American Taliban: It Is Happening Here

What was emerging from this and similar meetings was a political force — the so-called Religious Right — that injected into the Republican Party a new emphasis on the promotion of religious morality (an area of concern which most early Goldwater activists thought belonged in the private, not public, and not political, arena). The focus of this new religion-centered “conservatism” was not on liberty and limited government but on what Russell Kirk had called the “transcendent moral order.”

By 1989, the Moral Majority, the late Jerry Falwell’s organization which had emphasized religious values across sectarian lines and often shared a common purpose with many conservatives and orthodox Jews, had all but disappeared. In his place rose the Christian Coalition, led by Pat Robertson and Ralph Reed. The Religious Right had become the overtly Christian Right. (Edwards 2008: 40-41)

(….) Through aggressive grassroots activism the movement’s members and supporters won elections, took over party organizations, and dominated party conventions. Later, when George W. Bush’s political advisor Karl Rove would speak of “the Republican base,” this was who he had in mind. (Edwards 2008: 41)

The Christian Right was hardly the Republican base (the party’s voters were often much more moderate in their views than were Robertson and his followers), but because Robertson’s forces tended to dominate conventions and primaries in which voter turnout is often low, they exerted influence far beyond their numbers. In the process, they transformed the republican Party and indeed the conservative movement itself into an arm of religion, precisely the outcome the First Amendment of the Constitution was designed to prevent. There were instrumental in galvanizing the conservative opposition to death with dignity laws in Oregon, private medical decisions in Florida, and scientific advances in the nation’s medical laboratories. (Edwards 2008: 41)

A wall was erected between church and the state … as an extension of the founder’s experience with religious persecution in Europe. Placing religion in a position to dictate, or heavily influence, national policy had led to sanctions, torture, murder, and war. European battlefields were littered with the corpses of men sent to war on behalf of one religious sect or another. In a nation founded on Lockean principles of individual rights, there would be no place given to sectarian terror. (Edwards 2008: 64)

The wall between religion and statecraft serves an additional purpose. The enemy of civility (a necessary ingredient in the governance of a diverse society) is certitude. And nothing breeds certitude more than religious belief. Religion is often a positive force in the lives of individuals, but when the true believer feels compelled to impose upon the whole of the society the truths that have enriched his or her life, the threads that bind us as a nation begin to fray. (Edwards 2008: 123)

(….) Conservatism’s central philosophy has long been based on the regard for the individual rather than the collective. Yet today many are willing to support the imposition of the personal beliefs of some, be they a majority or a minority, on others, who do not subscribe to those beliefs. The title of Sinclair Lewis’s novel about a politician who rose to power on a wave of religious fervor was ironic: It Can’t Happen Here. Its message was: yes, it can. (Edwards 2008: 124)

Because the Constitution’s central premise is liberty — it is a document designed for a free people–it was created to prevent both the concentration of power in a few hands … and the ability of the majority to impose its will on the minority… The rule of law, not the rule of the masses or rulers, defines American constitutional government. But that is a lesson conservatives have forgotten. (Edwards 2008: 128-129)

The Constitution is for all Americans — Catholic, Protestant, Jewish, Buddhist, Muslim, and nonbeliever alike. We are free to practice or not as we deem fit. Religion is a personal thing; government is what we hold in common, and that distinction lies at the heart of American conservatism [opposed to extremist fundamentalism as exhibited above]. Community is not the same thing as government. The U.S. government is a secular institution, and its policy decisions should not be required to conform to religious doctrine. (Edwards, Mickey. Reclaiming Conservatism. Oxford: Oxford University Press; 2008; pp. 40-166) 

Now is the time to remember that our great religious traditions, notably Christianity, once upon a time could not even conceive of reducing religious engagement with public life to a narrow list of hot-button issues. They were too concerned with the whole person and the whole of society to limit their reach to a handful of questions. Now is the time to heed the call to social justice and social inclusion embedded deeply in in the scriptures. (E. J. Dionne Jr., Forward, in Lew Daly (2009) God’s Economy: Faith Based Initiatives & the Caring State. University of Chicago Press)

We Were Warned

America’s Founders and Abraham Lincoln warned us of the danger of elevating a demagogue such as Donald Trump to the highest office in the land and the virulent nature of the “angry mob” of sycophants easily manipulated by such demagogues making up the political base of such right-wing populism as we are witnessing today at the heart of the GOP:

In an 1838 address to the members of the Young Men’s Lyceum of Springfield, Illinois, Lincoln warned that since American democracy could never be overthrown by a foreign invader, the only enemy to be feared was one within: undisciplined passion. Pointing to several recent examples of frontier lynchings, Lincoln deplored “the increasing disregard for law which pervades the country; the growing disposition to substitute the wild and furious passions, in lieu of the sober judgment of the Courts; and the worse than savage mobs, for the executive ministers of justice.” (….) Lincoln warned the young men of his home town that during the generations to come ambitious demagogues would seek to prey upon the passions of the people, unless these were kept under stern control. “Passion has helped us” in rallying the people to the cause of the Revolution, Lincoln acknowledged, “but can do so no more. It will in future be our enemy.” He cautioned: “Reason, cold, calculating, unimpassioned reason, must furnish all the materials for our future support and defence.” Only by the control of passion could American democracy keep from degenerating into anarchy or demagogy. When Lincoln declared that America would stand or fall by “the capability of a people to govern themselves,” he meant this in both a political and a psychological sense.

Howe, Daniel Walker. Making the American Self: Jonathan Edwards to Abraham Lincoln (pp. 142-143). Oxford University Press. Kindle Edition.

Alexander Hamilton, James Madison, and John Hay were the authors of The Federalist Papers:

No document relating to the Constitution of the United States has received more attention than The Federalist Papers. The papers were written in 1787–88 for the purpose of persuading the people of the state of New York to elect a convention that would ratify the proposed Constitution of the United States…. The authors of The Federalist—Alexander Hamilton, James Madison, and John Jay—were practical men, writing under intense pressures, with a strong sense of the campaign strategy they were pursuing. They submerged their individual differences in the collective persona of Publius, who for our purposes may be treated as a single author (Howe 2009, 78-79)…. What Publius fears is that “a torrent of angry and malignant passions will be let loose,” frustrating all attempts at rational discourse. He himself will engage in rational argument, without impugning the motives of individuals…. However he may feel provoked, Publius will take his stance with Prospero in The Tempest:

“Though with their high wrongs I am struck to the quick,
Yet with my nobler reason ‘gainst my fury
Do I take part.” (Howe 2009, 85)

“In all very numerous assemblies, of whatever characters composed, passion never fails to wrest the scepter from reason,” and the more numerous the assembly, “the greater is known to be the ascendancy of passion over reason.” Once dominated by passion, an assembly became a “mob.” (Howe 2009, 86) (….) Publius complained that the Anti-federalists’ rhetoric suggested “an intention to mislead the people by alarming their passions, rather than to convince them by arguments addressed to their understandings.” (….) Publius’s line of argument was not unprecedented: the seventeenth-century English classical republican theorist James Harrington had argued that government should be designed to maintain the supremacy of reason over passion, and had blamed passion for the degeneration of monarchy into tyranny, aristocracy into oligarchy, or democracy into anarchy….. The Federalist quoted Jefferson with approval: “An elective despotism was not the government we fought for.” (….) The demagogue is a sinister figure in The Federalist. He lurks ready to exploit the passions and create a faction. He is the natural enemy of the statesman, who has virtue and the common interest at heart. The Constitution, Publius argues, will provide a context within which the statesman can defeat the demagogue. Fittingly, he both begins and ends his series of letters with warnings against demagogues.

Howe, Daniel Walker. Making the American Self: Jonathan Edwards to Abraham Lincoln (pp. 90-95). Oxford University Press. Kindle Edition.

This very generation is witnessing with the rise of Trumpism an insidious form of right-wing populist extremism and “faction” that seeks to gratify “private passion by public means.” This is the exact kind of “faction” (i.e., the elevation of personal bias and opinion into absolutist totalitarian rhetoric) our Founders feared; the collective expression of “some common impulse of passion, or of interest, adverse to the rights of other citizens, or to the permanent and aggregate interests of the community.” Factions stem from passions writ large inflamed by ambitious demagogues, as we witnessed during Trump’s many campaign rallies were he regularly incites violence, boasting he could murder someone in the street and the “angry mob” would still vote for him. Hamilton warned that in this form, passions become more dangerous than ever: “a spirit of faction” can lead men “into improprieties and excesses for which they blush in a private capacity.” (Howe 2009, 95)

Donald Trump’s demagoguery directed at his ignorant base is aimed at inciting their passions and is bent upon ripping apart the social fabric of our society and tearing down our democratic institutions. The GOP has mainstreamed extremism, and by so doing has signaled the death of any semblance of classical principled conservatism. In the words of the 18th Century Irish statesman Edmund Burke, “All drapery of life is to be rudely torn off… Their liberty is not liberal. Their [anti-science] is presumptuous ignorance. Their humanity is savage and brutal.” By elevating license over liberty, zealotry and extremism over moderation and reason, the GOP has set America on a course Lincoln warned us would happen when we lost our ability to reasonably govern our own passions and prejudices. Trumpism is an existential threat to the very existence of America’s Constitutional form of government and balanced separation of powers. Time is swiftly running out and if we don’t augment our political discourse with a heavy dose of wisdom we will plunge ourselves over the cliff into another “dark ages” of the interregnum of wisdom bearing witness to the inexorable consequences of confusing license for liberty.

Charmed by Dimensional Analysis

I was charmed when as a young student I watched one of my physics professors, the late Harold Daw, work a problem with dimensional analysis. The result appeared as if by magic without the effort of constructing a model, solving a differential equation, or applying boundary conditions. But the inspiration of the moment did not, until many years later, bear fruit. In the meantime my acquaintance with this important tool remained partial and superficial. Dimensional analysis seemed to promise more than it could deliver. (Lemons 2017, ix, emphasis added)

Dimensional analysis has charmed and disappointed others as well…. The problem for teachers and students is that … [t]he mathematics required for its application is quite elementary — of the kind one learns in a good high school course — and its foundational principle is essentially a more precise version of the rule against “adding apples and oranges.” Yet the successful application of dimensional analysis requires physical intuition — an intuition that develops only slowly with the experience of modeling and manipulating physical variables. (Lemons 2017, Preface ix, emphasis added)

A Mistake to Avoid

A model of a state or process incorporates certain idealizations and simplifications. Skill and judgement are required to decide which quantities are needed to describe the state or process and what idealizations and simplifications should be incorporated. Similar skill and judgement are required in dimensional analysis, for the analysis in dimensional analysis is the analysis of a model. And the model we adopt in a dimensional analysis is determined by the dimensional analysis variables and constants we adopt and the dimensions in terms of which they are expressed. (….) While a certain part of dimensional analysis reduces to the algorithmic, no algorithm helps us answer [certain physical questions]. Rather, our answers define the state or process we describe and the model we adopt. We will, on occasion, make mistakes. (Lemons 2017, 11, emphasis original)

Dimensional analysis makes it possible to analyze in a systematic way dimensional relationships between physical quantities defining a model (Higham 2015, 90-91, emphasis added). Dimensional analysis is a clever strategy for extracting knowledge from a remarkably simple idea, nicely stated by Richardson[,] “… that phenomena go their way independently of the units whereby we measure them.” Within its limits, it works excellently, and makes possible astonishing economies in effort. The limits are soon reached, and beyond them it cannot help. In that it is like a specialized tool in carpentry or cooking or agriculture, like the water-driven husking mill … which husks rice elegantly and admirably but cannot do anything else. (Palmer 2015, v, emphasis added)

Hubris leads some to claim to be engaged in modeless modeling despite evidence plainly to the contrary (e.g., stylised facts, etc.). Whether talking of applied mathematics or dimensional analysis, one is by the very nature of the process engaged in mathematical modeling. Those who are arrogant enough to abuse mathematics for polemic purposes are breaking mathematical sense and are often, to put it kindly, philosophically naïve blinded by their own mathematical pride.

Physical (material) things have quantitative relationships that are measurable. A dimensional model uses a number of dimensional variables (physical variables) and constants that describe the model. Dimensional analysis is not a straightforward task for it requires skill and judgment — the same kind of skill and judgment needed to construct a model of a physical state or process. Add the complexity of open social systems and this requires even more skill, judgment, and frankly, enough wisdom to know the difference between a physical quantitative fact and qualitative mind-value judgement.

Some blinded by mathematical pride and/or statistical egotism and/or confused by philosophical materialism/monistic reductionism, not to mention spiritual blindness, fail to make a distinction between quantitative and qualitative observations, both dependent upon concepts experienced in the mind of the scientist whose very supermaterial insight formulates such a misguided self-contradictory monistic and reductive metaphysics.

What is Applied Mathematics?

The Big Picture

Applied mathematics is a large subject that interfaces with many other fields. Trying to define it is problematic, as noted by William Prager and Richard Courant, who set up two of the first centers of applied mathematics in the United States in the first half of the twentieth century, at Brown University and New York University, respectively. They explained that:

Precisely to define applied mathematics is next to impossible. It cannot be done in terms of subject matter: the borderline between theory and application is highly subjective and shifts with time. Nor can it be done in terms of motivation: to study a mathematical problem for its own sake is surely not the exclusive privilege of pure mathematicians. Perhaps the best I can do within the framework of this talk is to describe applied mathematics as the bridge connecting pure mathematics with science and technology.

Prager (1972)

Applied mathematics is not a definable scientific field but a human attitude. The attitude of the applied scientist is directed towards finding clear cut answers which can stand the test of empirical observation. To obtain the answers to theoretically often insuperably difficult problems, he must be willing to make compromises regarding rigorous mathematical completeness; he must supplement theoretical reasoning by numerical work, plausibility considerations and son on.

Courant (1965)

Garrett Birkhoff offered the following view in 1977, with reference to the mathematician and physicist Lord Rayleigh (John William Strutt, 1842-1919):

Essentially, mathematics becomes “applied” when it is used to solve real-world problems “neither seeking nor avoiding mathematical difficulties” (Raleigh).

Rather than define what applied mathematics is, one can describe the methods used in it. Peter Lax stated of these methods, in 1989, that:

Some of them are organic parts of pure mathematics: rigorous proofs of precisely stated theorems. But for the greatest part the applied mathematician must rely on other weapons: special solutions, asymptotic description, simplified equations, experimentation both in the laboratory and on the computer.

Here, instead of attempting to give our own definition of applied mathematics we describe the various facets of the subject, as organized around solving a problem. The main steps are described in figure 1. Let us go through each of these steps in turn. (Higham 2015, 1)

Modeling a problem. Modeling is about taking a physical problem and developing equations—differential, difference, integral, or algebraic—that capture the essential features of the problem and so can be used to obtain a qualitative or quantitative understanding of its behavior. Here, “physical problem” might refer to a vibrating string, the spread of an infectious disease, or the influence of people participating in a social network. Modeling is necessarily imperfect and requires simplifying assumptions. One needs to retain enough aspects of the system being studied that the model reproduces the most important behavior but not so many that the model is too hard to analyze. Different types of models might be feasible (continuous, discrete, stochastic), and for a given type there can be many possibilities. Not all applied mathematicians carry out modeling; in fact, most join the process at the next step. (Higham 2015, 2)

Analyzing the mathematical problem. The questions formulated in the previous step are now analyzed and, ideally, solved. In practice, an explicit, easily evaluated solution usually cannot be obtained, so approximations may have to be made, e.g., by discretizing a differential equation, producing a reduced problem. The techniques necessary for the analysis of the equations or reduced problem may not exist, so this step may involve developing appropriate new techniques. If analytic or perturbation methods have been used then the process may jump from here directly to validation of the model.

Developing algorithms. It may be possible to solve the reduced problem using an existing algorithm—a sequence of steps that can be followed mechanically without the need for ingenuity. Even if a suitable algorithm exists it may not be fast or accurate enough, may not exploit available structure or other problem features, or may not fully exploit architecture of the computer on which it is to be run. It is therefore often necessary to develop new or improved algorithms.

Writing software. In order to use algorithms on a computer it is necessary to implement them in software. Writing reliable, efficient software is not easy, and depending on the computer environment being targeted it can be a highly specialized task. The necessary software may already be available, perhaps in a package or program library. If it is not, software is ideally developed and documented to a high standard and made available to others. In many cases the software stage consists simply of writing short programs, scripts, or notebooks that carry out the necessary computations and summarize the results, perhaps graphically.

Computational experiments. The software is now run on problem instances and solutions obtained. The computations could be numeric or symbolic, or a mixture of the two.

Validation of the model. The final step is to take the results from the experiments (or from the analysis, if the previous three steps were not needed), interpret them (which may be a nontrivial task), and see if they agree with the observed behavior of the original system. If the agreement is not sufficiently good then the model can be modified and the loop through the steps repeated. The validation step ma be impossible, as the system in question ma not yet have been built (e.g., a bridge or a building).

Other important tasks for some problems, which are not explicitly shown in our outline, are to calibrate parameters in a model, to quantify the uncertainty in these parameters, and to analyze the effect of that uncertainty on the solution of the problem. These steps fall under the heading of UNCERTAINTY QUANTIFICATION [II.34].

Once all the steps have been successfully completed the mathematical model can be used to make predictions, compare competing hypotheses, and so on. A key aim is that the mathematical analysis gives new insights into the physical problem, even though the mathematical model may be a simplification of it.

A particular applied mathematician is most likely to work on just some of the steps; indeed, except for relatively simple problems it is rare for one person to have the skills to carry out the whole process from modeling to computer solution and validation.

In some cases the original problem may have been communicated by a scientist in a different field. A significant effort can be required to understand what the mathematical problem is and, when it is eventually solved, to translate the findings back into the language of the relevant field. Being able to talk to people outside mathematics is therefore a valuable skill for the applied mathematician. (Higham 2015, 2)

On Letting it Slide

The paradox of believing your own bullshit parallels the paradox of self-deception.  If a deceiver by definition knows that the belief he induces is false, it’s hard to see how he can convince himself that the selfsame belief is true (Hardcastle et. al. 2006, 10) ….  In his book Self Deception Unmasked (Princeton: Princeton University Press, 2001), Alfred Mele argues that self deception should not be understood on the model of interpersonal deception. In interpersonal deception, the deceiver does not believe the claim that he hopes his victim will accept as true. If self deception were to fit the interpersonal model, then the self-deceived person would have to play both roles, both affirming and denying the same belief. Mele takes this consequence to show that the interpersonal model fails. For self deception happens quite frequently, and belief in outright logical contradictions rarely seems involved. (Kimbrough, Scott. On Letting It Slide. In Bullshit and Philosophy (editors Hardcastle, Gary L. and Reisch, George A.). Chicago: Open Court; 2006; p. 10.)

Self deceived individuals “mask the evidence” and engage in a “motivated misinterpretation of evidence and selective evidence gathering.” For reasons of courtesy, strategy, and good evidence, we should criticize the product, which is visible, and not the process, which is not. (Frankfurt, p. 336) Warmed over bullshit is not merely a stale imitation of the original, but a fresh deposit that compounds the methodological faults of the original. (Ibid., p. 12-14.)

[B]ullshit results from the adoption of lame methods of justification, whether intentionally, blamelessly or as a result of self-deception. The function of the term is to emphatically express that a given claim lacks any serious justification, whether or not the speaker realizes it. By calling bullshit, we express our disdain for the speaker’s lack of justification, and indignation for any harm we suffer as a result. (Ibid., p. 16.)

[B]ullshit’s indifference to truth and falsity, its hidden interest in manipulating belief and behavior, and the way one senses, as Frankfurt put it in his book [On Bullshit], that the “bullshitter is trying to get away with something.” The audience had come to see Stewart and his writers skewer current political events, after all, so few would have missed the obvious referents—the absence of weapons of mass destruction in Iraq and the admission that sources for these claims were, in retrospect, not credible—that made the book so apropos. (Ibid., pp. viii-ix)

I always love that kind of argument. The contrary of a thing isn’t the contrary; oh, dear me, no! It’s the thing itself, but as it truly is. Ask any die-hard what conservatism is; he’ll tell you that it’s true socialism. And the brewers’ trade papers: they’re full of articles about the beauty of true temperance. Ordinary temperance is just gross refusal to drink; but true temperance, true temperance is something much more refined. True temperance is a bottle of claret with each meal and three double whiskies after dinner.

Aldous Huxley, Eyeless in Gaza (London: Chatto and Windus, 1936) pp. 122–23.

Semantic Negligence

Bullshit is not the only sort of deceptive talk. Spurious definitions, such as those quoted above, are another important variety of bad reasoning. (Ibid., p. 151) …. Whereas the liar represents as true something he believes to be false, the bullshitter represents something as true when he neither knows nor cares whether it is true or false (On Bullshit, p. 55)…. [T]his indifference is much of what we find most objectionable about bullshit. The liar has a vested interest in the institution of truth-telling, albeit a parasitical one: he hopes that his falsehoods will be accepted as true. The bullshitter may also hope to be believed, but he himself is not much bothered whether what he says is true, hence his disregard for the truth is of a deeper and potentially more pernicious character. (Ibid., pp. 151-152)

Our outrage is conditioned on our being the objects of a deception. When we know what the bullshitter is up to we can be much more indulgent. As the comic novelist Terry Pratchett observes of two of his characters, “they believed in bullshit and were the type to admire it when it was delivered with panache. There’s a kind of big, outdoor sort of man who’s got no patience at all with prevaricators and fibbers, but will applaud any man who can tell an outrageous whopper with a gleam in his eye.” The gleam in the eye is essential here: it is this complicity between bullshitter and audience which constitutes the “bull session” (On Bullshit, p. 34). Only when it escapes from the bull session and masquerades as regular assertion is bullshit deceptive; however, the insidious nature of this deception degrades the commitment to truth upon which public discourse depends. (….) [The bullshitter’s] indifference as to the truth value of his statements, that is whether they are true or false, a meaning-related or semantic property, may thus be termed semantic negligence. (Ibid., p. 152)

Breaking Mathematical Sense

Introduction

Mathematicians, as far as I can see, are not terribly interested in the philosophy of mathematics. They often have philosophical views, but they are usually not very keen on challenging or developing them—they don’t usually consider this as worthy of too much effort. They’re also very suspicious of philosophers. Indeed, mathematicians know better than anyone else what it is that they’re doing. The idea of having a philosopher lecture them about it feels kind of silly, or even intrusive. (Roi 2017, 3)

So we turn to people who have something to do with mathematics in their professional or daily lives, but are not focused on mathematics. Such people often have some sort of vague, sometimes naïve, conceptions of mathematics. One of the most striking manifestations of these folk views is the following: If I say something philosophical that people don’t understand, the default assumption is that I use big pretentious words to cover small ideas. If I say something mathematical that people don’t understand, the default assumption is that I’m saying something so smart and deep that they just can’t get it. (Roi 2017, 3-4)

There’s an overwhelming respect for mathematics in academia and wider circles. So much so that bad, trivial, and pointless forms of mathematization are often mistaken for important achievements in the social sciences, and sometimes in the humanities as well. It is often assumed that all ambiguities in our vague verbal communication disappear once we switch to mathematics, which is supposed to be purely univocal and absolutely true. But a mirror image of this approach is also common. According to this view, mathematics is a purely mechanical, inhuman, and irrelevantly abstract form of knowledge. (Roi 2017, 4)

I believe that the philosophy of mathematics should try to confront such naïve views. To do that, one doesn’t need to reconstruct a rational scheme underlying the way we speak of mathematics, but rather paint a richer picture of mathematics, which tries to affirm, rather than dispel, its ambiguities, humanity, and historicity. (Roi 2017, 4)

(….) The uncritical idolizing of mathematics as the best model of knowledge, just like the opposite trend of disparaging mathematics as mindless drudgery, are both detrimental to the organization and evaluation of contemporary academic knowledge. Instead, mathematics should be appreciated and judged as one among many practices of shaping knowledge. (Roi 2017, 4-5)

A Vignette: Option Pricing and the Black-Sholes Formula

The point of the following vignette is to give a concrete example of how mathematics relates to its wider scientific and practical context. It will show that mathematics has force, and that its force applies even when actual mathematical claims to not quite work as descriptions of reality…. The context of this vignette is option pricing. An “option” is the right (but not the obligation) to make a certain transaction at a certain cost at a certain time. For example, I could own the option to buy 100 British pounds for 150 US dollars three months from today. If I own the option, and three months from today 100 are worth more than 150 dollars, I will most probably simply discard it. Such options could be used as insurance. The preceding option, for example, would insure me against a drop in the dollar-pound exchange rate, if I needed such insurance. It could also serve as a simple bet for financial gamblers. But what price should one put on this kind of insurance or bet? There are two narratives to answer this question. The first says that until 1973, no one really knew how to price such options, and prices were determined by supply, demand, and guesswork. More precisely, there existed some reasoned means to price options, but they all involved putting a price on the risk one was willing to take, which is a rather subjective issue. (Roi 2017, 6)

In two papers published in 1973, Fischer Black and Myron Sholes, followed by Robert Merton, came up with a reasoned formula for pricing options that did not require putting a price on risk. This feat was deemed so important that in 1977 Scholes and Merton were awarded the Nobel Prize in economics [see The Nobel Factor] for their formula (Black had died two years earlier). Indeed, “Black, Merton and Scholes thus laid the foundation for the rapid growth of markets for derivatives in the last ten years”—at least according to the Royal Swedish Academy press release (1977). (Roi 2017, 6-7)

But there’s another way to tell the story. This other way claims that options go back as far as antiquity, and option pricing has been studied as early as the seventeenth century. Option pricing formulas were established well before Black and Scholes, and so were various means to factor out putting a price on risk (based on something called put-call parity rather than the Nobel-winning method of dynamic hedging, but we can’t go into details here). Moreover, according to this narrative, the Black-Sholes formula simply doesn’t work and isn’t used (Derman and Taleb 2005; Haug and Taleb 2011).

If we wanted to strike a compromise between the two narratives, we could say that the Black-Scholes model was a new and original addition to existing models and that it works under suitable ideal conditions, which are not always approximated by reality. But let’s try to be more specific. (Roi 2017, 7)

The idea behind the Black-Scholes model is to reconstruct the option by a dynamic process of buying and selling the underlying assets (in our preceding example, pounds and dollars). It provides an initial cost and a recipe that tells you how to continuously buy and sell these dollars and pounds as their exchange rate fluctuates over time in order to guarantee that by the time of the transactions, that money one has accumulated together with the 150 dollars dictated by the option would be enough to buy 100 pounds. This recipe depends on some clever, deep, and elegant mathematics. (Roi 2017, 7)

This recipe is also risk free and will necessarily work, provided some conditions hold. These conditions include, among others, the capacity to always instantaneously buy and sell as many pounds/dollars as I want and a specific probabilistic model for the behavior of the exchange rate (Brownian motion with a fixed and known future volatility, where volatility is a measure of the fluctuations of the exchange rate). (Roi 2017, 7)

The preceding two conditions do not hold in reality. First, buying and selling is never really unlimited and instantaneous. Second, exchange rates do not adhere precisely to the specific probabilistic model. But if we can buy and sell fast enough, and the Brownian model is a good enough approximation, the pricing formula should work well enough. Unfortunately, prices sometimes follow other probabilistic models (with some infinite moments), where the Black and Scholes formula may fail to be even approximately true. The latter flaw is sometimes cited as an explanation for some of the recent market crashes—but this is a highly debated interpretation. (Roi 2017, 7-8)

Another problem is that the future volatility (a measure of cost fluctuations from now until the option expires) of whatever the option buys and sells has to be known for the model to work. One could rely on past volatility, but when comparing actual option prices and the Black-Sholes formula, this doesn’t quite work. The volatility rate that is required to fit the Black-Sholes formula to actual market option pricing is not simply past volatility. (Roi 2017, 8)

In fact, if one compares actual option prices to the Black-Sholes formula, and tries to calculate the volatility that would make them fit, it turns out that there’s no single volatility for a given commodity at a given time. The cost of wilder options (for selling or buying at a price far removed from the present price) reflects higher volatility than the more tame options. So something is clearly empirically wrong with the Black-Sholes model, which assumes a fixed (rather than a stochastic) future volatility for whatever the option deals with, regardless of the terms of the option. (Roi 2017, 8)

So the Black-Sholes formula is nice in theory, but needn’t work in practice. Haug and Taleb (2011) even argue that practitioners simply don’t use it, and have simpler practical alternatives. They go as far as to say that the Black-Sholes formula is like “scientists lecturing birds on how to fly, and taking credit for their subsequent performance—except that here it would be lecturing them the wrong way” (101, n. 13). So why did the formula deserve a Nobel prize? (Roi 2017, 8)

Looking at some informal exchanges between practitioners, one can find some interesting answers. The discussion I quote from the online forum Quora was headed by the question “Is the Black-Sholes Formula Just Plain Wrong?” (2014). All practitioners agree that the formula is not used as such. Many of them don’t quite see it as an approximation either. But this does not mean they think it is useless. One practitioner (John Hwang) writes:

Where Black-Sholes really shines, however, is as a common language between options traders. It’s the oldest, simplest, and the most intuitive option pricing model around. Every option trader understands it, and it is easy to calculate, so it makes sense to communicate implied volatility [the volatility that would make the formula fit the actual price] in terms of Black-Sholes…. As proof, the exchanges disseminate [Black-Sholes] implied volatility in addition to data.

Another practitioner (Rohit Gupta) adds that this “is done because traders have better intuition in terms of volatilities instead of quoting various prices.” In the same vein, yet another practitioner (Joseph Wang) added:

One other way of looking at this is that Black-Sholes provides something of a baseline that lets you compare the real world to a nonexistent ideal world…. Since we don’t live in an ideal world, the numbers are different, but the Black-Sholes framework tells us *how different* the real world is from the idealized world.

So the model earned its renown by providing a common language that practitioners understand well, and allowing them to understand actual contingent circumstances in relation to a sturdy ideal. (Roi 2017, 9)

Now recall that practitioners extrapolate the implied volatility by comparing the Black-Sholes formula to actual prices, rather than plug a given volatility into the formula to get a price. This may sound like data fitting. Indeed, one practitioner (Ron Ginn) states that “if the common denominator of the crowd’s opinion is more or less Black-Sholes … smells like a self fulfilling prophecy could materialize,” or, put in a more elaborate manner (Luca Parlamento):

I just want to add that CBOE [Chicago Board Options Exchange] in early ’70 was looking to market a new product: something called “options.” Their issue was that how you can market something that no one evaluate? You can’t! You need a model that helps people exchange stuff, turn[s] out that the BS formula … did the job. You have a way to make people easily agree on prices, create a liquid market and … “why not” generate commissions.

The tone here is more sinister: the formula is useful because it’s there, because it’s a reference point that allows a market to grow around it. (Roi 2017, 9)

But why did this specific formula attract the market, and become a common reference point, possibly even a self-fulfilling prophecy? Why not any of the other older or contemporary pricing practices, which are no worse? Why was this specific pricing model deemed Nobel worthy? (Roi 29017, 10)

The answer, I believe, lies in the mathematics. The formula depends on a sound and elegant argument. The mathematics it uses is sophisticated, and enjoys a record of good service in physics, which imparts a halo of scientific prestige. Moreover, it is expressed in the language of an expressive mathematical domain that makes sense to practitioners (and, of course, it also came at the right time).

This is the force of mathematics. It’s a language that the practitioners of the relevant niches understand and value. It feels well founded and at least ideally true. If it is sophisticated and comes with a good track record in other scientific contexts, it is assumed to be deep and somehow true. All this helps build rich practical networks around mathematical ideas, even when these ideas do not reflect empirical reality very well. (Roi 29017, 10)

(….) [I]f we want to understand the surprising force of mathematics demonstrated in this vignette, we need to engage in a more careful analysis of mathematical practice. (Roi 29017, 10)