Mathematics, Metaphysics, and Mystery, #2.

(Montessori Schools, 2018)

TABLE OF CONTENTS

1. Introduction: From Mind to Mathematics

2. The Nature of Mathematical Entities: Nothing Works

3. Set Theory: Should One Believe?

4. Metaphysics and Ontology

5. Mathematical Fictionalism

6. Mathematical Realism

7. Hanna’s Neo-Intuitionism as a Way Out of the Impasse?

8. Conclusion: From Mathematics to Mind

This essay will be published in four installments, two sections per installment; this is the second, and contains sections 3 and 4.

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

3. Set Theory: Should One Believe?

The core idea of logicism is that all mathematics can be defined in strictly logical terms, and more specifically, in set theoretical terms (Tiles, 1989; Maddy, 1990). Thus, regarding the natural numbers, for example, 0 is alleged to be defined as the set with zero elements, which sounds circular. But “zero elements” are taken to be the null set, so 0 = { } (Halmos, 1974: p. 44). From this, each natural number is equal to the set of its predecessors, that the successor of 7 is 8, but 7 is also a subset of 8, which may be “disturbing” (i.e. counter-intuitive) (Halmos, 1974: p. 45). Usually this is ignored. However, B.H. Slater has argued that numbers are not definable in terms of sets, an idea which is “based on a series of grammatical confusions” (Slater, 2006: p. 59). The general argument is not simply summarized, but his main point is that from grammatical considerations, the empty set is not the number zero; rather the number of elements in the empty set is zero, not that set itself. One argument that can be added here is that to make sense of the empty set, assuming the idea is coherent to start with, one needs to be able to characterize the set. It seems that there can be no non-circular characterization, since defining the set will require the number of zero elements at some point. Hence the set theoretical definition of zero, as the foundation of the natural numbers, will be circular.

And, what is the empty set anyway? What is a set, for that matter? Max Black, in his influential article, “The Elusiveness of Sets” (Black, 1971), notes that for most mathematicians, the term “set” is a primitive concept, implicitly defined by the set theoretical axioms of the system, such Zermelo-Fraenkel (ZF) set theory: whatever satisfies the axioms is a set. That is somewhat disappointing metaphysically, as it is possible that a wide range of objects with divergent ontological properties could be sets, ranging from physical things like animals (Halmos, 1974: p. 1), to abstract objects, even metaphysical, transcendental and theological entities (Black, 1971: p. 616). Of course, simply saying that a set is a collection, class, or aggregate will not do, as these accounts will be circular. Black examines a number of attempts to explicate the concept of a set, and finds them all circular, which is not surprising given how fundamental the concept of a set is, and explications must stop somewhere under pain of infinite regress.

Black favors an alternative account of sets as a form of plural talk, a

stand-in for plural referring expressions …the word ‘set,’ in its most basic use [is] ..and indefinite surrogate for lists and plural descriptions. (Black, 1971: p. 631)

Black believed that this plural conception made problematic empty and single-membered pluralities. As he wrote:

Of course, any transition from colloquial set talk to the idealized and sophisticated notion of making sense of a “null set” and of a “unit set” (regarded as distinct from its sole member) will cause trouble. From the standpoint of ordinary usage, such sets can hardly be regarded as anything else than convenient fictions (like the zero exponent in X⁰) useful for rounding off and simplifying a mathematical set theory. But they represent a significant extension of ordinary use. (Black, 1971: p. 633 )

Alex Oliver and Timothy Smiley (Oliver & Smiley, 2006) have also expressed skepticism about the notions of singletons and the empty set. They say:

Our own position is one of skeptical caution. The existence of an empty set should only be accepted if there are strong reasons for doing so. We have looked at the arguments in the literature, and found a dispiriting contrast between the technical virtuosity of set theorists when operating inside their fort, and the general poverty of the arguments offered to persuade others to enter it. (Oliver & Smiley, 2006: p. 127)

They show that many of the received attempts to show that there is an empty set are question begging, assuming at some point in the argument that there is an empty set (Oliver & Smiley, 2006: pp. 127-128).

Most of the philosophical perplexities with set theory lie with the use of not just the potential infinite, but the actual infinite, and the existence of infinite sets. This is one of the main objections to set theory by the Australian mathematician N.J. Wildberger (Wildberger, 2015). Wildberger is critical of the very notion of the infinite, accepting a strict finitism. While he does not discuss questions about the mathematical acceptability of the large cardinal hierarchy, such as inaccessible and hyper-inaccessible cardinals (Kanamori,2003), because he is skeptical about the existence of the ordinary infinities, and the continuum problem (which he views as a reductio of the idea of infinite sets), he would also object to the “higher infinities.”

Wildberger is skeptical of the coherence of most of the axioms of ZF set theory—they are “awash with difficulties” (Wildberger, 2015: p. 6)—particularly the axiom of infinity, that there exists an infinite set (Wildberger, 2015: p. 6). The axiom is more precisely stated to be: there exists a set containing 0 and the successor of each of its elements (Halmos, 1974: p. 44):

This axiom was controversial when it was introduced (Ramsey, 1926), but is now standardly accepted by most of those who accept that the concept of an infinite set is coherent. Strict finitists like Wildberger reject the axiom and may note with Zermelo himself that elementary number theory can be done without the axiom of infinity (Suppes, 1960: p. 139). As Wildberger says,

Do you really think you need to have all the natural numbers together in a set to define the function f (n) = n² + 1? Of course not—the rule itself, together with the specifications of the kinds of objects it inputs and outputs is enough. (Wildberger, 2015: p. 10)

Indeed, Wildberger is skeptical about an axiomatic approach to mathematics in general: “Axiomatic systems strongly misrepresents the practical reality of the subject” (Wildberger, 2015: p. 7). Wildberger writes:

The difficulty with the current reliance on “Axiom” arises from a grammatical confusion, along with the perceived need to have some (any) way to continue certain ambiguous practices that analysts historically have liked to make. People use the term “Axiom” when they really mean definition. Thus, the axioms of group theory are in fact just definitions. We say exactly what we mean by a group, that’s all. There are no assumptions anywhere. At no point do we or should we say, “Now we have defined an abstract group, let’s assume they exist.” Either we can demonstrate they exist by constructing some, or the theory becomes vacuous. Similarly, there is no need for “Axioms of Field Theory,” or “Axioms of Set theory,” or “Axioms” for any branch of mathematics—or for mathematics itself! (Wildberger, 2015: p. 8)

Wildberger’s anti-foundationalism in mathematics is similar in conclusion to Wittgenstein’s anti-foundationalism and finitism (Wittgenstein, 1967; Rodych, 2000), with both thinkers rejecting the actual infinite; there are no infinite sets such as the set of all natural numbers or the set of real numbers (Rodych, 2000: p. 286), and both are critical of transfinite set theory in particular. Thus, Wildberger writes regarding the Continuum Hypothesis (i.e., there is no infinite set whose cardinality is strictly between that of the integers and real numbers),

If you have an elaborate theory of “hierarchies upon hierarchies of infinite sets,” in which you cannot even in principle decide whether there is anything between the first and second “infinity” on your list, then it’s time to admit that you are no longer doing mathematics. (Wildberger, 2012: p. 9; see also Cohen, 1963)

Wildberger on this point is joined by Solomon Feferman, who does not believe that the Continuum Hypothesis is a definite mathematical problem since

the concept of an arbitrary set essential to its formulation is vague or undetermined and there is no way to sharpen it without violating what it is supposed to be about. (Feferman, 2011-2012: p 1)

Indeed, Robert Hanna has independently reached the same conclusion, but not with a negative take on set theory and the Continuum Hypothesis, rather that mathematics and logic cannot avoid philosophical commitments, and that metaphysical orientations may impact upon what is acceptable as being mathematics. He argues from a neo-Kantian position the following claim regarding the Continuum Hypothesis:

According to [my view], the real number structure is logico-mathematically a priori constructible from the set of all consciously experienceable points and stretches in spacetime, together with the set of all possible degrees of any consciously experienceable sensory quality, for each consciously experienceable point or stretch in spacetime. What I mean is that it is an a priori fact about the nature of human experience that any set of points or stretches of experienceable spacetime can instantiate any [possible] degree of some or another sense-experienceable quality. Building on that a priori fact, [my] proposal is that for each distinct point or stretch in sense experienceable spacetime, of which there are a denumerably infinite number, we can also find a denumerably infinite number of different [possible] degrees of some or another sense-experienceable quality. Then we can think of the latter cardinal number as an exponent of the former cardinal number in an operation that yields the former’s power set—the set of all its subsets. The cardinality of the result of that power set operation is the same as the first transfinite number, aleph1, which in turn has the same cardinality as the real number … Putting the same point in specifically Kantian terminology taken from the first Critique, [I’m proposing] that the basic structure of the continuum is the non-empirical extensive quantity structure as described in The Axioms of Intuition (CPR A162–166/B201–207) insofar as it is also exponentiated, according to the power set operation, by the non-empirical intensive quantity structure as described in The Anticipations of Perception (CPR A165–176/B207– 218). In this sense, the basic structure of the continuum is the Kantian synthesis of the extensive quantity structure and the intensive quantity structure. (Hanna, 2015: p. 391, bracketed material added)

This leads us to the next topic of investigation for controversy and “mystery,” the metaphysics and ontology of mathematical entities.

4.Metaphysics and Ontology

The standard debate in the metaphysics and ontology of mathematics is between realism and anti-realism. Mathematical realism “is the view that our mathematical theories are true descriptions of some real part of the world” (Balaguer, 2009: p. 36), however “world” is defined. Mathematical anti-realism is the position that mathematical realism is false (Balaguer, 2009). Platonism is the position that mathematical objects, relations, and structures exist, but not in space-time (Gödel, 1947; Resnik, 1997; Shapiro, 1997), and that “our mathematical theories are descriptions of an abstract mathematical realm (Balaguer, 2009: p. 40). Anti-Platonist realism, holds that mathematical theories are true descriptions of spatio-temporal objects, which could be a true description of mental objects (mathematical psychologism), or physical objects (mathematical physicalism) (Balaguer, 2009: p. 37).

The most commonly held anti-realist position is fictionalism/non-factualism (Balaguer, 2021), which holds that mathematical objects, structures and relations do not exist, so any existential mathematical statement is strictly false and mathematical singular terms are vacuous (Balaguer, 2009, 47, 2021; Field, 1980,1989; Hellman, 1989, 1996; Chihara, 1990; Sober, 1993; Yablo, 2001; Azzouni, 2004; Sanchez-Bennasar, 2014, 204; Maddy, 1997, 2005). Thus, as there is no such thing as the number 1, and the equation, 1 + 1 = 2, for example, is false.

Balaguer in Platonism and Anti-Platonism in Mathematics (Balaguer, 1998), and in a more recent overview paper (Balaguer, 2009), has given an outline of the principal arguments against both realism and anti-realism in the philosophy of mathematics, and he concludes with his own interesting, but we will argue, logically flawed position. He concludes that there is only one version of realism, full-blooded Platonism (FBP), which survives the standard criticisms, and one version of anti-realism (fictionalism), and that there are no good reasons for choosing one of these over the other. We will argue that this constitutes a case against both of these positions, and as he says, this may be thought to “undermine the two views” (Balaguer, 2009: p. 35).

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