Mathematics, Metaphysics, and Mystery, #1.

(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 first, and contains sections 1 and 2.

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.

Mathematics, Metaphysics, and Mystery

In perpetuating these notions, modern mathematics takes on many of the aspects of a religion. It has its essential creed – namely Set Theory, and its unquestioned assumptions, namely that mathematics is based on “Axioms,” in particular the Zermelo-Frankel “Axioms of Set Theory.” It has its anointed priesthood, the logicians, who specialize in studying the foundations of mathematics, a supposedly deep and difficult subject that requires years of devotion to master. Other mathematicians learn to invoke the official mantras when questioned by outsiders, but have only a hazy view about how the elementary aspects of the subject hang together logically. (Wildberger, 2015: p. 2)

1. Introduction: From Mind to Mathematics

Anyone who has an interest in the philosophy of mind and consciousness, would be acutely aware that mind and consciousness pose problems to the received “scientific” worldview, one where phenomena need to fit into the world as described by physics, chemistry and biology, the natural sciences which define the scope and limits of reality. Specifically, consciousness is a problem for a mechanistic worldview, one which holds that:

[E]verything in the world is fundamentally either a formal automaton or a natural automaton, operating strictly according to Turing-computable algorithms and/or time-reversible or time-symmetric deterministic or indeterministic laws of nature, especially the Conservation Laws (including the First Law of Thermodynamics) and the Second Law of Thermodynamics, which also imposes always-increasing entropy—i.e., the always-increasing unavailability of any system’s thermal energy for conversion into causal (aka “mechanical”) action or work—on all natural mechanisms, until a total equilibrium state of the natural universe is finally reached. (Hanna, 2024: p. 23)

It is a problem, the so-called “hard problem of consciousness,” to explain the subjective and qualitative aspects of human (and animal) consciousness within this physicalist/mechanistic materialist framework, that is, why there is “something it is like” to have a conscious experience (Nagel, 1974; Chalmers, 1996). We could conceive of philosophical “zombies,” being exactly like us, physically, but lacking conscious experience. Hence the physical account of conscious seems to be incomplete, but for those who embrace a mechanistic world view, this is a scandal, that consciousness itself becomes something of a metaphysical mystery (McGinn, 1991, 2000; Levine, 2001).

In this essay, we will argue that the concern with consciousness being a “mystery” within a mechanistic/physicalist framework is not a problem, since there is an even more troublesome problem of metaphysical mystery right in the heart of this framework. While there is a case that the very idea of the physical has its own metaphysical puzzlement, the issue discussed here concerns abstract essence, particularly the ontology of mathematical entities. It will be argued that there is a more fundamental problem of accounting for these within the mechanistic/physicalist framework, and indeed, within any framework. Hence, we are neck deep in “mysteries,” and up against the limits of the scientific world view, so we should not be unduly concerned that consciousness remains unexplained. There are more unexplained things in “heaven” and earth, Horatio, than are dreamt of in the scientific mechanistic/materialist worldview, and since this framework has mathematics as its basis, metaphysical mystery here undercuts the quest to reduce everything in reality, to the physically explicable.

2. The Nature of Mathematical Entities: Nothing Works

Hillary Putnam published an article called “Philosophy of Mathematics: Why Nothing Works,” in 1994, and a strong case can be made that this remains true today (Putnam, 1994). A philosophy of mathematics must answer a number of questions. One is, why does mathematics work in the sciences, and how is mathematics applicable to the world, which was discussed by physicist E. P. Wigner in his frequently cited article, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences (Wigner, 1960). Wigner saw the usefulness of mathematics in the natural sciences as something almost mysterious, and without rational explanation (Wigner, 1960: p. 14). R.W. Hamming has examined a number of responses to this problem, and has argued that these explanations are an inadequate response to the Wigner problem (Hamming, 1980).

To mention just one response, Max Tegmark (2014), proposes that the universe is mathematical, meaning that the physical world is isomorphic to mathematical structures. His Mathematical Universe Hypothesis (MUH) is: external reality is a mathematical structure, of abstract entities and their relationships. As well, he advances the External Reality Hypothesis (ERH): an external physical reality exists independently of humans. Presumably this external reality refers to our reality, and not some other dimension/universe of the multiverse. While the true mathematical structure which is isomorphic to the world has not been discovered, when it is, it will be revealed as a Platonic abstract entity existing outside of space-time. Thus, the world is not just described by mathematics, but is mathematics; as he says, that if two structures are isomorphic, then they are one and the same (Tegmark, 2007). He claims that evidence for ERH is also evidence for MUH. Wigner’s problem of how to explain the “unreasonable” effectiveness of mathematics is solve, Tegmark believes in his system, because if the world itself is a mathematical structure, then the discovery of mathematical patterns is evidence that the world is completely mathematical (Tegmark, 2007).

Tegmark’s position has been shown to be circular (Sullivan, 2014). However, the principal philosophical objection to it, in our opinion, is that if we take Tegmark as being literal in his claim that the world is not just described by mathematics, which is true (Hossenfelder, 2020), but is mathematics, then physics concerned with an external reality is impossible, since there is no external physical reality as such, being that which is in space-time. As he asserts, mathematical entities are abstract Platonic entities and structures, outside of space-time. Hence, there is then no such thing as conventional physics, which deals with causally efficacious entities and facts inside space-time. That is not to say that neo-Pythagoreanism is false as a metaphysical position, as all sorts of things can and have been argued for in philosophy. But Tegmark has not shown this, because an argument from the presence and success of mathematics in physics does out show that all is mathematics (Jannes, 2009). In passing, it is interesting to speculate that if Tegmark’s Platonistic position was accepted, it is still conceptually possible, as examined below, that on an independent philosophical basis, mathematical Platonism should be rejected in favour of, say, fictionalism, namely, the thesis that mathematical entities and structures do not exist, and that mathematical statements are false. It would seem then that nothing exists, an argument for ontological nihilism.

Another key issue in the philosophy of mathematics relates to the alleged necessity of mathematics, that mathematical statements are true in some sense quite independent of the existence of physical reality, and perhaps true by necessity, whatever that means (Pap, 1958). Thus, statements in Peano arithmetic, for example, would not be refutable by empirical matters, as such statements are not the sort of things which can be refuted by observations of the world. Yet, even here there is controversy, as argued by Peter J. Lewis, arithmetic seems to apply to macroscopic objects only as an approximation according to spontaneous collapse theories of quantum mechanics (SCTQM) (Lewis, 1997, 2003). The enumeration principle is alleged violated by SCTQM, this being that given n objects, if object O₁ is in a given spatial region, as well as all of O₂, O₃, … to On,then there are in total n objects in that region. Lewis argues that from the position of SCTQM, it can be shown by eigenstate analysis that “it is true of each marble that it is in the box, but it is not the case that there are n marbles in the box” (Lewis, 2003: p. 166).Naturally, this has led to a lively debate that now seems to have fizzled out (Clifton & Monton, 1999; Bassi & Ghirardi, 2001). The technicalities of this debate need not be entered into here; the example shows that it is at least conceivable that mathematical statements, true of the macroscopic world, could be false when applied to the quantum domain (Mortensen, 1989), just as has been proposed in the parallel debate on quantum mechanics violating classical logical principles such as distribution, and hence the need for a quantum logic (Gibbins, 2008).

As another challenge to the view that mathematical statements are necessarily true, or even true, the intuitionist L.E.J. Brouwer (Brouwer, 1967: p. 337) wrote a number of papers producing a critique of classical mathematics (Heyting, 1966: pp. 115-120). Thus, he claimed to be able to be able to prove the following:

(B) There is a real number R, which is not equal to 0, ~(R=0), as R=0 is contradictory, but it is not provable that R>0, or R<0. (Van Dantzig, 1949)

A number of other counter-examples were advanced, primarily based upon Brouwer’s rejection of the classical logical principle of the law of excluded middle (Brouwer, 1967). It is interesting to speculate about how much of classical mathematics would survive his intuitionist philosophy. Nevertheless, as will be discussed below, forms of neo-intuitionism (Hanna, 2022), could offer some progress in addressing the many problems discussed here, but this philosophical worldview will provide no comfort to the reductionist materialist.

Another argument raising doubts that mathematics consists of necessary truths, comes from the paradoxes of set theory, such as Russell’s paradox. Consider the set of all sets which are not members of themselves. Is this set a member of itself? If it is, then it is not. If it is not, then it is. There are a number of other set theoretical paradoxes such as the Burali-Forti paradox and Cantor’s paradox. Typically, the set theoretical paradoxes have been dealt with by modifying our naïve conception of a set through various formal set theories. Ingenious as these theories have been it would appear from a survey of the critical literature that a final resolution of these difficulties has not been accomplished (Priest, 2006).

For example, Grim argued for some time that the set of all truths, or, all true statements, conflicted with Cantor’s power set theorem (Grim, 1984). The power set is the set of all subsets of a given set, and if a set S has n elements, then the Power Set PS has 2ⁿ elements (Suppes, 1960: pp. 46-48). If we take the intuitive idea of a “set” to be a “collection of entities of any sort” (Suppes, 1960: p. 1), then we should be able to meaningfully deal with both the set of all apples, and the set of all true statements. Of course, in the light of the set theoretical paradoxes, logicians have restricted the objects of sets containing special set constituents, as in the set of all sets, and have made a distinction between sets and classes. But set theory should not yield paradoxes merely from considering elements such as sentences, which are ontologically distinct from sets. However, for the set of all truths, for each subset of this set, there will be a truth, and thus a corresponding statement, so there will be at least as many truths as there are elements of the power set, contrary to the power set theorem (or in some systems, power set axiom [Suppes, 1960: p. 46]). Thus, a counter-example is presented to a provable theorem.

Reflecting up this result and other paradoxes of totalities, Rescher and Grim state:

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)

If mathematics is taken to be reducible to set theory, then the paradoxes create havoc for the foundations of mathematics. But is this so? This leads us to our next major issue in the philosophy of mathematics

Download