)

Here’s your cool math for the day: a model showing that when one new thing happens, more will follow.

It seems like common sense. If you listen to a new artist on your online music engine for the first time, you’ll likely become aware of many other songs by that artist that you’ve never heard before. And by discovering one new artist's music, you’ll be opened up to songs that the algorithm finds to be similar to it.

The same occurs in biological evolution. Birds are a good example: As dinosaurs, they probably developed rudimentary feathers for warmth. But as those feathers got more complex, they opened up the possibility of flight. The power of flight put birds into a new evolutionary niche, and more novel adaptations followed. It's all part of what we call the adjacent possible - the new things that can only be thought up once something else exists. There was no use for a cellphone case before a cellphone, after all.

So to Cornell University mathematics professor (and popular math communicator) Steven Strogatz, the paper that he and his colleagues released in Scientific Reports looks at evolution in the broadest sense of the word. “We have lots of examples of this idea that one new thing leads to another,” Strogatz said, “but they’re all just stories. It’s hard to make a science of it.” The fingerprints of the effect must exist, he said, “but we don’t even know what those prints look like, so we don’t know that we’re not seeing them.”

To explain the mathematical prints they detected, Strogatz and colleagues use a model that’s common in mathematics: An urn is filled with different colored balls, and one is drawn at random. The selected ball is returned, along with several new balls in the same color. So when one color is selected, it increases its color's odds of further selection.

But the new model gets a bit more complex.

Instead of a color, the ball is labeled with a concept — the word “dance” for example. And when it’s added back, 10 more “dance” balls are added. But so are 10 balls with related concepts on them — ballet, hip-hop, music, and so on. The discovery of a new idea leads to the spread of related ideas soon after.

“They’re not cropping up by chance,” Strogatz said. "They’re happening in bursts.”

The mathematicians saw this pattern when looking at the edit events of Wikipedia pages, the emergence of tags in annotation systems, the sequence of words in texts, and songs listened to in online music catalogues.

“My hope is that scientists will look at this and realize that they have much better databases to observe this in,” Strogatz said. He’s particularly hopeful that experts in the history of technology will think of the model when looking at historical patents to see how technological innovation evolves.

“People listening to music, filing patents, evolution, it’s all the same thing to a mathematician,” Strogatz said. “We like to sort of fly above and take a bird’s eye view of many things that seem to be different. But we hope that people in other fields will realize that this has been under their noses all the time, and there’s science to be done here.”