Please enjoy this free excerpt from Lectures on the Philosophy of Mathematics. This essay appears in chapter 7, focused on Gödel’s incompleteness theorems.

Mathematical logic, as a subject, truly comes of age with Kurt Gödel's incompleteness theorems, which show that for every sufficiently strong formal system in mathematics, there will be true statements that are not provable in that system, and furthermore, in particular, no such system can prove its own consistency. The theorems are technically sophisticated while also engaged simultaneously with deeply philosophical issues concerning the fundamental nature and limitations of mathematical reasoning. Such a fusion of mathematical sophistication with philosophical concerns has become characteristic of the subject of mathematical logic—I find it one of the great pleasures of the subject. The incompleteness phenomenon identified by Gödel is now a core consideration in essentially all serious contemporary understanding of mathematical foundations.

In order to appreciate the significance of his achievement, let us try to imagine mathematical life and the philosophy of mathematics prior to Gödel. Placing ourselves in that time, what would have been our hopes and goals in the foundations of mathematics? By the early part of the twentieth century, the rigorous axiomatic method in mathematics had found enormous success, helping to clarify mathematical ideas in diverse mathematical subjects, from geometry to analysis to algebra. We might naturally have had the goal (or at least the hope) of completing this process, to find a complete axiomatization of the most fundamental truths of mathematics. Perhaps we would have hoped to discover the ultimate foundational axioms—the bedrock principles—that were themselves manifestly true and also encapsulated such deductive power that with them, we could in principle settle every question within their arena. What a mathematical dream that would be.

Meanwhile, troubling antinomies—contradictions, to be blunt—had arisen on the mathematical frontiers in some of the newly proposed mathematical realms, especially in the naive account of set theory, which exhibited enormous promise as a unifying foundational theory. Set theory had just begun to provide a unified foundation for mathematics, a way to view all mathematics as taking place in a single arena under a single theory. Such a unification allowed us to view mathematics as a coherent whole, enabling us sensibly, for example, to apply theorems from one part of mathematics when working in another; but the antinomies were alarming. How intolerable it would be if our most fundamental axiomatic systems of mathematics turned out to be inconsistent; we would have been completely mistaken about some fundamental mathematical ideas. Even after the antinomies were addressed and the initially naive set-theoretic ideas matured into a robust formal theory, uncertainty lingered. We had no proof that the revised theories were safe from new contradictions, and concern remained about the safety of some fundamental principles, such as the axiom of choice, while other principles, such as the continuum hypothesis, remained totally open. Apart from the initial goal of a complete account of mathematics, therefore, we might have sought at least a measure of safety, an axiomatization of mathematics that we could truly rely on. We would have wanted at the very least to know by some reliable finitary means that our axioms were not simply inconsistent.

The Hilbert program

Such were the hopes and goals of the Hilbert program, proposed in the early twentieth century by David Hilbert, one of the world's leading mathematical minds. To my way of thinking, these hopes and goals are extremely natural in the mathematical and historical context prior to Gödel. Hilbert expected, reflexively, that mathematical questions have answers that we can come to know. At his retirement address, Hilbert (1930) proclaimed:

Wir müssen wissen. Wir werden wissen. (We must know. We will know.)

Thus, Hilbert expressed completeness as a mathematical goal. We want our best mathematical theories ultimately to answer all the troubling questions. Hilbert wanted to use the unifying foundational theories, including set theory, but he also wanted to use these higher systems with the knowledge that it is safe to do so. In light of the antinomies, Hilbert proposed that we should place our mathematical reasoning on a more secure foundation, by providing specific axiomatizations and then proving, by completely transparent finitary means, that those axiomatizations are consistent and will not lead us to contradiction.

Formalism

Hilbert outlined his vision of how we might do this. He proposed that we should view the process of proving theorems, making deductions from axioms, as a kind of formal mathematical game—a game in which mathematical practice consists ultimately of manipulating strings of mathematical symbols in conformance with the rules of the system. We need not encumber our mathematical ontology with uncountable (or even infinite) sets, just because our mathematical assertions speak of them; rather, let us instead consider mathematics merely as the process of formulating and working with those assertions as finite sequences of symbols. Inherent in the Hilbert program, and one of its most important contributions, is the idea that the entire mathematical enterprise, viewed as a formal game in a formal axiomatic system, may itself become the focus of metamathematical investigation, which he had hoped to undertake by entirely finitary means.

According to the philosophical position known as formalism, this game is indeed all that there is to mathematics. From this view, mathematical assertions have no meaning; there are no mathematical objects, no uncountable sets, and no infinite functions. According to the formalist, the mathematical assertions we make are not about anything. Rather, they are meaningless strings of symbols, manipulated according to the rules of our formal system. Our mathematical theorems are deductions that we have generated from the axioms by following the inference rules of our system. We are playing the game of mathematics.

One need not be a formalist, of course, to analyze a formal system. One can fruitfully study a formal system and its deductions, even when one also thinks that those mathematical assertions have meaning—a semantics that connects assertions with the corresponding properties of a mathematical structure that the assertion is about. Indeed, Hilbert applies his formalist conception principally only to the infinitary theory, finding questions of existence for infinitary objects to be essentially about the provability of the existence of those objects in the infinitary theory. Meanwhile, with a hybrid view, Hilbert regards the finitary theory as having a realist character with a real mathematical meaning.

The Hilbert program has two goals, seeking both a complete} axiomatization of mathematics, one which will settle every question in mathematics, and a proof using strictly finitary means to analyze the formal aspects of the theory that the axiomatization is reliable. Hilbert proposed that we consider our possibly infinitary foundation theory T, perhaps set theory, but we should hold it momentarily at arm's length, with a measure of distrust; we should proceed to analyze it from the perspective of a completely reliable finitary theory F, a theory concerned only with finite mathematics, which is sufficient to analyze the formal assertions of T as finite strings of symbols. Hilbert hoped that we might regain trust in the infinitary theory by proving, within the finitary theory F, that the larger theory T will never lead to contradiction. In other words, we hope to prove in F that T is consistent. Craig Smorynski (1977) argues that Hilbert sought more, to prove in F not only that T is consistent, but that T is conservative over F for finitary assertions, meaning that any finitary assertion provable in T should be already provable in F. That would be a robust sense in which we might freely use the larger theory T, while remaining confident that our finitary conclusions could have been established by purely finitary means in the theory F.

Life in the world imagined by Hilbert

Let us suppose for a moment that Hilbert is right—that we are able to succeed with Hilbert's program by finding a complete axiomatization of the fundamental truths of mathematics; we would write down a list of true fundamental axioms, resulting in a complete theory T, which decides every mathematical statement in its realm. Having such a complete theory, let us next imagine that we systematically generate all possible proofs in our formal system, using all the axioms and applying all the rules of inference, in all possible combinations. By this rote procedure, we will begin systematically to enumerate the theorems of T in a list,

a list which includes all and only the theorems of our theory T. Since this theory is true and complete, we are therefore enumerating by this procedure all and only the true statements of mathematics. Churning out theorems by rote, we could learn whether a given statement φ was true or not, simply by waiting for φ or ¬φ to appear, complete and reliable. In the world imagined by the Hilbert program, therefore, mathematical activity could ultimately consist of turning the crank of this theorem-enumeration machine.

The alternative

However, if Hilbert is wrong, if there is no such complete axiomatization, then every potential axiomatization of mathematics that we can describe would either be incomplete or include false statements. In this scenario, therefore, the mathematical process leads inevitably to the essentially creative or philosophical activity of deciding on additional axioms. At the same time, in this situation, we must inevitably admit a degree of uncertainty, or even doubt, concerning the legitimacy of the axiomatic choices we had made, precisely because our systems will be incomplete and we will be unsure about how to extend them, and furthermore, because we will be unable to prove even their consistency in our finitary base theory. In the non-Hilbert world, therefore, mathematics appears to be an inherently unfinished project, perhaps beset with creative choices, but also with debate and uncertainty concerning those choices.

Which vision is correct?

Gödel's incompleteness theorems are bombs exploding at the very center of the Hilbert program, decisively and entirely refuting it. The incompleteness theorems show, first, that we cannot in principle enumerate a complete axiomatization of the truths of elementary mathematics, even in the context of arithmetic, and second, no sufficient axiomatization can prove its own consistency, let alone the consistency of a much stronger system. Hilbert's world is a mirage.

Meanwhile, in certain restricted mathematical contexts, a reduced version of Hilbert's vision survives, since some naturally occurring and important mathematical theories are decidable. Tarski proved, for example, that the theory of real-closed fields is decidable, and from this (as discussed in chapter 4), it follows that the elementary theory of geometry is decidable. Additionally, several other mathematical theories, such as the theory of abelian groups, the theory of endless, dense linear orders, the theory of Boolean algebras, and many others, are decidable. For each of these theories, we have a theorem-enumeration algorithm; by turning the crank of the mathematical machine, we can in principle come to know all and only the truths in each of these mathematical realms.

But these decidable realms necessarily exclude huge parts of mathematics, and one cannot accommodate even a modest theory of arithmetic into a decidable theory. To prove a theory decidable is to prove the essential weakness of the theory, for a decidable theory is necessarily incapable of expressing elementary arithmetic concepts. In particular, a decidable theory cannot serve as a foundation of mathematics; there will be entire parts of mathematics that one will not be able to express within it.

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Please enjoy this free short excerpt from Lectures on the Philosophy of Mathematics.

This discussion takes place in chapter 4, focused on philosophical issues arising in geometry, and appears directly after an account of Euclidean versus non-Euclidean geometry, including spherical geometry, elliptical geometry, hyperbolic geometry.

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The curvature of space

Let us explore a little more the differing character of non-Euclidean geometry. Spherical geometry, for example, has positive curvature, and part of what this means is that if one selects a point on the sphere and draws circles of increasing radius on the surface of the sphere, then the circumferences of these circles do not grow quite as fast as they do in the Euclidean plane. The circles are bent in slightly because they lie in the surface of the sphere. The formula for the circumference, therefore, is no longer 2πr, but rather something less. The circumference of a circle in spherical geometry is less than expected in Euclidean space; there are simply fewer locations within a given distance of you in spherical geometry than in Euclidean space.

Swing your arm around and contemplate the locations that exist at arm's length from you. In Euclidean space, this is a certain number of locations, having to do with the circumference of a circle, or in higher dimensions, with the area of the corresponding sphere. But in spherical geometry, the number of such locations at arm's length is somewhat less than in Euclidean space.

In hyperbolic space, in contrast, the opposite is true; there are more such locations at arm's length than in Euclidean space, and indeed, prodigiously more when the space is strongly negatively curved. Hyperbolic space is negatively curved, and so the circumference of a circle grows more rapidly with the radius than in Euclidean space.

If you are a resident of a city in hyperbolic space, then the number of shops and restaurants within a few blocks of your apartment would be enormous because in hyperbolic space, there are so many more locations that are this close to you. If the space is extremely negatively curved, then there could be whole undiscovered civilizations within walking distance, simply because a strong negative curvature means that there are such a vast number of locations quite nearby. Criminals escape easily in hyperbolic space, simply by walking away a short distance; it is too difficult to follow them far, since at every moment, one must choose from amongst so many further directions to continue. For the same reason, you or your loved ones may easily become lost in hyperbolic space, for it is so difficult to find your way exactly back home again; so please be careful and hold them close.

Let me emphasize again that both elliptical and hyperbolic geometry appear increasingly Euclidean on very small scales. If the geometric universe were vast, then a person living in such a space might believe the universe to be Euclidean based on their experience at comparatively small scale.

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Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.

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Infinitesimals revisited

For the final theme of this chapter, let us return to the infinitesimals. Despite the problematic foundations and Berkeley's criticisms, it will be good to keep in mind that the infinitesimal conception was actually extremely fruitful and led to many robust mathematical insights, including all the foundational results of calculus. Mathematicians today routinely approach problems in calculus and differential equations essentially by considering the effects of infinitesimal changes in the input to a function or system.

For example, to compute the volume of a solid of revolution y = f(x) about the x-axis, it is routine to imagine slicing the volume into infinitesimally thin disks. The disk at x has radius f(x) and infinitesimal thickness dx (hence volume πf(x)²dx), and so the total volume between a and b, therefore, is

Another example arises when one seeks to compute the length of the curve traced by a smooth function y = f(x). One typically imagines cutting it into infinitesimal pieces and observing that each tiny piece is the hypotenuse ds of a triangle with infinitesimal legs dx and dy. So by an infinitesimal instance of the Pythagorean theorem, we see

from which one “factors out” dx, obtaining

and therefore the total length of the curve is given by

Thus, one should not have a cartoon understanding of developments in early calculus, imagining that it was all bumbling nonsense working with the ghosts of departed quantities. On the contrary, it was a time of enormous mathematical progress and deep insights of enduring strength. Perhaps this situation gives a philosopher pause when contemplating the significance of foundational issues for mathematical progress — must one have sound foundations in order to advance mathematical knowledge? Apparently not. Nevertheless, the resolution of the problematic foundations with epsilon-delta methods did enable a far more sophisticated mathematical analysis, leading to further huge mathematical developments and progress. It was definitely valuable to have finally fixed the foundations.

Nonstandard analysis and the hyperreal numbers

What an astounding development it must have been in 1961, when Abraham Robinson introduced his theory of nonstandard analysis. This theory, arising from ideas in mathematical logic and based on what are now called the hyperreal numbers ℝ*, provides a rigorous method of handling infinitesimals, having many parallels to the early work in calculus. I look upon this development as a kind of joke that mathematical reality has played on both the history and philosophy of mathematics.

The way it works is as follows. The hyperreal number system ℝ* is an ordered field extending the real numbers, but having new infinite and infinitesimal numbers. In the figure, the real number line—meaning all of it, the entire real number line—is indicated in blue. The hyperreal number line, indicated in red, thus extends strictly beyond the end of the real number line. It may be difficult to imagine at first, but indeed this is what the construction of the hyperreal numbers produces. The hyperreal number N indicated in the figure, for example, is larger than every real number, and the number δ is positive, being strictly larger than 0 but smaller than every positive real number. One could imagine that δ = 1/N, since the reciprocal of an infinite number will be infinitesimal and conversely.

Every real number is surrounded by a neighborhood of those hyperreal numbers infinitesimally close to it, and so if one were to zoom in on a real number in the hyperreal numbers, one would ultimately find a window containing only that real number and the hyperreal numbers infinitely close to it. Every hyperreal z that is bounded by standard real numbers is infinitesimally close to a unique standard real number, called the standard part of z and denoted as std(z), as indicated in the figure.

Part of what makes the hyperreal numbers attractive for nonstandard analysis is their further remarkable property that any statement in the formal language of analysis that is true for the real numbers ℝ is also true for the hyperreal numbers ℝ*. The transfer principle asserts that every real number a and function f on the real numbers have nonstandard counterparts a* and f* in the hyperreal numbers, such that any assertion φ(a, f ) about a and f that is true in the real numbers ℝ is also true of the counterparts φ(a*, f *) in the hyperreal numbers ℝ*. Thus, the real numbers have an elementary embedding a ↦ a* into the hyperreal numbers ℝ ≺ ℝ*.

Because of the transfer principle, we know exactly what kinds of arithmetic and algebraic operations are allowed in the hyperreal numbers—they are the same as those allowed in the real number system. Because of the transfer principle, we will have concepts of infinite integers, infinite prime numbers, infinite powers of 2, and so on, and we can divide an interval in the real numbers into infinitely many subintervals of equal infinitesimal length. These are exactly the kinds of things that one wants to do in calculus, and nonstandard analysis provides a rigorous foundation for it.

Calculus in nonstandard analysis

Let us illustrate this by explaining how the derivative is treated in nonstandard analysis. One computes the derivative of a function f by using an infinitesimal hyperreal number, without any limit process or epsilon-delta arguments. To find the derivative of f(x) = x², for example, let δ be a positive infinitesimal hyperreal. We compute the rate-of-change quotient over the interval from x to x + δ,

with the same algebraic steps as performed earlier, and then simply take the standard part of the result, arriving at

Thus, the derivative of x² is 2x. The use of the standard-part operation in effect formalizes how one can correctly treat infinitesimals—it explains exactly how the ghosts depart! Thus, nonstandard analysis enables a fully rigorous use of infinitesimals.

Nonstandard analysis has now grown into a mature theory, providing an alternative conception parallel to the classical theory. There are several philosophical perspectives one might naturally adopt when undertaking work in nonstandard analysis, which I would like now to explain. Some of this material is technical, however, and so readers not familiar with ultrapowers, for example, might simply skip over it.

Classical model-construction perspective

In this approach, one thinks of the nonstandard universe as the result of an explicit construction, such as an ultrapower construction. In the most basic instance, one has the standard real number field structure ⟨ℝ, +, ·, 0, 1, ℤ⟩, and you perform the ultrapower construction with respect to a fixed nonprincipal ultrafilter U on the natural numbers (or on some other set if this were desirable). The ultrapower structure ℝ* = ℝ^ℕ/U is then taken as a conception of the hyperreal numbers, an ordered non-Archimedean field, which therefore has infinitesimal elements.

In time, however, one is led to want more structure in the pre-ultrapower model, so as to be able to express more ideas, which will each have nonstandard counterparts. One will have constants for every real number, a predicate for the integers ℤ, or indeed for every subset of ℝ, and a function symbol for every function on the real numbers, and so on. In this way, one gets the nonstandard analogue z* of every real number z, the set of nonstandard integers ℤ*, and nonstandard analogues f * for every function f on the real numbers, and so on. Before long, one also wants nonstandard analogues of the power set P(ℝ) and higher iterates. In the end, what one realizes is that one might as well take the ultrapower of the entire set-theoretic universe V → V^ω/U, which amounts to doing nonstandard analysis with second-order logic, third- order, and indeed α-order for every ordinal α. One then has the copy of the standard universe V inside the nonstandard realm V*, which one analyzes and understands by means of the ultrapower construction itself.

Some applications of nonstandard analysis have required one to take not just a single ultrapower, but an iterated ultrapower construction along a linear order. Such an ultrapower construction gives rise to many levels of nonstandardness, and this is sometimes useful. Ultimately, as one adds additional construction methods, this amounts just to adopting all model theory as one's toolkit. One will want to employ advanced saturation properties, or embeddings, or the standard system, and so on. There is a well developed theory of models of arithmetic that uses quite advanced methods. To give a sample consequence of saturation, every infinite graph, no matter how large, arises as an induced subgraph of a nonstandard-finite graph in every sufficiently saturated model of nonstandard analysis. This sometimes can allow you to undertake finitary constructions with infinite graphs, with the cost being a move to the nonstandard context.

Axiomatic approach

Most applications of nonstandard analysis, however, do not rely on the details of the ultrapower or iterated ultrapower constructions, and so it is often thought worthwhile to isolate the general principles that make the nonstandard arguments succeed. Thus, one writes down the axioms of the situation. In the basic case, one has the standard structure ℝ, and so on, perhaps with constants for every real number (and for all subsets and functions in the higher-order cases), with a map to the nonstandard structure ℝ*, so that every real number a has its nonstandard version a* and every function f on the real numbers has its nonstandard version f *. Typically, the main axioms would include the transfer principle, which asserts that any property expressible in the language of the original structure holds in the standard universe exactly when it holds of the nonstandard analogues of those objects in the nonstandard realm. The transfer principle amounts precisely to the elementarity of the map a ↦ a* from standard objects to their nonstandard analogues. One often also wants a saturation principle, expressing that any sufficiently realizable type is actually realized in the nonstandard model, and this just axiomatizes the saturation properties of the ultrapower. Sometimes one wants more saturation than one would get from an ultrapower on the natural numbers, but one can still achieve this by larger ultrapowers or other model-theoretic methods.

Essentially the same axiomatic approach works with the higher-order case, where one has a nonstandard version of every set-theoretic object, and a map V → V*, with nonstandard structures of any order. And similarly, one can axiomatize the features that one wants to use in the iterated ultrapower case, with various levels of standardness.

“The” hyperreal numbers?

As with most mathematical situations where one has both a construction and an axiomatic framework, it is usually thought better to argue from the axioms, when possible, than to use details of the construction. And most applications of nonstandard analysis that I have seen can be undertaken using only the usual nonstandardness axioms. A major drawback of the axiomatic approach to nonstandard analysis, however, is that the axiomatization is not categorical. There is no unique mathematical structure fulfilling the hyperreal axioms, and no structure that is “the” hyperreal numbers up to isomorphism. Rather, we have many candidate structures for the hyperreal numbers, of various cardinalities; some of them extend one another, and all of them satisfy the axioms, but they are not all isomorphic.

For this reason, and despite the common usage that one frequently sees, it seems incorrect, and ultimately not meaningful, to refer to “the” hyperreal numbers, even from a structuralist perspective. The situation is totally unlike that of the natural numbers, the integers, the real numbers, and the complex numbers, where we do have categorical characterizations. To be sure, the structuralist faces challenges with the use of singular terms even in those cases (challenges we discussed in chapter 1), but the situation is far worse with the hyperreal numbers. Even if we are entitled to use singular terms for the natural numbers and the real numbers, with perhaps some kind of structuralist explanation as to what this means, nevertheless those structuralist explanations will fail completely with the hyperreal numbers, simply because there is no unique structure that the axioms identify. We just do not seem entitled, in any robust sense, to make a singular reference to the hyperreal numbers.

And yet, one does find abundant references to “the” hyperreal numbers in the literature. My explanation for why this has not caused a fundamental problem is that for the purposes of nonstandard analysis, that is, for the goal of establishing truths in calculus about the real numbers by means of the nonstandard real numbers, the nonisomorphic differences between the various hyperreal structures simply happen not to be relevant. All these structures are non-Archimedean ordered fields with the transfer principle, and it happens that these properties alone suffice for the applications of these fields that are undertaken. In this sense, it is as though one fixes a particular nonstandard field ℝ*—and for the applications considered, it does not matter much which one—and then reference to “the” hyperreal numbers simply refer to that particular ℝ* that has been fixed. It is as though the mathematicians are implementing Stewart Shapiro's disguised bound quantifiers, but without a categoricity result.

I wonder whether this lack of categoricity may explain the hesitancy of some mathematicians to study nonstandard analysis; it prevents one from adopting a straightforward structuralist attitude toward the hyperreal numbers, and it tends to push one back to the model-theoretic approach, which more accurately conveys the complexity of the situation.

Meanwhile, we do have a categoricity result for the surreal numbers, which form a nonstandard ordered field of proper class size, one that is saturated in the model-theoretic sense—as in, every set-sized cut is filled. In the standard second-order set theories such as Gödel- Bernays set theory with the principle of global choice (a class well-ordering of the universe) or Kelley-Morse set theory, one can prove that all such ordered fields are isomorphic. This is a sense in which “the hyperreal numbers” might reasonably be given meaning by taking it to refer to the surreal numbers, at the cost of dealing with a proper class structure.

Radical nonstandardness perspective

This perspective on nonstandard analysis involves an enormous foundational change in one's mathematical ontology. Namely, from this perspective, rather than thinking of the standard structures as having analogues inside a nonstandard world, one instead thinks of the nonstandard world as the “real” world, with a “standardness” predicate picking out parts of it. On this approach, one views the real numbers as including both infinite and infinitesimal real numbers, and one can say when and whether two finite real numbers have the same standard part, and so on. With this perspective, we think of the “real” real numbers as what from the other perspective would be the nonstandard real numbers, and then we have a predicate on that, which amounts to the range of the star map in the other approach. So some real numbers are standard, some functions are standard, and so on.

In an argument involving finite combinatorics, for example, someone with this perspective might casually say, “Let N be an infinite integer” or “Consider an infinitesimal rational number.” (One of my New York colleagues sometimes talks this way.) That way of speaking may seem alien for someone not used to this perspective, but for those that adopt it, it is productive. These practitioners have drunk deeply of the nonstandardness wine; they have moved wholly into the nonstandard realm — a new plane of existence.

Extreme versions of this idea adopt many levels of standardness and nonstandardness, extended to all orders. Karel Hrbacek (1979, 2009) has a well developed theory like this for nonstandard set theory, with an infinitely deep hierarchy of levels of standardness. There is no fully “standard” realm according to this perspective. In Hrbacek's system, one does not start with a standard universe and go up to the nonstandard universe, but rather, one starts with the full universe (which is fundamentally nonstandard) and goes down to deeper and deeper levels of standardness. Every model of ZFC, he proved, is the standard universe inside another model of the nonstandard set theories he considers.

Translating between nonstandard and classical perspectives

Ultimately, my view is that the choice between the three perspectives I have described is a matter of taste, and any argument that can be formulated in one of the perspectives has analogues in the others. In this sense, there seems to be little at stake, mathematically, between the perspectives. And yet, as I argued in section 1.16, divergent philosophical views can lead one toward different mathematical questions and different mathematical research efforts.

One can usually translate arguments not only amongst the perspectives of nonstandard analysis, but also between the nonstandard realm and the classical epsilon-delta methods. Terence Tao (2007) has described the methods of nonstandard analysis as providing a smooth way to manage one's ε arguments. One might say, “This δ is smaller than anything defined using that ε.” It is a convenient way to undertake error estimation. Tao similarly points out how ultrafilters can be utilized in an argument as a simple way to manage one's estimates; if one knows separately that for each of the objects a, b, and c, there is a large measure (with respect to the ultrafilter) of associated witnesses, then one can also find a large measure of witnesses working with all three of them.

For some real analysts, however, it is precisely the lack of familiarity with ultrafilters or other concepts from mathematical logic that prevents them from appreciating the nonstandard methods. In this sense, the preference for or against nonstandard analysis appears to be in part a matter of cultural upbringing.

H. Jerome Keisler wrote an introductory calculus textbook, Elementary Calculus: An Infinitesimal Approach (1976), intended for first-year university students, based on the ideas of nonstandard analysis, but otherwise achieving what this genre of calculus textbook achieves. It is a typical thick volume, with worked examples on definite integrals and derivatives and optimization problems and the chain rule, and so on, all with suitable exercises for an undergraduate calculus student. It looks superficially like any of the other standard calculus textbooks used in such a calculus class. But if you peer inside Keisler's book, in the front cover alongside the usual trigonometric identities and integral formulas you will find a list of axioms concerning the interaction of infinite and infinitesimal numbers, the transfer principle, the algebra of standard parts, and so on. It is all based on nonstandard analysis and is fundamentally unlike the other calculus textbooks. The book was used for a time, successfully, in the calculus classes at the University of Wisconsin in Madison.

There is an interesting companion tale to relate concerning the politics of book reviews. Paul Halmos, editor of the Bulletin of the American Mathematical Society, requested a review of Keisler's book from Errett Bishop, who decades earlier had been his student but who was also prominently known for his constructivist views in the philosophy of mathematics—views that are deeply incompatible with the main tools of nonstandard analysis. The review was predictably negative, and it was widely criticized, notably by Martin Davis (1977), later in the same journal. In response to the review, Keisler remarked (see also Davis and Hausner, 1978), that the choice of Bishop to review the book was like “asking a teetotaler to sample wine.”

Criticism of nonstandard analysis

Alan Connes mounted a fundamental criticism of nonstandard analysis, remarking in an interview

At that time, I had been working on nonstandard analysis, but after a while I had found a catch in the theory.... The point is that as soon as you have a nonstandard number, you get a nonmeasurable set. And in Choquet's circle, having well studied the Polish school, we knew that every set you can name is measurable; so it seemed utterly doomed to failure to try to use nonstandard analysis to do physics. (Goldstern, Skandalis 2007, interview with A Connes)

What does he mean? To what is he referring? Let me explain.

“as soon as you have a nonstandard number, you get a nonmeasurable set.”

Every nonstandard natural number N gives rise to a certain notion of largeness for sets of natural numbers: a set X ⊆ ℕ is large exactly if N ∈ X*. In other words, a set X is large if it expresses a property that the nonstandard number N has. No standard finite set is large, and furthermore, the intersection of any two large sets is large and any superset of a large set is large. Thus, the collection 𝒰 of all these large sets X forms what is called a nonprincipal ultrafilter on the natural numbers. We may identify the large sets with elements of Cantor space 2^ℕ, which carries a natural probability measure, the coin-flipping measure, and so 𝒰 is a subset of Cantor space.

But the point to be made now is that a nonprincipal ultrafilter cannot be measurable in Cantor space, since the full bit-flipping operation, which is measure-preserving, carries 𝒰 exactly to its complement, so 𝒰 would have to have measure 1/2 , but 𝒰 is a tail event, invariant by the operation of flipping any finite number of bits, and so by Kolmogorov's zero-one law, it must have measure 0 or 1.

“in the Polish school, ...every set you can name is measurable.”

Another way of saying that a set is easily described is to say that it lies low in the descriptive set-theoretic hierarchy, famously developed by the Polish logicians, and the lowest such sets are necessarily measurable. For example, every set in the Borel hierarchy is measurable, and the Borel context is often described as the domain of explicit mathematics.

Under stronger set-theoretic axioms, such as large cardinals or projective determinacy, the phenomenon rises to higher levels of complexity, for under these hypotheses, it follows that all sets in the projective hierarchy are Lebesgue measurable. This would include any set that you can define by quantifying over the real numbers and the integers and using any of the basic mathematical operations. Thus, any set you can name is measurable.

The essence of the Connes criticism is that one cannot construct a model of the hyperreal numbers in any concrete or explicit manner, because one would thereby be constructing explicitly a nonmeasurable set, which is impossible. Thus, nonstandard analysis is intimately wrapped up with ultrafilters and weak forms of the axiom of choice. For this reason, it seems useless so far as any real-world application in science and physics is concerned.

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Lectures on the Philosophy of Mathematics, MIT Press 2021

Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.

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Abstraction in the function concept

The increasing rigor in mathematical analysis was also a time of increasing abstraction in the function concept. What is a function? In the naive account, one specifies a function by providing a rule or formula for how to compute the output value y in terms of the input value x. We all know many examples, such as y = x² + x + 1 or y = e^x or y = sin x. But do you notice that already these latter two functions are not expressible directly in terms of the algebraic field operations? Rather, they are transcendental functions, already a step up in abstraction for the function concept.

The development of proper tools for dealing with power series led mathematicians to consider other more general function representations, such as power series and Fourier series, as given here for the exponential function e^x and the sawtooth function s(x):

The more general function concept provided by these kinds of representations enabled mathematicians to solve mathematical problems that were previously troubling. One can look for a solution of a differential equation, for example, by assuming that it will have a certain series form and then solve for the particular coefficients, ultimately finding that indeed the assumption was correct and there is a solution of that form. Fourier used his Fourier series to find solutions of the heat equation, for example, and these series are now pervasive in science and engineering.

The Devil's staircase

Let us explore a few examples that might tend to stretch one's function concept. Consider the Devil's staircase, pictured below.

This function was defined by Cantor using his middle-thirds set, now known as the Cantor set, and serves as a counterexample to what might have been a certain natural extension of the fundamental theorem of calculus, since it is a continuous function from the unit interval to itself, which has a zero derivative at almost every point with respect to the Lebesgue measure, yet it rises from 0 to 1 during the interval. This is how the Devil ascends from 0 to 1 while remaining almost always motionless.

To construct the function, one starts with value 0 at the left and 1 at the right of the unit interval. Interpolating between these, one places constant value 1/2 on the entire middle-third interval; this leaves two intervals remaining, one on each side. Next, one places interpolating values 1/4 and 3/4 on the middle-thirds of those intervals, and so on, continuing successively to subdivide. This defines the function on the union of all the resulting middle-thirds sections, indicated in orange in the figure, and because of the interpolation, the function continuously extends to the entire interval. Since the function is locally constant on each of the middle-thirds intervals, the derivative is zero there, and those intervals add up to full measure one. The set of points that remain after omitting all the middle-thirds intervals is the Cantor set.

Space-filling curves

Next, consider the phenomenon of space-filling curves. A curve is a continuous function from the one-dimensional unit interval into a space, such as the plane, a continuous function c:[0,1] → ℝ². We can easily draw many such curves and analyze their mathematical properties.

A curve in effect describes a way of traveling along a certain path: one is at position c(t) at time t. The curve therefore begins at the point c(0); it travels along the curve as time t progresses; and it terminates at the point c(1). Curves are allowed to change speed, cross themselves, and even to move backward on the same path, retracing their route. For this reason, one should not identify the curve with its image in the plane, but rather the curve is a way of tracing out that path.

Since the familiar curves that are easily drawn and grasped seem to exhibit an essentially one-dimensional character, it was a shocking discovery that there are space-filling curves, which are curves that completely fill a space. Peano produced such a space-filling curve, a continuous function from the unit interval to the unit square, with the property that every point in the square is visited at some time by the curve. Let us consider Hilbert's space-filling curve, shown here, which is a simplification.

The Hilbert curve can be defined by a limit process. One starts with a crude approximation, a curve consisting of three line segments, as in the figure at the upper left; each approximation is then successively replaced by one with finer detail, and the final Hilbert curve itself is simply the limit of these approximations as they get finer and finer. The first six iterations of the process are shown here. Each approximation is a curve that starts at the lower left and then wriggles about, ultimately ending at the lower right. Each approximation curve consists of finitely many line segments, which we should imagine are traversed at uniform speed. What needs to be proved is that as the approximations get finer, the limiting values converge, thereby defining the limit curve.

One can begin to see this in the approximations to the Hilbert curve above. If we travel along each approximation at constant speed, then the halfway point always occurs just as the path crosses the center vertical line, crossing a little bridge from west to east (look for this bridge in each of the six iterations of the figure — can you find it?). Thus, the halfway point of the final limit curve h is the limit of these bridges, and one can see that they are converging to the very center of the square. So h(1/2) = (1/2, 1/2). Because the wriggling is increasingly local, the limit curve will be continuous; and because the wriggling gets finer and finer and ultimately enters every region of the square, it follows that every point in the square will occur on the limit path. So the Hilbert curve is a space-filling curve. To my way of thinking, this example begins to stretch, or even break, our naive intuitions about what a curve is or what a function (or even a continuous function) can be.

Conway base-13 function

Let us look at another such example, which might further stretch our intuitions about the function concept. The Conway base 13 function C(x) is defined for every real number x by inspecting the tridecimal (base 13) representation of x and determining whether it encodes a certain secret number, the Conway value of x. Specifically, we represent x in tridecimal notation, using the usual ten digits 0,...,9, plus three extra numerals ⊕, ⊖, and ⊙, having values 10, 11 and 12, respectively, in the tridecimal system. If it should happen that the tridecimal representation of x has a final segment of the form

where the a_i and b_j use only the digits 0 through 9, then the Conway value C(x) is simply the number

understood now in decimal notation. And if the tridecimal representation of x has a final segment of the form

where again the a_i and b_j use only the digits 0 through 9, then the Conway value C(x) is the number

in effect taking ⊕ or ⊖ as a sign indicator for C(x). Finally, if the tridecimal representation of x does not have a final segment of one of those forms, for example, if it uses the ⊕ numeral infinitely many times or if it does not use the ⊙ numeral at all, then we assign the default value C(x) = 0.

Some examples may help illustrate the idea:

The point is that we can easily read off the decimal value of C(x) from the tridecimal representation of x. On the left, we have the input number x, given in tridecimal representation, and on the right is the encoded Conway value C(x), in decimal form, with a default value of 0 when no number is encoded. Note that we ignore the tridecimal point of x in the decoding process.

Let us look a little more closely at the Conway function to discover its fascinating features. The key thing to notice is that every real number y is the Conway value of some real number x, and furthermore, you can encode y into the final segment of x after having already specified an arbitrary finite initial segment of the tridecimal representation of x. If you want the tridecimal representation of x to start in a certain way, go ahead and do that, and then simply add the numeral ⊕ or ⊖ after this, depending on whether y is positive or negative, and then list off the decimal digits of y to complete the representation of x, using the numeral ⊙ to indicate where the decimal point of y should be. It follows that C(x) = y. Because we can specify the initial digits of x arbitrarily, before the encoding of y, it follows that every interval in the real numbers, no matter how tiny, will have a real number x, with C(x) = y. In other words, the restriction of the Conway function to any tiny interval results in a function that is still onto the entire set of real numbers.

Consequently, if we were to make a graph of the Conway function, it would look totally unlike the graphs of other, more ordinary functions. Here is my suggestive attempt to represent the graph:

The values of the function are plotted in blue, you see, but since these dots appear so densely in the plane, we cannot easily distinguish the individual points, and the graph appears as a blue mass. When I draw the graph on a chalkboard, I simply lay the chalk sideways and fill the board with dust.

But the Conway function is indeed a function, with every vertical line having exactly one point, and one should imagine the graph as consisting of individual points ( x, C(x) ). If we were to zoom in on the graph, as suggested in the figure, we might hope to begin to discern these points individually. But actually, one should take the figure as merely suggestive, since after any finite amount of magnification, the graph would of course remain dense, with no empty regions at all, and so those empty spaces between the dots in the magnified image are not accurate. Ultimately, we cannot seem to draw the graph accurately in any fully satisfactory manner.

Another subtle point is that the default value of 0 actually occurs quite a lot in the Conway function, since the set of real numbers x that fail to encode a Conway value has full Lebesgue measure. In this precise sense, almost every real number is in the default case of the Conway function, and so the Conway function is almost always zero. We could perhaps have represented this aspect of the function in the graph with a somewhat denser hue of blue lying on the x-axis, since in the sense of the Lebesgue measure, most of the function lies on that axis. Meanwhile, the truly interesting part of the function occurs on a set of measure zero, the real numbers x that do encode a Conway value.

Notice that the Conway function, though highly discontinuous, nevertheless satisfies the conclusion of the intermediate value theorem: for any a < b, every y between f(a) and f(b) is realized as f(c) for some c between a and b.

We have seen three examples of functions that tend to stretch the classical conception of what a function can be. Mathematicians ultimately landed with a very general function concept, abandoning all requirements functions must be defined by a formula of some kind; rather, a function is simply a certain kind of relation between input and output: a function is any relation for which an input gives at most one output, and always the same output.

In some mathematical subjects, the function concept has evolved further, growing horns in a sense. Namely, for many mathematicians, particularly in those subjects using category theory, a function f is not determined merely by specifying its domain X, and the function values f(x) for each point in that domain x ∈ X. Rather, one must also specify what is called the codomain of the function, the space Y of intended target values for the function f : X → Y. The codomain is not the same as the range of the function, because not every y ∈ Y needs to be realized as a value f(x). On this concept of the function, the squaring function s(x) = x² on the real numbers can be considered as a function from ℝ to ℝ or as a function from ℝ to [0,∞), and the point would be that these would count as two different functions, which happen to have the same domain ℝ and the same value x² at every point x in that domain; but because the codomains differ, they are different functions. Such a concern with the codomain is central to the category-theoretic conception of morphisms, where with the composition of functions f ∘ g, for example, one wants the codomain of g to align with the domain of f.

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Lectures on the Philosophy of Mathematics, MIT Press 2021

Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.

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Indispensability of mathematics

What philosophical conclusion can we make from the fact that mathematical tools and vocabulary seem to lie at the very core of nearly every contemporary scientific theory? How remarkable that at every physical scale, from the microscopic to the cosmic, our best scientific theories are thoroughly mathematical. Why should this be? The laws of Newtonian physics are expressed in universal differential equations that explain the interaction of forces and motion, unifying our understanding of diverse physical phenomena, from the harmonic oscillations of a mass on a spring to the planetary motions of the heavenly bodies; our best theory of electromagnetism posits unseen electrical and magnetic fields that surround us all, enveloping the Earth; relativity theory explains the nature of space and time with exotic mathematical geometries; quantum mechanics uses Hilbert spaces; string theory uses still more abstract mathematical objects; and all the experimental sciences, including the social sciences, make fundamental use of mathematical statistics. Physicist Paul Dirac (1963) describes the situation like this:

It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.

(Footnote: One should not misunderstand Dirac's views on God, however, for he also said, “If we are honest—and scientists have to be—we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination.” Dirac, quoted by Werner Heisenberg (1971).)

Thus, mathematics appears to be indispensable for physics and other sciences.

On the basis of this, Hilary Putnam and Willard Van Orman Quine mount the indispensability argument for mathematical realism, arguing that we ought to have an ontological commitment to the objects that are indispensably part of our best scientific theories (see Putnam (1971) for a classic presentation). Just as a scientist finds grounds for the existence of unseen microscopic organisms on the basis of the well supported germ theory of disease, even in the absence of direct observations of those organisms, and just as a scientist might commit to the atomic theory of matter or the existence of electrons or a molten iron core in the Earth, or black holes, or wave functions, even when the evidence for them in our well supported theories is indirect, then similarly, according to the indispensability argument, we should find grounds for the existence of the abstract mathematical objects that appear in our best theories. Quine emphasizes a view of confirmational wholism, by which theories are confirmed only as a whole. If mathematical claims are an indispensable part of the theory, then they are part of what is confirmed.

Science without numbers

Attacking the indispensability argument at its heart, Hartry H. Field (1980) argues that the truth of mathematics is not actually indispensable for science. Defending a nominalist approach to mathematics, he argues that we do not require the actual existence of these abstract mathematical objects in order to undertake a successful scientific analysis.

Rather, Field points out that there is a kind of logical error underlying the indispensability argument. Namely, even if the mathematical theories are indispensable to the scientific analysis, this is not a reason to suppose that the mathematical theories are actually true. It would be sufficient, for example, if the mathematical claims formed merely a conservative extension} of the scientific theory. Specifically, as mentioned in chapter 1, a theory S is conservative over another theory T that it extends with respect to a class of assertions ℒ if, whenever S proves an ℒ assertion, this assertion is already provable in T. In other words, the stronger theory S tells us no new ℒ facts that we could not already know on the basis of T. This does not mean that the theory S is useless or unhelpful, however, for perhaps S unifies our knowledge somehow or is more explanatory or makes reasoning easier, even if ultimately, no new ℒ assertions will be proved in S.

In the case of the indispensability argument, our scientific theory S describes the nature of the physical world, but this theory includes mathematical claims making existence assertions about various mathematical objects. Let N be the nominalist fragment of the scientific theory, omitting the mathematical claims. If the full theory S were conservative over N concerning assertions about the physical world, then we could safely use the full theory S to make deductions about the physical world, whether or not the mathematical claims it makes are true. In the mathematically augmented theory, the scientist may safely reason as if the mathematical part of the theory were true, without needing to commit to the truth of those additional mathematical assertions.

Instances of this same pattern have arisen entirely in mathematics. Consider the early use of the complex numbers in mathematics, before the nature of imaginary numbers was well understood. In those early days, skeptical mathematicians would sometimes use the so-called imaginary numbers, using expressions like 1 + √(-5), even when they looked upon these expressions as meaningless, because in the end, the imaginary parts of their calculations would sometimes cancel out and they would find the desired real number solution to their equation. It must have been mystifying to see calculations proceeding through the land of nonsense, manipulating those imaginary numbers with ordinary algebra, and yet somehow working out in the end to a real number solution that could be verified independently of the complex numbers. My point here is that even if a mathematician did not commit to the actual existence of the complex numbers, the theory of complex numbers was conservative over their theory of real numbers, so far as assertions about the real numbers are concerned. So even a skeptical mathematician could safely reason as if imaginary numbers actually existed.

Field is arguing similarly for applications of mathematics in general. Ultimately, it does not matter, according to Field, whether the mathematical claims made as part of a scientific theory are actually true or false, provided that the theory is conservative over the nonmathematical part of the theory, so far as physical assertions are concerned, for in this case, the scientist can safely reason as if the mathematical claims were true.

Impressively, Field attempts to show how one can cast various scientific theories without any reference to mathematical objects, replacing the usual theories with nominalized versions, which lack a commitment to the existence of mathematical objects. He provides a nominalist account of Newtonian spacetime and of the Newtonian gravitational theory. One basic idea, for example, is to use physical arrangements as stand-ins for mathematical quantities. One may represent an arbitrary real number, for example, by the possible separations of two particles in space, and then refer to that number in effect by referring to the possible locations of those particles. Thus, one avoids the need for abstract objects.

Critics of Field point out that although he has strived to eliminate numbers and other abstract mathematical objects, nevertheless his ontology is rich, filled with spatiotemporal regions and other objects that can be seen as abstract. Shall we take physics to be committed to these abstract objects? Also, the nominalized theory is cumbersome and therefore less useful for explanation and insight in physics—isn't this relevant for indispensability?

Fictionalism

Fictionalism is the position in the philosophy of mathematics, according to which mathematical existence assertions are not literally true, but rather are a convenient fiction, useful for a purpose, such as the applications of mathematics in science. According to fictionalism, statements in mathematics are similar in status to statements about fictional events. An arithmetic assertion p, for example, can be interpreted as the statement, “According to the theory of arithmetic, p.” It is just as one might say, “According to the story by Beatrix (1906), Jeremy Fisher enjoys punting.” In his nominalization program for science, Field is essentially defending a fictionalist account of mathematics in science. Even if the mathematical claims are not literally true, the scientist can reason as if they were true.

I find it interesting to notice how the fictionalist position might seem to lead one to nonclassical logic in mathematics. Let us suppose that in the story of Jeremy Fisher, the cost of his punt is not discussed; and now consider the statement, “Jeremy Fisher paid more than two shillings for his punt.” It would be wrong to say, “According to the story by Beatrix Potter, Jeremy Fisher paid more than two shillings for his punt.” But it would also be wrong to say, “According to the story by Beatrix Potter, Jeremy Fisher did not pay more than two shillings for his punt.” The story simply has nothing to say on the matter. So the story asserts neither p nor ¬p. Is this a violation of the law of excluded middle?

As I see it, no, this is not what it means to deny the law of excluded middle. While one asserts neither p nor ¬p, still, one asserts p ∨ ¬p, since according to the story by Beatrix Potter, we may reasonably suppose that either Jeremy Fisher did pay more than two shillings for his punt or he did not, since Jeremy Fisher's world is presented as obeying such ordinary logic. Another way to see this point is to consider an incomplete theory T in classical logic. Since the theory is incomplete, there is a statement p not settled by the theory, and so in the theory, we do not assert p and we do not assert ¬p, and yet we do assert p ∨ ¬p, and we have not denied the law of excluded middle. Fictional accounts are essentially like incomplete theories, which do not require one to abandon classical logic.

The theory/metatheory distinction

In a robust sense, fictionalism is a retreat from the object theory into the metatheory. Let us make the theory/metatheory distinction. In mathematics, the object theory is the theory describing the mathematical subject matter that the theory is about. The metatheory, in contrast, places the object theory itself under mathematical analysis and looks into meta-theoretic issues concerning it such as provability and consistency. In the object theory—take ZFC set theory, for instance—one asserts that there are sets of all kinds, including well-orders of the real numbers and diverse uncountable sets of vast cardinality, while in the metatheory, typically with only comparatively weak arithmetic resources suitable for managing the theory, one might make none of those existence claims outright, and instead one asserts like the fictionalist merely that “According to the theory ZFC, there are well-orders of the real numbers and uncountable sets of such-and-such vast cardinality.” In this way, fictionalism amounts exactly to the metamathematical move. For this reason, there are affinities between fictionalism and formalism (see chapter 7), for the formalist also retreats into the metatheory, taking himself or herself not to have asserted the existence of infinite sets, for example, but asserting instead merely that, according to the theory at hand, there are infinite sets.

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The subject of real analysis can be founded upon the least-upper-bound principle, a version of Dedekind completeness. Taking this as a core principle, one proceeds to prove all the familiar foundational theorems, such as the intermediate value theorem, the Heine-Borel theorem and many others. In a sense, the least-upper-bound principle is to real analysis what the induction principle is to number theory.

Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.

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Least-upper-bound principle. Every nonempty set of real numbers with an upper bound has a least upper bound.

An upper bound of a set A ⊆ ℝ is simply a real number r, such that every element of A is less than or equal to r. The number r is the least upper bound of A, also called the supremum of A and denoted sup(A), if r is an upper bound of A and r ⩽ s whenever s is an upper bound of A.

Consequences of completeness

Let us illustrate how this fundamental principle is used by sketching proofs of a few familiar elementary results in real analysis. Consider first the intermediate-value theorem, which asserts that if f is a continuous function on the real numbers and d is an intermediate value between f(a) and f(b), then there is a real number c between a and b, with f(c) = d. In short, it asserts that for continuous functions, every intermediate value is realized.

To prove this from the least-upper-bound principle, assume that a < b and f(a) < d < f(b), and consider the set A of all real numbers x in the interval [a,b] for which f(x) < d. This set is nonempty because a is in A, and it is bounded by b. By the least-upper-bound principle, therefore, it has a least upper bound c = sup(A). Consider the value of f(c). If f(c) is too small, meaning that f(c) < d, then by the continuity of f, the values of f(x) for x near c will all also be less than d. But this would mean that A has some elements above c, contrary to c = sup(A). Therefore, f(c) ⩾ d. Similarly, if f(c) is too large, meaning that f(c) > d, then again by the continuity of f, the values of f(x) for x in a small neighborhood of c will all be above d, contradicting c = sup(A), since there must be elements of A as close as desired to c, and those elements x must have f(x) < d by the definition of A. Therefore, it must be that f(c) = d, and we have found the desired point c realizing the intermediate value d.

Consider next the Heine-Borel theorem, which asserts that the unit interval [0,1] is compact. What this means is that whenever 𝒰 is a set of open intervals covering the closed unit interval in the real numbers [0,1], then there are finitely many open intervals of 𝒰 that already cover [0,1]. Huh? For someone new to real analysis, the importance of this open-cover conclusion may not be readily apparent; perhaps it may even seem a little bizarre. Why would it be important that every open cover of a closed interval admits a finite subcover? The answer is that this property is vitally important, and indeed, it is difficult to overstate the importance of the compactness concept, which is the key to thousands of mathematical arguments. The underlying ideas with which it engages grow ultimately into the subject of topology. Let us have a small taste of it.

To prove the Heine-Borel theorem, fix the open cover 𝒰, consisting of open intervals in the real numbers, such that every number x ∈ [0,1] is in some U ∈ 𝒰. Consider the set B consisting of all x ∈ [0,1], such that the interval [0,x] is covered by finitely many elements of 𝒰. Thus, 0 ∈ B, since [0,0] has just the point 0, which is covered by some open set in 𝒰. Let b be the least upper bound of B in [0,1]. If b = 1, then we can cover [0,1] with finitely many elements of 𝒰, and we're done. So assume that b < 1. Since 𝒰 covers the entire interval, there is some open interval U ∈ 𝒰 with b ∈ U. Since U contains a neighborhood of b, which is the supremum of B, there must be a point x ∈ B with x ∈ U. Therefore, we can cover [0,x] with finitely many open sets from 𝒰, and the set U itself covers the rest of the interval [x,b], and so we have covered the interval [0,b] with finitely many open intervals from 𝒰. But because the open interval must spill strictly beyond b, there must be elements of B strictly larger than b, which contradicts the assumption that b is the least upper bound of B. So we have proved the theorem.

Another application of the least-upper-bound principle is the fact that every nested descending sequence of closed intervals has a nonempty intersection. That is, if

then there is a real number z inside every interval z ∈ [a_n,b_n]. One can simply let z = sup_n a_n. More generally, using the Heine-Borel theorem, one can prove that every nested descending sequence of nonempty closed sets in a real number interval has a nonempty intersection. For if there were no such real number z, then the complements of those closed sets would form an open cover of the original interval, with no finite subcover, contrary to the Heine-Borel theorem.

Continuous induction

Earlier I made an analogy between the least-upper-bound principle in real analysis and the principle of mathematical induction in number theory. This analogy is quite strong in light of the principle of continuous induction, which is an induction-like formulation of the least-upper-bound principle that can be used to derive many fundamental results in real analysis.

Principle of continuous induction. If A is a set of nonnegative real numbers such that

• 0 ∈ A;

• If x ∈ A, then there is δ > 0 with [x,x + δ) ⊆ A;

• If x is a nonnegative real number and [0,x) ⊆ A, then x ∈ A.

Then A is the set of all nonnegative real numbers.

In plain language, the principle has the anchoring assumption that 0 is in the set A, and the inductive properties that whenever all the numbers up to a number x are in A, then x itself is in A; and whenever a number x is in A, then you can push it a little higher, and all the numbers between x and some x + δ are in A. The conclusion, just as in the principle of induction, is that any such set A must contain every number (in this case, every nonnegative real number).

Yuen Ren Chao (1919) describes a similar principle like this:

The theorem is a mathematical formulation of the familiar argument from “the thin end of the wedge,” or again, the argument from “the camel's nose.”

Hyp. 1. Let it be granted that the drinking of half a glass of beer be allowable.

Hyp. 2. If any quantity, x, of beer is allowable, there is no reason why x + δ is not allowable, so long as δ does not exceed an imperceptible amount δ.

Therefore any quantity is allowable.

The principle of continuous induction can be proved from the least-upper-bound principle, by considering the supremum r of the set of numbers a for which [0,a] ⊆ A. This supremum must be in A because [0,r) ⊆ A; but if r ∈ A, then it should be at least a little larger than it is, which is a contradiction. The converse implication also holds, as the reader will prove in question 2.16, and so we have three equivalent principles: continuous induction, least upper bounds, and Dedekind completeness.

Mathematicians are sometimes surprised by the principle of continuous induction — by the idea that there is a principle of induction on the real numbers using the real-number order — because there is a widely held view, or even an entrenched expectation, that induction is fundamentally discrete and sensible only with well-orders. Yet here we are with an induction principle on the real numbers based on the continuous order.

The principle of continuous induction is quite practical and can be used to establish much of the elementary theory of real analysis. Let us illustrate this by using the principle to give an alternative inductive proof of the Heine-Borel theorem. Suppose that 𝒰 is an open cover of [0,1]. We shall prove by continuous induction that every subinterval [0,x] with 0 ⩽ x ⩽ 1 is covered by finitely many elements of 𝒰. This is true of x = 0, since [0,0] has just one point. If [0,x] is covered by finitely many sets from 𝒰, then whichever open set contains x must also stick a bit beyond x, and so the same finite collection covers a nontrivial extension [0,x + δ]. And if every smaller interval [0,r] for r < x is finitely covered, then take an open set containing x, which must contain (r,x] for some r < x; by combining that one open set with a finite cover of [0,r], we achieve a finite cover of [0,x]. So by continuous induction, every [0,x] has a finite subcover from 𝒰. In particular, [0,1] itself is finitely covered, as desired.

Continue reading more about this topic in the book:

Lectures on the Philosophy of Mathematics, MIT Press 2021

Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.

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Continuity

In natural language, one distinguishes between a continuous process, which is one that proceeds in an unbroken manner, without interruption, and a process performed continually, which means without ending. For example, you might hope that your salary payments arrive continually in the coming decades, but it is not necessary that they do so continuously, since it will be fine to receive a separate payment each month.

Informal account of continuity

In mathematics, a continuous function is one whose graph is unbroken in a sense. What is this sense? Perhaps an informal continuity concept suffices at first. In my junior high school days, my teachers would say:

A function is continuous if you can draw it without lifting your pencil.

This statement conveys the idea that a jump discontinuity, as occurs in the middle of the red function, should disqualify a function from being continuous, because you would have to lift your pencil to jump across the gap. But surely it is inadequate to support precise mathematical argument; and it is inaccurate in the fine detail, if one considers that the lead of a pencil has a certain nonzero width, and furthermore, the material coming off the pencil consists of discrete atoms of carbon. So let us take it as a suggestive metaphor rather than as a mathematical definition.

In introductory calculus classes, one often hears a slightly better statement:

A function f is continuous at c if the closer and closer x gets to c, the closer and closer f(x) gets to f(c).

This is an improvement, by suggesting that one can obtain increasingly good approximations to the value of a continuous function at a point by applying the function to increasingly good approximations to the input; we view f(x) as an approximation of f(c) when x is an approximation of c.

But the definition is still much too vague. Worse, it is not quite right. Suppose you were to walk through Central Park in New York, proceeding uptown from Central Park South. As you walk north, you would be getting closer and closer (if only slightly) to the North Pole. But you would not be getting close to the North Pole, since you would remain thousands of miles away from it. The problem with the definition above is that it does not distinguish between the idea of getting closer and closer to a quantity and the idea of getting close to it. How close does it get? How close is close enough? The definition does not tell us.

To make the same point differently, consider the elevation function of a hiker as she descends a gently sloped plateau toward its edge, where a dangerous cliff abruptly drops. As she approaches the cliff's edge, she is getting closer and closer to the edge, and her elevation gets closer and closer to the elevation of the valley floor (since she is descending, even if only slightly), but the elevation function is not continuous, since there is an abrupt vertical drop at the cliff's edge, a jump discontinuity, if she were to proceed that far.

The definition of continuity

A more correct definition should therefore not speak of “closer and closer,” but should rather concern itself with exactly how close x is to c and how close f(x) is to f(c), and how these degrees of closeness are related. This is precisely what the epsilon-delta definition of continuity achieves.

Definition. A function f on the real numbers is continuous at the point c if for every positive ε > 0, there is δ > 0 such that whenever x is within δ of c, then f(x) is within ε of f(c). The function overall is said to be continuous if it is continuous at every point.

In the figure, the y values within ε of f(c) are precisely those within the horizontal green band, while the x values within δ of c are those within the vertical red band. The diagram therefore illustrates a successful choice of δ, a situation (and you will explain precisely why in the exercise questions) where every x within δ of c has f(x) within ε of f(c).

We may express the continuity of f at c succinctly in symbols as follows:

The quantifier symbol ∀ is to be read as “for all” and the symbol ∃ as “there exists.” So a function f is continuous at a point c, according to what this says, if for any desired degree of accuracy ε, there is a degree of closeness δ, such that any x that is that close to c will have f(x) within the desired accuracy of f(c). In short, you can ensure that f(x) is as close to f(c) as you want by insisting that x is sufficiently close to c.

The continuity game

Consider the continuity game. In this game, your role is to defend the continuity of the function f. The challenger presents you with a value c and an ε > 0, and you must reply with a δ > 0. The challenger can then pick any x within δ of c, and you win, provided that f(x) is indeed within ε of f(c). In the exercise questions, you will show that the function f is continuous if and only if you have a winning strategy in this game.

Many assertions in mathematics have such alternating ∀∃ quantifiers, and these can always be given the strategic reading for the game, in which the challenger plays instances of the universal ∀ quantifier and the defender answers with witnesses for ∃. Mathematically complex assertions often have many alternations of quantifiers, and these correspond to longer games. Perhaps because human evolution took place in a challenging environment of essentially game-theoretic human choices, with consequences for strategic failures, we seem to have an innate capacity for the strategic reasoning underlying these complex, alternating-quantifier mathematical assertions. I find it remarkable how we can leverage our human experience in this way for mathematical insight.

Estimation in analysis

Let us illustrate the epsilon-delta definition in application by proving that the sum of two continuous functions is continuous. Suppose that f and g are both continuous at a point c, and consider the function f + g, whose value at c is f(c) + g(c). To see that this function is continuous at c, we shall make what is known as an ε/2 argument. Consider any ε > 0. Thus also, ε/2 > 0. Since f is continuous, there is δ₁ > 0 such that any x within δ₁ of c has f(x) within ε/2 of f(c). Similarly, since g is continuous, there is δ₂ > 0 such that any x within δ₂ of c has g(x) within ε/2 of g(c). Let δ be the smaller of δ₁ and δ₂. If x is within δ of c, therefore, then it is both within δ₁ of c and within δ₂ of c. Consequently, f(x) is within ε/2 of f(c) and g(x) is within ε/2 of g(c). It follows that f(x) + g(x) is within ε of f(c) + g(c), since each term has error less than ε/2, and thus we have won this instance of the continuity game. So f + g is continuous, as desired.

This argument illustrates the method of “estimation,” so central to the subject of real analysis, by which one delimits the total error of a quantity by breaking it into pieces that are analyzed separately. One finds not only ε/2 arguments, but also ε/3 arguments, breaking the quantity into three pieces, and ε/2ⁿ arguments, splitting into infinitely many pieces, with the error in the nth piece at most ε/2ⁿ. The point is that because

one thereby bounds the total error by ε, as desired. Let me emphasize that this use of the word estimate does not mean that one is somehow guessing how much the difference can be, but rather one is proving absolute bounds on how large the error could possibly be.

The analyst's attitude can be expressed by the slogan:

In algebra, it is equal, equal, equal.

But in analysis, it is less-than-or-equal, less-than-or-equal, less-than-or-equal.

In algebra, one often proceeds in a sequence of equations, aiming to solve them exactly, while in analysis, one proceeds in a sequence of inequalities, mounting an error estimate by showing that the error is less than one thing, which is less than another, and so on, until ultimately, it is shown to be less than the target ε, as desired. The exact value of the error is irrelevant; the point, rather, is that it can be made as small as desired.

Limits

The epsilon-delta idea enables a general formalization of the limit concept. Namely, one defines that the limit of f(x) as x approaches c is the quantity L, written like this:

if for any ε > 0, there is δ > 0 such that any x within δ of c (but ignoring x = c) has f(x) within ε of L.

But why all the fuss? Do limits and continuity require such an overly precise and detailed treatment? Why can't we get by with a more natural, intuitive account? Indeed, mathematicians proceeded with an informal, intuitive account for a century and half after Newton and Leibniz. The epsilon-delta conception of limits and continuity was a long time coming, achieving its modern form with Weierstrass and with earlier use by Cauchy and Bolzano, after earlier informal notions involving infinitesimals, which are infinitely small quantities. Let us compare that usage with our modern method.

Instantaneous change

In calculus, we seek to understand the idea of an instantaneous rate of change. Drop a steel ball from a great tower; the ball begins to fall, with increasing rapidity as gravity pulls it downward, until it strikes the pavement — watch out! If the height is great, then the ball might reach terminal velocity, occurring when the force of gravity is balanced by the force of air friction. But until that time, the ball was accelerating, with its velocity constantly increasing. The situation is fundamentally different from the case of a train traveling along a track at a constant speed, a speed we can calculate by solving the equation:

For the steel ball, however, if we measure the total elapsed time of the fall and the total distance, the resulting rate will be merely an average velocity. The average rate over an interval, even a very small one, does not quite seem fully to capture the idea of an instantaneous rate of change.

Infinitesimals

Early practitioners of calculus solved this issue with infinitesimals. Consider the function f(x) = x². What is the instantaneous rate of change of f at a point x? To find out, we consider how things change on an infinitesimally small interval — the interval from x to x + δ for some infinitesimal quantity δ. The function accordingly changes from f(x) to f(x + δ), and so the average rate of change over this tiny interval is

Since δ is infinitesimal, this result 2x + δ is infinitely close to 2x, and so we conclude that the instantaneous change in the function is 2x. In other words, the derivative of x² is 2x.

Do you see what we did there? Like Newton and Leibniz, we introduced the infinitesimal quantity δ, and it appeared in the final result 2x + δ, but in that final step, just like them, we said that δ did not matter anymore and could be treated as zero. But we could not have treated it as zero initially, since then our rate calculation would have been 0/0, which makes no sense.

What exactly is an infinitesimal number? If an infinitesimal number is just a very tiny but nonzero number, then we would be wrong to cast it out of the calculation at the end, and also we would not be getting the instantaneous rate of change in f, but rather only the average rate of change over an interval, even if it was a very tiny interval. If, in contrast, an infinitesimal number is not just a very tiny number, but rather infinitely tiny, then this would be a totally new kind of mathematical quantity, and we would seem to need a much more thorough account of its mathematical properties and how the infinitesimals interact with the real numbers in calculation. In the previous calculation, for example, we were multiplying these infinitesimal numbers by real numbers, and in other contexts, we would be applying exponential and trigonometric functions to such expressions. To have a coherent theory, we would seem to need an account of why this is sensible.

Bishop Berkeley (1734) makes a withering criticism of the foundations of calculus.

And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

Berkeley's mocking point is that essentially similar-seeming reasoning can be used to establish nonsensical mathematical assertions, which we know are wrong. For example, if δ is vanishingly small, then 2δ and 3δ differ by a vanishingly small quantity. If we now treat that difference as zero, then 2δ = 3δ, from which we may conclude 2 = 3, which is absurd. Why should we consider the earlier treatment of infinitesimals as valid if we are not also willing to accept this conclusion? It seems not to be clear enough when we may legitimately treat an infinitesimal quantity as zero and when we may not, and the early foundations of calculus begin to seem problematic, even if practitioners were able to avoid erroneous conclusions in practice. The foundations of calculus become lawless.

Modern definition of the derivative

The epsilon-delta limit conception addresses these objections and establishes a new, sound foundation for calculus, paving the way for the mature theory of real analysis. The modern definition of the derivative of a function f is given by

provided that this limit exists, using the epsilon-delta notion of limit we mentioned earlier. Thus, one does not use just a single infinitesimal quantity δ, but rather one in effect uses many various increments h and takes a limit as h goes to zero. This precise manner of treating limits avoids all the paradoxical issues with infinitesimals, while retaining the essential intuition underlying them — that the continuous functions are those for which small changes in input cause only small changes in the output, and the derivative of a function at a point is obtained from the average rate of change of the function over increasingly tiny intervals surrounding that point.

An enlarged vocabulary of concepts

The enlarged mathematical vocabulary provided by the epsilon-delta approach to limits expands our capacity to express new, subtler mathematical concepts, greatly enriching the subject. Let us get a taste of these further refined possibilities.

Strengthening the continuity concept, for example, a function f on the real numbers is said to be uniformly continuous if for every ε > 0, there is δ > 0 such that whenever x and y are within δ, then f(x) and f(y) are within ε. But wait a minute — how does this differ from ordinary continuity? The difference is that ordinary continuity is a separate assertion made at each point c, with separate ε and δ for each number c. In particular, with ordinary continuity, the value δ chosen for continuity at c can depend not only on ε, but also on c. With uniform continuity, in contrast, the quantity δ may depend only on ε. The same δ must work uniformly with every x and y (the number y in effect plays the role of c here).

Consider the function f(x) = x², a simple parabola, on the domain of all real numbers. This function is continuous, to be sure, but it is not uniformly continuous on this domain, because it becomes as steep as one likes as one moves to large values of x. Namely, for any δ > 0, if one moves far enough away from the origin, then the parabola becomes sufficiently steep so that one may find numbers x and y very close together, differing by less than δ, while x² and y² have changed by a huge amount. For this reason, there can be no single δ that works for all x and y, even when one takes a very coarse value of ε. Meanwhile, using what is known as the compactness property of closed intervals in the real number line (expressed by the Heine-Borel theorem), one can prove that every continuous function f defined on a closed interval in the real numbers [a,b] is uniformly continuous on that interval.

The uniform continuity concept arises from a simple change in quantifier order in the continuity statement, which one can see by comparing:

A function f on the real numbers is continuous when

whereas f is uniformly continuous when

Let us explore a few other such variations — which concepts result this way? The reader is asked to provide the meaning of these three statements in the exercise questions and to identify in each case exactly which functions exhibit the property:

Also requiring x ≠ y in the last example makes for an interesting, subtle property.

Suppose we have a sequence of continuous functions f₀, f₁, f₂, ..., and they happen to converge pointwise to a limit function f_n(x) → f(x). Must the limit function also be continuous? Cauchy made a mistake about this, claiming that a convergent series of continuous functions is continuous. But this turns out to be incorrect.

For a counterexample, consider the functions x, x², x³, ... on the unit interval, as pictured here in blue. These functions are each continuous, individually, but as the exponent grows, they become increasingly flat on most of the interval, spiking to 1 at the right. The limit function, shown in red, accordingly has constant value 0, except at x = 1, where it has a jump discontinuity up to 1. So the convergent limit of continuous functions is not necessarily continuous. In Cauchy's defense, he had a convergent series ∑_n f_n(x) rather than a pointwise convergent limit lim_nf_n(x), which obscures the counterexamples, although one may translate between sequences and series via successive differences, making the two formulations equally wrong. Meanwhile, Imre Lakatos (1976) advances a more forgiving view of Cauchy's argument in its historical context.

One finds a correct version of the implication by strengthening pointwise convergence to uniform convergence f_n ⇉ f, which means that for every ε > 0, there is N such that every function f_n for n ⩾ N is contained within an ε tube about f, meaning that f_n(x) is within ε of f(x) for every x. The uniform limit of continuous functions is indeed continuous by an ε/3 argument: if x is close to c, then f_n(x) is eventually close to f(x) and to f_n(c), which is close to f(c). More generally, if a merely pointwise convergent sequence of functions forms an equicontinuous family, which means that at every point c and for every ε > 0, there is a δ > 0 that works for every f_n at c, then the limit function is continuous.

What I am arguing is that the epsilon-delta methods do not serve merely to repair a broken foundation, leaving the rest of the structure intact. We do not merely carry out the same old modes of reasoning on a shiny new (and more secure) foundation. Rather, the new methods introduce new modes of reasoning, opening doors to new concepts and subtle distinctions. With the new methods, we can make fine gradations in our previous understanding, now seen as coarse; we can distinguish between continuity and uniform continuity or among pointwise convergence, uniform convergence, and convergence for an equicontinuous family. This has been enormously clarifying, and our mathematical understanding of the subject is vastly improved.

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Given the real numbers, one proceeds to the complex numbers ℂ, motivated by the enticing, yet perhaps terrifying, possibility that the imaginary unit i = √-1 exists as an actual number. One wants to consider complex numbers of the form a + bi, where a and b are real. What is a complex number?

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We can easily construct a natural presentation of the complex field by means of the complex plane. Specifically, since complex numbers are to have the form a + bi for real numbers a and b, let us think of the number a + bi as represented by the pair (a,b), a point in the plane. We may define the usual coordinate-wise addition operation (a,b) + (c,d) = (a + c, b + d), but we use a certain strange multiplication operation, defined by (a,b) · (c,d) = (ac - bd, ad + bc). This definition exactly implements the distributive consequences of i² = -1 in the product

By identifying a + bi with the point (a,b), we have realized the complex numbers as points in the plane, now called the complex plane, and we understand the complex arithmetic simply as certain elementary operations defined on those points. So there is no terrifying mystery after all in the complex numbers. We can construct them from the real numbers.

Platonism for complex numbers

We formed the complex numbers by extending the real numbers with a solution of the equation z² + 1 = 0, using the solution z = i. It is a remarkable fact, known as the fundamental theorem of algebra, that by adding this one solution, the complex numbers thereby become algebraically closed: every nontrivial polynomial equation over the complex numbers has a full solution there. The complex numbers are the algebraic closure of the real numbers.

What is a complex number, actually? Imagine that at your death, you are astonished to meet God in Heaven, who informs you, “Yes, you were completely right about platonism for the real numbers—there they are!” He points across the way, and there you see them—the real numbers, each of them a perfect platonic ideal of its kind. You find the numbers π, e, √2, each where you expect them. “But,” God continues, “you were wrong about platonism for the complex numbers; you have to construct them from the real numbers as pairs (a,b), with the parentheses and comma and everything.”

The situation is absurd because we expect that our mathematical ontology should treat similar kinds of mathematical objects similarly; if the real numbers are real, then the complex numbers should be as well. Is this a slippery slope for platonism? Once one admits a real existence for one kind of mathematical object or structure, why not more? Soon, we shall find ourselves in plenitudinous platonism. But what of mathematical structures that might differ in their level of abstraction? Some philosophers propose that the natural numbers have a more definite existence than real numbers, and that while platonism is correct for the natural numbers, it is not for the real numbers. Is the allegory relevant for them? Perhaps not; perhaps the difference in abstraction makes natural numbers and real numbers fundamentally different in kind, unlike the real and complex numbers of the allegory.

Categoricity for the complex field

Like the real numbers, the complex field ℂ admits a categorical characterization. Namely, the complex field is uniquely characterized up to isomorphism as being the algebraic closure of a complete ordered subfield, the real numbers. Any two fields like that are isomorphic, since their real subfields will be isomorphic and this isomorphism will extend to the algebraic closure. The complex field is also characterized up to isomorphism as the unique algebraically closed field of characteristic 0 having size continuum. One can express the concept of having size continuum in second-order logic by asserting that there is a bijection with a subset that is a real continuum.

Thus, each of our familiar number systems—the natural numbers ℕ, the integer ring ℤ, the rational field ℚ, the real field ℝ and the complex field ℂ—admit categorical characterizations. Precisely because of these categorical accounts, we are able to pick out and refer to these structures simply by describing what is true in them, rather than by having to exhibit sample instances of the structures. We don't need to present a particular constructed copy of ℂ to refer to the complex field, because we can just say that we are referring to the algebraically closed field of characteristic 0 having size continuum. To my way of thinking, this ability to refer to structures without needing to exhibit particular instances is a core part of the deep connection between categoricity results in mathematics and the philosophy of structuralism.

A complex challenge for structuralism?

Although one conventionally describes i as “the square root of negative one,” nevertheless one might reply to this, “Which one?” in light of the fact that -i also is such a root:

Indeed, the complex numbers admit an automorphism, an isomorphism of themselves with themselves, induced by swapping i with -i—namely, complex conjugation:

The conjugation map preserves the field structure, since

and therefore the complex field is not a rigid mathematical structure. Since conjugation swaps i and -i, it follows that i can have no structural property in the complex numbers that -i does not also have. So there can be no principled, structuralist reason to pick one of them over the other. Is this a problem for structuralism? It does seem to be a problem for singular terms, since how do we know that the i appearing in my calculations this week is the same number as what will appear in your calculations next week? Perhaps my i is your -i, and we do not even realize it.

If one wants to understand mathematical objects as abstract positions within a structure, as in abstract structuralism, then one must grapple with the fact that in light of the conjugation automorphism, the numbers i and -i play exactly the same roles in this structure (see Shapiro, 2012). The numbers i and -i have the same isomorphism orbit with respect to the complex field, and so in this sense, although distinct, they each play exactly the same structural role in ℂ. This would seem to undermine the idea that mathematical objects are abstract positions in a structure, since we want to regard these as distinct complex numbers.

Furthermore, there is nothing special about the numbers i and -i in this argument. For example, the numbers √2 and -√2 also happen to play the same structural role in the complex field ℂ, because there is an automorphism of ℂ that swaps them (although one uses the axiom of choice to prove this). Contrast this with the real field ℝ, where √2 and -√2 are of course discernible, since one is positive and the other is negative, and the order is definable from the field operations in ℝ via

It follows that the real number field is not definable in the complex field by any assertion in the language of fields. In fact, there is an enormous diversity of automorphisms of the complex field; one may move the real cube root of 2, for example, to one of the nonreal cube roots of 2, such as

Therefore, the numbers

are indiscernible in the complex field—there is no property expressible in the language of fields that will distinguish them. Indeed, except for the rational numbers, every single complex number is part of a nontrivial orbit of automorphic copies, from which it cannot be distinguished in the field structure. So the same issue as with i and -i occurs with every irrational complex number. For this reason, it is problematic to try to identify complex numbers with the abstract positions or roles that the numbers play in the complex field.

Meanwhile, one recovers the uniqueness of the structural roles simply by augmenting the complex numbers with additional natural structure. Specifically, once we augment the complex field ℂ with the standard operators for the real and imaginary parts:

then the expanded structure ⟨ℂ,+,·,Re,Im⟩ becomes rigid, meaning that it has no nontrivial automorphisms. Thus, every complex number plays a unique structural role in this new structure, which is Leibnizian. This additional structure is implicit in the complex plane conception of the complex numbers, which is part of why the number i appears fine as a singular term—it refers to the point (0,1) in the complex plane—whereas -i refers to (0,-1). The complex plane is not merely a field, for it carries along its coordinate information by means of the real-part and imaginary-part operators, making it rigid. In the complex plane, every complex number plays a different role.

Structure as reduct of rigid structure

This situation, where a natural nonrigid structure is made rigid by natural additional structure, is extremely common in mathematics. Examples abound. The additive group of integers ⟨ℤ,+⟩ admits an automorphism by negation, but is made rigid with the multiplicative structure ⟨ℤ,+,·⟩ or the order structure ⟨ℤ,+,<⟩. The rational order ⟨ℚ, < ⟩ is a countable endless dense linear order and therefore highly nonrigid—every point looks the same, and indeed any two finite sets of the same size are order-automorphic—but becomes rigid with the field structure ⟨ℚ,+,·,<⟩. The complex field ⟨ℂ,+,·⟩ has

many automorphisms, but is made rigid by incorporating the coordinate structure. Every group G with at least three elements is nonrigid, but elements are distinguished when the group is given a particular presentation, such as by means of generators and relations or as permutations of a particular set.

The pattern is that a particular nonrigid structure is realized as a reduct substructure of another structure that is rigid, thereby resolving the problem of reference, since we may refer to the objects of the nonrigid structure by reference to their roles in the expanded structure. I claim that this pattern is inherent in mathematical practice. The reason is that precisely because of the reference problem, it is difficult for us ever actually to present or specify a nonrigid structure, except by presenting it as a reduct substructure of a structure in which the objects are individuated. How else are we coherently to specify the structure on those objects in the first place? We don't start with a naked copy of ℂ and then seek to impose an orientation on it that will enable us to resolve i from -i. Rather, we proceed oppositely: instances of mathematical structures are obtained from richer contexts where the objects were already individuated. We might build a copy of ℂ from ordered pairs of real numbers, for example, where we can discern (0,1) from (0,-1) and therefore i from -i in this particular copy of ℂ. Every particular copy of ℂ and indeed every particular mathematical structure of any kind arises similarly from a context in which the objects are individuated.

When using ZFC set theory as a foundation of mathematics, this philosophical observation becomes a mathematical theorem: every set is a reduct substructure of a rigid structure, a structure in which every individual plays a distinct structural role. The reason is that every set is a subset of a transitive set, and every transitive set is rigid with respect to the ∈ membership relation. Indeed, the set-theoretic universe ⟨V,∈⟩ as a whole is rigid—any two objects in the set-theoretic universe are therefore distinguishable as sets and play different set-theoretic roles (see the argument on page 286. Therefore, every mathematical structure that can be realized in set theory at all can be realized as a reduct substructure of a rigid structure. We can refer to distinct individuals in the original structure by the distinct structural roles they play in the larger context.

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Let us consider the real continuum. The classical discovery of irrational numbers reveals gaps in the rational number line: the place where √2 would be, if it were rational, is a hole in the rational line. Thus, the rational numbers are seen to be incomplete. One seeks to complete them, to fill these holes, forming the real number line ℝ.

Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.

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Dedekind cuts

Dedekind (1901, I.3) observed how every real number cuts the line in two and found in that idea a principle expressing the essence of continuity:

If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. —Dedekind, 1901

For Dedekind, the real numbers are what we now call Dedekind complete: every cut is filled. In the rational line, some cuts, determined by a rational number, are already filled; but other cuts correspond to holes in the rational line, not yet filled. For any such unfilled cut, Dedekind proposes that we may imagine or “create” an irrational number in thought precisely to fill it. In this way, we shall realize the real number line as the Dedekind-completion of the rational number line.

And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it continuous; this filling up would consist in a creation of new point-individuals and would have to be effected in accordance with the above principle. —Dedekind, 1901

Theft and honest toil

Russell explains how one may undertake this creation process explicitly, building the real numbers as a mathematical structure that fulfills Dedekind's completeness property. In a truly elegant construction, he forms the Dedekind-completion of the rational line from the set of all Dedekind cuts themselves, viewing each cut as constituting a single new point. A Dedekind cut in the rational line is a bounded nonempty initial segment of the rationals with no largest element. The no-largest-element requirement ensures that rational numbers are represented uniquely, since otherwise we could place the rational limit point on either side, forming two distinct cuts where only one is wanted.

Toiling under Russell's direction, we form the set of all Dedekind cuts, viewing each as a single new point; we define the natural order upon them (it is just the inclusion order ⊆ on the cuts); we prove easily that this new order is Dedekind complete (the union of any bounded set of cuts is itself a cut that is the least upper bound); we extend the field operations from the rational numbers to the set of cuts, defining what it means to add any two cuts or multiply them; and we prove that these operations and order make the set of cuts into an ordered field. Thus, we construct the real numbers as Dedekind cuts, forming a Dedekind-complete ordered field.

Although one can imagine the Dedekind cuts as arising from the real numbers, to do so is precisely the inverse of the intended logic. Rather, we seek to use the cuts to define what the real numbers are, or at least what they could be. According to this account, a real number is a Dedekind cut in the rational numbers. Indeed, Russell (1919, p. 71) makes a withering criticism of Dedekind's axiomatic approach, by which one postulates that the real numbers are Dedekind complete.

The method of “postulating” what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil. —Russell, 1919

Russell's “honest toil” was to construct the real numbers via Dedekind cuts as described here, proving that the resulting structure is Dedekind complete, rather than merely postulating that the real numbers are already Dedekind complete.

Cauchy real numbers

An alternative continuity concept is provided by Augustin-Louis Cauchy, who was inspired by the idea that every real number is the limit of the various rational sequences converging to it. A sequence of real numbers is a Cauchy sequence if the points in the sequence become eventually as close as desired to one another. The continuity of the real numbers is expressed by Cauchy completeness, the property that every Cauchy sequence converges to a limiting real number.

The rational line, of course, is not Cauchy complete, for there are Cauchy sequences converging to where √2 would be, but there is no rational number there as the limit of this sequence. And it is similar for the other irrational numbers. But one may form the Cauchy completion of the rational numbers by considering all possible Cauchy sequences on them. Two such sequences are equivalent if their members eventually become as close to each other as desired, and we may form the real numbers as the collection of equivalence classes of Cauchy sequences. This admits a natural ordered field structure; it is Archimedean, which means that the finite sums 1 + 1 + ··· + 1 are unbounded; and it is Cauchy complete. According to this account, a real number is an equivalence class of Cauchy sequences.

Real numbers as geometric continuum

The ancient Greek conception of the continuum, in contrast, persisting through the ages, was inherently geometric: a real quantity is a length, area, or volume. According to the classical number line conception of number, advanced by René Descartes and taught in primary schools everywhere, a real number is a point on the number line, specified by an origin and a unit length.

One problem with this conception is that if a real number x is a length, a product xy is an area and xyz is a volume, then how are we to conceptualize expressions such as x + xy + xyz, which mix quantities of different dimensions? Can we add a length to an area or a volume? Quadratic expressions ax² + bx + c become problematic. We all agree that 2 × 3 = 6, but if 2 × 3 is an area and 6 is a length, what does that mean? One can solve this, of course, by considering 6 = 1 × 6 also as an area, and similarly in higher dimensions.

Another problem is that one wants to express the idea that the geometric continuum itself is continuous. Dedekind does this by means of his cuts, asserting that every cut is filled.

Categoricity for the real numbers

David Hilbert identified the essential natural properties that we want to be true of the real numbers, which, it turns out, characterize the field of real numbers up to isomorphism. He specified that the real numbers are a maximal Archimedean ordered field—maximal in the sense that they cannot be extended to a larger Archimedean ordered field. This is a form of completeness precisely because the Dedekind completion of any Archimedean ordered field remains Archimedean. In modern terminology, the definition amounts to saying that the real numbers are a complete ordered field, using the Dedekind formulation of completeness, since the least-upper-bound property implies the Archimedean property, as I shall argue in the proof of Huntington’s theorem below. Indeed, one can prove that the real numbers construed as Dedekind cuts or as equivalence classes of Cauchy sequence are complete ordered fields and thereby fulfill Hilbert's axioms.

What is a real number? What is the number π, for example, as a mathematical object? Is it a certain Dedekind cut? Is it an equivalence class of Cauchy sequences? A geometric length? Something else? The structuralist answers these questions by pointing to the categoricity result, asserting that there is only one complete ordered field up to isomorphism. The real numbers are a complete ordered field, and all such fields are isomorphic.

Theorem. (Huntington, 1903) All complete ordered fields are isomorphic.

Proof sketch. I claim first that every complete ordered field R is Archimedean—there is no number in R that is larger than every finite sum 1 + 1 + ··· + 1. If there were such a number, then by completeness, there would have to be a least such upper bound b to these sums; but b - 1 would also be an upper bound, which is a contradiction. So every complete ordered field is Archimedean.

Suppose now that we have two complete ordered fields, ℝ₀ and ℝ₁. We form their respective prime subfields, that is, their copies of the rational numbers ℚ₀ and ℚ₁, by computing inside them all the finite quotients ±(1 + 1 + ··· + 1)/(1 + ··· + 1). This fractional representation itself provides an isomorphism of ℚ₀ with ℚ₁, indicated below with blue dots and arrows:

Next, by the Archimedean property, every number x in ℝ₀ determines a cut in ℚ₀, indicated in yellow, and since ℝ₁ is complete, there is a counterpart x̄ in ℝ₁ filling the corresponding cut in ℚ₁, indicated in violet. Thus, we have defined a map π:x↦x̄ from ℝ₀ to ℝ₁. This map is surjective, since every y in ℝ₁ determines a cut in ℚ₁, and by the completeness of ℝ₀, there is an x in ℝ₀ filling the corresponding cut. Finally, the map π is a field isomorphism since it is the continuous extension to ℝ₀ of the isomorphism of ℚ₀ with ℚ₁. □

This result characterizes the structure of the real numbers in the same way that Dedekind's arithmetic axioms characterize the structure of the natural numbers. We found the fundamental principles for the real continuum and proved that they determine that structure up to isomorphism. Thus, we have identified the real numbers ℝ as a mathematical structure.

According to structuralism, it is not necessary to pick out a particular complete ordered field, an official copy, since the only mathematically relevant property of the real numbers—the only property that should be used in a mathematical argument—is that they constitute a complete ordered field. Individual real numbers are comprehended by their roles within such a structure. As we noted earlier, √2 is the unique object in whichever complete ordered field you have selected, that happens to be positive and to square to the number 2 in that field, where 2 is the number 1 + 1 in that field, where 1 is the unique multiplicative identity in that field. This is the structural role played by √2. In any complete ordered field, every rational number is algebraically definable, and every real number is characterized by the cut that it makes in the rational numbers. It follows that the real field ℝ is a Leibnizian structure: any two real numbers are discernible in the language of fields.

Kevin Buzzard (2019) highlights the question of structuralism by inquiring: How do we know that a theorem proved using the Dedekind-cut real numbers is also true of Cauchy-completion real numbers? Why is it that a mathematical assertion involving the real numbers, even if only incidentally, when true for the Dedekind real numbers, must also be true when one uses the Cauchy real numbers? There would seem to be an enormous pile of mathematical material that would have to be proved isomorphism-invariant in order to make such sweeping general conclusions, and has this work actually been done?

As a community, mathematicians in current practice are highly structuralist, often insistently so. It would be considered very strange to prove a theorem involving the real numbers by insisting that one is using the Dedekind real numbers as opposed to the Cauchy real numbers, for example, unless one were specifically concerned with the additional structural features that those formulations of the real numbers involved. Because of this widespread practice, the vast bulk of mathematical development is indeed structuralist and follows the structuralist imperative with regard to the central mathematical structures, including the natural numbers, the integers, the real numbers, and so on. Therefore, the enormous pile of isomorphism-invariant material that Buzzard claims must be undertaken has in fact already been undertaken—this is the standard practice of normal mathematics—and this is why we may deduce that mathematical statements involving the real numbers do not depend on which particular copy of the real numbers we are using.

Categoricity for the real continuum

We characterized the real numbers above by the fact that they form a complete ordered field and all such fields are isomorphic. This categoricity argument, therefore, uses the algebraic properties of the real numbers—the fact that they form an ordered field—as an essential part of the characterization. It turns out, however, that we may also characterize the real number line purely by its order properties rather than its algebraic properties as an ordered field.

Specifically, let us consider the real number line ⟨ℝ, < ⟩ with only the order structure. Viewed as a topological space under the order topology, this is known as the real continuum. What can we say about it? Well, this is a linear order, of course, since any two real numbers are comparable; and it is endless, meaning that there is neither a largest nor a smallest real number; and it is densely ordered, meaning that between any two real numbers, there is another; and it is Dedekind complete, meaning that every cut in the real number line is filled. Thus, the real number line is an endless, complete, dense linear order. This is not yet enough to characterize the real number line, however, for there are other such orders, not isomorphic to the real number line, such as the endless long line, for those who are familiar with it.

One additional property, however, will enable a characterization to go through. It suffices to add that the real number line has a countable dense subset. That is, there is a subset ℚ ⊆ ℝ, the set of rational numbers, which is (1) dense in the real number line in the sense that every nontrivial interval (a,b) of real numbers contains elements of the subset; and (2) the subset ℚ is countable, as discussed in chapter 3. All these properties together now determine the real numbers order up to isomorphism.

Theorem. Any two complete endless dense linear orders with countable dense sets are order isomorphic.

The essence of the proof is Cantor's back-and-forth method, which shows that the two countable dense suborders are isomorphic, for indeed, Cantor shows that any two countable, endless, dense linear orders are isomorphic. One can then lift this isomorphism from the suborders to the whole order using the completeness of the orders, just as we did in the case of the complete ordered fields.

I mention this categoricity result in part because a fascinating foundational issue arises when one considers a small variation of it, weakening the countable-dense-set requirement to what is called the countable chain condition, which asserts that every family of nonoverlapping intervals is countable. The real number line has the countable chain condition, since if we have a family of nonoverlapping intervals in the real number line, then inside each one we may pick a rational number, and we will never pick the same rational number twice since they do not overlap; so the family must have been countable.

Question. Are all complete endless dense linear orders with the countable chain condition isomorphic?

In other words, do these properties characterize the real number line? The answer is subtle and fascinating. A positive answer is known as Suslin's hypothesis, while a counterexample order, a complete endless dense linear order with the countable chain condition, but which is not isomorphic to the real number line, is called a Suslin line. The extremely interesting situation is that this question cannot be settled using the standard axioms of set theory; Suslin's hypothesis is an independent statement, neither provable nor refutable from the axioms of set theory; it is consistent either way. In particular, the question of whether the real numbers are categorically characterized by the property of Suslin's hypothesis is itself independent, neither provable nor refutable from the axioms of set theory. We shall discuss the independence phenomenon at length in chapter 8.

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Lectures on the Philosophy of Mathematics, MIT Press 2021

Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach grounded in mathematics, arising organically from mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.

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Structuralism

Now that we have established Dedekind's categoricity result for arithmetic, let us discuss the philosophical position known as structuralism, in one form perhaps the most widely held philosophical position amongst mathematicians today. Contemporary structuralist ideas in mathematics tend to find their roots in Dedekind's categoricity result and the other classical categoricity results characterizing the central structures of mathematics, placing enormous importance on the role of isomorphism-invariance in mathematics. Much of the philosophical treatment of structuralism, meanwhile, grows instead out of Benacerraf's influential papers (1965, 1973).

The main idea of structuralism is that it just does not matter what numbers or other mathematical objects are, taken as individuals; what matters is the structures they inhabit, taken as a whole. Numbers each play their structural roles within a number system, and other mathematical objects play structural roles in their systems. The slogan of structuralism, according to Shapiro (1996, 1997), is that “mathematics is the science of structure.”

A defining structural role played by the number zero in any copy of the ring of integers ℤ is that it is the additive identity. It also happens to be the unique additive idempotent, the only number z for which z + z = z; it is the only additively self-inverse number z = -z; and it is the smaller of the two multiplicative idempotents. So in general, there can be many ways to characterize the role played by a mathematical object. In the rational numbers ℚ, the number 1/2 is the only number whose sum with itself is the multiplicative identity 1/2 + 1/2 = 1. In the real field ℝ, the defining structural role played by √2 is that it is positive and its square is equal to 2, which is 1 + 1, where 1 is the multiplicative identity.

Yet, one should not confuse structural roles with definability. Tarski's theorem on real-closed fields, after all, implies that the number π, being transcendental, is not definable in the real field ℝ by any property expressible in the language of ordered fields. Yet it still plays a unique structural role, determined, for example, by how it cuts the rational numbers into those below and those above; only it makes exactly that same cut.

Definability versus Leibnizian structure

Let me elaborate. An object a in a structure M is definable in that structure if it has a property φ(a) in M that it alone has — a property expressible in terms of the structural relations of M, which picks out this object a uniquely. This is relevant for structuralism, because the definition φ specifies explicitly the structural role played by the object in that structure. A structure is pointwise definable if every object in it is definable in this way.

In the directed graph pictured here, for example, node 2 is the unique node that is pointed at by a node that is not pointed at by any node (namely, 2 is pointed at by 1, which is not pointed at at all); and node 4 is the unique node that is pointed at by a node, yet does not itself point at any node. In fact, every node in this graph is characterized by a property expressible in terms of the pointing-at relation, so this graph is pointwise definable. A mathematical structure is Leibnizian, in contrast, if any two distinct objects in the structure can be distinguished by some property. In other words, a Leibnizian structure is one that fulfills Leibniz's principle on the identity of indiscernibles with respect to properties expressible in the language of that structure.

Every pointwise definable structure is Leibnizian, since the defining properties of two different objects will distinguish them. But the notions are distinct. For example, the real ordered field ⟨ℝ, + , · ,<,0,1⟩ is Leibnizian, since for any two distinct real numbers x < y, there is a rational number p/q between them, and x has the property that x < p/q, while y does not, and this property is expressible in the language of ordered fields. But this structure is not pointwise definable because there are only countably many possible definitions to use in this structure, but uncountably many real numbers, so they cannot all be definable.

Every Leibnizian structure must be rigid, meaning that it admits no nontrivial automorphism, because automorphisms are truth-preserving — any statement true of an individual in a structure will also be true of its image under any automorphism of the structure. If all individuals are discernible, therefore, then no individual can be moved to another. Because of this, we should look upon the Leibnizian property as a strong form of rigidity. These two concepts are not identical, however, because there can be rigid structures that are not Leibnizian. Every well-order structure, for example, is necessarily rigid, but when an order is sufficiently large — larger than the continuum is enough — then not every point can be characterized by its properties, simply because there aren't enough sets of formulas in the language to distinguish all the points, and so it will not be Leibnizian. Indeed, for any language ℒ, every sufficiently large ℒ-structure will fail to be Leibnizian for the same reason.

The rigid relation principle, introduced and investigated by Hamkins and Palumbo (2012), is the mathematical principle asserting that every set carries a rigid binary relation. This is a consequence of the well-order principle, because well-orders are rigid, but it turns out to be strictly weaker; it is an intermediate weak form of the axiom of choice, neither equivalent to the axiom of choice nor provable in ZF set theory without the axiom of choice.

Role of identity in the formal language

The nature of Leibnizian structures is often sensitive to the question of whether one has included the equality or identity relation x = y in the formal language. In contemporary approaches to model theory and first-order logic, it often goes without saying that equality is included as a logical relation in every language and interpreted in every model as actual equality. This is ultimately a convention, of course, and one can easily and sensibly undertake a version of model theory without treating equality in this special manner.

When one omits equality from the language, then every model is elementarily equivalent to a model that violates the Leibnizian principle on the identity of indiscernibles. Specifically, for any model M in a language without equality, consider a new model M* obtained by adding any number of duplicate elements for any or all of the elements of M, defining the atomic relations for the duplicates in the new structure M* in accordance with the original structure. For example, in the rational order ⟨ℚ, ⩽ ⟩, we might consider the order ⟨ℚ*, ⩽ ⟩ in which every rational number has two copies, each less-than-or-equal to the other and ordered with the other elements as one would expect, so that both copies of 0, for example, are less than any of the copies of positive elements, and so on. It follows inductively that any equality-free statement φ(a₀,...,a_n) true of individuals in M will also be true in M* of any of their duplicates φ(a₀*,...,a_n*). In particular, the structure M* will not be able to discern an individual from its copies, and so this structure will not be Leibnizian if indeed any nontrivial duplication occurred. Furthermore, the two models have exactly the same equality-free truth assertions; they are elementarily equivalent in that language. (Meanwhile, with equality we can distinguish the models, since ⩽ is anti-symmetric in ℚ, meaning that

but this is not true in ℚ*.)

The philosophical point to make about this is that one can never expect a theory to give rise to the Leibnizian principle of identity of indiscernibles, unless the language includes the equality relation explicitly. In particular, nothing you say about a nonempty structure can possibly ensure that it is Leibnizian, unless equality is explicitly mentioned, since the structure has all the same equality-free assertions as the corresponding structure in which every individual has been duplicated.

Isomorphism orbit

While definability and even discernibility are sufficient for capturing the structural roles played by an object, they are not necessary, and a fuller account will arise from the notion of an isomorphism orbit. Specifically, two mathematical structures A and B are isomorphic if they are copies of one another, or more precisely, if there is an isomorphism π:A → B between them, a one-to-one correspondence or bijective map between the respective domains of the structures that respects the salient structural relations and operations. For example, an order isomorphism of linear orders ⟨L₁, ⩽₁⟩ and ⟨L₂, ⩽₂⟩ is a bijection π:L₁ → L₂ between the domains of the orders that preserves the order structure from one to the other, meaning that x ⩽₁ y if and only if π(x) ⩽₂ π(y). An isomorphism of arithmetic structures ⟨Y,+,·⟩ and ⟨Z,⊕,⊗⟩ is a bijection τ:Y → Z, for which τ(a + b) = τ(a) ⊕ τ(b) and τ(a · b) = τ(a) ⊙ τ(b), translating the structure from Y to Z. Every mathematical structural conception is accompanied by a corresponding isomorphism concept.

The isomorphism concept is intricately linked with that of formal language, which is a way of making precise exactly which mathematical structure one is considering. Whether a given one-to-one correspondence is an isomorphism depends crucially, after all, on which structural features are deemed salient. Is one considering the rational numbers only as an order, or as an ordered field? A given bijection may preserve only part of the structure.

Structural roles are respected by isomorphism, and indeed, they are exactly what is respected by isomorphism. An object a in structure A plays the same structural role as object b in structure B exactly when there is an isomorphism of A with B carrying a to b.

So let us consider the isomorphism orbit of an object in a structure, the equivalence class of the object/structure pair (a,A) with respect to the same-structural-role-as relation.

This orbit tracks how a is copied to all its various isomorphic images in all the various structures isomorphic to A. And whether or not these objects are definable or discernible in their structures, it is precisely the objects appearing in the isomorphism orbit that play the same structural role in those structures that a plays in A.

Categoricity

A theory is categorical if all models of it are isomorphic. In such a case, the theory completely captures the structural essence of what it is trying to describe, characterizing that structure up to isomorphism. Dedekind, for example, had isolated fundamental principles of arithmetic and proved that they characterized the natural numbers up to isomorphism; any two models are isomorphic. In other words, he proved that his theory is categorical.

The analogous feat has long been performed for essentially all the familiar mathematical structures, and we have categorical characterizations not only of the natural numbers ℕ, but of the ring of integers ℤ, the field of rational numbers ℚ, the field of real numbers ℝ, the field of complex numbers ℂ, and many more (see sections 1.11 and 1.13). Daniel Isaacson (2011) emphasizes the role that categoricity plays in identifying particular mathematical structures. Namely, we become familiar with a structure; we find the essential features of that structure; and then we prove that those features axiomatically characterize the structure up to isomorphism. For Isaacson, this is what it means to identify a particular mathematical structure, such as the natural numbers, the integers, the real numbers, or indeed, even the set- theoretic universe.

Categoricity is central to structuralism because it shows that the essence of our familiar mathematical domains, including ℕ, ℤ, ℚ, ℝ, ℂ, and so on, are determined by structural features that we can identify and express. Indeed, how else could we ever pick out a definite mathematical structure, except by identifying a categorical theory that is true in it? Because of categoricity, we need not set up a standard canonical copy of the natural numbers, like the iron rod kept in Paris that defined the standard meter; rather, we can investigate independently whether any given structure exhibits the right structural features by investigating whether it fulfills the categorical characterization.

Invariably, for deep reasons, these categorical characterizations use second-order logic, meaning that their fundamental axioms involve quantification not only over the individuals of the domain, but also over arbitrary sets of individuals or relations on the domain. Dedekind's arithmetic, for example, asserts the induction axiom for arbitrary sets of natural numbers, and we shall see in section 1.11 that Dedekind's completeness axiom for the real numbers involves quantifying over arbitrary bounded sets of rational numbers.

The deep reasons are that a purely first-order theory, one whose axioms involve quantification only over the domain of individuals in a structure rather than over arbitrary sets of individuals, can never provide a categorical characterization of an infinite structure. This is a consequence of the Löwenheim-Skolem theorem, which shows that every first-order theory that is true in an infinite model is also true in models of diverse infinite cardinalities, which therefore cannot be isomorphic. Meanwhile, the Löwenheim-Skolem theorem does not apply to second-order theories, and it should be no surprise to find second-order axioms in the categorical theories characterizing our familiar mathematical structures.

Some philosophers object that we cannot identify or secure the definiteness of our fundamental mathematical structures by means of second-order categoricity characterizations. Rather, we only do so relative to a set-theoretic background, and these backgrounds are not absolute. The proposal is that we know what we mean by the structure of the natural numbers—it is a definite structure—because Dedekind arithmetic characterizes this structure uniquely up to isomorphism. The objection is that Dedekind arithmetic relies fundamentally on the concept of arbitrary collections of numbers, a concept that is itself less secure and definite than the natural-number concept with which we are concerned. If we had doubts about the definiteness of the natural numbers, how can we assuaged by an argument relying on the comparatively indefinite concept of “arbitrary collection”? Which collections are there? The categoricity argument takes place in a set-theoretic realm, whose own definite nature would need to be established in order to use it to establish definiteness for the natural numbers.

Structuralism in mathematical practice

I should like to contrast several forms of structuralism, distinguishing first a form of structuralism that is widespread amongst mathematicians—a form which I call structuralism in practice. Structuralism in practice involves an imperative about how to undertake mathematics, a view about which kinds of mathematical investigations will be fruitful. According to the structuralist-in-practice, mathematics is about mathematical structure, and mathematicians should state and prove only structuralist theorems in a structuralist manner. All of one's mathematical concepts should be invariant under isomorphisms.

The structuralist imperative. For mathematical insight, investigate mathematical structure, the relations among entities in a mathematical system, and consider mathematical concepts only as being invariant under isomorphism. Therefore, do not concern yourself with the substance of individual mathematical objects, for this is mathematically fruitless as structure arises with any kind of object.

According to the structuralist imperative, it would be mathematically misguided to state theorems about particular instantiations of mathematical structure; a theorem involving the real numbers, for example, should do so in a way that it becomes invariant under isomorphism; one should be able to replace the real numbers with any other complete ordered field, while preserving the truth of the theorem. The structuralist-in-practice dismisses questions about the “true nature” of mathematical objects—about what numbers “actually” are—as mathematically irrelevant.

It would accord with the structuralist imperative, for example, to prove a theorem about the countable random graph if one takes this term to refer to any countable graph with the finite pattern property, a feature that characterizes these graphs up to isomorphism. One might prove, for example, that the countable random graph is homogeneous or that it has diameter two or an infinite chromatic number; what this really means is that all such countable graphs with the finite pattern property have these features. Because the hypothesis is invariant under isomorphism, we do not care which particular copy of the countable random graph we are using, and this is the heart of structuralism.

It would be antistructuralist, in contrast, to state those theorems specifically only about the Rado graph if this is taken to refer to the specific graph relation on the natural numbers with an edge between n and m, where n < m, if the nth binary digit of m is 1; this graph happens to exhibit the finite pattern property, and therefore it is a specific instance of the countable random graph (once one has fixed a copy of the natural numbers).

Notice that it would be fine, logically, to prove something about the countable random graph by proving it specifically about the Rado graph, even using specific features of the Rado graph, provided that the theorem itself was invariant under graph isomorphisms, for then it would transfer from the Rado graph to all copies of the countable random graph. In this sense, it might seem that structuralism in practice requires only that one's theorems are structuralist, that is, that they are properly invariant under isomorphism.

Yet, the structuralist imperative recommends against that style of proof, against using nonstructural details of one's specific interpretation instances, even when they might seem convenient. The reason is that those details never lie at the core of the mathematical phenomenon—they are always a distraction—precisely because they cannot matter for the structuralist conclusion of the theorem. If you have a proof of an isomorphism-invariant theorem that uses incidental details of a specific instantiation, then the mathematical structuralist will say that you have a poor argument; you have missed the essential point; and your argument will not produce mathematical insight. In this sense, the structuralist imperative is a recommendation about mathematical efficacy. Namely, by undertaking structural arguments, we will stay closer on the trail of mathematical truth.

Meanwhile, it would be structuralist to prove theorems about the Rado graph if one was concerned with some of the extra structure inherent in that particular presentation of this graph. For example, the edge relation of the Rado graph is a computably decidable relation on the natural numbers, but not every copy of the countable random graph on the natural numbers is computably decidable. In this case, one is not really studying the countable random graph, with only its graph structure, but rather one is studying computable model theory, looking at the computational complexity of presentations of this graph. This is again structural, but with different additional structure beyond pure graph theory.

Consider a structuralist analogy with computer programming. A structuralist approach to programming treats its data objects only with respect to the structural features explicitly in the defining data types; this way of programming is often portable to other operating systems and implementations of the programming language. It would be antistructuralist for a program to use details of how a particular piece of data is represented on a particular system, to peek into the internal coding of data in the machine; this sneaky way of getting at the data might work fine at first, but it often causes portability issues because the methods can fail when one changes to a different machine, which might represent the data differently “under the hood,” so to speak.

The structuralist imperative tends to lead one away from the junk-theorem phenomenon, for junk is often particularly objected to, specifically because it is antistructuralist (but consider question 1.19 as a counterpoint). The structuralist-in-practice dismisses the Julius Caesar objection as misguided, for it does not matter what the cardinal numbers are, so long as they obey the Cantor-Hume principle, and so we do not care if any of them are Julius Caesar or not. Indeed, we can easily define an interpretation of number in which Julius Caesar is the number 17, or not, and everything in our theory will work fine either way. There is nothing mathematical at stake in it.

Some mathematicians have emphasized that in some of the category-theoretic foundations, such as in ETCS, the formal languages provided for these systems are necessarily invariant under isomorphisms. When working in those languages, therefore, one cannot help but follow the structuralist imperative.

Eliminative structuralism

What I am calling structuralism in practice is closely related to eliminative structuralism (also called post-rem structuralism), defended by Benacerraf (1965)—namely, the view that mathematical structure is simply that which is instantiated in particular structures. Eliminative structuralism includes the nominalist claim that there is no abstract object that is the mathematical structure itself, beyond representations in particular instantiations. There is no abstract thing that is “the number 3”—any object can play that role in a suitable structure—and so talk of numbers and other particular mathematical objects is merely instrumental. Shapiro (1996) says, “Accordingly, numerals are not genuine singular terms, but are disguised bound variables.” A reference to the number 3 really means: in the model of Dedekind arithmetic at hand, the successor of the successor of the successor of zero.

One difference between structuralism in practice and eliminative structuralism, however, is that the structuralist-in-practice drops the elimination claim, the nominalist ontological claim that abstract structural objects do not exist; rather, the structuralist-in-practice simply follows the structuralist imperative to pursue isomorphism-invariant mathematics, whether abstract structural objects exist or not. And since elimination is not part of the view, it would seem wrong to call it eliminative structuralism.

Another form of eliminative structuralism is the view of modal structuralism, also called in-re structuralism and defended by Geoffrey Hellman, according to which assertions about mathematical objects are to be understood modally as necessary claims about their possible instantiations. According to this view, mathematical structures are ontologically dependent on the systems that exemplify them. In extreme form, one might hope to reduce mathematical structure ultimately to concrete physical systems. And there is also a relative form of eliminative structuralism, which reduces structure to sets. Namely, according to set-theoretic reductionism, an extreme form of set-theoretic foundationalism, structure does not exist apart from its set-theoretic realizations, such as by means of the isomorphism orbit. This is different from merely using set theory as a foundation of mathematics, since one can propose set theory as a foundation simply by interpreting mathematical structure within set theory, without insisting that there is no structure outside of set theory.

Abstract structuralism

There is something a little puzzling about the structuralist mathematician, who follows the structuralist imperative, yet happily refers to the natural numbers and the number 17 and the real number π. If we only care about the natural numbers up to isomorphism, after all, then there is not any longer a unique mathematical object or structure corresponding to these terms, and so what is meant by “the” here? It would seem that the structuralist-in-practice should be referring instead to a natural numbers or a number 17. But mathematicians do not generally talk that way, even when they are structuralist. Structuralism seems to face a serious problem with singular reference.

To be sure, most mathematicians, when pressed about their singular references, articulate the structuralist-in-practice view. They say that it does not matter to them which copy of the natural numbers we use, and that by 17, they mean to refer to the object playing that role in whichever version we currently have. Thus, they have dutifully inserted Shapiro's disguised quantifiers.

But some philosophers aim for a more robust solution to the problem of singular reference. According to abstract structuralism, also known as ante-rem structuralism, defended by Stewart Shapiro (1997), Michael Resnik (1988), and others, the objects of mathematics, including numbers, functions, and sets, are inherently structural; they exist as purely structural abstract objects, positions within a structure, locations in a pattern of arrangement that might be realized in diverse instantiations. The quarterback is a position on an American football team, the role played by the person who calls the play, receives the hike and makes the passes. Each individual quarterback is a person rather than a position — a person who plays the role of quarterback on a particular team. Similarly, the natural number 3 is the third successor “location” in the natural number structure — the position that any particular copy of 3 fills in any particular instantiation of the natural numbers. On this view, mathematical structure exists independently of the particular systems instantiating that structure.

Abstract structuralism provides a direct account of the reference of singular terms in mathematics, explaining how the number 3 and the natural numbers can sensibly refer, even from a structuralist perspective, to the purely structural object or the abstract structural role played by these entities. One undertakes the Fregean process of abstraction from the same-structural-role relation, whose equivalence classes are precisely the isomorphism orbits. Every isomorphism orbit leads one by abstraction to a corresponding abstract structural role. Shapiro also argues, much like Maddy in the egg carton argument (mentioned on page 21), that abstract structuralism offers a solution to Benacerraf's epistemological problem: we gain access to finite instances of abstract structures and then proceed by abstraction to the structure itself.

The abstract structuralist is providing a structuralist account of mathematics by realizing mathematical objects as purely structural. Yet, the structuralist-in-practice will say that this form of abstract structuralism is not structuralist at all—it violates the structuralist imperative—precisely because it is concerned with what the mathematical objects are, even if the answer it provides is that they are purely structural. According to the structuralist-in-practice, such concerns are misguided; they never elucidate a mathematical phenomenon and are irrelevant for mathematical insight. The structuralist-in-practice will happily consider any classification invariant of the isomorphism orbit relation, such as the orbit itself (akin to Frege taking numbers as equinumerosity classes), without the need for a purely structural abstract object representing the structural role.

Yet, the abstract structuralist may reply, “Fine, we do not pursue abstract structuralism for mathematical insight, but rather as a philosophical investigation in mathematical ontology, aiming to understand what mathematical structure really is.” The abstract structuralist seeks to identify and elucidate the essential nature of mathematical objects, a philosophical effort rather than a mathematical one. The abstract structuralist seeks to give an account of what structure is—the thing that the mathematicians take to be so fundamental.

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Jump into the middle of one of my Oxford lectures where I discuss this topic: (go to 33:20s for structuralism)

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Interpreting arithmetic

Mathematicians generally seek to interpret their various theories within one another. How can we interpret arithmetic in other domains? Let us explore a few of the proposals.

Numbers as equinumerosity classes

We have just discussed (see the earlier post on logicism) how Frege, and later Russell, defined numbers as equinumerosity equivalence classes. According to this account, the number 2 is the class of all two-element sets, and the number 3 is the class of all three-element sets. These are proper classes rather than sets, which can be seen as set-theoretically problematic, but one can reduce them to sets via Scott's trick (due to Dana Scott) by representing each class with its set of minimal-rank instances in the set-theoretic hierarchy.

Numbers as sets

Meanwhile, there are several other interpretations of arithmetic within set theory. Ernst Zermelo represented the number zero with the empty set and then successively applied the singleton operation as a successor operation, like this:

One then proceeds to define in purely set-theoretic terms the ordinary arithmetic structure on these numbers, including addition and multiplication.

John von Neumann proposed a different interpretation, based upon an elegant recursive idea: every number is the set of smaller numbers. On this conception, the empty set ∅ is the smallest number, for it has no elements and therefore, according to the slogan, there are no numbers smaller than it. Next is { ∅ }, since the only element of this set (and hence the only number less than it) is ∅. Continuing in this way, one finds the natural numbers:

The successor of any number n is the set n ∪ { n }, because in addition to the numbers below n, one adds n itself below this new number. In the von Neumann number conception, the order relation n < m on numbers is the same as the element-of relation n ∈ m, since m is precisely the set of numbers that are smaller than m.

Notice that if we unwind the definition of 3 above, we see that

But what a mess! And it gets worse with 4 and 5, and so on. Perhaps the idea is unnatural or complicated? Well, that way of looking at the von Neumann numbers obscures the underlying idea that every number is the set of smaller numbers, and it is this idea that generalizes so easily to the transfinite, and which smoothly enables the recursive definitions central to arithmetic. Does the fact that

mean that the number 1 is complicated? I do not think so; simple things can also have complex descriptions. The von Neumann ordinals are generally seen by set theorists today as a fundamental set-theoretic phenomenon, absolutely definable and rigid, forming a backbone to the cumulative hierarchy, and providing an interpretation of finite and transfinite arithmetic with grace and utility.

One should not place the Zermelo and von Neumann number conceptions on an entirely equal footing, since the Zermelo interpretation is essentially never used in set theory today except in this kind of discussion comparing interpretations, whereas the von Neumann interpretation, because of its convenient and conceptual advantages, is the de facto standard, used routinely without remark by thousands of set theorists.

Numbers as primitives

Some mathematicians and philosophers prefer to treat numbers as undefined primitives rather than interpreting them within another mathematical structure. According to this view, the number 2 is just that—a primitive mathematical object—and there is nothing more to say about what it is or its more fundamental composition. There is a collection of natural numbers 0, 1, 2, and so on, existing as urelements or irreducible primitives, and we may proceed to construct other, more elaborate mathematical structures upon them.

For example, given the natural numbers, we may construct the integers as follows: We would like to think of every integer as the difference between two natural numbers, representing the number 2 as 7-5, for example, and - 3 as 6-9 or as 12-15. The positive numbers arise as differences between a larger number and a smaller number, and negative numbers conversely. Since there are many ways to represent the same difference, we define an equivalence relation, the same-difference relation, on pairs of natural numbers:

Notice that we were able to express the same-difference idea by writing a + d = c + b rather than a - b = c - d, using only addition instead of subtraction, which would have been a problem because the natural numbers are not closed under subtraction. Just as Frege defined numbers as equinumerosity classes, we now may define the integers simply to be the equivalence classes of pairs of natural numbers with respect to the same-difference relation. According to this view, the integer 2 simply is the set

and -3 is the set

We may proceed to define the operations of addition and multiplication on these new objects. For example, if [(a,b)] denotes the same-difference equivalence class of the pair (a,b), then we define

If we think of the equivalence class [(a, b)] as representing the integer difference a-b, these definitions are natural because we want to ensure in the integers that

There is a subtle issue with our method here; namely, we defined an operation on the equivalence classes [(a, b)] and [(c, d)], but in doing so, we referred to the particular representatives of those classes when writing a + c and b + d. For this to succeed in defining an operation on the equivalence class, we need to show that the choice of representatives does not matter. We need to show that our operations are well defined with respect to the same-difference relation—that equivalent inputs give rise to equivalent outputs. If we have equivalent inputs

and

then we need to show that the corresponding outputs are also equivalent, meaning that

and similarly for multiplication. And indeed our definition does have this well defined feature. Thus, we build a mathematical structure from the same-difference equivalence classes of pairs of natural numbers, and furthermore, we can prove that it exhibits all the properties that we expect in the ring of integers. In this way, we construct the integers from the natural numbers.

Similarly, we construct the rational numbers as equivalence classes of pairs of integers by the same-ratio relation. Namely, writing the pair (p, q) in the familiar fractional form p / q, and insisting that q ≠ 0, we define the same-ratio equivalence relation by

Note that we used only the multiplicative structure on the integers to do this. Next, we define addition and multiplication on these fractions:

and verify that these operations are well defined with respect to the same-ratio relation. This uses the fraction p / q as a numeral, a mere representative of the corresponding rational number, which is the same-ratio equivalence class

Thus, we construct the field of rational numbers from the integers, and therefore ultimately from the natural number primitives.

The program continues to the real numbers, which one may define by various means from the rational numbers, such as by the Dedekind cut construction or with equivalence classes of Cauchy sequences explained in the section on real numbers, and then the complex numbers as pairs of real numbers a + bi, as in the section on complex numbers. And so on. Ultimately, all our familiar number systems can be constructed from the natural number primitives, in a process aptly described by the saying attributed to Leopold Kronecker:

Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk

(God made the integers, all the rest is the work of man.)

Kronecker (1886), quoted in Weber (1893)

Numbers as morphisms

Many mathematicians have noted the power of category theory to unify mathematical constructions and ideas from disparate mathematical areas. A construction in group theory, for example, might fulfill and be determined up to isomorphism by a universal property for a certain commutative diagram of homomorphisms, and the construction might be identical in that respect to a corresponding construction in rings, or in partial orders. Again and again, category theory has revealed that mathematicians have been undertaking essentially similar constructions in different contexts.

Because of this unifying power, mathematicians have sought to use category theory as a foundation of mathematics. In the elementary theory of the category of sets (ETCS), a category-theory-based foundational system introduced by F. WilliamLawvere (Functorial semantics of algebraic theories, PhD thesis, 1963), one works in a topos, which is a certain kind of category having many features of the category of sets. The natural numbers in a topos are represented by what is called a natural-numbers object, an object ℕ in the category equipped with a morphism z : 1 → ℕ that serves to pick out the number zero, where 1 is a terminal object in the category, and another morphism s : ℕ → ℕ that satisfies a certain universal free-action property, which ensures that it acts like the successor function on the natural numbers. The motivating idea is that every natural number is generated from zero by the successor operation, obtained from the composition of successive applications of s to z:

In this conception, a natural number is simply a morphism n : 1 → ℕ. The natural numbers object is unique in any topos up to isomorphism, and any such object is able to interpret arithmetic concepts into category theory.

Numbers as games

We may even interpret numbers as games. In John Conway's account, games are the fundamental notion, and one defines numbers to be certain kinds of games. Ultimately, his theory gives an account of the natural numbers, the integers, the real numbers, and the ordinals, all unified into a single number system, the surreal numbers. In Conway's framework, the games have two players, Left and Right, who take turns making moves. One describes a game by specifying for each player the move options; on their turn, a player selects one of those options, which in effect constitutes a new game starting from that position, with turn-of-play passing to the other player. Thus, games are hereditarily gamelike: every game is a pair of sets of games,

where the games in G_L are the choices available for Left and those in G_R for Right. This idea can be taken as a foundational axiom for a recursive development of the entire theory. A player loses a game play when they have no legal move, which happens when their set of options is empty.

We may build up the universe of games from nothing, much like the cumulative hierarchy in set theory. At first, we have nothing. But then we may form the game known as zero,

which has no options for either Left or Right. This game is a loss for whichever player moves first. Having constructed this game, we may form the game known as star,

which has the zero game as the only option for either Left or Right. This game is a win for the first player, since whichever player's turn it is will choose 0, which is then a loss for the other player. We can also form the games known as one and two:

which are wins for Left, and the games negative one and negative two:

which are wins for Right. And consider the games known as one-half and three-quarters:

Can you see how to continue?

Conway proceeds to impose numberlike mathematical structure on the class of games: a game is positive, for example, if Left wins, regardless of who goes first, and negative if Right wins. The reader can verify that 1, 2, and 1/2 are positive, while -1 and -2 are negative, and 0 and ✱ are each neither positive nor negative. Conway also defines a certain hereditary orderlike relation on games, guided by the idea that a game G might be greater than the games in its left set and less than the games in its right set, like a Dedekind cut in the rationals. Specifically, G ≤ H if and only if it never happens that h ≤ G for some h in the right set of H, nor H ≤ g for some g in the left set of G; this definition is well founded since we have reduced the question of G ≤ H to lower-rank instances of the order with earlier-created games. A number is a game G where every element of its left set stands in the ≤ relation to every element of its right set. When games are constructed transfinitely, this conception leads to the surreal numbers. Conway defines sums of games G + H and products G × H and exponentials G^H and proves all the familiar arithmetic properties for his game conception of number. It is a beautiful and remarkable mathematical theory.

Junk theorems

Whenever one has provided an interpretation of one mathematical theory in another, such as interpreting arithmetic in set theory, there arises the junk-theorem phenomenon, unwanted facts that one can prove about the objects in the interpreted theory which arise because of their nature in the ambient theory rather than as part of the intended interpreted structure. One has junk theorems, junk properties, and even junk questions.

If one interprets arithmetic in set theory via the von Neumann ordinals, for example, then one can easily prove several strange facts:

The P notation here means “power” set, the set of all subsets. Many mathematicians object to these theorems on the grounds that we do not want an interpretation of arithmetic, they stress, in which the number 2 is an element of the number 3, or in which it turns out that the number 2 is the same mathematical object as the set of all subsets of the number 1. These are “junk” theorems, in the sense that they are true of those arithmetic objects, but only by virtue of the details of this particular interpretation, the von Neumann ordinal interpretation; they would not necessarily be true of other interpretations of arithmetic in set theory. In the case of interpreting arithmetic in set theory, many of the objections one hears from mathematicians seem concentrated in the idea that numbers would be interpreted as sets at all, of any kind; many mathematicians find it strange to ask, “What are the elements of the number 7?” In an attempt to avoid this junk-theorem phenomenon, some mathematicians advocate certain alternative non-set-theoretic foundations.

To my way of thinking, the issue has little to do with set theory, for the alternative foundations exhibit their own junk. In Conway's game account of numbers, for example, it is sensible to ask, “Who wins 17?” In the arithmetic of ETCS, one may speak of the domain and codomains of the numbers 5 and 7, or form the composition of 5 with the successor operation, or ask whether the domain of 5 is the same as the domain of the real number π. This counts as junk to mathematicians who want to say that 17 is not a game or that numbers do not have domains and cannot be composed with morphisms of any kind. The junk theorem phenomenon seems inescapable; it will occur whenever one interprets one mathematical theory in another.

Interpretation of theories

Let us consider a little more carefully the process of interpretation in mathematics. One interprets an object theory T in a background theory S, as when interpreting arithmetic in set theory or in the theory of games, by providing a meaning in the language of the background theory S for the fundamental notions of the object theory T. We interpret arithmetic in games, for example, by defining which games we view as numbers and explaining how to add and multiply them. The interpretation provides a translation

of assertions φ in the object theory T to corresponding assertions φ* in the background theory S. The theory T is successfully interpreted in S if S proves the translations φ* of every theorem φ proved by T. For example, Peano arithmetic (PA) can be interpreted in Zermelo-Fraenkel set theory (ZFC) via the von Neumann ordinals (and there are infinitely many other interpretations).

It would be a stronger requirement to insist on the biconditional—that S proves φ* if and only if T proves φ—for this would mean that the background theory S was no stronger than the object theory T concerning the subject that T was about. In this case, we say that theory S is conservative over T for assertions in the language of T; the interpreting theory knows nothing more about the object theory than T does.

But sometimes we interpret a comparatively weak theory inside a strong universal background theory, and in this case, we might expect the background theory S to prove additional theorems about the interpreted objects, even in the language of the object theory T. For example, under the usual interpretation, ZFC proves the interpretation of arithmetic assertions not provable in PA, such as the assertion Con(PA) expressing the consistency of PA (see later chapter on the incompleteness theorem). This is not a junk theorem, but rather reflects the fact that the background set theory ZFC simply has stronger arithmetic consequences than the arithmetic theory PA. Meanwhile, other nonstandard interpretations of PA in ZFC, such as those obtained by interpreting arithmetic via certain nonstandard models, are not able to prove the interpretation of Con(PA) in ZFC.

The objectionable aspect of junk theorems are not cases where the foundational theory S proves additional theorems beyond T in the language of T, but rather where it proves S-features of the interpreted T-objects. A junk theorem of arithmetic, for example, when interpreted by the von Neumann ordinals in set theory is a theorem about the set-theoretic features of these numbers, not a theorem about their arithmetic properties.

What numbers could not be

Paul Benacerraf (What numbers could not be, 1965) tells the story of Ernie and Johnny, who from a young age study mathematics and set theory from first principles, with Ernie using the von Neumann interpretation of arithmetic and Johnny using the Zermelo interpretation (one wonders why Benacerraf did not use the names the other way around, so that each would be a namesake). Since these interpretations are isomorphic as arithmetic structures—there is a way to translate the numbers and arithmetic operations from one system to the other—naturally Ernie and Johnny agree on all the arithmetic assertions of their number systems. Since the sets they used to interpret the numbers are not the same, however, they will disagree about certain set-theoretic aspects of their numbers, such as the question of whether 3 ∈ 17, illustrating the junk-theorem phenomenon.

Benacerraf emphasizes that whenever we interpret one structure in another foundational system such as set theory, then there will be numerous other interpretations, which disagree on their extensions, on which particular sets are the number 3, for example. Therefore, they cannot all be right, and indeed, at most one—possibly none—of the interpretations are correct, and all the others must involve nonnecessary features of numbers.

Normally, one who identifies 3 with some particular set does so for the purpose of presenting some theory and does not claim that he has discovered which object 3 really is. [III.B]

Pressing the point harder, Benacerraf argues that no single interpretation can be the necessarily correct interpretation of number.

To put the point differently—and this is the crux of the matter—that any recursive sequence whatever would do suggests that what is important is not the individuality of each element but the structure which they jointly exhibit... [W]hether a particular “object”—for example, {{{∅}}}—would do as a replacement for the number 3 would be pointless in the extreme, as indeed it is. “Objects” do not do the job of numbers singly; the whole system performs the job or nothing does. I therefore argue, extending the argument that led to the conclusion that numbers could not be sets, that numbers could not be objects at all; for there is no more reason to identify any individual number with any one particular object than with any other (not already known to be a number). [III.C]

The epistemological problem

In a second influential paper, Mathematical truth (1973), Benacerraf identifies an epistemological problem with the platonist approach of taking mathematical objects as being abstract. If mathematical objects exist abstractly or in an ideal platonic realm, totally separate in causal interaction from our own physical world, how do we interact with them? How can we know of them or have intuitions about them? How are we able even to refer to objects in that perfect platonic world?

W. D. Hart (Benacerraf's-dilemma, 1991) describes Benacerraf's argument as presenting a dilemma—a problem with two horns, one metaphysical and the other epistemological. Specifically, mathematics is a body of truths about its subject matter—numbers, functions, sets—which exist as abstract objects. And yet, precisely because they are abstract, we are causally disconnected from their realm. So how can we come to have mathematical knowledge?

It is at least obscure how a person could have any knowledge of a subject matter that is utterly inert, and thus with which he could have no causal commerce. And yet by the first horn of the dilemma, the numbers, functions and sets have to be there for the pure mathematics of numbers, functions and sets to be true. Since these objects are very abstract, they are utterly inert. So it is at least obscure how a person could have any knowledge of the subject matter needed for the truth of the pure mathematics of numbers, functions and sets. As promised, Benacerraf's dilemma is that what seems necessary for mathematical truth also seems to make mathematical knowledge impossible. (p. 98)

Penelope Maddy (Realism in mathematics, 1992), grabbing the bull firmly by the horns, addresses the epistemological objection by arguing that we can gain knowledge of abstract objects through experience with concrete instantiations of them. You open the egg carton from the refrigerator and see three eggs; thus you have perceived, she argues at length, a set of eggs. Through this kind of experience and human evolution, humans have developed an internal set detector, a certain neural configuration that recognizes these set objects, much as we perceive other ordinary objects, just as a frog has a certain neural configuration, a bug detector, that enables it to perceive its next meal. By direct perception, she argues, we gain knowledge of the nature of these abstract objects, the sets we have perceived. However, see the evolution of her views expressed in Maddy (Naturalism in Mathematics, 1997) and subsequent works.

Barbara Gail Montero (1999, 2020) deflates the significance of the problem of causal interaction with abstract objects by pointing out that we have long given up the idea that physical contact is required for causal interaction: consider the gravitational attraction of the Sun and the Earth, for example, or the electrical force pushing two electrons apart. But further, she argues, objections based on the difficulty of causal interactions with abstract objects lack force in light of our general failure to give an adequate account of causality of any kind. If we do not have a clear account of what it means to say that A causes B even in ordinary instances, then how convincing can an objection be based on the difficulty of causal interaction with abstract objects? And this does not even consider the difficulty of providing a sufficient account of what abstract objects are in the first instance, before causality enters the picture.

Sidestepping the issue of causal interaction, Hartry H. Field (Realism, mathematics, and modality, 1988) argues that the essence of the Benacerraf objection can be taken as the problem of explaining the reliability of our mathematical knowledge, in light of the observation that we would seem to have exactly the same mathematical beliefs, even if the mathematical facts had turned out to be different, and this undermines those beliefs. Justin Clarke-Doane (What is the Benacerraf problem? 2017), meanwhile, argues that it is difficult to pin down exactly what the Benacerraf problem is in a way that exhibits all the features of the problem that are attributed to it.

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Logicism

Pursuing the philosophical program known as logicism, Gottlob Frege, and later Bertrand Russell and others at the dawn of the twentieth century, aimed to reduce all mathematics, including the concept of number, to logic. Frege begins by analyzing what it means to say that there are a certain number of things of a certain kind. There are exactly two things with property P, for example, when there is a thing x with that property and there is another distinct thing y with that property, and furthermore, anything with the property is either x or y. In logical notation, therefore, “There are exactly two Ps” can be expressed like this:

The quantifier symbol ∃ is read as “There exists” and ∀ as “For all,” while ∧ and ∨ mean “and” and “or,” and → means “implies.” In this way, Frege has expressed the concept of the number 2 in purely logical terms. You can have two sheep or two apples or two hands, and the thing that is common between these situations is what the number 2 is.

Equinumerosity

Frege's approach to cardinal numbers via logic has the effect that classes placed in a one-to-one correspondence with each other will fulfill exactly the same Fregean number assertions, because the details of the truth assertion transfer through the correspondence from one class to the other. Frege's approach, therefore, is deeply connected with the equinumerosity relation, a concept aiming to express the idea that two sets or classes have the same cardinal size. Equinumerosity also lies at the core of Georg Cantor's analysis of cardinality, particularly the infinite cardinalities discussed in a later chapter.

Specifically, two classes of objects (or as Frege would say: two concepts) are equinumerous—they have the same cardinal size—when they can be placed into a one-to-one correspondence. Each object in the first class is associated with a unique object in the second class and conversely, like the shepherd counting his sheep off on his fingers.

So let us consider the equinumerosity relation on the collection of all sets. Amongst the sets pictured here, for example, equinumerosity is indicated by color: all the green sets are equinumerous, with two elements, and all the red sets are equinumerous, and the orange sets, and so on.

In this way, equinumerosity enables us systematically to compare any two sets and determine whether they have the same cardinal size.

The Cantor-Hume principle

At the center of Frege's treatment of cardinal numbers, therefore, is the following criterion for number identity:

Cantor-Hume principle. Two concepts have the same number if and only if those concepts can be placed in a one-to-one correspondence.

In other words, the number of objects with property P is the same as the number of objects with property Q exactly when there is a one-to-one correspondence between the class { x | P(x) } and the class { x | Q(x) }. Expressed in symbols, the principle asserts

where #P and #Q denote the number of objects with property P or Q, respectively, and the symbol ≃ denotes the equinumerosity relation.

The Cantor-Hume principle is also widely known simply as Hume's principle, in light of Hume's brief statement of it in A Treatise of Human Nature (1739, I.III.I), which Frege mentions (the relevant Hume quotation appears in a later chapter). Much earlier, Galileo mounted an extended discussion of equinumerosity in his Dialogues Concerning Two New Sciences (1638), considering it as a criterion of size identity, particularly in the confounding case of infinite collections, including the paradoxical observation that line segments of different lengths and circles of different radii are nevertheless equinumerous as collections of points. Meanwhile, to my way of thinking, the principle is chiefly to be associated with Cantor, who takes it as the core motivation underlying his foundational development of cardinality, perhaps the most successful and influential, and the first finally to be clear on the nature of countable and uncountable cardinalities. Cantor treats equinumerosity in his seminal set-theoretic article, Über eine Eigenschaft des Inbegriffs aller reelen algebraischen Zahlen (1874), and states a version of the Cantor-Hume principle in the opening sentence of Ein Beitrag zur Mannigfaltigkeitslehre (1878). In an 1877 letter to Richard Dedekind, he proved the equinumerosity of the unit interval with the square, the cube, and indeed the unit hypercube in any finite dimension, saying of the discovery, “I see it, but I don't believe it!” We shall return to this example in a later section.

The Cantor-Hume principle provides a criterion of number identity, a criterion for determining when two concepts have the same number. Yet it expresses on its face merely a necessary feature of the number concept, rather than identifying fully what numbers are. Namely, the principle tells us that numbers are classification invariants of the equinumerosity relation. A classification invariant of an equivalence relation is a labeling of the objects in the domain of the relation, such that equivalent objects get the same label and inequivalent objects get different labels. For example, if we affix labels to all the apples we have picked, with a different color for each day of picking, then the color of the label will be an invariant for the picked-on-the-same-day-as relation on these apples. But there are many other invariants; we could have written the date on the labels, encoded it in a bar code, or we could simply have placed each day's apples into a different bushel.

The Cantor-Hume principle tells us that numbers—whatever they are—are assigned to every class in such a way that equinumerous classes get the same number and nonequinumerous classes get different numbers. And this is precisely what it means for numbers to be a classification invariant of the equinumerosity relation. But ultimately, what are these “number” objects that get assigned to the sets? The Cantor-Hume principle does not say.

The Julius Caesar problem

Frege had sought in his logicist program an eliminative definition of number, for which numbers would be defined in terms of other specific concepts. Since the Cantor-Hume principle does not tell us what numbers are, he ultimately found it unsatisfactory to base a number concept solely upon it. Putting the issue boldly, he proclaimed

we can never—to take a crude example—decide by means of our definitions whether any concept has the number Julius Caesar belonging to it, or whether that same familiar conqueror of Gaul is a number or not.

Frege, Foundations of arithmetic (1884)

The objection is that although the Cantor-Hume principle provides a number identity criterion for identities of the form

comparing the numbers of two classes, it does not provide an identity criterion for identities of the form

which would tell us which objects x were numbers, including the case

On its face, this objection is strongly antistructuralist, a position we shall discuss at length in a later section, for it is concerned with what the numbers are, rather than with their structural features and roles. Yet, Greimann (2003) and others argue, nevertheless, that there is a nuanced structuralism in Frege.

Numbers as equinumerosity classes

In order to define a concept of number suitable for his program, Frege undertakes a process of abstraction from the equinumerosity relation. He realizes ultimately that the equinumerosity classes themselves can serve as numbers.

Specifically, Frege defines the cardinal “number” of a concept P to be the concept of being equinumerous with P. In other words, the cardinal number of a set is the class of all sets equinumerous with it, the equinumerosity class of the set:

For Frege, the number 2 is the class of all two-element collections; the number 3 is the class of all three-element collections, and so on. The number 0 is precisely {∅}, since only the empty set has no elements. Frege's definition fulfills the Cantor-Hume principle because sets have the same equinumerosity class if and only if they are equinumerous.

Frege proceeds to develop his arithmetic theory, defining zero as the cardinal of the empty class and defining that a number m is the successor cardinal of another n if m is the cardinal of a set obtained by adding one new element to a set of cardinal n; the natural numbers are the cardinals generated from 0 by successive applications of the successor operation, or more precisely, the numbers having every property of 0 that also is passed from every cardinal to its successor. If one regards equinumerosity and these other notions as logical, then this approach seems ultimately to reduce arithmetic to logic.

At first independently of Frege, but later responding to and building upon Frege's work, Bertrand Russell also sought to found mathematics in logic. Russell was impressed by Peano’s (1889) formalization of arithmetic, but he viewed Frege's foundation as still deeper in providing purely logical accounts of the arithmetic notions of “zero,” “successor,” and “natural number.” In their subsequent monumental work, Principia Mathematica, Russell and Alfred North Whitehead wove both themes into their account of arithmetic.

Neologicism

The logicist programs of Frege and Russell came widely to be viewed as ultimately unsuccessful in their attempts to reduce mathematics to logic. Frege's foundational system was proved inconsistent by the Russell paradox (discussed in a later chapter), and Russell's system is viewed as making nonlogical existence assertions with the axiom of infinity and the axiom of choice. Nevertheless, logicism is revived by Bob Hale and Crispin Wright and others in the neologicist program, reconstruing Frege's approach to logicism in a way that avoids the pitfall of inconsistency, aiming once again to reduce mathematics to logic.

Specifically, Frege’s Conception of Numbers as Objects (Wright 1983) aims to found Frege's arithmetic directly upon the Cantor-Hume principle, establishing the Peano axioms without need for the problematic foundational system. Equinumerosity, expressed in second-order logic, is taken as logical. (Some philosophers object at this point, regarding second-order principles as inherently set-theoretic and mathematical, but let us proceed with the neologicist perspective.) The neologicists address the Julius Caesar problem by seeking a purely logical account of the number-assignment problem, building upon suggestions of Frege regarding the process of abstraction. Frege explains, for example, how we refer to the direction of a line, having initially only a criterion of same-directionality, namely, parallelism. Similarly, we might speak of the order of a sequence of chess moves, or the pattern of an arrangement of colored tiles, or perhaps the value of a certain item. In each case, we refer to an abstract entity with a functional expression f, defined only implicitly by a criterion of the form

with an equivalence relation R. The direction of line ℓ₁ is the same as the direction of line ℓ₂, for example, exactly when ℓ₁ is parallel to ℓ₂; one tiling pattern is the same as another if the tiles can be associated to match color and adjacency from one arrangement to the other. The Fregean process of abstraction takes such an abstraction principle to define an abstract function f. The main point, of course, is that the Cantor-Hume principle itself is such an abstraction principle, where we refer to the number of elements in a class, and two classes have the same number if and only if the classes are equinumerous. By the process of abstraction, we therefore achieve numbers as abstract objects.

Meanwhile, there are logical problems with taking every abstraction principle as legitimate, as some of them lead to inconsistency. The general comprehension principle, for example, can arise via Fregean abstraction. Thus, the principle of abstraction is said to consort with “bad company”; known-to-be-false principles arise as instances of it. This is merely a gentle way of saying, of course, that abstraction is wrong as a principle of functional definition. We do not usually say that the fallacy of denying the antecedent consorts with “bad company” because it has false instances; rather, we say that it is fallacious.

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