Maths and Music

An excellent way to kill a conversation is to say you are a mathematician. Tell others you are also a musician, however, and they will be hooked. Although there are obvious similarities between mathematical and musical activity, there is no direct evidence for the kind of magical connection many people seem to believe in.

I’m partly referring here to the “Mozart effect”, where children who have been played Mozart compositions are supposedly more intelligent, including at maths, than other children. It is not hard to see why such a theory would be popular: we would all like to become better at maths without putting in any effort. But the conclusions of the experiment that expressed the belief in the Mozart effect were much more modest. If you want your brain to work better, you clearly have to put in hard work. As for learning to play the piano, it also takes effort.

Surely a connection is quite reasonable. Both maths and music deal with abstract structures, so if you become good at one, then it is likely that you become good at something more general that helps you with the other. If this is correct, it would show a connection between mathematical and musical ability. It would be more like the connection between abilities at football and tennis. To become better at one, you need to improve your fitness and coordination (协调). That makes you better at sport and probably helps with the other.

Abstract structures don’t exist only in maths and music. If you learn a language then you need to understand its abstract structures like grammar. Yet we don’t hear people asking about a connection between mathematical and linguistic (语言的) ability. Maybe this is because grammar feels mathematical, so it wouldn’t be surprising that mathematicians were better at learning grammar. Music, however, is strongly tied up with feelings and can be enjoyed even by people who know little about it. As such, it seems different from maths, so there wouldn’t be any connection between the two.

Let’s see how we solve problems of the “A is to B as C is to D” kind. These appear in intelligence tests but they are also central to both music and maths. Consider the opening of Mozart’s Eine Kleine Nachtmusik (小夜曲). The second phrase is a clear answer to the first. The listener thinks: “The first phrase goes upward and uses the notes of a G major chord (和弦); what would be the corresponding phrase that goes downward and uses the notes of a D7?” Music is full of puzzles like this. If you are good at them, expectations will constantly be set up in your mind. The best moments surprise you by being unexpected, but we need the expectations in the first place.

A company called Neuralink has shared a video where a monkey is playing a video game. That' s fairly unusual, but what makes the video even stranger is that the monkey is playing the video game with just his mind.

The monkey in the video is called Pager who has two of Neuralink's special "Link" devices(装置)inside his brain. The devices planted in Pager's brain are connected to 2,048 wires which lead to the parts of Pager's brain that control movements of the arms and hands.

Scientists taught Pager to play a video game. At first, Pager controlled the video game using a joystick it, which is a normal gaming controller. But as Pager played, his Link devices wirelessly sent out information about the signals his brain was using to control his arms and hands. Neuralink's scientists recorded all of these signals.

Then they used computers to match the signals from Pager's brain to the movements that his hands were actually doing. This was the most difficult work and the scientists counted on artificial intelligence ( AI) to help them decode(解码)Pager's brain signals.

The final step was to have a computer make moves in the video game as if Pager had actually moved the joystick. If Pager thought about moving the joystick up, the computer would send an “up” signal to the video game.

At first, the researchers let Pager keep moving the joystick with his hand, even though it was no longer connected to the computer. But soon Pager was able to play the video game using just his brain.

Even though Neuralink's work right now focuses on animals and video games, there's a very serious purpose behind it. Neuralink wants to make it possible for humans who have lost the ability to make physical movements to interact with the world around them.

Aristotle thought the face was a window onto a person’s mind. Cicero agreed. Two thousand years passed, and facial expressions are still commonly thought to be a universally valid way to judge other people’s feelings, irrespective of age, sex and culture. A raised eyebrow suggests confusion. A smile indicates happiness.

Or do they? An analysis of hundreds of research papers that examined the relationship between facial expressions and underlying ( 潜在的) emotions has uncovered a surprising conclusion: there is no good scientific evidence to suggest that there are such things as recognizable facial expressions for basic emotions which are universal across cultures. Just because a person is not smiling, the researchers found, does not mean that person is unhappy.

This may raise questions about the efforts of information-technology companies to develop artificial-intelligence algorithms (算法) which can recognize facial expressions and work out a person’s underlying emotional state. Microsoft, for example, claims its “Emotion API” is able to detect what people are feeling by examining video footage of them. Another of the study’s authors, however, expressed scepticism. Aleix Martinez, a computer engineer at Ohio State University, said that companies attempting to obtain emotions from images of faces have failed to understand the importance of context.

For a start, facial expression is but one of a number of non-verbal ways, such as body posture, that people use to communicate with each other. Machine recognition of emotion needs to take account of these as well. But context can reach further than that. Dr Martinez mentioned an experiment in which participants were shown a close-up picture of a man’s face, which was bright red with his mouth open in a scream. Based on this alone, most participants said the man was extremely angry. Then the whole picture was shown. It was a football player with his arms outstretched, celebrating a goal. His angry-looking face was, in fact, a show of pure joy.

Given that people cannot guess each other’s emotional states most of the time, Dr Martinez sees no reason computers would be able to. “There are companies right now claiming to be able to do that and apply this to places I find really scary and dangerous, for example, in hiring people,” he says. “Some companies require you to present a video resume, which is analyzed by a machine-learning system. And depending on your facial expressions, they hire you or not, which I find really shocking.”