How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

The two rails are parallel. But the two planes are not.

If two different lines are both horizontal, they are parallel; and yet, in projective geometry, they meet, at the point at infinity.

Shannon, in the paper that launched the theory of information, identified the basic tradeoff that engineers still grapple with today: the more resistant to noise you want your signal to be, the slower your bits are transmitted. The presence of noise places a cap on the length of a message your channel can reliably convey in a given amount of time; this limit was what Shannon called the capacity of the channel.

After Hamming and Shannon, it sufficed to make errors rare enough that the flexibility of the error-correcting code could counteract whatever noise got through. Error-correcting codes are now found wherever data needs to be communicated quickly and reliably.

Math moves faster than the patent office:

Hamming distance, which was adapted to the new mathematics of information just as the distance Euclid and Pythagoras understood was adapted to the geometry of the plane. Hamming’s definition was simple: the distance between two blocks is the number of bits you need to alter in order to change one block into the other.

You can make language more efficient—but when you do, you hit the same hard tradeoff Shannon discovered.

So the problem of constructing error-correcting codes has the same structure as a classical geometric problem, that of sphere packing: how do we fit a bunch of equal-sized spheres as tightly as possible into a small space, in such a way that no two spheres overlap? More succinctly, how many oranges can you stuff into a box?

The most customary choice, called the face-centered cubic lattice, has the nice property that every layer has the spheres placed directly over the spheres three layers below. According to Kepler, there is no denser way to pack spheres in space. And in the face-centered cubic lattice, each sphere touches exactly twelve others.

Codes that have a lot of structure, like the Hamming codes, tend to be easy to decode. But these very special codes, it turns out, are usually not as efficient as the completely random ones that Shannon studied! And in the decades between then and now, mathematicians have tried to ride that conceptual boundary between structure and randomness, laboring to construct codes random enough to be fast, but structured enough to be decodable.

One popular account was offered by our old buddies Milton Friedman and Leonard Savage, who proposed that lottery players follow a squiggly utility curve, reflecting that people think about wealth in terms of classes, not numerical amounts. If you’re a middle-class worker who spends five bucks a week on the lottery, and you lose, that choice costs you a little money but doesn’t change your class position; despite the loss of money, the negative utility is pretty close to zero. But if you win, well, that moves you into a different stratum of society. You can think of this as the “deathbed” model—on your deathbed, will you care that you died with a little less money because you played the lottery? Probably not at all.

This is regret optimizing

Economics isn’t like physics and utility isn’t like energy. It is not conserved, and an interaction between two beings can leave both with more utility than they started with.

Complete freedom to enter trade and the continuance of competition mean the perpetuation of mediocrity.

To be fair, Darwin might have been biased, being Galton’s first cousin.

But quantifying the heredity of “genius” wasn’t so easy: how, exactly, was one to measure just how notable his notable Englishmen were? Undeterred, Galton turned to human characteristics that could be more easily placed on a numerical scale, like height. As Galton and everyone else already knew, tall parents tend to have tall children. When a six-foot-two man and a five-foot-ten woman get married, their sons and daughters are likely to be taller than average. But now here is Galton’s remarkable discovery: those children are not likely to be as tall as their parents. The same goes for short parents, in the opposite direction; their kids will tend to be short, but not as short as they themselves are. Galton had discovered the phenomenon now called regression to the mean. His data left no doubt that it was real.

Excellence doesn’t persist; time passes, and mediocrity asserts itself.*

Tall people are tall because their heredity predisposes them to be tall, or because external forces encourage them to be tall, or both. And the taller a person is, the likelier it is that both factors are pointing in the upward direction.

regression to the mean is “theoretically a necessary fact.”

When the son’s height is completely unrelated to those of the parents, as in the second scatterplot above, Galton’s ellipses are all circles, and the scatterplot looks roughly round. When the son’s height is completely determined by heredity, with no chance element involved, as in the first scatterplot, the data lies along a straight line, which one might think of as an ellipse that has gotten as elliptical as it possibly can.

If two measurements are highly correlated (like the length of the left foot and the length of the right) there’s little point in taking the time to record both numbers. The best measurements to take are the ones that are uncorrelated with each of the others.

In modern terms we would say that the more strongly correlated the measurements, the less information, in Shannon’s precise sense, a Bertillon card conveys.

it’s precisely this fact that makes it possible* to have compression, the critical mathematical technology that allows images, videos, music, and text to be stored in much smaller spaces than you’d think. The presence of correlation makes compression possible;

Mathematics is a way not to be wrong, but it isn’t a way not to be wrong about everything.

In the post-Galton world, you could talk about an association between two variables while remaining completely agnostic about the existence of any particular causal relationship, direct or indirect. In its way, the conceptual revolution Galton engendered has something in common with the insight of his more famous cousin, Charles Darwin. Darwin showed that one could meaningfully talk about progress without any need to invoke purpose. Galton showed that one could meaningfully talk about association without any need to invoke underlying cause.

Mathematicians ever since Descartes have enjoyed the wonderful freedom to flip back and forth between algebraic and geometric descriptions of the world. The advantage of algebra is that it’s easier to formalize and to type into a computer. The advantage of geometry is that it allows us to bring our physical intuition to bear on the situation, particularly when you can draw a picture.

That’s the dirty little secret of advanced geometry. It may sound impressive that we can do geometry in ten dimensions (or a hundred, or a million . . .), but the mental pictures we keep in our mind are two- or at most three-dimensional. That’s all our brains can handle. Fortunately, this impoverished vision is usually enough.

Why study geometry that isn’t realized in the universe? One answer comes from the study of data, currently in extreme vogue. Remember the digital photo from the four-megapixel camera: it’s described by 4 million numbers, one for each pixel. (And that’s before we take color into account!) So that image is a 4-million-dimensional vector; or, if you like, a point in 4-million-dimensional space. And an image that changes with time is represented by a point that’s moving around in a 4-million-dimensional space, which traces out a curve in 4-million-dimensional space, and before you know it you’re doing 4-million-dimensional calculus, and then the fun can really start.

Important real life example

this is Pearson’s formula, in geometric language. The correlation between the two variables is determined by the angle between the two vectors. If you want to get all trigonometric about it, the correlation is the cosine of the angle. It doesn’t matter if you remember what cosine means; you just need to know that the cosine of an angle is 1 when the angle is 0 (i.e., when the two vectors are pointing in the same direction) and −1 when the angle is 180 degrees (vectors pointing in opposite directions). Two variables are positively correlated when the corresponding vectors are separated by an acute angle—that is, an angle smaller than 90 degrees—and negatively correlated when the angle between the vectors is larger than 90 degrees, or obtuse. It makes sense: vectors at an acute angle to each other are, in some loose sense, “pointed in the same direction,” while vectors that form an obtuse angle seem to be working at cross purposes. When the angle is a right angle, neither acute nor obtuse, the two variables have a correlation of zero; they are, at least as far as correlation goes, unrelated to each other.

Correlation is not transitive.

If correlation were transitive, medical research would be a lot easier than it actually is.

Galton’s notion of correlation is limited in a very important way: it detects linear relations between variables, where an increase in one variable tends to coincide with a proportionally large increase (or decrease) in the other. But just as not all curves are lines, not all relationships are linear relationships.

As the voters get more informed, they don’t get more Democratic or more Republican, but they do get more polarized: lefties go farther left, right-wingers get farther right, and the sparsely populated space in the middle gets even sparser. In the lower half of the graph, the less-informed voters tend to adopt a more centrist stance.

Remember: the expected value doesn’t represent what we literally expect to happen, but rather what we might expect to happen on average were the same decision to be repeated again and again.

one thing’s for certain: refraining from making recommendations at all, on the grounds that they might be wrong, is a losing strategy. It’s a lot like George Stigler’s advice about missing planes. If you never give advice until you’re sure it’s right, you’re not giving enough advice.

The “majority rules” system is simple and elegant and feels fair, but it’s at its best when deciding between just two options. Any more than two, and contradictions start to seep into the majority’s preferences.

mistake is the same one that constantly trips up attempts to make sense of public opinion: the inconsistency of aggregate judgments.

Where does irrationality come from? We’ve seen already that the apparent irrationality of popular opinion can arise from the collective behavior of perfectly rational individual people. But individual people, as we know from experience, are not perfectly rational.

This is a story told in mathematics again and again: we develop a method that works for one problem, and if it is a good method, one that really contains a new idea, we typically find that the same proof works in many different contexts,

More generally, formalism in the law manifests itself as an adherence to procedure and the words of the statutes, even when—or especially when—they cut against what common sense prescribes. Justice Antonin Scalia, the fiercest advocate of legal formalism there is, puts it very directly: “Long live formalism. It is what makes a government a government of laws and not of men.” In Scalia’s view, when judges try to understand what the law intends—its spirit—they’re inevitably bamboozled by their own prejudices and desires. Better to stick to the words of the Constitution and the statutes, treating them as axioms from which judgments can be derived by something like logical deduction.

Even Justice Scalia has occasionally conceded that when the literal words of the law seem to require an absurd judgment, the literal words have to be set aside in favor of a reasonable guess as to what Congress must have meant.

What you learn after a long time in math—and I think the lesson applies much more broadly—is that there’s always somebody ahead of you, whether they’re right there in class with you or not.

by 2050, the noise had engulfed the signal.

With a few uncertainties here and there, and given a long eough time period, the signal is always always drowned by the noise. Thats why you need to keep it simple ans straightforward. And always keep an open mind

Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.