With the U.S. in the final throes of a presidential election, my professional mathematician's mind naturally turned to the decidedly tricky matter of election math. Voting provides a great illustration of how mathematics - which rules supreme, yielding accurate and reliable answers to precise questions, in the natural sciences and engineering - can lead us astray when we try to apply it to human and social activities.
A classic example is how we count votes in an election, the topic of an article I wrote for the Mathematical Association of America, back in November 2000. In that essay, I looked at how different ways to tally votes could affect the outcome of the then-imminent Bush v Gore election, at the time blissfully unaware of how chaotic would be the process of counting votes and declaring a winner on that particular occasion. The message there was, particularly in the kinds of tight race we typically see today, the different ways that votes can be tallied can lead to very different results.
Everything I said back then remains just as valid and pertinent today (mathematics is like that), so in this essay I will look at another perplexing aspect of election math: Why do we make the effort to vote? After all, while elections are sometimes decided by a small number of votes, it is unlikely in the extreme that an election on the scale of a presidential election will hang on the decision of a single voter, and even if it did, that would be well within the range of procedural error, so it makes no difference if any one individual votes or not.
To be sure, if a large number of people decide to opt out, that can affect the outcome. But there is no logical argument that takes you from that observation to it being important for a single individual to vote.
This state of affairs is known as the Paradox of Voting, or sometimes Downs Paradox, after Anthony Downs, a political economist whose 1957 book An Economic Theory of Democracy examined the conditions under which (mathematical) economic theory could be applied to political decision-making.
On the face of it, Downs' analysis does lead to a paradox. Economic theory tells you that rational beings make decisions based on expected benefit (a notion that can be made numerically precise). That approach works well for analyzing, say, why people buy insurance year after year, even though they may never submit a claim. The theory tells you that the expected benefit is greater than the cost; so it is rational to buy insurance. But when you adopt the same approach to an election, you find that, because the chance of exercising the pivotal vote in an election is minute compared to any realistic estimate of the private individual benefits of the different possible outcomes, the expected benefits of voting are less than the cost. So you should opt out. [The same observation had in fact been made much earlier, in 1793, by Nicolas de Condorcet - mentioned in my 2000 MAA essay cited above - but without the theoretical backing that Downs brought to the issue.]
Yet, many otherwise sane, rational citizens do not opt out. Indeed, society as a whole tends to look down on those who do not vote, saying they are not "doing their part." (In fact, many countries make participation in a national election obligatory, but that is a separate, albeit related, issue.)
So why do we (or at least many of us) bother to vote? I can make the question even more stark, and personal. Suppose you have intended to "do your part" and vote. You wake up on election morning with a sore throat, and notice that it is raining heavily. Being numerically able (I will assume), you say to yourself, "It cannot possibly affect the result if I just stay at home and nurse my throat. I was intending to vote, after all. And my changing my mind about voting at the last minute cannot possibly influence anyone else. Especially if I don't tell anyone." The math and the logic, surely, are rock solid. Yet, professional mathematician as I am, I would struggle out and cast my vote. And I am sure many people reading these words would too - perhaps most of them.
So what is going on? How is it that people who can do the math and are good logical thinkers do not act according to that reasoning? Must we concede that mathematics actually isn't that useful? The answer is no. Math is useful. But only when applied with a specific purpose in mind, and chosen/designed in a way that makes it appropriate for that purpose.
Which brings me to my main point.
To make it, let me switch for a moment from elections to an old staple of mathematics writing: the Golden Ratio. In April 2015, the magazine Fast Company Design published an article titled The Golden Ratio: Design's Biggest Myth, pointing out that most of the many claims made about the appearance of that number in art, music, architecture, and even the human body, were in fact outright false - urban legends dating back to the 19th Century. The article quoted me as saying, "Strictly speaking, it's impossible for anything in the real-world to fall into the golden ratio, because it's an irrational number." A high school math teacher wrote to me recently saying he had read the piece, and was sure I had been misquoted. I had not. From what he said, it was clear to me that the teacher had drunk not just Golden Ratio Kool Aid, but Math Kool Aid as well.
In the interest of full disclosure, let me admit that, in the early part of my career as a mathematics expositor, I was as guilty as anyone of distributing both Golden Ratio Kool Aid and Math Kool Aid, to whoever would drink it. But, as a committed scientist, when presented with evidence to the contrary, I re-examined my thinking, admitted I had been wrong, and started to push better, more honest products, which I will call Golden Ratio Milk and Mathematical Milk. The Fast Company article actually describes what I am calling Golden Ratio Milk - namely fascinating, yet sensible and valid, facts about the number. (There are plenty, particularly in the botanical world.) But what is Mathematical Milk?
The reason why the Golden Ratio's irrationality prevents its use in, say architecture, is that the issue at hand involves measurement. Measurement requires fixing a unit of measure - a scale. It doesn't matter whether it is meters or feet or whatever, but once you have fixed it, that is what you use. When you measure things, you do so to an agreed degree of accuracy. Perhaps one or two decimal places. Almost never to more than maybe twenty decimal places, and that only in a few instances in subatomic physics. So in terms of actual, physical measurement, or manufacturing, or building, you never encounter objects to which a numerical measurement has more than a few decimal places. You simply do not need a number system that has fractions with denominator much greater than, say, 1,000,000, and generally much less than that.
Even if you go beyond physical measurement, to the theoretical realm where you imagine having an unlimited number decimal places available, you will still be in the domain of the rational numbers. Which means the Golden Ratio does not arise. Irrational numbers arise to meet mathematical needs, not the requirements of measurement. They live not in the physical world but in the human imagination. (Hence my Fast Company quote.) It is important to keep that distinction clear in our minds.
The point is, when we abstract from our experiences of the world around us, to create mathematical models, two important things happen. A huge amount of information is lost; and there is a significant gain in precision. The two are not independent. If we want to increase the precision, we lose more information, which means that our model has less in common with the real world it is intended to represent. Moreover, when we construct a mathematical model, we do so with a particular question, or set of questions in mind.
In astronomy and physics, and related domains such as engineering, all of this turns out to be not too problematic. For example, the simplistic model of the Solar System as a collection of point-masses orbiting around another, much heavier, point-mass, is extremely useful. We can formulate and solve equations in that model, and they turn out to be very useful. At least they turn out to be useful in terms of the goal questions, initially in this case predicting where the planets will be at different times of the year. The model is not very helpful in telling us what the color of each planet's surface is, or even if it has a surface, both of which are certainly precise, scientific questions.
When we adopt a similar approach to model money supply or other economic phenomena, we can obtain results that are, mathematically, just as precise and accurate, but their connection to the real world is far more tenuous and unreliable - as has been demonstrated several times in recent years when those mathematical results have resulted in financial crises, and occasionally disasters.
So what of the paradox of voting? The paradox arises when you start by assuming that people vote to choose, say, a president. Yes, we all say that is what we do. But that's just because we have drunk Election Kool Aid. We don't actually behave in accordance with that statement. If we did, then as rational beings we would indeed stay at home on election day.
Time then to throw out the Kool Aid and go out to buy a gallon jug of far more beneficial Election Milk: (Presidential) elections are about a society choosing a president. Where that purpose impacts the individual voter is not who we vote for, but in providing social pressure to be an active member of that society.
That this is what is actually going on is illustrated by the fact that US society created, and millions of people wear, "I have voted" badges on election day. The focus, and the personal reward, is not "Who I voted for" but "I participated in the process." [For an interesting perspective on this, see the recent article in the Smithsonian magazine, Why Women Bring Their "I Voted" Stickers to Susan B. Anthony's Grave.]
To be sure, you can develop mathematical models of group activities, like elections, and they will tend to lead to fewer problems (and "paradoxes") than a single-individual model will, but they too will have limitations. All mathematical models do. Mathematics is not reality; it is just a model of reality (or rather, it is a whole, and constantly growing, collection of models).
When we develop and/or apply a mathematical model, we need to be clear what questions it is designed to help us answer. If we try to apply it to a different question, we may get lucky and get something useful, but we may also end up with nonsense, perhaps in the form of a "paradox".
With both measurement and the election, as is so often the case, one benefit we get from trying to apply mathematics to our world and to our lives is we gain insight into what is really going on.
Attempting to use numbers to model the acts of measuring physical objects leads us to recognize the dependency on the physical activity of measurement.
Likewise, grappling with Downs paradox leads us to acknowledge what elections are really about - and to recognize that choosing a leader is a societal activity. In a democracy, who each one of us votes for is inconsequential; that we vote is crucial. That's why I did not just spend a couple of hours yesterday making my choices and filling in my ballot and leaving it at that. I also went out earlier today - in light rain as it happens (and without a sore throat) - and put my ballot in the mailbox. Yesterday, I acted as an individual, motivated by my felt societal obligation to participate in the election process. Today I acted as a member of society.