At a dinner I attended some years ago, the distinguished differential geometer Eugenio Calabi volunteered to me his tongue-in-cheek distinction between pure and applied mathematicians. A pure mathematician, when stuck on the problem under study, often decides to narrow the problem further and so avoid the obstruction. An applied mathematician interprets being stuck as an indication that it is time to learn more mathematics and find better tools.
I have always loved this point of view; it explains how applied mathematicians will always need to make use of the new concepts and structures that are constantly being developed in more foundational mathematics. This is particularly evident today in the ongoing effort to understand “big data” — data sets that are too large or complex to be understood using traditional data-processing techniques.
Our current mathematical understanding of many techniques that are central to the ongoing big-data revolution is inadequate, at best. Consider the simplest case, that of supervised learning, which has been used by companies such as Google, Facebook and Apple to create voice- or image-recognition technologies with a near-human level of accuracy. These systems start with a massive corpus of training samples — millions or billions of images or voice recordings — which are used to train a deep neural network to spot statistical regularities. As in other areas of machine learning, the hope is that computers can churn through enough data to “learn” the task: Instead of being programmed with the detailed steps necessary for the decision process, the computers follow algorithms that gradually lead them to focus on the relevant patterns.
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Ever since you were little, you’ve seen triangles everywhere. You find them in jack-o’-lantern eyes, corporate logos, and grilled cheese sandwiches halved diagonally. They crop up in all kinds of construction projects: the pyramids, the supports beneath the Golden Gate Bridge, the tracks of roller coasters. When architects and engineers want a shape that’s sturdy and dependable, they turn to the triangle. There’s only one problem.
Triangles don’t exist.
I don’t mean to alarm you, and I hope I’m not spoiling any fond childhood memories of geometric forms. But triangles are like Santa Claus, the tooth fairy, and Beyoncé: too strange and perfect to exist in the actual world.
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Buffett has taken the criticism from these fellow giants of finance in his stride, responding with trademark wit and humour. He even compared himself to an orang-utan flipping coins. Joking aside, this is a testable hypothesis: Is Buffett’s performance better than chance? To test it, we will stand on the shoulders of another giant: Jacob Bernoulli.
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Again it is a very small number, but we can use our formula to calculate its value:
The expected value is much smaller than 1, so we can conclude that Buffett is a better investor than the luckiest orang-utan. If stock returns really do follow a random process – as Eugene Fama asserted – then Warren Buffett is more than just lucky. Compared with his competitors in the S&P 500, he’s brilliant.
From fluid dynamics to finance, creatures like the Weierstrass function have challenged our ideas about the relationship between mathematics and the natural world. Mathematicians around the time of Weierstrass used to believe that the most useful mathematics was inspired by nature, and that Weierstrass’ work did not fit into that definition. But stochastic calculus and Mandelbrot’s fractals have proven them wrong. It turns out that in the real world—the messy, complex real world—monsters are everywhere. “Nature has played a joke on the mathematicians,” as Mandelbrot put it. Even Weierstrass himself fell victim to the trick. He created his function to argue that mathematics should not be based only on physical observations. His followers believed that Newton had been constrained by real-life intuition and that, once free of these limitations, there were vast, elegant new theories to be discovered. They thought that mathematics would no longer need nature. Yet Weierstrass’ monster has revealed the opposite to be true. The relationship between nature and mathematics runs deeper than anyone ever imagined.
Illustration : Alessandro Gottardo
Ultimately, it all comes back to counting things, and counting is a universally handy tool. Which in turn makes the encyclopedia handy, too. “Suppose you are working on a problem in one domain, say, electronics, and while solving a problem you encounter a sequence of integers,” said Manish Gupta, a coding theorist by training who runs a lab at the Dhirubhai Ambani Institute of Information and Communication Technology. “Now you can use the encyclopedia and search if this is well known. Many times it happens that this sequence may have appeared in a totally unrelated area with another problem. Since numbers are the computational output of nature, to me, these connections are quite natural.”