Unrealistic Models in Mathematics

1. Unrealistic Models In Mathematics

Let me begin with some comments about the scope of my study and its rationale.

First, the subject matter of this paper — mathematics and models — might bring to mind the branch of mathematical logic known as model theory. For the model theorist, a model is a structure that satisfies a given set of sentences in a specified formal language under an interpretation. I am not primarily interested in this special sense of ‘model’ but rather in the broader scientific meaning of the term. In this broader sense, I take it, a model is any object M that is used to represent some other phenomenon, system, or body of information P. There are no a priori restrictions here on the nature of M or its relationship to P; in particular, there is no requirement that M satisfy some set of sentences associated with P.

Another clarification. It is in the nature of a model to take some liberties with its target phenomenon — “all models are wrong”, as the saying goes — but different models do so in different ways and to different degrees. Some abstract away from irrelevant details but are otherwise largely realistic; their elements represent only real features of the target phenomenon, all the most important features are represented, and these representations are more or less accurate. Moreover, such models can often be “de-idealized” even further without fundamentally changing their character (by adding in missing details or relaxing simplifying assumptions, for example).

In other cases, however, the relationship between surrogate and reality is less tidy. Many models explicitly and essentially misrepresent key aspects of their target phenomena and hence are nonveridical in a deeper sense. In these cases, no simple de-idealization procedure is available; the models are what they are, and function as they do, precisely on account of the distortions they contain. This latter sort of case is what I mean by an unrealistic model.¹

To get a clearer sense of the distinction, consider a schematic street map or a simple lunar model of the tides. Both models omit some features of their targets: the map may not depict the relative widths of the streets or the locations of alleys and unpaved drives, while the model of the tides neglects the gravitational influence of the Sun and the effects of Earth’s rotation. Nevertheless, both accurately represent the most important features of their target systems without introducing major ontological or ideological distortions.

Compare Bohr’s model of the atom or Schelling’s model of housing segregation. It is essential to Bohr’s model that it portrays electrons as moving in well-defined orbits around their nuclei, when in fact they do no such thing. The electron orbitals, as Alisa Bokulich puts it, are fictions, which “[cannot] be properly thought of as an ‘idealization’ of the true quantum dynamics” (Bokulich 2011, 43). Meanwhile, the Schelling model represents an agent’s housing choices as completely determined by two factors: their preference to be surrounded by a certain percentage of neighbors from their own group and the current composition of their immediate neighborhood. Cost considerations and other factors of obvious real-world importance are absent from the model, which is therefore usually viewed as a toy model: a “strongly idealized” and “extremely simple” representation that omits most of the factors on which the target phenomenon actually depends (cf. Reutlinger et al. 2018). Unrealistic models raise some especially interesting questions, and their role in mathematics will be my focus below.

2. The Cramér Random Model

One phenomenon often studied via models is the distribution of the prime numbers. In this section, I describe one of the most important and widely used of these: Cramér’s random model of the primes. Since Cramér introduced the model in 1932, numerous refinements, spinoffs and variants have emerged. Some of these are more accurate or useful than the original model for certain purposes. I focus mostly on the original here, because it is the simplest and remains in frequent use.

Let me start with some background. Famously, and perhaps to a greater degree than any other branch of mathematics, number theory is rife with simple and natural questions that have proven very hard to answer. Among the most well-known examples are the four Landau problems²:

1. Goldbach’s conjecture: Is every even integer greater than 2 the sum of two primes?

2. Twin Primes conjecture: Are there infinitely many pairs of prime numbers of the form p, p + 2?

3. Legendre’s conjecture: Is there a prime between n² and (n + 1)² for every positive integer n?

4. Fourth Landau conjecture: Are there infinitely many primes of the form n² + 1?

Settling these conjectures requires understanding the arrangement of the primes among the natural numbers. This is no easy task, because “the series of prime numbers exhibits great irregularities of detail” (Ingham 1932, 1) and “do[es] not follow any apparent pattern” (Koukoulopoulous 2019, 1).³ The first three questions have thus remained open for 170 years or more.⁴ Although existing technology still does not seem up to the challenge of solving the Landau problems, we have learned enough to make some headway.

The very first relevant discovery was Euclid’s theorem that there are infinitely many prime numbers. This is a precondition for the conjectures’ possible truth, but not helpful for their resolution, because it does not provide any information about the distribution of the primes. For this, we need the much more recent prime number theorem (PNT; 1896, proved independently by Hadamard and de la Vallée Poussin). Where log x is the natural logarithm and π(x) is the prime-counting function (giving the number of primes less than or equal to x), the PNT says that

Unfortunately, this still does not provide enough information about the distribution of the primes to settle Landau’s problems. (A proof of the Riemann Hypothesis would help, because it can be viewed as an improvement on the PNT, bounding how far off π(x) can be from x / log x. But such a proof seems unlikely to be forthcoming any time soon.)

In view of these difficulties, the Swedish mathematician Harald Cramér proposed a new way of approaching the distribution problem. Rather than directly studying the primes themselves, he constructed a more tractable surrogate, now known as “Cramér’s model” or the “random model” of the primes. The idea, set out in Cramér (1936), is to build a subset of the natural numbers by independently choosing to include each n > 2 with probability 1 / log n. The resulting sequence might look something like this:⁶

3, 4, 6, 11, 12, 25, 26, 28, 32, 34, 35, 36, 43, 57, 66, 68, 80, 83, 87, 93, …

100005, 100006, 100008, 100018, 100045, 100055, 100074, 100094, 100096, 100106, …

Think of this set as a model of the real sequence of primes. By the consequence of the prime number theorem mentioned above, the “primes” in Cramér’s model will have the same asymptotic density in the natural numbers as the real primes (with probability 1) — one can observe, for instance, that the larger terms in the sample sequence are spaced out somewhat more than the smaller ones. So, it is reasonable to hope that other statistical and distributional properties of the real prime sequence will also resemble those of the model. (Note, by the way, that “the model” here refers to an arbitrary sequence generated by Cramér’s procedure, not to any definite sequence in particular. Correspondingly, claims of the form “P is true in the model” mean that P holds for an arbitrary such sequence with probability 1, perhaps with finitely many exceptions.⁷)

Given that the “Cramér primes” and the real primes are similarly distributed in ℕ, what is the benefit of working with the former instead of the latter? As it turns out, it is much easier to study the statistics of distributions with strong joint independence properties, like those of the random model.⁸ Consequently, we know a lot about the behavior of the surrogate primes.

Some of these facts were independently known or strongly believed to be true of the actual primes, while other claims are considered to have gained support from the fact that they hold in the model (e.g. the Riemann Hypothesis and the Landau conjectures). Yet other hypotheses were originally motivated by observations about the model itself. Among these is the important Cramér conjecture on the sizes of gaps between primes,⁹ introduced in Cramér’s original paper on the model, which even in recent years “does not seem to be attackable by other methods” (Granville 1995b, 391).

I want to make one observation and two more substantive claims about this case. The observation is that the Cramér model is manifestly not a model of the theory of the primes in the model-theoretic sense. There are many sentences true of the real primes that are false of the Cramér primes (for instance, “exactly one even number is prime”).¹⁰ So, model theory is not the proper framework for thinking about this situation, as per the remarks in §1.

The first substantive claim I want to defend is that Cramér’s model (and similar random models) have significantly improved our understanding of the distribution of the primes. The model makes several kinds of epistemic contribution.

To start with, mathematicians take the model seriously because it correctly predicts many known facts about the primes. Kannan Soundararajan notes, for instance, that “the Cramér model makes accurate predictions for the distribution of primes in [very] short intervals” (Soundararajan 2007a, 64). (Note that this is not just a trivial consequence of the fact that the model gets the asymptotic density right; two sequences can have similar long-run behavior without looking alike at small scales.¹¹) More generally, Andrew Granville writes that “the probabilistic model usually gives one a strong indication of the truth” (Granville 1995b, 391). In Tao’s words, “[we] have a number of extremely convincing and well supported models for the primes … the most accurate [of these] in practice are random models” (Tao 2015).

Since Cramér-type models have proven generally reliable in regimes where their predictions can be verified, number theorists view them as useful guides to unknown territory. Their contributions along these lines fall into at least three categories: (1) increasing or decreasing our confidence in conjectures derived from independent sources; (2) motivating entirely new conjectures; and (3) suggesting novel methods of proof.

Let us start with (1). As mentioned above, a number of fundamental statements in number theory are known to hold in the Cramér model but are not yet known to hold for the actual primes. These include the Riemann Hypothesis (RH) and the four Landau conjectures. Each of these hypotheses was suspected to be true before the advent of the Cramér model. But the results from the model served to increase mathematicians’ confidence, in some cases significantly. For instance, van der Poorten claims that “the most compelling” evidence in favor of RH is the fact that it holds in a related random model of the primes, the Hawkins model (van der Poorten 1996, 147).¹² Similarly, according to Patterson’s textbook on the zeta function, the validation of RH by random models “represents one of the more reassuring reasons for expecting the Riemann Hypothesis to be true” (Patterson 1988, 75).

The credence calibration provided by Cramér-style models extends much further, as elaborated on by Tao. In the setting of these models, he writes, “many difficult conjectures on the primes reduce to relatively simple calculations … Indeed, the models are so effective at this task that analytic number theory is in the curious position of being able to confidently predict the answer to a large proportion of the open problems in the subject, whilst not possessing a clear way forward to rigorously confirm these answers!” (Tao 2015).

Let us now turn to (2), new conjectures. In addition to bolstering confidence in independently motivated hypotheses, “the probabilistic heuristic, in which independence is assumed, provides a useful means of constructing conjectures” (Montgomery & Vaughan 2007, 57).¹³ The most famous of these is the Cramér conjecture on prime gaps, mentioned above, which Cramér arrived at by way of the model. Random models have also led to progress in other parts of number theory. One example is the theory of “lucky numbers” (the sequence 1, 3, 7, 9, 13, 15, 21, 25, …, generated by a certain sieve process).¹⁴ On the basis of his random model of the primes, Hawkins conjectured (in Hawkins 1957) and was later able to prove (in Hawkins & Briggs 1957) a PNT-type result for lucky numbers, to the effect that their asymptotic density in the natural numbers is also 1 / log n. Random models continue to be deployed on the front lines of research, sometimes in novel ways. For example, Lozano-Robledo (2020) “propose[s] a new probabilistic model for the distribution of ranks of elliptic curves … in the spirit of Cramér’s model for the prime numbers” (2), which is used to generate predictions about the number of elliptic curves of a given rank. In general, then, Cramér-type models “[give] a clearer indication of what results one expects to be true, thus guiding one to fruitful conjectures” (Tao 2015).

Finally, let us consider (3), novel methods of proof. As we have seen, models of the primes are generally used for heuristic purposes rather than as tools for proving theorems. Nevertheless, in the view of the number theorist János Pintz, “probabilistic models can help or could have helped not only to conjecture but also prove results about primes” (Pintz 2007, 362). Pintz goes on to show how a particular result — Maier’s theorem about the number of primes in small intervals — could have been established much earlier using a modified Cramér model.

In addition to these three applications, Tao mentions several other uses of random models: “providing a quick way to scan for possible errors in a mathematical claim (e.g. by finding that the main term is off from what a model predicts …); gauging the relative strength of various assertions (e.g. classifying some results as ‘unsurprising’ [and] others as ‘potential breakthroughs’ …); or setting up heuristic barriers … that one has to resolve before resolving certain key problems” (Tao 2015). In view of these various uses, benefits and insights, I conclude that number theorists have gained significant understanding from Cramér-type models.

One could try to push back against this claim by noting the lack of philosophical consensus around the notion of understanding. In the absence of a widely accepted explicit theory, which criteria are being used to judge cases such as this? And why should we think those criteria are appropriate?

It is true that philosophers disagree about understanding. For instance, some equate understanding a phenomenon with having an explanation of it (Strevens 2013). Others link understanding with the possession of certain abilities (Delarivière & Van Kerkhove 2021), with suitably structured knowledge (Kelp 2015), or with the disposition to generate new knowledge from a minimal core (Wilkenfeld 2019). I do not take sides in this debate here (although I do argue against the explanation account in §4, below). My approach is different, and it has two components. First, I claim that the Cramér model ought to count as a source of understanding on any reasonable view — the same goes for the function field model discussed in the next section. Second, I offer the appraisals of mathematicians themselves, which I take to count at least as strongly as philosophical arguments in this context.¹⁵

The Cramér model, as just shown, has strengthened number theorists’ confidence in some important hypotheses and has played a key role in generating others. It has led to a clearer overall picture of the phenomena. It helps mathematicians organize, justify, and check their reasoning. It would be a tendentious and implausible theory that regarded these achievements as insufficient for improving understanding. In particular, the model evidently confers abilities associated with understanding, lends valuable structure to number theorists’ knowledge, and allows much novel information to be spun out from a compact representational core. So, theories in the spirit of the last three mentioned above will count the model as a source of understanding. This seems correct.

What is more, the same conclusion has been reached by number theorists who are intimately familiar with the model and its uses. Granville, for instance, refers to “Cramér’s probabilistic approach [to] understanding the distribution of prime numbers, which underpins most of the heuristic reasoning still used in the subject today” (Granville 1995a, 15). Absent compelling reasons to do otherwise, good methodology recommends taking such judgments at face value.¹⁶

I conclude from these considerations that the Cramér model is a source of understanding. (To be precise, it contributes to understanding the distribution of the prime numbers. I take this to be a case of understanding a phenomenon, as opposed to, say, a case of understanding-why.)

The second main claim of this section is that Cramér’s model is quite unrealistic, in the sense discussed in §1 above. That is, rather than a mild idealization that merely abstracts away from inessential details, the model involves an explicit and extensive misrepresentation of its subject matter.

One major distortion is that the Cramér primes are chosen probabilistically, but the actual primes are not in any sense random. Rather, as George Pólya says, whether or not a number is prime “can be decided by the ‘definite rules’ of arithmetic — where and how could chance enter the picture?” (Pólya 1959, 376). Although the assumption of randomness is unrealistic, it is essential to all Cramér-type models. There is no prospect of de-idealizing to remove this assumption without discarding the model framework entirely.

Cramér’s model also fails to capture the important multiplicative structure of the actual primes — for instance, the fact that if p is prime then n ∙ p cannot be. (Recall that the Cramér primes are chosen independently, so the selection of one number has no effect on the probability of choosing any other number.) Hence the model generates infinitely many even primes, pairs of consecutive primes, and other absurdities — for example, in the run of the Cramér algorithm given above, 34, 35, and 36 are all chosen. Some modifications of the simple Cramér model reintroduce basic aspects of the actual multiplicative structure of the primes, such as by forbidding even primes greater than 2. But going much further in the direction of realism would be counterproductive, because the joint independence of the surrogate primes is exactly the feature that makes the models more tractable than the real primes.

Thus, “[d]espite its predictive power, Cramér’s model is a vast oversimplification” (Klarreich 2018a, 25) — indeed, a distortion — of its target, the prime number sequence. This is interesting for a number of reasons, most obviously because it shows that mathematicians, like empirical scientists, make serious use of unrealistic models and rely on them to gain understanding. I defer discussion of further philosophical consequences to §4 below, after discussing my second example in the next section.

3. The Function Field Model of the Integers

This section discusses so-called dyadic models of linear structures, in particular the model of the integers as polynomials over a finite field. This is a further example of an unrealistic model in widespread use as a source of mathematical understanding. This second case also bears consequentially on the questions about modeling mentioned at the start of the paper.

Dyadic models in mathematics take on a variety of forms depending on the settings in which they are deployed, which range from differential equations and harmonic analysis to combinatorics and number theory. But a common motivation for using such models is the desire to avoid spillover between scales exhibited by the integers, real numbers, cyclic groups, and other linearly structured sets.

I will focus here on the integers. One manifestation of the spillover phenomenon in this domain is the need to carry digits when adding numbers together. In the sum 28 + 75 = 103, for example, the addition of 8 and 5 in the units place spills over to affect the values in the tens and hundreds places. This kind of interaction between fine and coarse scales can be inconvenient. For instance, when adding many integers together, an accumulation of tiny (“fine-scale”) errors can significantly distort the final (“coarse-scale”) result.

The most common dyadic model of the integers is the ring of polynomials F[t] over a finite field F. This is known as the function field model of the integers.¹⁷ The elements of F[t] are polynomials with coefficients from F. The role of positive integers in the model is played by monic polynomials, i.e. polynomials of the form $t^n+a_{n-1}{t}^{n-1}+\dots +a_{1}t+a_0$ with leading coefficient 1.

(Some relevant definitions: a field is an algebraic structure with commutative addition and multiplication operations in which every non-zero element has both an additive inverse and a multiplicative inverse. The real numbers are a familiar example. A finite field is a field with finitely many elements. In fact, a finite field always has pⁿ elements, with p prime and n ≥ 1. The simplest examples are the fields $\mathbb{\text{F}}$_p, whose elements are {0, 1, 2, …, p – 1}. Addition in $\mathbb{\text{F}}$_p works like addition modulo p. For example, in the field with seven elements $\mathbb{\text{F}}$₇, we have 5 + 6 = 11 (“mod” 7) = 4.)

In the function field model, arithmetical operations on fine-scale terms do not affect the values of coarse-scale terms and vice versa. Compare adding (t² + 2t + 5) + (t + 6) in $\mathbb{\text{F}}$₇[t], for example, with the analogous 125 + 16 in .ℤ¹⁸ The latter exhibits spillover, because the units-place sum (5 + 6 = 11) contributes 1 to the tens-place result. But not so in the model. In $\mathbb{\text{F}}$₇[t], as noted above, 5 + 6 equals 4, and so the sum (t² + 2t + 5) + (t + 6) = t² + 3t + 4 is spillover-free.

Another important feature of the function field model is that its surrogate “integers”, being polynomials, have non-trivial derivatives. (The derivative of an ordinary integer is, of course, always 0.) I will say more about why this is useful below.

F[t] might, at first glance, seem like a strange model for the integers. Why would representing numbers as polynomials be appropriate, and how might it be useful? As Michael Rosen writes (in his textbook on the subject, Number Theory in Function Fields):

Early on … it was noticed that ℤ has many properties in common [with F[t]] … Both rings are principal ideal domains … both rings have infinitely many prime elements, and both rings have finitely many units. Thus, one is led to suspect that many results which hold for ℤ have analogues [in F[t]]. This is indeed the case. (Rosen 2002, vii)

In other words, the two structures have importantly similar algebraic properties. Hence, “number theory in F[t]” should (and does) resemble ordinary number theory to a significant degree. The study of function fields as a source of number-theoretic insight goes back at least to Dedekind and Weber’s 1882 paper “Theory of Algebraic Functions of One Variable” (Dedekind & Weber 2012). As the translator John Stillwell notes, “the paper revealed the deep analogy between number fields and function fields — an analogy that continues to benefit both number theory and geometry today” (vii).

Indeed, the function field model has proven fruitful in many ways. Several of its uses resemble those of Cramér’s model of the primes: increasing our confidence in independent hypotheses, suggesting new conjectures, and offering novel methods of proof. The function field model is also unique in at least one important way: it suggests to many mathematicians that there ought to exist a novel kind of object, the “field with one element”, to complete certain aspects of the correspondence between F[t] and ℤ. If efforts to make sense of this notion prove successful, momentous developments in algebra, number theory and geometry are expected to ensue.

Let me fill in some details, starting with the first items mentioned above. A famous open problem in number theory is the abc conjecture of Oesterlé and Masser. The conjecture is roughly as follows: Let a,b,c be relatively prime integers such that a + b = c, and let D denote the product of the distinct prime factors of abc. Then c is significantly bigger than D in only finitely many cases. Many other major conjectures are known to be true conditional on the truth of abc; it would also yield a simple proof of Fermat’s Last Theorem. So abc is of great interest to number theorists.¹⁹

It is therefore significant that the counterpart of the abc conjecture, known as the Mason–Stothers theorem, is known to hold in the function field model. The Mason–Stothers theorem concerns relatively prime polynomials a(t), b(t), c(t), not all constant and with a + b = c. Where D is the degree of the product of the distinct irreducible factors of a, b, and c, the theorem asserts that D is significantly bigger than the maximum among the degrees of a, b, and c. The Mason–Stothers theorem has an elementary proof in F[t] based on taking derivatives of a, b, and c (Snyder 2000) — a trick unavailable for proving abc in the integers. This is an example of the aforementioned usefulness of derivatives in the function field model.

The abc conjecture is just one hypothesis on which the function field model sheds light. As Rudnick notes, a variety of classic problems “which are currently viewed as intractable over the integers, have recently been addressed in the function field context … and the resulting theorems can be used to check existing conjectures over the integers, and to generate new ones” (Rudnick 2014, 443). (Rudnick’s paper discusses five such problems.)

In some cases, results in the function field model can be used to directly prove the corresponding statements in ordinary number theory. One example is the Ax–Kochen theorem, an important result concerning the zeroes of certain polynomials over the p-adic numbers. In the standard proof of Ax–Kochen, the first step is to show that the analogous claim holds in the function field model. Using the transfer principle technique from model theory, it is then possible to import the function field statement back to the original p-adic context, thus proving the theorem.

A final way in which the function field model has advanced number theory is by motivating research around the notional “field with one element” F₁. In standard algebra, a field with one element is impossible, because fields, by definition, have an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1. So, the quest for F₁ can be seen as an exercise in conceptual engineering; the task is to build a coherent theory in which an F₁-like object exists and has certain desirable properties.²⁰

Impetus for this quest comes from several sources, a major one being the success and promise of the function field model. Work on F₁ is technical and describing it in adequate detail would be unduly lengthy, so I will give only a brief sketch here.

The starting point is André Weil’s 1948 proof of the Riemann Hypothesis for function fields (Weil 1948). As Oliver Lorscheid writes, “[t]he analogies between number fields and function fields led to the hope that one can mimic these methods for ℚ and approach the [standard] Riemann hypothesis” (Lorscheid 2018, 94). (Here, ℚ denotes the set of rational numbers.)

Weil’s proof starts with a “global” function field F, that is, a finite field extension of the rational function field $\mathbb{\text{F}}$_p(t). (See footnote ¹⁷ for more on $\mathbb{\text{F}}$_p(t).) It turns out that F can be interpreted as the function field of a curve C over the base field $\mathbb{\text{F}}$_p. One can then define a zeta function ζ_C for this curve and use the tools of algebraic geometry to prove the analogue of the Riemann Hypothesis for ζ_C. In particular, Weil’s proof counts the number of intersection points of C with a “twisted” version of itself inside the fiber product C × _{Spec$\mathbb{\text{F}}$p} C.²¹

Many mathematicians have hoped, as per Lorscheid’s remarks above, to obtain a proof of the standard Riemann Hypothesis by translating Weil’s proof from the function field model to the integers. The first step in this process is to identify the curve C such that ζ_C is the ordinary Riemann zeta function that features in RH. This curve turns out to be the spectrum of the integers, Spec ℤ. The remaining task is to specify the base field over which Spec ℤ is to be viewed as a function field. It is at this point that the field with one element enters the scene:

The analogy between number fields and function fields finds a basic limitation with the lack of a ground field. One says that Spec ℤ is … like a (complete) curve; but over which field? In particular, one would dream of having an object like

$\text{Spec\hspace{0.17em}}\mathbb{Z}{\times }_{\text{Spec\hspace{0.17em}\hspace{0.17em}}{\mathbb{\text{F}}}_1}\text{Spec\hspace{0.17em}}\mathbb{Z},$

since Weil’s proof of the Riemann hypothesis for a curve over a finite field makes use of the product of two copies of this curve. (Soulé 1999, 1–2)

The field with one element $\mathbb{\text{F}}$₁, then, should be an object over which it makes sense to view Spec ℤ as a curve.

Using $\mathbb{\text{F}}$₁ to prove the Riemann Hypothesis is just one motivation for its study. Broadly speaking, the idea of doing geometry with the integers over $\mathbb{\text{F}}$₁ “emerged from certain heuristics in combinatorics, number theory and homotopy theory that could not be explained in the framework of Grothendieck’s scheme theory” (Lorscheid 2018, 83). Given that scheme theory has served as the foundation for algebraic geometry for over half a century, a fully realized theory of the field with one element would necessitate a major rethinking of a large body of mathematics.

Although we still lack a definition of $\mathbb{\text{F}}$₁ that seems likely to yield a proof of RH, an impressive collection of mathematicians have undertaken extensive exploratory theory-building: Jacques Tits (credited with first suggesting $\mathbb{\text{F}}$₁ in Tits (1957)), Alain Connes, Caterina Consani, Yuri Manin, and Christophe Soulé, to name a few particularly influential contributors. Thas (2016) and Lorscheid (2018) are a recent essay collection and a survey paper, respectively. Even if RH remains elusive, work on F₁ goes on, and has already produced a richer picture of the relationship between number theory and geometry.

My final claims in this section will come as no surprise: I want to argue that mathematicians have gained significant understanding from the function field model despite its unrealistic character. The considerations from the last section apply here also in support of this claim. As with the Cramér model, the experts best positioned to assess the merits of the function field model describe it as a source of understanding. Here, for instance, is Tao (recall that the dyadic models are a family that includes the function field model, as noted at the beginning of this section):

In some areas [dyadic constructions] are an oversimplified and overly easy toy model; in other areas they get at the heart of the matter by providing a model in which all irrelevant technicalities are stripped away; and in yet other areas they are a crucial component in the analysis of the non-dyadic case. In all of these cases, though, it seems that the contribution that dyadic models provide in helping us understand the non-dyadic world is immense. (Tao 2008, 68)

Finally, the function field model is an unrealistic representation of the integers. Most obviously and importantly, there is no natural notion of order on the elements of F[t], so the function field model completely lacks the linear structure of ℤ. This difference has far-reaching consequences. For instance, mathematical induction does not make sense in F[t], whereas the availability of induction is often taken to be a characteristic property of the whole numbers. (Cf. Stewart and Tall’s Foundations of Mathematics: “What is a number? … The first step [in finding the answer] was to characterise natural numbers. It turned out that their most important defining feature wasn’t counting, or arithmetic: it was the possibility of proving theorems using mathematical induction” (Stewart & Tall 2015, 159).)

In addition, the metric on F[t] is non-Archimedean, meaning that the familiar triangle inequality d(x,z) ≤ d(x,y) + d(y,z) takes the stronger “ultrametric” form d(x,z) ≤ max{d(x,y), d(y,z)}. This implies, for instance, that all triangles in F[t]ⁿ are isosceles, that every point on the interior of a ball is its center, and that for any two intersecting balls, one is contained inside the other. These properties are, of course, very much unlike those of ℤⁿ equipped with the usual metric.

Despite sharing some algebraic features, then, the function field model differs from the integers in fundamental ways. As with the Cramér model, these differences go beyond mere elisions of small or unimportant details. Nor is it possible to recover the key features of the integers by any straightforward process of de-idealization. F[t] is an unrealistic model.

4. Morals: Understanding, Explanation, Counterfactuals

At the beginning of the paper, I listed three pressing questions about unrealistic scientific models. The previous sections have shown that mathematicians engage in modeling, and that some widely used models in pure mathematics are unrealistic. So, these cases are relevant to the questions at issue. What can we learn from them?

I have already argued that the Cramér model of the primes and the function field model of the integers are sources of understanding. So, the first of the three questions — can we gain genuine understanding from unrealistic models? — has an affirmative answer.

The use of unrealistic models in mathematics has gone unappreciated by philosophers, and cases like those I have described are note-worthy for that reason. But this answer is not otherwise surprising. Various kinds of epistemically salutary unrealistic models have been studied extensively in recent years, and the mere fact of their existence no longer seems especially controversial. See, for example, Batterman and Rice (2014), Bokulich (2011), de Regt (2015), Hindriks (2013), Mäki (2009), Morrison (2015), Rice (2016), and the papers in Synthese’s recent collection “What to Make of Highly Unrealistic Models”: Boesch (2019), Knuuttila and Koskinen (2020), Papayannopoulos (2020), and van Eck and Wright (2020).

The remaining questions, though, are very much under debate. The mathematical cases I have described can help to resolve both.

5. Conclusion

This paper has argued for three main claims. First: that unrealistic models have important uses in pure mathematics and their epistemic benefits include improving our understanding of their target phenomena. Second: that the understanding gained from these models (and hence from unrealistic models in general) need not flow from explanations of the target phenomena. Third: that it need not flow from counterfactual knowledge either.

Future work on the philosophy of modeling could benefit from further examination of mathematical cases. Consider the metaphysics of models. One popular view holds that models are artifacts (Thomasson 2020; Thomson-Jones 2020); a related view holds that models in general, or perhaps unrealistic models in particular, are fictional entities of some sort (Bokulich 2011; Frigg 2010; Salis 2021). It would seem to follow from such views that the polynomial ring F[t], for example, is an artifact or a fiction. But F[t] is also a piece of ordinary mathematics, whose ontological status is presumably the same as that of other mathematical objects. Artifactualism about models therefore seems to imply artifactualism about mathematics in general. Depending on one’s metaphysical commitments, this may be either a welcome consequence or a damning reductio of artifactualist views.

Some other important questions are broadly epistemological. For example, there is a tradition of viewing (some) model-based inference as a kind of analogical reasoning (Bartha 2009; Hesse 1963), and mathematical models may offer some unique data. Mathematics should also join the conversation about thought experiments and imagination in science (Brown 2010; Murphy 2022) and the relationship between these activities and modeling practices (Arfini 2006).

These are just a few of the ways in which philosophy stands to benefit from taking mathematics seriously. The kingdom will prosper when the sequestered queen of the sciences is allowed to return to court.²⁹