Mathematical Skepticism: Infinity and Real Numbers, #1.

(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 first, contains sections 1-3.

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Mathematical Skepticism: Infinity and Real Numbers

In terms of the foundations of mathematics, my position (point of view) is based on the following two main principles (or opinions): (1) No matter which semantics is applied, infinite sets do not exist (both in practice and in theory). More precisely, any description about infinite sets is simply meaningless. (2) However, we still need to conduct mathematical research as we have used to. That is, in our work, we should still treat infinite sets as if they realistically exist. (Robinson, 1964)

While others are still trying to buttress the shaky edifice of set theory, the cracks that have opened up in it have strengthened my disbelief in the reality, categoricity or objectivity, not only of set theory but also of all other infinite mathematical structures including arithmetic. (Robinson, 1973: p. 514)

1. Introduction

In this paper, we review the positions of some mathematicians and logicians who are skeptical about the existence of real numbers as standardly defined, along the lines of the opening quotes by Abraham Robinson, primarily due to the problem of the actual infinite. We are not concerned here in detail with the foundations of set theory, particularly transfinite set theory, which was subjected to a controversial critique by Wittgenstein (Rodych, 2000), nor with the more rigorous critique than Wittgenstein’s by Rescher and Grim (2011) (see also (Feferman, 1989, 1998). Rescher and Grim noted that set theory still faces paradoxes insofar as the power set theorem, or in some systems, power set axiom, while provable, can be subjected to counter-examples which cannot be ruled out as items such as the set of all sets can. Thus, the set of all truths: for each subset of this set there will be a truth, so there will be at least as many truths (defined as not sets) as there are elements of the power set, contradicting the power set theorem. Likewise for the set of thoughts and facts, which should form coherent sets as much as the set of all apples. As Rescher and Grim reflect:

Set theory was born in paradox, was shaped by paradox, and continues to carry the threat of paradox into its current adolescence. Properly understood … the threat of contradiction is not merely formal and is not to be evaded by merely formal techniques. The fact that there can be no set of all non-self-membered sets might be shrugged aside as a minor logical surprise. Beyond Russell’s paradoxical set, however, there are serious philosophical difficulties of coherently conceptualising a set of all things, the realm of unrestricted quantification (or even the sense of restricted quantification), the totality of all events, all facts, all propositions, or all that is true. Sets are structurally incapable of handling any of these. (Rescher & Grim, 2011: p. 6)

We have discussed philosophical problems with sets in another paper (Smith & Stocks,2024) (see also, controversially, (Lin et al., 2008; Zhu et al., 2008a, 2008b, 2008c, 2008d, 2008e, 2008f, 2008g, 2008h, 2008i, 2008j, 2008k; Wildberger, 2015).

Finitism views the very presence of infinity in number theory as problematic, primarily on philosophical grounds, rather than being proven to be explicitly contradictory (Van Dantzig, 1956; Isles, 1992; Van Bendegem, 1994, 2000, 200, 2003 Ultra-finitists reject even potential infinities, such as the notion of a set of natural numbers, holding that numbers must be physically realizable (Yessenin-Volpin, 1970, 1981; Zeilberger, 2015)) and that mathematics can be “reduced to manipulations with a (finite!) set of symbols” (Zeilberger, 2015). In this spirit there have been some fertile attempts to reconstruct classical mathematics, including geometry, metric space theory, complex analysis, Hilbert spaces, analysis, and number theory without the use of the concept of infinity, within the framework of mathematical naturalism and nominalism (Parikh, 1971; Shepard, 1973; Mycielski, 1981; Lavine, 1995; Ye, 2011; Maudlin, 2014). As far as we are aware, a complete reworking of classical mathematics has not been fully completed, so technically it may be said that the spectre of infinity still haunts classical mathematics. That being said, let us now examine some theorists who see difficulties with the classical mathematical conception of real numbers, as suggested in the opening quotes of this paper from Robinson, seeing the notion of infinity raising problems for structures such as the real numbers for example.

2. Chaitin and Borel

Gregory Chaitin has argued that here are computational difficulties with real numbers (Chaitin, 2004). This, he claims, refutes the notion that real numbers underlie the physical furniture of the world. Physical reality may be digital and computational (Chaitin, 2005). The first objection Chaitin cites was made by E. Borel in 1927, Borel’s “amazing know-it-all number.” Thus, if real numbers are infinite sequences of digits, then all of humanity’s knowledge can be encoded in a single number (Borel, 1950). The number is written in binary with the n^th bit of the binary expansion giving an answer in a natural language to the question, yes=1, no =0. Borel thought that such a number was “unnatural” and not a real number at all because of its artificiality. So, he concluded, there is no reason to believe that such a number existed (Borel, 1952). That may well be so, but it shows only that the know-it-all number does not exist, not that there is therefore a problem with real number theory itself. Further, the claim of the number being merely “unnatural,” is not a telling objection to the existence of real numbers, if that is Chaitin’s aim. What is “natural”?

Chaitin observed that Borel mentioned another alleged paradox, that of “inaccessible numbers.” He asserted that real numbers only exist if they can be defined and expressed in a finite number of words, using a natural language such as French or English. This yields a countable infinity of tests for possible expression. But he then argues, there is a supposed denumerable infinity of reals having measure zero. Thus, there will be via diagonalization, reals that cannot be described, contrary to Borel’s existence assumption.

Chaitin develops this argument for uncomputable reals. The set of all possible computer programs is countable. So, the set of computable reals is countable, measure zero. By diagonalization, an uncomputable real can be constructed. The key issue here, Chaitin says, is this: if such real numbers are unknowable, why believe in them? Chaitin reinforces this claim with a version of Richard’s paradox, by diagonalization over all nameable reals to produce a nameable, but also unnameable real. The set of reals is uncountable, the set of all possible texts in a natural language such as English is countable. Hence the set of all possible mathematical questions being formulated in a natural language is countable too. So, there are by diagonalization real numbers that cannot be defined and are unnameable. However, this is itself a definition or name of the number, hence a version of Richard’s paradox.

In response to this, one counter-argument would be to deny that the set of all possible texts in a natural language such as French is countable, or that there is a countable infinity of texts for possible expression in a natural language. The real numbers occur in a natural language, and we are writing about them now. Thus, nothing prevents natural language sentences being given a real number index, say adding “index expression R,” to any sentence, in order to produce a 1-1 correspondence between the reals and these indexed natural numbers, hence defeating he Borel-Chaitin argument which requires natural language sentences to be countable, while the reals are uncountable.

Further, from the perspective of classical mathematics which accepts actual infinities, it could be maintained that the Borel-Chaitin argument is question begging as unknowable reals are nothing more than a product of the infinity of the reals, with the finite nature of mathematicians. For classical mathematics, this is all part of the course … of course.

3. B.H. Slater

Western Australian philosopher and logician, B. H. Slater has rejected the idea that numbers are sets, seeing this position “based on a series of grammatical confusions” (Slater, 2006: p. 59). Thus, the empty set is not the number zero, but rather the number of elements in the empty set is zero, not the set itself. To characterize the empty set requires prior recognition that the set is empty, having no members, which means that the number of elements in the set is zero. So, defining zero in terms of the empty set will be circular, and this is the foundation of the natural numbers according to a number of positions in the philosophy of mathematics. Indeed, Slater believes that set theory does not give a correct account of the use of collective terms in general, terms such as “flock” and “groups” (Slater, 1998: pp.144-156).

Slater rejects the idea that there is a determinate, let alone infinite number of natural numbers. He believes that two alleged “infinite sets,” even if put in a supposed 1-1 correspondence, may not have the same cardinality as they may have no determinate number at all (Slater, 2002: p.34). There is no number of the natural numbers, the reals, or of the continuum, (Slater, 2002: pp. 35-39), and no irrational numbers:

[I]f we define them not in terms of impossible Platonic limits but merely convergent sequences of rational numbers, then we are identifying “irrational” numbers with certain functions, since sequences are functions from the natural numbers. But the description “number” is then strictly a misnomer since a function is not a number, even if each of its values is one. (Slater, 2002: p.38)

Slater has also made an attack upon the Weierstrass classic definition of the derivative in the differential calculus, because it presupposes a non-finitist definition of the real numbers involving infinity (Slater, 2002: pp. 171-177). While these remarks are provocative, we believe that the same points have been argued for in more depth by fellow Australian mathematician N.J. Wildberger, whose work will now be considered.

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