Mathematical Skepticism: Infinity and Real Numbers, #2.

(Scientific American, 2023)

TABLE OF CONTENTS

1. Introduction

2. Chaitin and Borel

3. B.H. Slater

4. N.J. Wildberger

5. A Benacerraf-Inspired Critique of Real Numbers

6. Critique of Other Accounts of Real Numbers

7. Infinite Decimals and a Contradiction

The following essay will be published in four installments; this installment, the second, contains section 4.

But you can also download and read or share a .pdf of the complete text of this essay, including the REFERENCES, by scrolling down to the bottom of this post and clicking on the Download tab.

4. N.J. Wildberger

N.J. Wildberger has given a comprehensive critique of classical mathematics in a number of lectures posted on YouTube. As this breaks somewhat academic conventions, which favors the printed word, referencing is difficult in this format, and URLs for YouTube are annoying to type, we find. Thus, it is more convenient to list the titles of the main lectures in the main text, which an interested reader could type into the YouTube search engine. The lectures of relevance include: “A Socratic Look at Logical Weaknesses in Modern Mathematics”; “Mathematical Space and Basic Duality in Geometry”; “Mathematics without Real Numbers”; “The Mostly Absent Theory of Real Numbers”; “The Decline in Rigour in Modern Mathematics”; “Logical Weaknesses in Modern Mathematics”; “Deflating Modern mathematics; The Problem with ‘Functions’”; “Reconsidering ‘Function’ in Modern Mathematics”; “Modern Set Theory—Is It a Religious Belief System?” “The Continuum, Zeno’s Paradox, and the Price We Pay for Coordinates.”

Wildberger argues that due to the use of the concept of infinity in classical mathematics, “fundamental concepts of calculus, such as continuity, the derivative and integral, rest on the idea of “completing infinite processes” and/or performing an infinite number of tasks” (Wildberger, 2015a, 2015b; see also 2006, 2012, 2021). His position is that conventional real number theory has conceptually insecure foundations, as real numbers do not exist and neither do infinite sequences. The theory of real numbers was heavily influenced by Cantor’s theory of infinite sets, and he sees no justification for the postulation of infinite sets.

Real numbers have been viewed in three main ways: infinite decimals, Dedekind cuts, and Cauchy sequences of rational numbers, and Wildberger believes that all of these views have insuperable problems. Starting with the position that real numbers are infinite decimals, Wildberger’s main criticism, apart from seeing the concept of infinity as inherently problematic, is that there is a major problem of how to do operations such as multiplication, division, addition, and subtraction for two (or more) non-periodic infinite decimals. Usually, irrationals such as √2 and transcendental numbers such as e, can be manipulated as symbols and left as that, but in general it is not possible to decide if given statements involving operations on infinite decimals is correct, by a program, algorithm or function, and “infinite patterns” may not be characterized by a finite rule.

Wildberger does not mention that the formal construction of real numbers as infinite decimals was undertaken by mathematicians such as Karl Weierstrass and Otto Stolz, but not solved. There are more recent attempts (Fardin & Li, 2021; Richman, 1999; Klazar, 2009; Hua, 2012; Gower, n.d.), which seem to have solved the formal issue of definition, but not the practical issue raised by Wildberger, of how actually to do the calculations with non-trivial examples. Therefore, while it seems that there can be a formal account of real numbers as infinite decimals, it remains problematic from an applied position. However, in section 7 below we will argue that use of infinite decimals and processes can generate contradictions.

Wildberger rejects the Dedekind cut account of the real numbers. This approach defines a cut of the rationals as an ordered pair <A, B> of sets such that:

(1) A and B are not the null set.

(2) A È B= the set of rational.

(3) If x ∈ A and y ∈ B, then x < y.

And some add:

(4) A has no greatest element (for any a in A, there exists a’ in A such that a < a’).

A is the lower class and B the upper class, with every element of A preceding every element of B. Real numbers are sections of the rationals (Suppes, 1960: p. 160). A real number is then defined as a Dedekind cut, where the cut represents the boundary between A and B. For rational real numbers, B has a least element. For irrational numbers, B has no least element, as the cut corresponds to a “gap” in the rationals.

Wildberger’s primary objection to the Dedekind cut account of real numbers is that Dedekind cuts rely on infinite sets, which he considers ill-defined. A Dedekind cut partitions the rational numbers into two infinite sets, A and B, where A has no greatest element, and every element of A is less than every element of B. Wildberger argues that specifying such infinite sets requires an infinite process, which is not constructively feasible. He asserts that mathematics should be restricted to finite, computable objects, and infinite sets like those in Dedekind cuts lack a concrete, verifiable basis.

He further criticizes the practical utility of Dedekind cuts, particularly for transcendental numbers like π or e. While the cut can be readily given for relatively “straightforward” numbers, defining cuts for numbers without simple algebraic properties involves complex, uncomputable specifications. Wildberger asserts that this makes Dedekind cuts “undecidable” in practice, since one cannot algorithmically determine whether a given rational belongs to A or B for arbitrary cuts. This, he argues, undermines their claim to define real numbers rigorously.

Wildberger also questions the philosophical underpinning of Dedekind cuts, claiming that they simply assume the continuum they aim to construct. He echoes concerns raised by some mathematical philosophers that the “gap” between A and B presupposes a real number line, introducing circularity. In Wildberger’s view, the reliance on axiomatic set theory—for example,, ZFC—to justify infinite sets is a “sleight-of-hand,” avoiding the need explicitly to construct mathematical objects before using them.

However, Wildberger’s critique is open to counterarguments. Mainstream mathematicians would argue that Dedekind cuts are a logically consistent construction within set theory, requiring no prior assumption of the real number line. The total order of the rationals suffices to define cuts, and the resulting set of cuts satisfies the axioms of a complete ordered field, uniquely characterizing the reals up to isomorphism. Critics can argue that his rejection of infinite sets is a philosophical stance, not a proof of inconsistency. Wildberger’s critique of Dedekind cuts reflects his broader finitist philosophy, emphasizing computability and rejecting infinite objects. While his arguments highlight challenges in specifying infinite sets, they do not invalidate Dedekind cuts within the framework of classical mathematics, since classical mathematics would reject his finitism, which they would argue, begs the question against them.

Much the same critique can be made to the constructivist critique of Dedekind cuts. L.E.J. Brouwer and Errett Bishop challenged Dedekind cuts for their non-constructive nature. Constructivism requires that mathematical objects be explicitly computable or constructible in finitely many steps. A Dedekind cut, as a pair of infinite sets, often cannot be algorithmically specified, especially for transcendental numbers like e. For example, determining whether a rational q belongs to set A or B for the cut representing e requires an infinite amount of information about e’s decimal expansion, which is not computable in practice. Intuitionism, as developed by Brouwer, rejects the law of the excluded middle and emphasizes mathematics as a mental construction. Intuitionists have questioned Dedekind cuts for assuming a completed infinity of rational numbers. In intuitionist logic, a set is not a fixed entity but a process of construction, and the infinite partition of rationals into A and B presupposes a “finished” continuum. For instance, Brouwer argued that the real number line is not a pre-existing object but an evolving construct, and cuts imply a static view incompatible with this philosophy (Heyting, 1971). The problem here is that classical mathematicians will simply reject intuitionism and constructivism, seeing them as limiting mathematics, and claiming that there is no no-circular reason for accepting these positions.

The circularity objection is that the cut definition of the reals assumes a total order on the rational numbers and again the existence of infinite sets, which some finitists argue implicitly relies on a conception of the continuum that the cuts are meant to construct (Parsons, 1990). However, this is not a formal circularity in ZFC set theory, as the circularity concern is mitigated by noting that the rational numbers’ total order is sufficient to define cuts without assuming the reals. While it does raise questions about whether Dedekind cuts truly explain the reals or merely formalize an intuitive continuum, the classical mathematician will not be disturbed by this. We will return to a critique of the Dedekind account of real numbers after a discussion of real numbers as Cauchy sequences, also rejected by Wildberger.

The Cauchy sequence approach defines real numbers as equivalence classes of Cauchy sequences of rational numbers, a method developed by Georg Cantor.

Rational Numbers: Let Q denote the set of rational numbers (numbers that can be written as a/b, where a and b are integers and b ≠ 0).

Sequences: A sequence of rational numbers is an ordered list of numbers from Q, written as {a_n}, where a_n is the n-th term (n = 1, 2, 3, …).

Cauchy Sequence: A sequence {a_n} of rational numbers is called a Cauchy sequence if the terms get arbitrarily close to each other as n increases. Formally, for every positive rational number ε > 0, there exists a positive integer N such that for all m, n > N, the absolute difference |a_m – a_n| < ε.

Equivalence Relation: Two Cauchy sequences {a_n} and {b_n} of rational numbers are equivalent if the sequence of their differences converges to zero. That is, {a_n} ~ {b_n} if for every positive rational number ε > 0, there exists a positive integer N such that for all n > N, |a_n – b_n| < ε.

Equivalence Classes: An equivalence class of a Cauchy sequence {a_n} is the set of all Cauchy sequences {b_n} that are equivalent to {a_n} under the relation ~.

We denote this equivalence class by [{a_n}].

Real Numbers: The set of real numbers, denoted R, is the set of all equivalence classes of Cauchy sequences of rational numbers under the equivalence relation ~.

For example, the real number 2/3 is the equivalence class of all Cauchy sequences of rational numbers converging to 2/3. Examples of such sequences include:

{2/3, 2/3, 2/3, …}

{0.666, 0.6666, 0.66666, …}

{2/3, 4/6, 6/9, 8/12, …}

and so on, for “infinity.”

The equivalence class [{2/3, 2/3, 2/3, …}] contains all sequences {a_n} such that for every ε > 0 in Q, there exists N (a positive integer) such that for all n > N, |a_n – 2/3| < ε.

The set of all such equivalence classes, equipped with appropriate definitions of addition, multiplication, and order, forms a complete ordered field, identified as the real numbers. Completeness ensures that every Cauchy sequence of real numbers converges to a real number, resolving the “gaps” in the rational numbers.

Wildberger’s objections to Cauchy sequences mirror his critique of Dedekind cuts but focus on specific issues with sequences and their equivalence classes.

Infinite Processes and Non-Constructivity: Wildberger argues that Cauchy sequences rely on infinite processes, which are not constructively feasible in a finitist framework. Verifying that a sequence is Cauchy requires checking infinitely many terms to ensure all pairs beyond some N satisfy the epsilon condition. For example, a sequence approximating the square root of 2 may appear Cauchy, but confirming this property involves an infinite task, which Wildberger deems mathematically suspect. He insists that mathematical objects must be finitely specifiable and computable, and Cauchy sequences fail this test for most real numbers, especially transcendentals like π or e.

Equivalence Classes as Abstract and Ill-Defined: Wildberger criticizes the use of equivalence classes to define real numbers. Grouping all Cauchy sequences converging to the same limit (for example, different sequences approximating the square root of 2) into a single real number involves an infinite collection of infinite objects. He argues that this abstraction is philosophically problematic, as it assumes the existence of uncountably many such classes without explicitly constructing them. In his view, this reliance on set-theoretic machinery (for example, ZFC) obscures the lack of a concrete foundation for real numbers, making the construction “a house of cards.”

Practical Impracticality for Transcendental Numbers: Wildberger highlights the difficulty of defining Cauchy sequences for transcendental numbers like π or e. While algebraic numbers like can be approximated by algorithms, transcendental numbers often lack simple recursive definitions. Specifying a Cauchy sequence for π requires an infinite amount of information about its digits, which Wildberger argues is not practically achievable. He contends that this renders the Cauchy sequence construction “undecidable” in practice, undermining its claim to rigor.

Philosophical Objection to Completeness: Wildberger questions the need for a complete number system that Cauchy sequences aim to achieve by ensuring every sequence converges to a real number. He argues that rational numbers, supplemented by algebraic extensions (for example, treating the square root of 2 as a symbol satisfying x² = 2), suffice for most mathematical purposes. The insistence on completeness, he claims, introduces unnecessary complexity and philosophical baggage, driven by an unproven assumption that infinite limits are meaningful. Wildberger proposes alternatives like working strictly with rational numbers or finite algebraic fields, as in his “rational trigonometry,” which avoids real numbers altogether. He believes these approaches are more concrete and computationally well-grounded.

Mainstream Counterarguments: Mainstream mathematicians defend Cauchy sequences as a logically consistent construction within set theory, requiring no infinite computation to define the equivalence classes formally. The completeness property is essential for analysis, enabling results like the Intermediate Value Theorem, they contend. Critics may argue that Wildberger’s finitism sacrifices expressive power for philosophical purity, limiting mathematics’ ability to model continuous phenomena. The reliance on equivalence classes is seen as a standard abstraction, not a flaw, and computational approximations (for example, decimal expansions) align well with Cauchy sequences in practice. Infinite processes are seen as foundational, so once more the classical mathematician would reject Wildberger’s critique based upon finitism, as question begging.

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