6
Simplicity as a Surrogate1

6.1. Introduction

The idea is found almost everywhere, from the most prosaic to the most abstruse settings. Choosing the simpler option speeds you to the truth. In ordinary life, when the lights go out, we choose the simpler hypothesis that the electrical power has failed. We discard the more complicated hypothesis that every light bulb malfunctioned at the same time and, worse, that each malfunctioned for an independent reason. In cosmology, we choose the simpler hypothesis that the same physical laws obtain here as in distant places and epochs, even though we cannot rule out that they may differ in parts quite remote from us.

Do these judgments implement a universal principle of inductive inference? It says:

If two hypotheses are each adequate to the phenomena, the simpler is more likely true.

My goal in this chapter is to deny the efficacy of any such universal principle of inductive inference. For the material theory of induction entails that no such rules are efficacious. To explain the popularity of appeals to simplicity, I will argue that good invocations of simplicity are really veiled references to background facts or assumptions whose content functions to license the relevant inductive inference. The apparently singular appeal to simplicity actually masks an appeal to such a diversity of context-dependent facts that no univocal meaning can be attached to it.

This is the sense in which simplicity is a surrogate. In so far as it is epistemically efficacious, the short and snappy invocation of simplicity is really a surrogate for background facts or assumptions. These background facts do the real epistemic work and, commonly, are much harder to capture in a comparably short slogan. There will be cases in which these backgrounds resemble one another so that a common idea of simplicity appears to be invoked. However, the extent of these cases will always be limited. As we move farther afield, we will encounter cases in which the backgrounds differ sufficiently for the similarity to fail. In general, there is no well-specifiable, universally applicable, epistemically efficacious principle of simplicity in inductive inference.

The analysis of this chapter is a deflationary analysis of simplicity that runs counter to the celebration of simplicity in the scientific literature. The analysis does have a small pedigree in the philosophical literature. It is the view of simplicity long defended by Elliott Sober. Sober summarizes his view (in emphasized text) accordingly:

Whenever a scientist appeals to parsimony to justify the conclusion that one hypothesis is more reasonable than another in the light of observational data, substantive assumptions about the world must be involved. In practice, parsimony cannot be “purely methodological.” (1988, p. 40)

And more compactly: “Appeal to simplicity is a surrogate for stating an empirical background theory” (p. 64).

The following section, 6.2, provides a brief illustration of how apparently epistemically efficacious invocations of simplicity are really indirect appeals to background facts. Section 6.3 brackets off two cases in which simplicity offers only pragmatic gains. These are the cases in which simplicity is urged as an efficient search heuristic and in which simplicity is demanded merely to give a compact summary of past experiences.

The two sections that follow develop and deflate two primary senses of simplicity. The first principle, discussed in Section 6.4, expresses simplicity in a count of entities or causes. The classic statement is Ockham’s razor: “Entities must not be multiplied beyond necessity.” It fails as a principle of parsimony, I will argue, since there is no clear way to count the number of things to be minimized. The principle is reinterpreted as a truism of evidence, that one should not infer to more entities than the evidence warrants, where this evidential warrant is understood materially. The second principle of parsimony, discussed in Section 6.5, requires us to infer to hypotheses whose description is simple. This principle fails as an independent principle since modes of description vary. These variations greatly affect the descriptive simplicity of the hypotheses. This form of the principle can only guide us if we fix the mode of description, and the guidance will be good only if that mode is properly adapted to the prevailing facts.

Section 6.6 will examine in more detail the most popular illustration of simplicity in the philosophical literature: curve fitting. The invocation of simplicity in standard curve fitting, I argue, is a surrogate for specific background facts. These facts include the obtaining of a particular model of how error in data confounds some true curve; that the parameterization used is suitably adapted to the background facts; and that, in the strongest cases of this adaptation, the hierarchy of functional forms fitted corresponds to background assumptions on the presence, likelihood, and strength of certain processes. Ascending the hierarchy is not authorized by some abstract principle that tells us to proceed from the simpler to the more complex. Rather, it is a successive accommodation of the curves fitted to the most likely or strongest processes and then to those less so. The concluding two sections, 6.7 and 6.8, illustrate this last adaptation of the hierarchy in the examples of fitting orbits to observed positions in astronomy and the harmonic analysis of tides.

6.2. How It Works: The Birds

Just how can simplicity serve as a surrogate for background facts? Here is an easy illustration. Imagine that you are walking on a sandy beach washed smooth by the ocean waves. As you walk over a clear expanse of smooth sand, you notice a track left by a bird (Fig. 6.1).

The prints are clear and unmistakable. You can just see how the bird waddled over the sand—left, right, left, right—leaving the prints. But why assume that it was just one bird? Perhaps the left foot prints were made by a one-legged bird that hopped awkwardly over the sand. Then a second one-legged bird, this time having only a right leg, pursued it, leaving the right footprints in just the right place to simulate the waddle of a single two-legged bird. Or perhaps there was a large flock of one-legged birds, each of which touched down on the sand just once, all perfectly coordinated to leave the track.

Each hypothesis explains the track; however, we do not take the various, one-legged bird hypotheses seriously. How might we defend this judgment? The one-bird hypothesis is by far the simplest. In ordinary discourse, merely declaring it might be a sufficient defense. If our methodology is at issue, then just declaring that it is the simplest might not be enough to secure it. If we need more, we can turn to the great Isaac Newton. At the start of Book 3 of his magisterial Principia, he asserted four “Rules of Reasoning in Philosophy” that would guide subsequent analysis. The first two rules are

1. We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.

And

2. Therefore to the same natural effects we must, as far as possible, assign the same causes. As to respiration in a man and in a beast; the descent of stones in Europe and in America; the light of our culinary fire and of the sun; the reflection of light in the earth, and in the planets. (Newton [1726] 1962, p. 398; emphasis in original)

These two rules remain the clearest and firmest pronouncement of a methodological principle of parsimony in science.

Applied to the birds, Rule 1 tells us immediately that we should use the one-bird hypothesis, for it is a truth that there are two-legged birds and their behavior is sufficient to explain the tracks. We do not need the many-bird hypothesis, so it should not be admitted. Thus, we conform with Rule 2 by assigning the same cause, a single bird, to the many footprints.

So far, all is well. Simplicity has provided the principled justification for our intuitive judgment. That will not last, however. We now proceed farther down the beach and come to a place where the smooth sand is criss-crossed by very many tracks (Fig. 6.2).

We would normally posit that many birds alighted on the sand, each leaving just one track. However, there is another hypothesis: the tracks were left by just one, very busy bird. It waddled over the sand; flew to another spot; waddled some more; and so on, until the final set of tracks was formed.

A mechanical application of Newton’s rules leads us directly to the one-busy-bird hypothesis. We are, as before, assigning the same cause, one bird, to the same effects, the one set of tracks. Few of us would accept this outcome. We would be satisfied with the one-bird hypothesis for the single track but expect a good analysis to return a many-bird hypothesis for the case of many tracks. We would surely reason something like this. In the case of the single track, we rule out the many, one-legged bird hypothesis because we know that one-legged birds are rare and, if they were on the beach, it is unlikely that they would follow each other around in just the way needed to produce a single track. For the case of many tracks, we know that it is possible for one bird to be very busy and produce the multiple tracks. However, we rarely if ever see such a lone, busy bird, whereas flocks of birds producing tracks like this are quite common.

These further reflections show that our initial analysis was not merely based on the invocation of simplicity. We chose the one-bird hypothesis for the single track on the basis of our knowledge of birds. It is a shared knowledge, so we generally feel no need to explain in tedious detail why we rule out other possible but unlikely hypotheses: two one-legged birds hopping, many one-legged birds alighting at once, a mutant four-legged bird, and so on. We can dismiss all of these far-fetched notions with a breezy wave towards the simplest hypothesis.

In short, what we offer as a conclusion governed by some general principle of parsimony is really a conclusion dictated by our knowledge of background facts. We use an appeal to simplicity as a convenient way of circumventing the need to explain in detail these background facts, whose details can become quite convoluted. My claim here is that all epistemically efficacious invocations of simplicity have this character.

6.3. Pragmatic Justifications of Simplicity

Let us return to the standard view that treats a preference for simplicity as a methodological principle of universal character. What justifies it? What precisely does the principle assert? My interest in simplicity is restricted to the case in which simplicity functions epistemically as a marker of truth; we are to choose the simpler hypothesis or theory because, we are assured, it is more likely to be true. I will argue below that a principle of this form has no precise content and no good justification. However, before we proceed, we need to dispense with two distracting special cases that lie outside our immediate concern. In these cases, simplicity is sought merely for pragmatic reasons.

6.3.1. Simplicity for Economy of Search

In seeking to understand some new phenomenon, scientists commonly deal with many hypotheses or theories. How should they go about searching among them and testing them? A common recommendation is that they should start with the simplest hypothesis or theory. They are the easiest to deal with and, if incorrect, likely to be refuted by new evidence sooner than a more complicated one.

In the 1920s, it was found that distant galaxies recede from us at a speed that increases with distance. In 1929, Hubble proposed that the speed of recession was linearly proportional to the distance. In principle, he could have fitted a complicated, tenth-order polynomial function to his data. The linear dependency, however, was easier to deal with formally. If it was the wrong relation, new data would likely show the error much faster than with a more complicated function. A tenth-order polynomial is able to contort itself to fit a larger range of data so that considerably more data may be needed to refute it.

This sort of circumstance is common. One of the simplest hypotheses concerning an ailment is that it is caused by a specific pathogen. Famously, in the mid-nineteenth century, John Snow was able to trace the cause of a cholera outbreak in London to tainted drinking water drawn from a public water pump on Broad Street. More recently, the cause of AIDS—acquired immune deficiency syndrome—has been identified as the HIV virus. Once the simple hypothesis was pursued, it was readily affirmed. Were definite pathogens not responsible, the simple hypothesis could likely have been ruled out fairly quickly by the appearance of cases in which no exposure to the conjectured pathogen was possible. Matters are quite different with ailments such as cancer. Multiple factors can make a cancer more likely, including carcinogenic chemicals, ionizing radiation, certain viruses, and even specific genes. Dealing with this multiplicity of causal factors and discerning which are operative when is considerably more difficult.

These simple observations were incorporated into analyses of scientific discovery. Karl Popper (1968, chap. 7) argued that science proceeds through a continuing cycle of the conjecture of new hypotheses and their refutation. He identified the simpler hypotheses with the more falsifiable. It follows that the cycle advances faster if the scientists investigate more falsifiable hypotheses—that is, simpler hypotheses. A mathematically more sophisticated analysis of the role of simplicity in heuristic search has been provided by Kevin Kelly (2007). In the context of a formal learning theoretic analysis of the evidence-guided search for hypotheses, he showed that favoring simpler hypotheses is a more efficient way of getting to the truth.

How are these considerations relevant to our present concerns? One might ground the favoring of simplicity in searching in two related suppositions: that nature is ontically simple or that nature is descriptively simple in our languages. In both cases, further discussion must be deferred to later sections of this chapter, where I argue that both suppositions are epistemically efficacious only in so far as they make indirect appeals to background assumptions.

These assumptions, however, are not needed to ground the heuristic recommendation. It is still good advice to investigate the simplest hypothesis first in a world that is indifferent to the simplicity of hypotheses. Whether the world is more likely to give us a linear function or a tenth-order polynomial, the linear function will still be dealt with more easily and more quickly. Whether ailments are more likely to be caused by a single pathogen or by many factors, it is still most expeditious to check the single pathogen hypothesis first.

In short, simplicity can remain a good heuristic in hypothesis searching without a need for nature to be governed by a general principle of simplicity or parsimony.

6.3.2. Simplicity as Mere Economy of Expression

Ernst Mach famously held the view that scientific laws were merely compact summaries of our experience. In an 1882 address to the Imperial Academy of Sciences in Vienna, he said:

“the goal which it [the intellect] has set itself is the simplest and most economical abstract expression of facts” (Mach 1898, p. 207; emphasis in original).

The idea can be put crudely as follows. Galileo asserted that the distance fallen by a given body varies in accordance with the square of time of fall. In Mach’s view, all that Galileo is allowed to assert is that each pair of distances and times we have measured for falling bodies conforms to this relation.

If this is all that is asserted, then the role of simplicity is merely one of convenience. One seeks the least troublesome way of summarizing the facts at hand. In more modern terms, the exercise is essentially one of data compression. We could report all the numerical data pertaining to the fall of many bodies; or we could report merely that these data all conform to Galileo’s relation without loss of anything that matters.

This may seem like an extreme view that would be well outside the philosophical mainstream nowadays. But much engineering practice still conforms to it. This is because engineering commonly deals with systems dependent on many variables, and the systems are sufficiently complicated that a fundamental analysis is precluded. To deal with this problem, the behavior of the system is measured experimentally under widely varying circumstances and the collected data reduced to as compact a form as possible.

One of the best-known examples is the treatment of fluid flow in pipes. Even this simple problem involves many variables: the fluid’s speed, density, and viscosity; the pressure drop in the pipe; and the pipe’s diameter and surface roughness. Once the flow becomes turbulent, this empirical approach is the only tractable one. Moody (1944) presented a now famous chart summarizing the outcomes of many experiments (see Figure 6.3).

In this one chart, one can read the pressure drop associated with the flow of fluid of specified speed, density, and viscosity in a pipe of a specified diameter and surface roughness.

In so far as the chart merely summarizes the outcomes of experiments already performed, it is free of entanglement with the concerns of this chapter. One need make no reference to background facts when one reports that a simple formula generates a curve that happens to pass through the data points that are near enough. I will complain shortly of the ambiguity in the notion of simplicity. This ambiguity is untroubling here. We can use any formula that generates a curve that fits the data well enough. The choice is purely pragmatic.

But this purely pragmatic use of simplicity is an extreme case. I believe that it is rarely—and possibly never—realized in all purity. The examples above do not realize it fully. The Moody chart is a summary of past experience. But it is also a great deal more. Its primary use is as an instrument of prediction. The presumption is that if an engineer constructs a piping system with flow rates, fluid densities, and so on, which match conditions covered by the chart, then the relevant curve will reflect the pressure drop that will be found. That can only happen if the chart is properly adapted to broader facts about fluid flow in pipes in general.

These facts have the character of simplicity assumptions. We must assume that the variables included are all that matter. Temperature does not enter into the chart explicitly; it is assumed that thermal effects are fully captured by the unrepresented dependence of density and viscosity on temperature. We must assume that the curves fitted to the data points interpolate correctly between them so that the chart makes good predictions for cases whose precise combination of variables have never been observed.

To the extent that the descriptions seek to go beyond past experience, they seek the type of epistemic success to which the subsequent discussion applies.

6.4. Principles of Parsimony: Ontic Simplicity

The notion that parsimony can successfully guide us epistemically has many expressions, and one might despair of categorizing them all successfully. There is, however, a broad division between ontic simplicity and descriptive simplicity. I will discuss ontic simplicity first and then turn to descriptive simplicity.

In the ontic version of the principle, we are guided to the truth by favoring accounts that posit the fewest entities or processes in the world. The locus classicus of this notion is “Ockham’s razor.” Its now universal formulation is that “Entities must not be multiplied beyond necessity” (entia non sunt multiplicanda praeter necessitatem). Curiously, this formulation is not found in the writings of the fourteenth-century scholastic, William of Ockham. His closely related pronouncements include “It is useless to do with more what can be done with fewer” and “A plurality should not be assumed without necessity.”3 It has been a historical puzzle to locate the source of the popular formulation.4 Another puzzle is that Ockham’s name should be so exclusively attached to this maxim of simplicity, for it was an idea that, according to Maurer (1999, p. 121), was used commonly from the thirteenth century, after being gleaned from Aristotle.

The greater puzzle is why modern thinkers would look to a fourteenth-century scholastic for this sort of guidance. His understanding of the demand of parsimony was rather different from its modern scientific use. In Ockham’s view, it was not binding on God. As Maurer (1999, p. 120) reported, Ockham felt that “God is not bound by it; he does many things by more means which he could do by fewer, and yet this is not done uselessly, because it is God’s will.”5

The better-formulated statement of the sentiments of Ockham’s razor are Newton’s two rules of reasoning quoted in 6.2. The notion is advanced explicitly as a rule of reasoning; and Newton provided justifications for it: “Nature does nothing in vain” and “Nature is pleased with simplicity, and affects not the pomp of superfluous causes” (p. 398). The justifications are dressed in anthropomorphic garb. Nature, surely, is never literally pleased or displeased. If we abstract the anthropomorphism, Newton is enjoining us to infer to the simpler case of fewer causes because the world is simpler and harbors fewer causes. This is a factual claim about the world.

These justifications seem routine. We find something similar reported in Aquinas (1945, p. 129) (writing before Ockham’s birth): “If a thing can be done adequately by means of one, it is superfluous to do it by means of several; for we observe that nature does not employ two instruments where one suffices.”

6.4.1. The Difficulties of Ontic Simplicity

The ontic version of the principle of parsimony outlined above faces many difficulties. The most immediate is that we have no general prescription for how to count entities, processes, or causes to which the principle is applied. It is not hard to find a sufficiently severe ambiguity as to compromise the principle.

How can we count entities when we compare a continuum and a molecular theory of gases? The continuum theory represents the gas as a single, continuous fluid. The molecular theory represents it as a collection of numerous molecules, of the order of 1024 for ordinary samples of gases. Do we count one entity for the continuum gas and 1024 for the molecular gas so that the molecular gas posits many more entities? Or do we invert the count? A continuum is indefinitely divisible into infinitely many parts.6 The molecular gas consists of finitely many molecular parts. Has the continuum now infinitely many more parts than the molecular gas?

This discussion in terms of entities can be converted into the causal conception of Newton’s rules. What causes the pressure exerted by a gas jet impinging on a surface? Do we count the impact of the continuum gas as one cause? Or do we count infinite causes for the infinitely many impacts of its infinitely many parts?

What of the justification for this ontic principle? Newton asserts the world is simpler in employing fewer causes. This assertion is empty in so far as the counting of causes is ill-defined. However, even setting that concern aside, the claim is unsustainable. Nature is not simple. Traditional alchemical theories posited three or four elements in an attempt to account for chemical appearances. We now know that this count is far too low. A tractable model of chemistry requires over ninety elements. Perhaps nature is pleased with chemistry, but surely not for the simplicity of the count of elements.

The existence of isotopes is especially damaging to Newton’s justification. For one can explain the chemistry of carbon quite well just by assuming that there is one element, carbon. Hence, according to Newton’s rules, we should infer to there being just one carbon element since nature “affects not the pomp of superfluous causes.” That is, we should infer to the one element and not to the existence of multiple types of chemically identical carbon, because that is the way nature is. Yet that is not the way nature is. Carbon exists in multiple, chemically identical isotopes, carbon-12, carbon-13, and carbon-14. The recommendation to infer to just one type of carbon may well be good advice as far as the chemistry of carbon is concerned. I do not wish to impugn this recommendation or to suggest that an inductive rule is defective because it sometimes leads us astray. This is part of the risk we take whenever we carry out inductive inference. Rather, the issue here is the justification. While the recommendation may be good, it cannot be justified by a supposition that factually there is just one type of carbon. Factually, this is false.

6.4.2. Rescue by the Material Theory: The Principle as an Evidential Truism

This ontic form of the principle of parsimony is troubled. Yet it has figured and continues to figure prominently in successful science. There must be something right about it. The clue to what is right lies in the ambiguous qualifications found in every formulation. We are not to multiply entities “beyond necessity”; we are to admit no more causes than are “sufficient to explain…”; we are to assign the same cause “as far as possible.” These qualifications mean that the principle is not self-contained. Something more supplies the sense of necessity, possibility, and sufficiency.

Newton’s formulation gives us the clearest indication of what that something is. We are not to proceed beyond that which is “sufficient to explain the[ir] appearances.” This gives us some flexibility. We can add to the causes we admit as long as they are sufficient to explain the appearances. We are not to go beyond that. Since we routinely infer inductively to that which we count as explaining the appearances, this amounts to telling us to infer to no more than that for which we have inductive authorization. Understood this way, the principle is revealed to be a truism of inductive inference, which says, “We should not infer to more than that for which we have good evidence.” It is a corollary of another truism: “We should infer inductively only to that for which we have good evidence.”

How did an inductive truism become enmeshed with the muddle of the metaphysics of simplicity? The key relevant fact is that the truism is not an independent inductive principle; it is a meta-inductive principle. That is, it is not a principle of an inductive logic. Rather, it is a principle about how other inductive logics should be used. That this is so is harder to see if one conceives of inductive inference formally. The principle is entangled with assertions about how the world is factually. If one understands inductive inference materially, however, this entanglement is expected. Moreover, it clarifies how the original principle can be a good epistemic guide.

We can see this entanglement in Newton’s first use of his Rules 1 and 2. In Book 3 of Principia, Proposition 4 Theorem 4 asserts that the force of gravity that draws objects near the earth’s surface is the same force that holds the moon in its orbit. He assumes that the force acting on the moon intensifies with a decreasing orbital radius according to an inverse square law, as it does with other celestial objects. It follows that were the moon to be just above the earth’s surface, it would fall to earth with the same motion as ordinary bodies fall by gravity. On this point, he wrote:

And therefore the force by which the Moon is retained in its orbit becomes, at the very surface of the Earth, equal to the force of gravity which we observe in heavy bodies there. And therefore (by Rule 1 & 2) the force by which the Moon is retained in its orbit is that very same force which we commonly call gravity; for were gravity another force different from that, then bodies descending to the Earth with the joint impulse of both forces would fall with a double velocity… ([1726] 1962, p. 408)

Newton here invokes his rules to complete the inference. However, the inference is already fully controlled by a factual presumption: that the matter of the moon is the same as the matter of the earth and, if brought to the surface of the earth, would behave like ordinary terrestrial matter.

This factual assumption authorizes Newton’s conclusion, and he gives the reason. Were there to be some additional celestial force acting on the matter of the moon but not on ordinary terrestrial matter, then the moon would fall with double the motion of ordinary terrestrial matter. This contradicts the assumption that the matter of the moon behaves just like that of the earth. This is a kind of simplicity assumption: contrary to ancient tradition, there is no difference between terrestrial and celestial matter. But the comprehension and use of this assumption make no appeal to abstract metaphysical notions of simplicity. The assumption is a specific factual statement and it powers the inductive inference, as the material theory requires.

We also see this entanglement in the illustrations Newton supplies for his rules. To illustrate Rule 2, he considered the “light of our culinary fire and of the sun” (p. 398). We are to assign the same cause to both. We now know that this is an erroneous conclusion. Culinary fires generate light from combustion; the sun generates light by a different process, nuclear fusion. What makes the inference appear unproblematic for Newton is that he is really relying on a tacit background assumption: that it is very unlikely that there is a process that produces intense light other than combustion. This fact powers the inductive inference.

In short, the ontic formulation of the principle of parsimony fails as a universal principle of inductive inference. It is too vague to be applied univocally, and efforts to give it a foundation in a supposedly general, factual simplicity of the world flounder. Its successes, however, can be understood to the extent that it is the meta-inductive principle that one should infer inductively to no more than that for which one has good evidence. The assertions of simplicity are veiled invocations of relevant facts that authorize the inductive inference, in accord with the material theory.

6.5. Principles of Parsimony: Descriptive Simplicity

The descriptive versions of the principle of parsimony do not directly address the numerical simplicity of entities or causes. Instead, we are enjoined to favor the simplest description of processes pertaining to them. It may not be possible to effect an absolute separation between the descriptive and the ontic versions of the principles of parsimony. For descriptive simplicity is tacitly supposed to reflect some sort of ontic simplicity. However, the explicit focus on language introduces sufficient complications to necessitate a separate analysis.

The best-known application of descriptive simplicity is curve fitting. In its simplest form, we are given a set of data points—that is, many measured values of a variable x and a variable y. These are presented as points on a graph, as shown below. We then seek the curve that fits them best. It is routine in curve fitting to start with a constant relation, then a linear one, then a quadratic, and so on, evaluating the fit of the curve as we proceed to higher order polynomials, as shown in Figure 6.4.

The fit will always improve as we increase the order. The higher the polynomial order, the larger the repertoire of curves available, and hence the more likely we are to come close to the data points.

Eventually, however, this greater flexibility will cause trouble. For the data is routinely assumed to be a compound of the true curve sought and confounding error. If the true law sought is merely a linear curve, the error will scatter the data around the true straight line. Higher order polynomial curves will have little trouble adapting to the random deviations due to noise. This will lead the fitted curve to deviate from the true curve as it responds to the vagaries of the noise. Figure 6.5 shows the best fit of linear and eighth-order polynomial curves to a data set.

The apparently better fit of the higher-order curves is spurious. This phenomenon is known as “overfitting.”

The primary burden in curve fitting is to find a balance of the two effects: the simplicity of the curves that fit less well must be balanced against the better fit of more complicated curves that overfit. The simplicity of a curve is derived from its description. The polynomial family consists of smaller, nestled families of curves:

Constant: y = a

Linear: y = a + bx

Quadratic: y = a + bx + cx2

Cubic: y = a + bx + cx2 + dx3

Quartic: y = a + bx + cx2 + dx3 + ex4

And so on.

That is, the formulae that describe the curves have more parameters as we proceed to the less simple, higher-order polynomials. The constant curve has one parameter, a. The linear curve has two parameters, a and b. The quadratic curve has three parameters, a, b, and c. And so on. Built into the practice of curve fitting is a simplicity postulate: favor those curves whose descriptions require fewer parameters.

The preference for simpler descriptions has been applied more broadly. Perhaps its best-credentialed proponent is Albert Einstein. The laws of the fundamental theories of physics employ constants. Newton’s theory of gravitation employs the gravitational constant G. Einstein’s special theory of relativity employs the speed of light c, and his general theory employs both c and G. Quantum theory employs Planck’s constant h, as well as numerous quantities characteristic of the various particle interactions, such as the charge of the electron e. The standard model of particle physics now employs nearly twenty such constants. Some of these constants can be absorbed into the system of units used. The speed of light c can be suppressed merely by measuring distance in light years; then the speed of light reduces to unity.

Einstein grounded his hope for a physics free of all further constants in a belief in the simplicity of nature:

If one considers this [suppression] done, then only “dimension-less” constants could occur in the basic equations of physics. Concerning such, I would like to state a theorem which at present cannot be based upon anything more than upon a faith in the simplicity, i.e., intelligibility, of nature: there are no arbitrary constants of this kind; that is to say, nature is so constituted that it is possible logically to lay down such strongly determined laws that within these laws only rationally completely determined constants occur (not constants, therefore, whose numerical value could be changed without destroying the theory). (1949, p. 61–63)

While the freedom from these constants reflects something factual in the structure of the world, Einstein expressed it in terms of the descriptions of that structure—that is, in terms of the constants appearing in the equations that describe it. Just as curve fitting should favor smaller numbers of parameters, Einstein favored laws with the fewest arbitrary parameters.

These sentiments come from Einstein later in his life. By then, he had abandoned his earlier allegiance to positivistic approaches. He had become a mathematical Platonist, and that was a doctrine, he assured us, he had learned from his experiences in physics.7 His 1933 Herbert Spencer lecture, “On the Methods of Theoretical Physics,” offered an explicit and powerful manifesto:

Our experience hitherto justifies us in believing that nature is the realisation of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. ([1933] 1954, p. 274)

Einstein continued to detail how we could use mathematical constructions to make discoveries in physics:

The physical world is represented as a four-dimensional continuum. If I assume a Riemannian metric in it and ask what are the simplest laws which such a metric can satisfy, I arrive at the relativistic theory of gravitation in empty space. If in that space I assume a vector-field or an anti-symmetrical tensor-field which can be derived from it, and ask what are the simplest laws which such a field can satisfy, I arrive at Maxwell’s equations for empty space.

The recipe is one of descriptive simplicity. In each context, one writes the simplest admissible equations and thereby recovers the law.8

6.5.1. The Difficulties of Descriptive Simplicity

The difficulty with any principle expressed in terms of descriptions is that it can be rendered incoherent by merely altering the descriptive system used. In so far as the simplicity principle of curve fitting only requires us to favor the curves with fewer parameters, it is unsustainable. Simply rescaling the variables used can overturn its judgments completely, as will be illustrated in the following section.

The idea that we get closer to the truth by writing mathematically simpler laws has the imprimatur of Einstein. However, it is unsustainable for the same reasons that trouble curve fitting. Judgments of just what is descriptively simple are too malleable. Einstein’s own general theory of relativity illustrates the problem. When the theory first came to public notice after the eclipse tests of 1919, it was notorious for its abstruse difficulty. The eminent astronomer George Ellery Hale confided in a correspondence:9

I confess that the complications of the theory of relativity are altogether too much for my comprehension. If I were a good mathematician I might have some hope of forming a feeble conception of the principle, but as it is I fear it will always remain beyond my grasp.

The New York Times of 19 November 1919 reported an incredible tale, reflected in the partial headline “A book for 12 wise men”:

When he [Einstein] offered his last important work to the publishers he warned them there were not more that twelve persons in the whole world who would understand it, but the publishers took it anyway.

The fable took root. It is repeated in the publisher’s introduction to Hendrik Lorentz’s (1920, p. 5) popularizations of relativity theory.

As the decades passed, general relativity was absorbed into mainstream physics and opinions began to migrate. By the 1970s, the standard textbook for the theory came to a startlingly different conclusion:

“Nature likes theories that are simple when stated in coordinate-free, geometric language.” […] According to this principle, Nature must love general relativity, and it must hate Newtonian theory. Of all theories ever conceived by physicists, general relativity has the simplest, most elegant geometric foundation […] By contrast, what diabolically clever physicist would ever foist on man a theory with such a complicated geometric foundation as Newtonian theory? (Misner et al. 1973, pp. 302–03)

How is a reversal of this magnitude possible? The key is the centrality of “coordinate-free, geometric language.” One finds general relativity to be the simpler theory when one adopts the appropriate language. As a result, the principle of descriptive simplicity, as enunciated by Einstein, is incomplete. Without a specification of the right language to be used, it can give no direction at all.

Perhaps one might hope that mathematics would provide the natural descriptive language for our science and, in particular, for our physics. A cursory glance at the interplay of mathematics and science shows things to be different. There is no unique language for physical theories. New physical theories commonly appear mathematically difficult and even messy. This is followed by efforts by mathematicians and the scientists themselves to simplify the presentation and manipulations of the theory. As I have argued elsewhere (Norton 2000, pp. 166–68), what results are new mathematical methods and new formulations of the theories that become successively simpler.

Newton’s development of his mechanics employed the ancient methods of Euclidean geometry. His contemporaries required considerable insight and facility in geometry to follow and emulate his difficult demonstrations and proofs. Over the centuries, Newton’s theory was re-expressed in terms of algebraic symbols and calculus. Many of what were once abstruse results became natural and simple. Quantum mechanics developed in the first quarter of the twentieth century. The theory that resulted in the late 1920s was a complicated mess of different approaches and techniques: matrix mechanics, wave mechanics, Dirac’s c- and q-numbers. Subsequent efforts showed all of these theories to be variant forms of a single theory that found its canonical mathematical formulation in von Neumann’s 1932 classic, Mathematical Foundations of Quantum Mechanics. Even Einstein’s general relativity benefited from this reformulation. His original methods did not include the now key notion of parallel displacement. This notion was introduced in response to the completion of the theory by the mathematician Tullio Levi-Civita in 1917.

6.5.2. Rescue by the Material Theory: Adaptation of Language as a Factual Judgment

As discussed above, we must acknowledge that there is something right in the idea that descriptive simplicity is a guide to the truth. The real epistemic work is done in finding and developing a language or descriptive apparatus appropriate to the system under investigation. What makes an apparatus appropriate is precisely that the truths concerning the system find simple expression in it. Then, it is automatic that seeking simple assertions in the language or descriptive apparatus leads us to the truth.

That is, the principle of descriptive simplicity guides us to the truth in so far as the language we use is properly adapted to the background facts. Hence, what is really guiding us is not some abstract notion of simplicity but merely the background facts as reflected in our choice of descriptive language. This inductive guidance from background facts is, of course, precisely what is called for by the material theory of induction. This idea will be illustrated with the example of curve fitting in the next section.

6.6. Curve Fitting and the Material Theory of Induction

As a mode of discovery, curve fitting is based on the idea that fitting a simple curve to data can guide us to the truth. The material theory of induction requires that these inductive inferences are warranted by background facts. Here, I will describe in greater detail the character of these background facts. We will see that the vague and incomplete injunctions to seek the simpler curve translate into more precise constraints, expressed in terms of these background facts. The characteristics of background facts found in many but not all cases of curve fitting can be grouped under three headings, as below.

6.6.1. The Error Model

When curve fitting is used to seek some underlying truth or law, the presumption is that the data to which the curve is fitted have been generated by a standard error model of the form:

Error laden data = true curve + error noise.

That curve fitting operates with data produced by this model is so familiar that it is easy to overlook its importance. The techniques of curve fitting are designed to strip away confounding noise. Thus, the assumption of the standard error model must be true if these techniques are to guide us towards the truth.

A quick way to see its importance is to consider the curve-fitting problem shown in Figure 6.6. We seek the value of the quantity a that gives the best fit of y = 1/[log(x) − a] to the data.

The optimum value turns out to be the value shown, a = 0.41.

Superficially, the problem looks traditional. However, this curve-fitting problem does not conform to the standard error model. The data represent the density of prime numbers between x = 10 and x = 100. The first datum at x = 10 reports the number of primes between 1 and 10. There are four of them: 2, 3, 5, and 7, so that y = 4/10 = 0.4. The prime number theorem assures us that the density y(x) approaches 1/log(x) for large x. A corollary is that the density also approaches 1/[log(x) – a] for some constant a, for the two quantities, 1/log(x) and 1/[log(x) − a] approach one another for large x. The curve-fitting problem is to ascertain which value of a gives the best-fitting curve for values of x in the range 10 to 100 covered by the data. The result is 0.41, as shown.10

Instead of the standard error model, this problem conforms to a non-standard error model in which truth and error are permuted:

True data = error laden curve + error noise.

This means that, epistemically, the exercise is different. We are not seeking truth. We already have the complete truth in the data that report the true density of prime numbers. Instead, we are seeking a summary that has the least deviation from the truth, where the notion of “least deviation” is one we can choose conventionally. In this case, I chose a fit that minimized the sum of squared deviations.

Engineering applications, such as the Moody diagram above, illustrate a second way that we may deviate from the standard error model. In so far as we are merely seeking a compact summary of past experience, there is no real error model in use at all, for there is no hidden truth sought. Our ambitions, however, are rarely so limited. For example, as noted above, the Moody diagram is typically not intended merely as a compact historical report. It is also intended as a predictive tool. To the extent that this is the case, the standard error model is presumed.

But the practice is somewhat protected from the full rigors of the model since engineering practice rarely requires perfectly exact prediction of pressure drops in pipes. A prediction correct to within a few percent is more than sufficient for most applications. This affords great protection when we seek predictions for conditions that interpolate between those used to create the original chart. Fitting just about any family of curves will interpolate to the requisite level of accuracy. In effect, we are conforming the data to a weaker model:

Error laden data = near enough to true curve + error noise.

Near enough to true is good enough for interpolation.

This protection is lost when we seek to extrapolate to new conditions outside those used to create the diagram. For then two curves that each interpolate among the data equally well may diverge markedly when we extend beyond the condition in which the data were collected. Then we need to find which is the true curve on pain of uncontrolled errors entering our predictions. This divergence is illustrated in Figure 6.4 above. The polynomials in the figure interpolate the data comparably well in the range of x = 0 to x = 10. They diverge rapidly outside the range in which the data were collected, giving markedly different results in the range x = 10 to x = 12.

6.6.2. The Parameterization

Descriptive simplicity can only be a good epistemic guide to the truth, I have argued, if the language of description is chosen so that the truths correspond to simple assertions. In the case of curve fitting, this condition translates into a matching with background facts of the parameterization used and the family of its functions from which the curves are drawn.

We are free to rescale the quantities used to describe our measurements; and we do. We may compare cars by their speeds or, equivalently, by the times they take to cover some fixed distance. Since one parameter is the inverse of the other, fitting some family of curves to speed will generally give different results from fitting the same family of curves to times. This reparameterization is common. In acoustics, we measure loudness of sounds in decibels, which is a logarithm of the power of the sound. In astronomy, we measure the apparent magnitude of the stars on a scale that is the logarithm of the intensity (that is, the energy flow per unit area).

To see how the parameterization we choose makes a difference, we will develop an example in which the data are generated by a true linear relation y = x, as in Figure 6.7. The data have been simulated with very little noise, so the best-fitting straight line11 comes close to y = x.

Rescaling the x variable to z = sin−1(x) means that the true curve will be rescaled to y = sin(z). However, a polynomial curve fit between y and z will never return this curve, for y = x is equivalent to a polynomial of infinite order in z:

A curve-fitting algorithm that proceeds up the family of polynomials in z will necessarily halt at some finite order and so cannot return the true curve. Finding polynomials of the best fit for the rescaled data of Figure 6.7 shows how poorly the polynomial fit performs. The best-fitting linear, quadratic, cubic, and quartic polynomial curves interpolate the data well. However, as Figure 6.8 shows, they fail immediately on extrapolation beyond the domain x = 0 to x = 0.9 in which the data was generated.12

The problem is the same in the reverse direction. If y = z is the true curve, then this true curve corresponds to an infinite polynomial if we parameterize the data using x, for

for |x| < 1. Once again, ascending the family of finite polynomials will never return the true curve.

Choosing the right parameterization and family of curves amounts to properly adapting them to the background facts. If the attenuation of the intensity I(r) of some signal with distance r is due to an absorptive medium, then the signal attenuates as I(r) = I(0) exp(−Ιr), for some constant Ι. The exponential dependence of I(r) amounts to another infinite order polynomial in r. If we rescale, the relation reduces to a simple linear dependence of the logarithm of I(r) on r, for then the attenuation follows log I(r) = constant – Ιr. If, however, the attenuation is due to spreading in space, signal intensity will attenuate according to I(r) = A/r2, for some constant A. This corresponds to log I(r) = A − 2 log(r), which once again corresponds to an infinite order polynomial of r. However, if we use both log I(r) and log(r) as our parameters, then the true curve is linear and its slope, −2, conveys the fact that the attenuation follows an inverse square law.

Perhaps the clearest example of this adaptation of the parameters and curves to the background facts arises when we have processes that are periodic in time. We should then use the time t as the parameter. The family of curves to be fitted should not be polynomials, since they are not periodic. Rather, we should use the family of periodic trigonometric functions, sin(t + a), sin(2t + b), sin(3t + c), etc., where the a, b, c, … are constant phase factors. We learn from Fourier analysis that this family is sufficiently rich to represent all periodic curves of likely interest to curve fitters.

6.6.3. The Order Hierarchy

We must have an adaptation of the descriptive language to background facts if descriptive simplicity is to be an effective guide to truth. In important cases, the adaptation can be especially tight. Curve fitting proceeds with some collections of families of curves, such as the constant, linear, quadratic, etc. In these important cases, the families of curves fitted correspond directly to particular processes. Then, fitting a curve from a family farther up the hierarchy corresponds to the inclusion of more processes in the account developed of the phenomena. Further, the adaptation has to be such that curves fitted earlier in the procedure correspond to stronger or more probable processes.

This adaptation will be illustrated in the following two sections with the cases of fitting trajectories to celestial objects and the harmonic analysis of the tides.

6.7. Fitting Orbital Trajectories

The standard method of curve fitting is to find the curve that minimizes the sum of the squares of deviations of the curve from the data. This “least-squares” technique was introduced around the start of the nineteenth century in astronomy to assist the fitting of orbits to celestial objects in our solar system. This application illustrates a tight adaptation of the curve-fitting method to the background assumptions that are the surrogates of simplicity. The family of curves fitted reflects the particular trajectories that background assumptions select. Moreover, ascending the order hierarchy reflects a pursuit of trajectories according to their likelihood and the strength of the processes that form them.

6.7.1. Ellipses, Parabolas, and Hyperbolas

When a new celestial object—a new planet or comet, for example—is sighted, an astronomer’s first task is to find the orbit that fits the positions seen. Astronomers do not follow the generic curve-fitting procedure of seeking first to fit a straight line and then proceeding up through higher order polynomials. Rather, the family of curves chosen is provided by gravitation theory. The initial model is provided by the “one-body problem”: the motion of a free body attracted to a central point by a force that varies inversely with the square of distance r to the point. That is, the attracting force is k/r2 for a suitable constant k. The familiar result, given early in any text on celestial mechanics,13 is that the trajectory is one of the three conic sections: an ellipse, a parabola, or a hyperbola.

Select polar coordinates (r, θ) in the plane of the orbit with the origin r = 0 at the center of force in the sun and set θ = 0 at the perihelion, the point of closest approach to the sun in the orbit. A single formula that covers all three curves is

where e is the eccentricity and L is the semi-latus rectum that, loosely speaking, fixes the width of the figure. (More precisely, it is the distance from a focus to the curve along a line perpendicular to the major axis.) We pass among the conic sections with equal semi-latus recta by changing the eccentricity e. A circle is e = 0, an ellipse is 0 < e < 1, a parabola is e = 1, and a hyperbola is e > 1.14

Figure 6.9 shows trajectories of the three types of conic sections with equal semi-latus rectum:

Once the semi-latus rectum L and eccentricity e are fixed, so too is the orbital motion. Kepler’s area law affirms that the radial line from the center of force to the object sweeps area at a constant rate with time t. That is the areal velocity h = r2dθ/dt is constant. This areal velocity is related to L byh = √(kL) . Hence, the angular speed of the object in its orbit, dθ/dt, is fixed once the trajectory is fixed.

Consider once again the default straight line of generic curve fitting. This familiar default is precluded by the background assumptions of gravitation theory. It can arise for objects in the solar system if they are moving at speeds much greater than those normally encountered. Then, the object will follow a hyperbolic trajectory with a very large eccentricity that is practically indistinguishable from a straight line.

The three conic sections provide the basis for the family of curves employed. When we allow for perturbations produced by gravitational attractions from other objects in the solar system, as we shall see below, the family is enlarged by allowing a slight motion in the curve. For example, the major axis of the ellipse along which the object moves may rotate slowly. Representing that slow rotation introduces further parameters and provides the full family of curves used in virtually all accounts of orbital motion.

Fitting an orbit to a celestial object involves moving up this hierarchy of curves until a suitably close fit is obtained. One might try describing this ascent as guided by some principle of parsimony that would require starting with the simplest curve and then moving up to more complicated ones; however, it is hard to see just which abstract notion of simplicity here would lead us to identify conic sections as the simplest case. Straight lines, the common starting point in general hierarchies of simple curves, are an extreme case never implemented in this celestial mechanics. Fortunately, no such notions of simplicity are needed to explicate the procedures used. The selection of curves and their order are guided by background assumptions on the likelihood of certain trajectories and on the likelihood and strength of processes that modify them in prescribed ways.

6.7.2. Comets

A simple illustration is provided by the methods used to fit orbits to newly discovered comets—that is, it is a simple illustration if we limit ourselves to the methods routinely used in the nineteenth century. The methods then customary were described by James Watson (1861, pp. 163–69). They depend essentially on the following background fact: comets typically have very eccentric orbits, and we get to observe them when they are in the vicinity of the sun. As Figure 6.9 above suggests, it becomes quite difficult there to separate the ellipses and hyperbolas with eccentricity close to unity from each other and from a parabola.15 This fact leads to the procedure described by Watson:

It is therefore customary among astronomers, when a comet has made its appearance unpredicted, to compute its orbit at first on the supposition that it is a parabola; and then, by computing its place in advance, find from a comparison of the actual observations, whether this hypothesis is the correct one. Should it be found to be impossible to represent the observed positions of the comet by a parabola, an ellipse is next computed and when this also fails, recourse is had to the hyperbola, which, provided the previous computations are correct in every particular, will not fail to represent the observations within the limits of their probable errors. (1861, p. 164)

In other words, the known fact of the high eccentricity of comets directs a choice of a parabola to fit the initial data. Astronomers then collect more data and move to a nearby ellipse and then hyperbola if the deviations from the parabola are sufficient.

The sequence of shifts reflects a definite physical assumption about the energy of the comet. Adopting an ellipse amounts to assuming that the comet’s total energy—kinetic plus potential—is negative so that it is bound to the sun and can never escape. Adopting the hyperbola amounts to assuming a positive energy sufficient to enable the comet to escape the sun’s gravity. The case of the parabola is the intermediate case of zero energy, which is the minimum level at which escape from the sun’s gravity is possible. Watson does not mention it in his text, but I believe the decision to try an ellipse after a parabola rather than a hyperbola reflects the prevalence of comets bound in elliptical orbits. Such comets will return periodically and thus are more likely to be seen by us. Adopting the hyperbola amounts to assuming that the comet will pass just once through our solar system, so that this is our one chance to see it. If the orbit is elliptical, we will have many chances.

There is a small element of arbitrariness in the procedure. Instead of fitting a parabola initially, astronomers could have chosen an ellipse with an eccentricity imperceptibly different from unity. (Such a trajectory would be near indistinguishable from a parabola in the vicinity of the sun.) Whichever is chosen as the first curve, the choice is driven by the background physical assumption that the comet is just on the energetic edge of being gravitationally bound to the sun permanently. Further data then directs a decision to one side or the other, within or beyond the edge. The selection of the curves fitted reflects this background physics.

6.7.3. Perturbed Orbits and New Planets

The conic sections discussed so far are grounded physically in Newton’s law of gravity through the one-body problem. However, they are not the complete family of curves fitted to bodies moving in our solar system. Very careful measurements show that a planet will trace out an almost perfectly elliptical orbit only over the shorter term. If it is tracked over the longer term, however, deviations will appear. They are sufficiently small that they can be represented as changes in the elliptical orbit that the planet is following. If, for example, the axis of the ellipse rotates in the plane of the orbital motion of a planet, then the orbit actually traced out takes on the flower-petal shape seen in Figure 6.10. It is an advance of the planet’s perihelion, its point of closest approach to the sun, with each orbit of the sun. Or at least this is the shape traced out if we depict an advance that is unrealistically fast for any planet.

In general, the small deviations from the perfect elliptical orbits are represented by slow changes in the six Keplerian elements that characterize each elliptical orbit. The first two elements are the most familiar: the semi-major axis and eccentricity specify the shape of the ellipse. The remaining elements locate that ellipse in space and place the planet on it at the appropriate time.16 The families of curves associated with these perturbed elliptical orbits are the ones fitted to the observed positions of the planets.17

These perturbed ellipses are initially fitted to the planets because they are found to fit. However, from the earliest days of Newtonian gravitation theory, the challenge has been to locate the physical cause of the perturbation, which is almost invariably sought in the perturbing gravitational influence of bodies other than the sun. Jupiter, the most massive of the planets, is a common source of perturbations. It exerts a large perturbing influence on Mercury for example. The axis of Mercury’s orbit advances and the ellipse is more eccentric when Jupiter is in line with this axis. The axis regresses and is less eccentric when Jupiter is perpendicular to the axis.18

The need to give a physical foundation for the perturbed ellipses fitted is uncompromised. One might initially find that some perturbed ellipse fits the motion. However, that fit remains tentative until the physical basis is located. Only then can astronomers know how well the perturbed ellipse will continue to fit the planet’s motion. More importantly, the perturbations to the ellipse can be adjusted according to the subsequent movements of the perturbing body.

Perhaps the most vivid illustration of the need for a physical basis for the changing elements of a planet’s ellipse arises in the celebrated discovery of the planet Neptune. The need for the physical basis is inverted to become a means for discovery, in this case, of a new planet. By the early nineteenth century, the orbit and perturbations of the planet Uranus had been established. However, not all of the perturbations could be explained by the gravitational action of known planets. In 1845, John Adams and Urbain Le Verrier independently pursued the possibility of another hitherto unknown planet outside the orbit of Uranus that would be responsible for the perturbations. They predicted the position of this planet. After an easy telescopic search in 1846, the planet was found and was eventually given the name Neptune.

That astronomers require the variant curve forms to have a physical foundation is seen most clearly when these efforts fail. The orbit of Mercury was also well established in the nineteenth century, and the bulk of its perturbations could be accounted for by the gravitational effects of the other planets. However, they could not be accounted for completely. Recalling the success with Neptune, Le Verrier (1859) proposed that these further perturbations could be accounted for by another new planet orbiting closer to the sun than Mercury. The new planet, which had come to be known as Vulcan, was never found.

This failure was discouraging. Nonetheless, astronomers could not abandon the idea that the perturbations were generated by some attractive mass somewhere. By the end of the century, many proposals were under investigation. Simon Newcomb’s (1895) treatment became the authoritative analysis. Its sixth chapter assessed a list of possible locations for new masses that might account for the anomalous motion of Mercury. They included masses located in a slight non-sphericity of the sun, in rings of masses or groups of planetoids inside Mercury’s orbit, or planetoids between the orbits of Mercury and Venus, and the possibility of masses associated with the zodiacal light, a diffuse glow seen around the sun.

More intriguing was a proposal by the astronomer Asaph Hall (1894). If the force of gravity does not dilute as the inverse square 1/r2 with distance r, but dilutes slightly faster, then the orbit of a planet would trace an ellipse that would advance slightly, as Mercury’s was observed to do. Hall noted that a very slight adjustment to the exponent in the inverse square was all that was needed to accommodate the anomalous motion of Mercury. He found that 1/r2.00000016 would suffice. Newcomb (1895, pp. 118–21) gave a more precise 1/r2.0000001574. None of these proposals survived Newcomb’s scrutiny and that of other astronomers.19

What is interesting for our purposes in Hall’s hypothesis is that it altered the default repertoire of curves to be fitted to planetary motions. The one-body problem no longer gives a fixed conic section as the simplest curve. Rather, under Hall’s modified law of attraction, it gives very slowly rotating ellipses for bound orbits. These become the default curves to be fitted to planetary motions. The choice has a physical grounding in Hall’s modified law of attraction.

While Hall’s hypothesis did not survive scrutiny, that a law slightly different from Newton’s prevails in the solar system soon proved to be the way to accommodate the anomalous motion of Mercury. In November 1915, as Einstein was completing his general theory of relativity, he discovered to his jubilation that the new theory predicted precisely the anomalous advance of the perihelion of Mercury. He computed an advance of forty-three seconds of arc per century, noting that the astronomers’ values lay in the range of 40–50 seconds. With the adoption of Einstein’s theory, it became automatic to include a relativistic correction to the Newtonian orbits; that is, under the physical grounding of Einstein’s theory, the default curves to be fitted to planetary motions became very slowly rotating ellipses.

6.8. Harmonic Analysis of Tides

On an oceanic coast, the sea level rises and falls periodically, with about two high tides and two low tides each day. Beyond this crude description are many variations. There is some variability in the timing of highs and lows; and there is considerable variation in just how high and low the tides get. This variability is also somewhat periodic over longer time scales, but the exact cycles are hard to pin down precisely by casual inspection of some portion of a tide’s history.

Accurate tidal prediction is important and even essential for coastal navigation. Since the ebb and flow of the tide can produce considerable currents in coastal estuaries and bays, reliable advance knowledge of the tides can be the difference between easy and hard exits from a port. Reliable tidal prediction can make the difference between a successful return to one’s home port or running aground in unexpected low water.

These factors make reliable long-term tidal prediction highly desirable. Since the behavior of the tides varies so much from place to place, the problem of prediction is best tackled as a curve-fitting problem. Start with a good history of tides at each place on a coast. For each, find the curve that best fits the history and use it for prediction. Since the tides are periodic phenomena, one would expect that the families of functions to be fitted are based on the periodic trigonometric functions: sines and cosines of time. The natural method is straightforward Fourier analysis, which is the celebrated mathematical method for representing periodic functions in terms of sine and cosine harmonic constituents. To apply it, we would assume that the dependence of the water height on time t is given by the series:

with the series continued as far as needed. We know from the theory of the Fourier series that any credible behavior of the tides over some nominated time can be represented arbitrarily closely by this expression. We merely need a suitable scaling for t, a suitable selection of the a and b coefficients, and the inclusion of enough terms from the series.

While this is the obvious approach, in the past century and a half of work on tides, I have found no serious effort to provide this sort of analysis. The core difficulty, I conjecture, is that the dominant harmonic constituents present do not have the frequencies 1, 2, 3, … of the generic Fourier analysis. Combinations of these dominant constituents could still be captured by Fourier analysis with components of frequency 1, 2, 3, …. However, a large number of these components would be needed to represent accurately the summation of a few dominant harmonics whose frequencies are not in this set.

Instead, from the first moments, a physical basis has always been demanded for the harmonic constituents fitted to observed tidal histories. William Thomson (later Lord Kelvin) introduced the method of fitting harmonic constituents to tidal histories in 1867. Writing in his report to the British Association (Thomson 1869), he noted that previous methods had merely recorded the times of high and low water. He proposed that fuller records be kept to which harmonic constituents would be fitted. The particular constituents he proposed were drawn directly from the background theory of the origin of the tides in the gravitational interaction of the earth, sun, and moon.

The elements of this theory are widely known. The moon’s gravity pulls on the waters of the earth’s oceans. The pull is stronger than the average on the side of the earth nearer the moon and weaker on the side farther from the moon. The net effect is an elongation of the oceans into a spheroid that bulges away from the earth on both sides in line with the earth-moon axis, as shown in Figure 6.11.

The earth rotates daily under this bulged shape. As a location on earth passes a bulge, the location registers a high tide. Since there are two bulges, each location registers two high tides and two low tides each day. The cycle is only roughly daily since it completes when a point on the earth returns to its original position in relation to the moon. The moon orbits the earth once a month and moves in the same direction as the earth rotates. So to return to its starting position in relation to the moon, the earth must rotate slightly more than the full rotation of twenty-four hours. It requires roughly twenty-four hours and fifty minutes. In this time, two tide cycles are completed. Half of this process gives us the most important harmonic constituent: the “principal lunar semidiurnal [= half-daily],” written as M2, where the 2 denotes two cycles per day. It has a period of about twelve hours and twenty-five minutes.

Superimposed on this semidiurnal cycle is another semidiurnal cycle. It results from the gravitational attraction of the sun on the waters of the oceans. The sun’s attraction also distorts the ocean waters into a spheroid elongated along the line of the earth-sun axis, or so it would if there were not greater distortions due to the moon’s gravity. The bulge produced would be a little less than half that raised by the moon. It takes twenty-four hours exactly for a point on the earth to return to a position with the same relation to the sun. There are once again two bulges passed in this time, so we cycle between them each twelve hours. This contributes another harmonic constituent, the “principal solar semidiurnal” S2, whose period is twelve hours.

That these two harmonic constituents have periods that differ by about twenty-five minutes is of the greatest consequence for the tides. At the full or new moon, when the sun and moon align, the two bulges add and we have especially high tides, known as the “spring” tides. They are so named since more waters are imagined as springing forth. The effect is shown in Figure 6.12.

Each twelve hours, the high water of lunar semidiurnal cycle will lag behind that of the solar semidiurnal cycle by about twenty-five minutes. This lag accumulates. After about a week, at the time of the half moon, the tidal bulges of the moon and sun are aligned roughly perpendicularly. The outcome is a lowering of the high tide and an elevation of the low tide, producing the more modest “neap” tides of Figure 6.13.

The combining of the two cycles to produce this further cycle of spring and neap tides is shown in Figure 6.14.

This is one of the most important outcomes of the combining of the tidal harmonic constituents.

One might expect that there is little more to the harmonic analysis of the tides. What we have seen so far is adequate for an informal understanding of the gross behavior of the tides. It associates tidal levels with the phases of the moon in a qualitatively correct way. However, it falls far short of what is needed for successful quantitative prediction of the tides. Many more physical factors must be considered.

The sun and moon also move to the north and south, carrying their tidal bulges with them. In the course of a solar year, the sun completes one cycle around the ecliptic, moving between 23.5o north and 23.5o south of the stellar equator. The moon’s monthly motion carries it along a plane that is inclined at about 5o to the plane of the ecliptic. Its resultant motions carry it north and south of the stellar equator, between maximum elongations from the stellar equator that vary from 18.5o to 28.5o. As the sun and moon change their longitude, they carry with them the tidal bulges that they raise. This affects the heights of the tides and does it differently at each location on the earth.

If the sun and moon are directly over the equator, the two tidal bulges will pass symmetrically over some fixed terrestrial location on the equator in the course of a day. In so far as these processes are concerned, successive tides will have equal height. If, however, the sun and moon have moved together to a position far to the north, then the two tidal bulges will be shifted towards different hemispheres. One will be massed in the northern hemisphere and the other in the southern hemisphere. As a result, a point on the earth away from the equator will meet with deeper and shallower portions of the successive bulges, adding a diurnal (daily) cycle to the semidiurnal cycles so far mentioned. In extreme cases, the sun and moon pass overhead sufficiently far from the equator that, at some locations, one bulge may be missed entirely. These locations experience a single high tide per day. That is, their tides are on a diurnal cycle.

There are further complications. The size of a tidal bulge depends on the distance of the earth from both the sun and moon. Since the orbits are elliptical, the sun and moon approach and recede from the earth as they complete their cycles, annually and monthly, respectively. In addition, this cyclic effect is compounded by the perturbations induced by the sun on the moon’s orbit. These perturbations alter the eccentricity of the moon’s ellipse, which introduce further variation in the distance of the moon from the earth. All of these astronomical effects happen on regular cycles, readily predictable in advance. They are incorporated into tidal analysis by adding more harmonic constituents of the appropriate form.

These astronomical effects may seem overwhelming. However, they are merely the most reliably regular of the influences on the tides. If the earth were a perfectly smooth spheroid, a tidal bulge of the ocean would wash over it as a uniform wave. However, the earth is not a perfectly smooth body and all sorts of irregularities in its surface obstruct the uniform passing of the tidal wave. These obstructions are great in coastal areas, which is precisely where we are seeking predictions. The problems are compounded if we wish to predict tides in bays and estuaries. For the rising and falling of a tide will be delayed by the need for water to flow in and out of the bay as the tidal wave passes. Enclosed bodies of water have their own natural frequencies with which water oscillates to and fro within them. The coming and going of tidal waves couples with these oscillatory processes, all of which are represented by further harmonic constituents.

The shallow-water constituents are of two types: overtides and compound tides. The first are the analog of harmonic overtones in music. For example, the principal lunar semidiurnal M2 consists of two high tides per day. It raises shallow-water overtides M4 and M6 with four and six peaks each day. Compound tides arise with a frequency that is the sum or difference of the components from which they are derived. The shallow-water terdiurnal MK3 is derived from the principal lunar semidiurnal M2 and the lunar diurnal K1. It sums their two and one peaks per day to give three peaks.

Finally, meteorological facts can have a major influence on tides. Strong winds can materially affect them. These factors, however, are the hardest to address. Accurate weather prediction is difficult even a day in advance, whereas tide tables are prepared years in advance. There is some small effort to allow for these meteorological effects by means of the solar components, SA, SSA, and S1; that is, the solar annual, solar semi-annual, and solar diurnal, which have periods of a year, half a year, and a day.20

In sum, the harmonic analysis of tides is complicated and difficult, even when we seek a sound physical basis for the harmonic constituents. Many are needed. This was already apparent to William Thomson (1869, p. 491), who initially listed twenty-three constituents. Many more can be needed. The most difficult locations for prediction are complex estuaries, such as Anchorage, Alaska and Philadelphia, Pennsylvania. An adequate analysis requires over one hundred harmonic constituents (Hicks 2006, p. 40). The United States National Oceanic and Atmospheric Administration (NOAA) employs a standard set of thirty-seven constituents for its tidal predictions for coastal regions in the US. Here is an illustration of their use.

Table 6.1 shows the harmonic constituents used by NOAA for Annapolis, Maryland, in the Chesapeake Bay:21

Table 6.1. Harmonic constituents used by NOAA for tidal predictions at Annapolis, Maryland.

These thirty-seven constituents fix the family of thirty-seven component functions whose sum is to be fitted to the tidal history in Annapolis. Each consists of a cosine wave whose amplitude, phase, and speed are to be determined either from background assumptions or by fitting to the tidal history. The resulting parameters, given in the last three columns of the table, are used to compute NOAA’s tidal prediction. Figure 6.15 shows the result of combining them for the week of 7 August 2014.22

References

Airy, George B. 1884. Gravitation: An Elementary Explanation of the Principal Perturbations in the Solar System. 2nd ed. London: MacMillan and Co.

Aquinas, Thomas. (1945) 1997. Basic Writings of Saint Thomas Aquinas. Edited by Anton C. Pegis. Random House. Reprint, Indianapolis: Hackett.

Brown, Ernest W. 1896. An Introductory Treatise on the Lunar Theory. Cambridge: Cambridge University Press.

Clark, Ronald W. 1984. Einstein: the Life and Times. New York: Avon.

Einstein, Albert. (1933) 1954. “On the Methods of Theoretical Physics.” In Ideas and Opinions, pp. 270–76. New York: Bonanza.

———. 1949. “Autobiographical Notes.” In Albert Einstein: Philosopher-Scientist, edited by P. A. Schilpp. Evanston, IL: Library of Living Philosophers.

Hall, Asaph. 1894. “A Suggestion in the Theory of Mercury.” The Astronomical Journal 14: pp. 49–51.

Hicks, Steacy D. 2006. “Understanding Tides.” Center for Operational Oceanographic Products and Services, National Oceanic and Atmospheric Administration, http://www.co-ops.nos.noaa.gov/publications/Understanding_Tides_by_Steacy_finalFINAL11_30.pdf

Kelly, Kevin. 2007. “A New Solution to the Puzzle of Simplicity.” Philosophy of Science 74: pp. 561–73.

Le Verrier, Urbain. 1859. “Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète.” Comptes rendus hebdomadaires des séances de l’Académie des sciences (Paris) 49: pp. 379–83.

Lorentz, Hendrik A. 1920. The Einstein Theory of Relativity: A Concise Statement. 3rd ed. New York: Brentano’s.

Mach, Ernst. 1898. “The Economical Nature of Physical Inquiry.” In Popular Scientific Lectures, translated by T. J. McCormack, pp. 186–213. 3rd ed. Chicago: Open Court.

Maurer, Armand A. 1999. The Philosophy of Ockham in the Light of Its Principles. Toronto: Pontifical Institute of Mediaeval Studies.

Misner, Charles W., Kip S. Thorne, and John Archibald Wheeler. 1973. Gravitation. San Francisco: W. H. Freeman

Moody, Lewis. F. 1944. “Friction factors for pipe flow.” Transactions of the American Society of Mechanical Engineers 66(8): pp. 671–84.

Newcomb, Simon. 1895. The Elements of the Four Inner Planets and the Fundamental Constants of Astronomy. Washington: Government Printing Office.

Newton, Isaac. (1726) 1962. Mathematical Principles of Natural Philosophy. 3rd ed. Translated by Andrew Motte; revised by Florian Cajori. Berkeley: University of California Press.

Norton, John D. 2000. “‘Nature in the Realization of the Simplest Conceivable Mathematical Ideas’: Einstein and the Canon of Mathematical Simplicity.” Studies in the History and Philosophy of Modern Physics 31, pp. 135–70.

Popper, Karl. 1968. Logic of Scientific Discovery. New York: Harper and Row.

Shureman, Paul. 1958. Manual of Harmonic Analysis of Tides. Washington: United States Printing Office.

Sober, Elliott. 1988. “The Philosophical Problem of Simplicity.” In Reconstructing the Past: Parsimony, Evolution, and Inference. Cambridge, MA: Bradford, MIT Press.

Sterne, Theodore E. 1960. An Introduction to Celestial Mechanics. New York: Interscience Publishers.

Thomson, William. 1869. “Committee for the Purpose of Promoting the Extension, Improvement, and Harmonic Analysis of Tidal Observations.” In Report of the Thirty-Eighth Meeting of the British Association for the Advancement of Science, Norwich August 1868, pp. 489–510. London: John Murray, Albemarle St.

Thorburn, W. M. 1918. “The Myth of Ockahm’s Razor.” Mind 27: pp. 345–53.

Watson, James C. 1861. A Popular Treatise on Comets. Philadelphia: James Challen & Son.