Tag: Mathematics

Quotes & Commentary #73: Keynes

Quotes & Commentary #73: Keynes

It is a great fault of symbolic pseudo-mathematical methods of formalizing a system of economic analysis . . . that they expressly assume strict independence between the factors involved and lose all their cogency and authority if this hypothesis is disallowed.

—John Maynard Keynes

I ended my last commentary by swearing to leave off thinking about the coronavirus. Alas, I am weak. The situation is bleak and depressing; it has affected nearly every aspect of my life, from my free-time to my work, my exercise routines and my relationships; but it is also, if one can be excused for saying so, quite morbidly absorbing.

What especially occupies me is how those in charge will weigh the costs and benefits of their policies. Because the threat posed by coronavirus is so novel, because these decisions involve human life, and because it is difficult not to feel afraid, I think there is a certain moral repugnance that many feel toward this kind of thinking. However, as I argued in my previous post, I think truly moral action will require a thorough appraisal of all of the many potential consequences of action and inaction. This will make any choice that much more difficult, and I do not envy those who will have to make it. 

As anyone familiar with the famous trolley problem knows, moral dilemmas often involve numbers. If the actor has to choose between a lower and a higher number of victims, one must choose the lower number. However, there are several refinements of the problem which show the limitations of our moral intuition. For example, respondents are willing to divert a runaway trolley onto a track where it will kill one person rather than five; but respondents are unwilling to push an enormously fat bystander onto the tracks to save five people. We seem to be willing to think in purely numerical terms only about those involved ‘in the situation’ and unwilling to do so with those we perceive as ‘outside the situation.’

Well, in case we are facing, virtually everyone is ‘inside the situation’; so this leads us to a numerical treatment. But of course this is not so simple. What should we be measuring and comparing, exactly? I raised the question in my last post about this calculus of harm, and how it seems impossible to compare different types and levels of harm. As hospitals get overwhelmed, however, and care begins to be rationed, doctors are forced to make difficult choices along these lines, giving treatment to patients with the highest chances of recovery. Politicians are now faced with a kind of society-level triage.

One obvious basis of comparison is the number of lives lost. This is how we think of the trolley problem. But I think there is a case for also considering the number of lived years lost. What is ethically preferable: allowing the death of one person, or allowing the lifespans of 10 people to be reduced by 20 years each? I cannot answer this question, but I do think that the answer is not easy or self-evident. Reducing somebody’s lifespan may not be ethically on a par with letting someone die, but it is still quite a heavy consideration.

Further down the line, ethically speaking, is quality of life. Though it seems egregious to weigh death against quality of life issues, in practice we do it all the time. Smoking, drinking, and driving carry a risk, and a certain number of people will die per year by engaging in these activities; but we accept the cost because, as a society, we apparently have decided that it is “worth it” in terms of our quality of life. But of course, this comparison is not exactly appropriate for the case of coronavirus, since we ourselves make the decision to smoke or drive, whereas the risk of coronavirus is not voluntary. Thus, to save lives we should be willing to accept a greater loss in quality of life in this case, since we cannot control our exposure to the risk.

How exactly we choose to weigh or balance these three levels of damage—lives lost, lives shortened, and lives made worse—is not something I am prepared to put into numbers. (I suppose some economist is already doing so.) But I think we are obligated to try to at least take all of them into account.

Now, the other set of variables we must consider are empirical. On the medical side, these are: the lethality of the virus and the percentage of the population likely to get infected. On the economic side there are obvious factors like unemployment and loss in GDP and so forth. There are also factors such as loss in standard of living, homelessness, and the poverty rate; and still more difficult to calculate variables like the rate of suicide and drug addiction likely to result.

One major problem is that we know all of these variables imperfectly, and in some cases very imperfectly. To take an obvious datum, there is the virus’s lethality rate. From the available numbers, in Italy the fatality rate appears as high as 8%, while in Germany it is as low as 0.5%. This huge range contains a great deal of uncertainty. On the one hand, there is a good case that Germany gives a more accurate picture of the virus’s lethality, since they have done the most testing, about 120,000 a day; and logically more testing gives a more accurate result. However, we should remember that the virus’s lethality rate is not a single, static number. It affects different demographics differently, and it also depends on the availability of treatment. All of these factors need to be taken into account to establish the virus’s risk.

Complicating the uncertainty is the fact that the virus can create mild or even no symptoms, thus leaving open the question of the total number of cases—a number that must be known to determine the lethality rate. Asked to offer an estimate of the total number of infected people in Spain (the registered number is about 45,000 as of now), mathematicians offered estimates ranging from 150,000 to 900,000—and, of course, these are little more than educated guesses. If the former figure is correct, it would put the lethality at around 2%, while if the latter is correct the lethality is about 0.4%: another big range. 

Now that Spain is receiving a massive shipment of tests from China, our picture of the virus will likely become much more accurate in the coming days and weeks. (Actually, many of these tests are apparently worthless, so nevermind.*) However, one crucial datum is still missing from our knowledge: the total number who have already had the virus. To ascertain this, we will need to test for antibodies. It appears we will begin to have information on this front soon, as well, since the UK has purchased a great deal of at-home antibody tests. I believe other countries are following suit. Not only is this data crucial to accurately estimating the virus’s threat, but it is also of practical value, since those with antibodies will be in far less danger either of catching or of spreading the disease. (In the movie Contagion, those with antibodies are given little bracelets and allowed to travel freely.)

The New York Times has created an interesting tool for roughly estimating the potential toll of the virus. By adjusting the infection and fatality rate, we can examine the likely death toll. Of course, these rough calculations are limited in that they make the mistake Keynes highlights above—they assume an independence of variables. For example, the calculator shows how the coronavirus would match up with expected cancer and heart disease deaths. But of course more coronavirus deaths would likely mean fewer deaths from other causes, since many who would have died from other causes would succumb to coronavirus. (Other causes of death like traffic accidents may also go down because of the lockdown.) The proper way to make a final estimate, I believe, would be to see how many total deaths we have had in a year, and then compare that total with what we would reasonably expect to have had without the coronavirus.

As you can see, the problem of coming up with a grand calculation is difficult in the extreme. Even if we can ultimately ascertain all of the information we need—medical, economic, sociological—we will still have only an imperfect grasp of the situation. Indeed, Keynes’s warning is quite pertinent here, since every factor will be influencing every other. Unemployment affects access to health care, an overwhelmed health care system will be less effective across the board, and the fear of the virus alone has economic consequences. This makes the ‘trolley problem’ model misleading, since there are no entirely independent tracks that the trolley can be moving on. Any decision will affect virtually everyone in many different ways; and this makes the arithmetical approach limited. 

Trump has said that the cure cannot be worse than the disease. Obviously, however, the decision is not a simple choice between economic and bodily well-being. This is what makes the decision so very subtle and complicated. Not only must we weigh sorts of damage in our ethical scales, but we also must be able to think synthetically about the whole society—the many ways in which its health and wealth are bound up together—in order to act appropriately.

Once again, I do not envy those who will have to make these choices.

Review: The Works of Archimedes

Review: The Works of Archimedes

The Works of Archimedes by Archimedes

My rating: 5 of 5 stars

In fact, how many theorems in geometry which have seemed at first impracticable are in time successfully worked out!

Many of the most influential and ingenious books ever written possess the strange quality of being simultaneously exhilarating and quite boring. Unless you are among that rare class of people who enjoy a mathematical demonstration more than a symphony, this book will likely possess this odd duality. I admit this is the case for me. Reading this book was a constant exercise in fighting the tendency for my eyes to glaze over. But I am happy to report that it is worth the trouble.

Archimedes lived in the 3rd century BCE, somewhat after Euclid, in Syracuse on the island of Sicily. Apart from this, not much else can be said with certainty about the man. But he is the subject of many memorable stories. Everybody knows, for example, the story of his taking a bath and then running through the streets naked, shouting “Eureka!” We also hear of Archimedes using levers to move massive boats, and claiming that he could move the whole earth if he just had a place to stand on. Even his death is the subject of legend. After keeping the invading Romans at bay using ingenious weapons—catapults, cranes, and even mirrors to set ships afire—Archimedes was killed by a Roman soldier, too preoccupied with a mathematical problem to care for his own well-being.

True or not, good stories tend to accumulate around figures who are worthy of our attention. And Archimedes is certainly worthy. Archimedes did not leave us any extended works, but instead a collection of treatises on several topics. The central concern in these different works—the keystone to Archimedes’s method—is measurement. Archimedes set his brilliant mind to measuring things that many have concerned impossible to reckon. His work, then, is an almost literal demonstration of the human mind’s ability to scan, delimit, and calculate things far outside the scope of our experience.

As a simple example of this, Archimedes established the ratios between the surface areas and volumes of spheres and cylinders—an accomplishment the mathematician was so proud of that he apparently asked for it to be inscribed on his tombstone. Cicero describes coming across this tombstone in a dilapidated state, so perhaps this story is true. Archimedes also set to work on giving an accurate estimation of the value of pi, which he accomplished by inscribing and circumscribing 96-sided polygons around a circle, and calculating their perimeters. If this sounds relatively simple to you, keep in mind that Archimedes was operating without variables or equations, in the wholly-geometrical style of the Greeks.

Archimedes’s works on conoids, spheroids, and spirals show a similar preoccupation with measurement. What all of these figures have in common is, of course, that they are composed of curved lines. How to calculate the areas contained by such figures is not at all obvious. To do so, Archimedes had to invent a procedure that was essentially equivalent to the modern integral calculus. That is, Archimedes used a method of exhaustion, inscribing and circumscribing ever-more figures composed of straight lines, until an arbitrarily small gap remained between his approximations and what he was attempting to measure. To employ such a method in an age before analytic geometry had even been invented is, I think, an accomplishment difficult to fully appreciate. When the calculus was finally invented, about two thousand years later, it was by men who were “standing on the shoulders of giants.” In his time, Archimedes had few shoulders to stand on.

The most literal example of Archimedes’s concern with measurement is his short work, The Sand Reckoner. In this, he attempts to calculate the number of grains of sand that would be needed to fill up the whole universe. We owe to this bizarre little exercise our knowledge of Aristarchus of Samos, the ancient astronomer who argued that the sun is positioned at the center of the universe. Archimedes mentions Aristarchus because a heliocentric universe would have to be considerably bigger than a geocentric one (since there is no parallax observed of the stars); and Archimedes wanted to calculate the biggest universe possible. He arrives at a number is quite literally astronomical. The point of the exercise, however, is not in the specific number arrived at, but in formulating a way of writing very large numbers. (This was not easy in the ancient Greek numeral system.) Thus, we partly owe to Archimedes our concept of orders of magnitude.

Archimedes’s contributions to natural science are just as significant as his work in pure mathematics. Indeed, one can make the case that Archimedes is the originator of our entire approach to the natural sciences; since it was he who most convincingly demonstrated that physical relationships could be described in purely mathematical form. In his work on levers, for example, Archimedes shows how the center of gravity can be found, and how simple principles can explain the mechanical operation of counterbalancing weights. Contrast this with the approach taken by Aristotle in his Physics, who uses wholly qualitative descriptions and categories to give a causal explanation of physical motion. Archimedes, by contrast, pays no attention to cause whatever, but describes the physical relationship in quantitative terms. This is the exact approach taken by Galileo and Newton.

Arguably, the greatest masterpiece in this collection is On Floating Bodies. Here, Archimedes describes a physical relationship that still bears his name: the relationship of density and shape to buoyancy. While everyone knows thpe story of Archimedes and the crown, it is possible that Archimedes’s attention was turned to this problem while working on the design for an enormous ship, the Syracusia, built to be given as a present to Ptolemy III of Egypt. This would explain Book II, which is devoted to finding the resting position of several different parabolas (more or less the shape of a ship’s hull) in a fluid. The mathematical analysis is truly stunning—so very far beyond what any of his contemporaries were capable of that it can seem even eerie in its sophistication. Even today, it would take a skilled physicist to calculate how a given parabola would rest when placed in a fluid. To do so in ancient times was simply extraordinary.

Typical of ancient Greek mathematics, the results in Archimedes’s works are given in such a way that it is difficult to tell how he originally arrived at these conclusions. Surely, he did not follow the steps of the final proof as it is presented. But then how did he do it? This question was answered quite unexpectedly, with the discovery of the Archimedes Palimpsest in the early 1900s. This was a medieval prayer book that contained the remains of two previously unknown works of Archimedes. (Parchment was so expensive that scribes often scraped old books off to write new ones; but the faded impression of the original work is still visible on the manuscript.) One of these works was the Ostomachion, a collection of different shapes that can be recombined to form a square in thousands of different ways (and it was the task of the mathematician to determine how many).

The other was the Method, which is Archimedes’s account of how he made his geometrical discoveries. Apparently, he did so by clever use of weights and balances, imagining how different shapes could be made to balance one another. His method of exhaustion was also a crucial component, since it allowed Archimedes to calculate the areas of irregular shapes. A proper Greek, Archimedes considered mechanical means to be intellectually unsatisfactory, and so re-cast the results obtained using this method into pure geometrical form for his other treatises. If it were not for the serendipitous discovery of this manuscript, and the dedicated work of many scholars, this insight into his method would have been forever lost to history.

As I hope you can see, Archimedes was a genius among geniuses, a thinker of the rarest caliber. His works are exhilarating demonstrations of the power of the human mind. And yet, they are also—let us admit it—not the most exciting things to read, at least for most of us mere mortals. Speaking for myself, I would need a patient expert as a guide if I wanted to understand any of these works in detail. Even then, it would be hard work. Indeed, I have to admit that, on the whole, I find mathematicians to be a strange group. For the life of me I cannot get excited about the ratio of a sphere to a cylinder—something that Archimedes saw as the culmination of his entire life.

Archimedes is the very embodiment of the man absorbed in impractical pursuits—so obsessed with the world of spirals and curves that he could not even avoid a real sword thrust his way. And yet, if subsequent history has shown anything, it is that these apparently impractical, frigid, and abstract pursuits can reveal deep truths about the universe we live in—much deeper than the high-flown speculations of our philosophers. I think this lesson is worth suffering through a little boredom.



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