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The unreasonable effectiveness of mathematics

A recent lecture by Prof. Alan Huckleberry made me revisit this old essay by Eugene Wigner, and prepare an episode for Engines about it:

The 16th century astronomer Galileo Galilei claimed that the language of nature is mathematics. As far as we can see, Galileo was right. But why is math the language of nature? And why is that language understandable to us?

These questions puzzled the physicist Eugene Wigner. Wigner and other scientists who were part of the revolution that brought us quantum mechanics and relativity, used math to express the laws of nature. To their surprise, they found that mathematicians had already invented the language they needed. They’d developed this language without thinking of whether or how their ideas could be applied.

This made Wigner ask why math is so unreasonably effective in describing nature. Much is hidden behind this question: First, why do general laws of nature exist? Even if we assume that without such laws all would be chaos, it is still a wonder that we can discover and understand them. But let us accept that that the nature is humanly comprehensible.

What still puzzled Wigner is that we have a language of our own making ready at hand to describe the world around us. The words, phrases and ideas of mathematics we need to talk about the laws of nature are often available when we need them. In math ideas are developed because they naturally flow from previous theories, and because mathematicians find them beautiful. It is then somewhat of a miracle that some of these ideas can be applied not just in physics, but in most other sciences.

Some have argued that math is not as useful as it seems. Maybe we focus too much on problems where math happened to be of great help. In fields like medicine, economics, and the social sciences general laws have been harder to come by. For example, try to find a short list of rules for the English language. Many have tried and failed.

Some data scientists have therefore argued that we need to embrace the complexity of such systems. We should forget the elegance of math for more pragmatic approaches, and let the data guide us. Yet in practice most of these statistical and machine learning approaches still rely on math.

Mathematics may not be able to unlock all mysteries. But it is still very useful – it lets us describe and understand things infinitesimally small, unimaginably large, and events far in the past and the future. Math gives us a glimpse into realms that we can’t directly experience, and where our intuitions are of no use.

Wigner concludes by saying that mathematics is a wonderful gift which we neither understand, nor deserve. He expresses a hope that it will continue to be useful to our continued surprise. I do believe that we will increasingly rely on computers to help us make sense of the world around us. Yet, I am sure that math will remain the language we will keep using in this conversation.

Notes:

Wigner’s original essay can be found here. Although over 50 years old, it asks questions that we are not much closer to answering. There have been many follow-ups to Wigner’s essay. I have drawn from the ideas of the mathematician R. W. Hamming, and the engineer Derek Abbott.

The precise quote from Galileo “[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.” More quotes here

I thank Prof. Alan Huckleberry for emphasizing that there is a natural flow of ideas and developments in mathematics that become apparent in retrospect. Yet, I would like to add that there is also a cultural component to these developments. It may thus be better to view the co-development of physics and mathematics as part of a larger cultural evolution of ideas. Perhaps in this larger context the relation between mathematics and physics is more natural. I leave this discussion to historians of science.

Libration points and space travel

Science fiction writers imagine that our descendants will use wormholes to travel instantaneously across galaxies without using much energy. We do not yet know how to cross vast distances instantaneously. However, using celestial mechanics – the mathematics that describes the motion of celestial bodies – astrophysicists have found ways of traversing space using very little fuel.

To understand how this is possible, think first of the Earth and the Moon. Each exerts a gravitational pull on objects, while they spin around each other. Connect the centers of the Earth and the Moon with an imaginary line. Somewhere along this line, the pull of the Earth equals that of the Moon, but in the opposite direction. Placing a spacecraft at this point is like placing it on a knife’s edge: A small nudge will break the balance of forces and send it drifting towards the Earth or the Moon. This position is called a “libration point”. The mathematicians Leonhard Euler and Joseph-Louis Lagrange showed that there are five libration points in a system of two rotating bodies such as the Earth and the Moon. At these points the combined gravitational forces of the two celestial bodies give exactly the right pulls to keep an object orbiting in an unchanging position.

Two of these libration points are stable – satellites placed at these points will remain there forever. Indeed, many asteroids can be found at the two stable libration points of the Sun-Jupiter system. These are known as the Trojan asteroids: One libration point is home to the Greek camp, while the other houses the Trojan camp.

However, the remaining three libration points are unstable, and these are key to low energy space travel. Imagine placing a marble exactly at the ridge of a saddle. The marble will roll down the ridge towards the saddle’s center, but a small nudge can make it veer down either side. The unstable libration points are similar to the center of the saddle – a spacecraft at the right position going at the right velocity travels along something like the saddle’s ridge approaching the libration point. A small nudge can make it veer down different paths on either side of the saddle. If this is done just right, the paths leading away from the ridge can take the spacecraft along just the orbit we want it to follow.

Designing a low energy orbit is like rolling a marble along saddle ridges, and providing the right nudges at the right time to leave the ridge along the desired exit trajectory. However, in the Solar System everything is in motion, so the saddles and the ridges keep spinning and swerving through space! Essential to mission design are good mathematical models of the solar system, and the computational tools to solve these models accurately.

Traversing the resulting trajectories can take a long time. However, their benefits can be great. The spacecraft Genesis traveled along such a trajectory to sample the solar wind while using only a fraction of the fuel expended by similar spacecraft in the past.

For thousands of years humans have tried to understand the motion of the planets. The language of mathematics can lead to such an understanding. But mathematics has also provided much more – it has told us how to traverse the Solar system, and may some day lead us far beyond.

Notes:

I am grateful to independent astrodynamics consultant Daniel Adamo, retired NASA planetary scientist Wendell Mendell, and astronaut Michael Barratt for their help in preparation of this episode.

I simplified the explanation of the trajectories between libration points here. They are in fact not straight lines, but take the form of spirals on the surface of tubes. These tubes approach closed orbits that surround the libration points. We are accustomed to the closed orbits of planets around the Sun (or any object that has mass) described by Kepler’s Laws. But libration points have no mass. It may therefore be surprising that we have closed orbits around points of no mass! A discussion of the mathematics of interplanetary travel along the orbits I discussed can be found here.  A number of further references, including more technical ones are provided in this article. Note that the feasibility of interplanetary missions using the same principles remains controversial.

More about Lagrangian (or Lagrange) points can be found here. The Wikipedia page also has an intuitive explanation of why they appear, and it also has a nice picture of the Jupiter Trojans.

Libration points are sometimes referred to as Lagrange points. Joseph-Louis Lagrange was an interesting figure himself. Read the entry for an overview  of his rich life that spanned the French Revolution.

Review of “Statistics Done Wrong” by Alex Reinhart

This is a book review for the applied math journal SIAM Review.  Comments are welcome.  An short version of the book can be found here.

Most of us accept that statistics is not applied mathematics. The goal of statistics is to obtain answers from data, and mathematics just happens to be an exceptionally useful tool for doing so. However, for many applied mathematicians
statistical analysis is an integral part of daily work. Experimental studies often motivate our research, and we use data to develop and validate our models. To understand how experimental outcomes are interpreted, and to communicate with scientists in other fields, a knowledge of statistics is indispensable.

One issue that we need to take seriously is that misapplications of statistics have lead to false conclusions in much, maybe even most published studies [1]. Although the “soft sciences” have received most scrutiny in this regard [3], the “hard sciences” suffer from closely related problems [2]. Anybody who uses the results of statistical data analysis – and this includes most applied mathematicians – needs to be aware of these issues.

As the title suggests, Alex Reinhart’s “Statistics Done Wrong” [4] is not a textbook. Rather, its aim is to explain some of
the ways in which data analysis can, and often does go wrong. The book is related to Darrel Huff’s classic “How to Lie with Statistics” [5] which covers many topics that are now part of freshman courses. Huff, a journalist (and later consultant to the tobacco industry) provided a lively discussion that alerted general readers to the misuses of statistics by the media and politicians. Since the first edition of Huff’s book in 1954 computational power has increased immensely. But our increased ability to collect and analyze data has also made it easier to misapply statistics. Reinhart’s book aims to introduce the present consumers of statistics to the resulting problems, and suggests ways to avoid them.

The book starts with a review of hypothesis testing, p-values and statistical power. Here Reinhart introduces a recurrent topic of the book: the errors and “truth inflation” due to the preference of most journals and scientists to publish positive results. The ease with which we can analyze data makes the problems of multiple comparisons and double dipping particularly important. The book provides a number of thoughtful examples to illustrate these issues. The last  few chapters provide good guides to data sharing, publication, and reproducibility in research. Each chapter ends with a list of useful tips.

Most of these issues are more subtle than those discussed by Huff [5]. While not heavy on math, the book presents arguments that require reflection. The ideas are frequently illustrated using well chosen examples, making for an entertaining read. The book is thus informative, yet easy to read.

Reinhart predominantly discusses issues resulting from the misuse of frequentist statistics. This is understandable, as the frequentist approach is currently dominant in most sciences. However, it is worth noting that Bayesian approaches make it easier to deal with some of the main problems discussed in the book. Bayesian statistics makes it easier to deal with multiple comparisons, and replaces p-values with measures that are easier to interpret. However, it is not a magic bullet – as Bayesian approaches become more common over the next decades, we may need another volume describing their misuses.

What is the audience for this book? Many of the topics need to be familiar to anybody doing science today. The book could also provide good supplementary material for a second course in statistics.

Doing statistics can be tricky. Finding the right experimental design requires a careful consideration of the question to be answered. The interpretation of the results requires a good understanding of the methods that are used. All statistical models are by necessity approximate. Knowing how to verify that the underlying assumptions are  reasonable, and choosing an appropriate way to analyze data is essential. A central point here is that the statistical analysis deserves as much
attention as the conclusions we draw from it. And perhaps the most important lessons of this book is that questions of statistical analysis should be addressed when the research is planned.

Reinhart’s book is not a comprehensive list of the different ways in which misuses of statistics can lead us astray. It provides no foolproof answers on how to detect problems in statistical analysis. However, it does an excellent job of introducing a range of common pitfalls, and provides sensible tips on hows to avoid them. Doing statistics means accepting that we will be wrong some of the time. The best we can do is to maximize our chances of being right, and understand how likely it is that we are not.

References:

  1. Ioannidis, J. P. A. Why most published research findings are false. PLoS Medicine, 2:8, (2005) e124. http://doi.org/10.1371/journal.pmed.0020124
  2. Button, K. S., et al. Power failure: why small sample size undermines the reliability of neuro- science, Nat Neuroscience, 14, p. 365-376 112. (2013) http://doi.org/10.1038/nrn3475
  3. Open Science Collaboration. Estimating the reproducibility of psychological science. Science 349:6251,p. 943. (2015) http://doi.org/10.1126/science.aac4716
  4.  Reinhart, A. Statistics done wrong. No Starch Press (2014).
  5. Huff, D. How to Lie with Statistics. Norton, W. W. & Company, Inc. (1954).

Statistics and clinical trials

In the night of December 13, 1799, George Washington woke his wife Martha to tell her that he was feeling ill. Following the medical practice of the day, he was bled repeatedly and given an assortment of medicines, some of which contained mercury. By the time Washington died, half of his blood had been removed. He may have lived longer had the doctors simply done nothing.

Washington’s case is not unique. Doctors of the past did not know whether their medicines worked. Some, like quinine, were real cures. Most others, like lead and mercury, did more harm than good. Until the second half of the 20th century physicians did not have the tools to decide which medicines help patients. And it was not microscopes or sophisticated lab equipment they were lacking. Rather, they did not know how to reliably test and compare different treatments.

Suppose you want to test whether a drug helps people sleep better. You could give it to many patients, and ask them if their sleep has improved. But how can you tell whether the results are due to chance, or the placebo effect? And how many people do you need to ask to conclude that the drug works? One answer is to divide patients randomly into two groups. Give the drug to those in the test group, and not to those in the so-called control group. If those in the test group fare better, the drug is effective.

This may sound simple, but it is not. Some people in both groups will still sleep better by chance. Fortunately, if you have sufficiently many patients, such differences even out. How many patients do you need, and how confident can you be in your conclusions? The mathematical theory of probability and statistics provide precise answers to these questions.

The ethical questions are more difficult. In such clinical trials we give the drug to some people and not to others. How can we withhold a potentially life saving drug from some dying patients, and give it to others? But remember – before a treatment is proven to work, it is possible that doing nothing is the better option. The reason we test treatments is that we only suspect that they work. Doctors, like the rest of us, are not clairvoyant. They do not know what treatment is best until it is tested.

As an example, antiarrhythmic drugs were given to patients for decades to stop irregular heartbeats. Doctors argued that it was self-evident that these drugs saved lives. The drugs were eventually tested in clinical trials, but only because doctors believed they should be used more widely. However, a statistical analysis showed that these supposedly life saving medicines were killing patients each year. The drugs may have been responsible for more than 50,000 deaths in the US alone. What seemed obvious and intuitive turned out to be very wrong.

We laugh at the medieval use of leaches, and other remedies that help “balance the humors”. But even today we sometimes base our decision on intuition rather than evidence. Many avoid vaccinations which have been proven to be safe and effective. On the other doctors and patients frequently demand expensive medical tests when none are needed. Clinical trials and statistics can tell us which treatments work. It is up to us to make use of that knowledge.

Some notes:

  1. How much doctors rely on proven medicines, and how much they go by instinct and guesses is a matter of debate.
  2. I recommend Druin Burch’s book “Taking the Medicine” for a look into the history of clinical trials.
  3. Much has been written about the modern anti-vaccine movement. Here an older, but still relevant article in Wired 
  4. The ethics of randomized clinical trials is a complicated subject. I have written about a particular case here.  You will find links to more detailed discussions here.
  5. Here is a good article about over treatment.

How to guide graduate students to good research projects?

I think it is great when graduate students come up with problems to work on by themselves. There is probably no better preparation for a research career.  Unfortunately, grad students in math only have about 3-4 years to learn about a field, produce new results, publish their research and write a thesis. This does not leave a lot of time for exploration. The role of the advisor is to try help select the most fruitful directions. Here are some of the questions that I found helpful:

1) Is the problem new, or has it been answered in some form before?

This is essential since you don’t want to have a student writing a thesis or a paper on something that is already known – and if it is known, it is good to find how others did it. This can then naturally lead to further questions.

2)  Is the question interesting?

This is harder. One way is to have students think about how they could convince other graduate students that this problem is worth studying. After they can think about how to convince non-scientists.

3) Is it the question related to what we are doing in the lab/research group? Can it be answered in a reasonable time?

This is a more practical point. More senior graduate students that already have the majority of the thesis written, can have more leeway.

I am sure there are a number of other criteria here. In his TED talk, Uri Alon says that he approaches research like improv theater. This is great once a question is clearly articulated, and the group is looking for a way forward. Picking the right problem to work on is tricky. What other questions can help guide graduate students to a good problem, while encouraging them to take a role in designing their own research?

Compressive Sensing

In his short story “On Exactitude in Science” Jorge Luis Borges describes a map so detailed that it is the exact size of the empire it represents. Every bridge, road and house on the map is the size of their real counterpart. Of course, such a map is absurd. Maps are useful because they show the most important features of a location – they give us only the information we need. Even a map that is more than a few feet across is useless for navigation. The job of a mapmaker is then to discard enormous amounts of detail, and show us only what we need to know to get from one place to another.

This is not only true for maps. Our megapixel cameras are similar to the cartographers in Borges’ story. Each photo we take contains enormous amounts of data. However, the information that we need to reconstruct a good approximation of each image is much smaller. This is why photos are almost always compressed when stored. Indeed, most compression works by discarding some of the unnecessary information in the original pictures.

But if much of the information in a photo is thrown away afterwards, then why do we need megapixel cameras? Why not just record the essential information about the scene in front of us to start with. If we did so, we would not have to compress the files later.

How to do this has long puzzled scientists and mathematicians. The problem is that one approach may work for some pictures, but fail for others. How can we make a camera that will always only make measurements we need to reconstruct a picture?

The mathematician Emmanuel Candés stumbled upon the answer almost by chance while trying to remove noise from an image. He knew that an exact algorithm that would identify only relevant information would require far too much time to run. He therefore tried an approach that at first sight should not have worked well. To his surprise he recovered almost all the information in the original image – the process worked like magic. Candés said later: “It was as if you gave me the first three digits of a 10-digit bank account number — and then I was able to guess the next seven.”

This was the first step in the development of what is now called “compressive sensing”: A way to take good pictures by taking very few measurements. Surprisingly, rather than carefully planning which measurements to take, it is best to take measurements at random. When this was first proposed, many thought it must be wrong. How could taking measurements at random work better than a sophisticated algorithm? Later mathematical analysis answered these doubts, however. As a result, compressive sensing is used in different areas of medical imaging, and its applications are growing.

We are awash in information. But the main problem is not how to acquire more. Like cartographers, we need to find what matters, and discard the rest. And here, like in so many other matters, it is mathematics that shows the way.

Some notes:

Here is an interview with one of the discoverers of compressive sensing, Emmanuel Candés. Follow the link to a more technical review article.

Here is a good article in Wired that gives a more detailed overview of the origins of compressive sensing  Here is another overview with a bit more math, but also very understandable. Here is also an understandable lecture.

Simpson’s paradox

Here is the text of another episode for Engines. For a great illustration of this idea, see this excellent web page created by undergrads at UC, Berkeley.  I will try to write a second  on compressive sensing, and record them together. Comments are welcome, of course:

In 1973 the University of California at Berkeley was sued for sex discrimination in graduate student admissions. The case seemed clear cut: only 35% of female compared to 44% of male applicants were admitted to graduate programs across the university. However, when statisticians looked at the data in more detail they found a surprise. When looking at the admission rates of individual departments, the apparent bias disappeared. Across departments, women were either more likely to be admitted or about equally likely to be admitted as men. Indeed, at the level of individual departments, women seemed to fare slightly better.

This is an example of Simpon’s paradox – a paradox that can affect averages whenever we combine, or pool, data. Here is another example involving two New York Yankees players, Derek Jeter and David Justice. In both 1995 and 1996 David Justice had a higher batting average than Derek Jeter. However, when we compute the batting average over both seasons, then Derek Jeter is ahead of David Justice. Again, pooling the data gives a different picture than when looking at smaller chunks.

How is this possible? Let’s look at the case of graduate school applicants to the University of California at Berkley. It turns out that more women applied to departments in the humanities, while men tended to apply in higher numbers to engineering and science departments. Humanities departments had fewer available slots, and rejected more applicants. Thus female applicants applied mostly to departments which admitted fewer students, whether male or female. As a result, the overall fraction of women admitted was lower than that of men. A bias may have existed, but it was not a bias in the rate of admissions. Rather, it was a bias in the number of women who chose to apply graduate studies in technical fields.

Simpson’s paradox can have important consequences. For example medical researchers compared a less invasive treatment for kidney stones to the established surgical methods, and found the new treatment to be better overall. However, the less invasive treatment was more frequently applied to small kidney stones. Since smaller kidney stones are easier to treat, this gave an advantage to the new, less invasive method. When the treatments were compared separately on small kidney stones and large kidney stones, the traditional treatment proved to be more successful. Taking into account kidney stone size completely changed the conclusion about which treatment is better.

The outcomes of lawsuits, promotions, and our choice of medical treatments are frequently based on numerical evidence. Yet our intuition can easily mislead us when we think about numbers. Mathematics and statistics can help – they can give us answers to the question we are asking. But it is up to us to make sure that we are asking the right questions.

Notes:

The Wikipedia article on Simpson’s Paradox has a number of other good examples.

The mathematician John Tukey is credited with saying that “Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.”

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