Smolin on mathematics

by Massimo Pigliucci

I have continued on with my critical reading of Roberto Unger and Lee Smolin’s thought provoking The Singular Universe and the Reality of Time: A Proposal in Natural Philosophy [1], about which I have already published one essay here at Scientia Salon [2], focused on the general premise of the book and on the first half of the volume, which presents the more obviously philosophical argument in support of the authors’ theses, and is written by Unger.

My original idea was to eventually publish a second commentary, focused on Smolin’s half of the book, which is written from a more overtly scientific perspective. While I have not actually finished the book yet, I decided otherwise. Smolin’s contribution is definitely worth reading in its entirety, but some of his general points, of course, are the same as Unger’s, complemented by his knowledge of physics and cosmology. So I am instead focusing here on one of Smolin’s most intriguing individual chapters: his treatment of mathematics (chapter II.5 of the book).

Before we proceed, however, a brief reminder of the three fundamental theses that Unger and Smolin present and defend in their book. Quoting from pp. x-xii, where they first lay out what they are up to:

“The first idea is the singular existence of the universe. … There is only one universe at a time, with the qualifications that we discuss. The most important thing about the natural world is that it is what it is and not something else. This idea contradicts the notion of a multiverse — of a plurality of simultaneously existing universes — which has sometimes been used to disguise certain explanatory failures of contemporary physics as explanatory successes. … The second idea is the inclusive reality of time. Time is real. Indeed, it is the most real feature of the world, by which we mean that it is the aspect of nature of which we have most reason to say that it does not emerge from any other aspect. Time does not emerge from space, although space may emerge from time. … The third idea is the selective realism of mathematics. (We use realism here in the sense of relation to the one real natural world, in opposition to what is often described as mathematical Platonism: a belief in the real existence, apart from nature, of mathematical entities.) Now dominant conceptions of what the most basic natural science is and can become have been formed in the context of beliefs about mathematics and of its relation to both science and nature. The laws of nature, the discerning of which has been the supreme object of science, are supposed to be written in the language of mathematics.”

This essay is concerned precisely with this third point [3].

Smolin begins by acknowledging that some version of mathematical Platonism — the idea that “mathematics is the study of a timeless but real realm of mathematical objects,” is common among mathematicians (and, I would add, philosophers of mathematics), though by no means universal or uncontroversial. The standard dichotomy here is between mathematical objects (a term I am using loosely to indicate any sort of mathematical construct, from numbers to theorems, etc.) being discovered (Platonism) vs being invented (nominalism [4]).

Perhaps the most original and intriguing contribution by Smolin to this debate is to reject the above choice as a case of false dichotomy: it is simply not the case that either mathematical objects exist and are therefore discovered, or that they do not exist prior to the intervention of human minds and are therefore invented. Smolin presents instead a table with four possibilities:

	existed prior? yes	existed prior? no
has rigid properties? yes	discovered	evoked
has rigid properties? no	fictional	invented

By “rigid properties” here Smolin means that the objects in question present us with “highly constrained” choices about their properties, once we become aware of such objects.

Let’s begin with the obvious entry in the table: when objects exist prior to humans thinking about them, and they have rigid properties. All scientific discoveries fall into this category: planets, say, exist “out there” independently of anyone being able to verify this fact (pace extreme postmodernists and radical skeptics), so when we become capable of verifying their existence and of studying their properties we discover them.

Objects that had no prior existence, and are also characterized by no rigid properties include, for instance, fictional characters. Sherlock Holmes did not exist until the time Arthur Conan Doyle invented (the appropriate term!) him, and his characteristics are not rigid, as has been (sometimes painfully) obvious once Holmes got into the public domain and different authors could pretty much do what they wanted with him (and I say this as a fan of both Robert Downey Jr. and Benedict Cumberbatch).

Smolin, unfortunately, doesn’t talk about the “fictional” category, comprising objects that had prior existence and yet are not characterized by rigid properties. Anyone wishs to submit examples?

The crucial entry in the table, of course, is that of “evoked” objects: “Why could something come to exist, which did not exist before, and, nonetheless, once it comes to exist, there is no choice about how its properties come out? Let us call this possibility evoked. Maybe mathematics is evoked” (p. 422).

Smolin goes on to provide an uncontroversial class of evocation: “For example, there are an infinite number of games we might invent. We invent the rules but, once invented, there is a set of possible plays of the game which the rules allow. We can explore the space of possible games by playing them, and we can also in some cases deduce general theorems about the outcomes of games. It feels like we are exploring a pre-existing territory as we often have little or no choice, because there are often surprises and incredibly beautiful insights into the structure of the game we created. But there is no reason to think that game existed before we invented the rules. What could that even mean?”

Interestingly, Smolin includes forms of poetry and music into the evoked category: once someone invented haiku, or blues, then others were constrained by certain rules if they wanted to produce something that could reasonably be called haiku poetry, or blues music.

The obvious example that is most close to mathematics (and logic?) itself is provided by board games: “When a game like chess is invented a whole bundle of facts become demonstrable, some of which indeed are theorems that become provable through straightforward mathematical reasoning. As we do not believe in timeless Platonic realities, we do not want to say that chess always existed — in our view of the world, chess came into existence at the moment the rules were codified. This means we have to say that all the facts about it became not only demonstrable, but true, at that moment as well … Once evoked , the facts about chess are objective, in that if any one person can demonstrate one, anyone can. And they are independent of time or particular context: they will be the same facts no matter who considers them or when they are considered” (p. 423).

This struck me as very powerful. Smolin isn’t simply taking sides in the old Platonist / nominalist debate, he is significantly advancing that debate by showing that there are two other cases missing from the pertinent taxonomy, and that moreover one of those cases provides a positive account of mathematical (and similar) objects, rather than just a rejection of Platonism.

But in what sense is mathematics analogous to chess? Here is Smolin again: “There is a potential infinity of formal axiomatic systems (FASs). Once one is evoked it can be explored and there are many discoveries to be made about it. But that statement does not imply that it, or all the infinite number of possible formal axiomatic systems, existed before they were evoked. Indeed, it’s hard to think what belief in the prior existence of an FAS would add. Once evoked, an FAS has many properties which can be proved about which there is no choice — that itself is a property that can be established. This implies there are many discoveries to be made about it. In fact, many FASs once evoked imply a countably infinite number of true properties, which can be proved” (p. 425).

But Smolin’s positive argument doesn’t end there. He recognizes that he has to come up with an alternative account for what has been called the “unreasonable effectiveness of mathematics” [5], or with an answer to the closely related “no miracles” argument for mathematical realism put forth by Quine and Putnam [6]. It does so by a dual, in my mind compelling, strategy: he wants to show that the effectiveness of mathematics in physics is actually somewhat overrated, and then proceeds to propose a multiple-stage account of the development of mathematics as a discipline.

In terms of the first point, Smolin observes that mathematical objects are actually seldom, if ever, a perfect match with objects in the real world, which is to be expected if one thinks of mathematics as dealing in part with abstractions from the real world. Also, mathematical models are grossly underdetermined by physical systems, in the sense that most mathematical laws do not actually have a physical counterpart, or do not uniquely model the physical systems they are intended to account for [7].

As for the second point, I can provide only the highlights here, but the chapter is well worth a full reading. According to Smolin, we can think of mathematics as having developed along the following stages:

“At the first stage, there is the study of the structure of our world, by examination of examples and relations between them, coming from the properties of physical objects or processes and their relations … The second stage is the organization of the knowledge acquired in the naturalistic phase. One makes the discovery that all the knowledge gathered by examination of cases in nature can be reproduced by deduction from a small set of axioms. This is the phase of the formalization of natural knowledge … At the next, or third, stage in the development of mathematics, several mechanisms of growth of mathematical knowledge come into effect which are internal to mathematics, as they no longer require the study of examples in nature to proceed … [then] More non-trivial examples of varying the natural case are found by altering one of the postulates. Famously , modification of the fifth postulate gave rise to the non-Euclidean geometries. This is the fourth stage, that of the evocation and study of variations on the natural case … A fifth stage of development is the invention and development of new modes of thought, new concepts and new methodologies in the study of an area. These can greatly progress an area as new kinds of facts become definable and discussable … Once there are a variety of cases developed by variation of the natural case, a sixth stage of development can play a role, which is to define new kinds of objects by unification of diverse cases. For example, the different Euclidean and non-Euclidean geometries are all unified within Riemannian geometry … mathematics [further] develops through two more kinds of discoveries, one external and one internal. The first is that a construction, example or case developed in the path flowing out of one of the core concerns can turn out to illuminate or apply to knowledge in another stream of development. Developments in geometry can illuminate problems in number theory and vice versa … Lastly, examples, cases or modes of reasoning invented due to the internal development of mathematics can surprisingly turn out to be applicable to the study of nature” (pp. 432-441).

And here is Smolin’s conclusion for that chapter: “the main effectiveness of mathematics in physics consists of these kinds of correspondences between records of past observations or, more precisely, patterns inherent in such records, and properties of mathematical objects that are constructed as representations of models of the evolution of such systems … Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will; neither has any transcendental existence. Both are static, not in the sense of existing outside of time, but in the weak sense that, once they come to exist, they don’t change” (pp. 445-446).

As should be clear by now, I find Smolin’s view intriguing, but not because it answers all the questions about the nature of mathematics and its relationship with the natural sciences. Frankly, nobody else has come even close to providing such a comprehensive account anyway, so it would be asking a bit too much of Smolin (and Unger) within the context of the much broader project with which they are primarily concerned.

But reading chapter II.5 of The Singular Universe and the Reality of Time did something that rarely happens to me: it provided me both with a fresh perspective on an old problem, and it sketched out tantalizing new answers to that problem. That chapter is worth the price of the book in and of itself, and the rest of the volume ain’t a slacker either.

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Massimo Pigliucci is a biologist and philosopher at the City University of New York. His main interests are in the philosophy of science and pseudoscience. He is the editor-in-chief of Scientia Salon, and his latest book (co-edited with Maarten Boudry) is Philosophy of Pseudoscience: Reconsidering the Demarcation Problem (Chicago Press).

[1] The Singular Universe and the Reality of Time: A Proposal in Natural Philosophy, by R.M. Unger and L. Smolin, Cambridge University Press, 2014.

[2] The Singular Universe and the Reality of Time, by M. Pigliucci, Scientia Salon, 24 March 2015.

[3] For a look at my changing opinions about mathematical Platonism, see: On mathematical Platonism, Rationally Speaking, 14 September 2012. / Mathematical Universe? I ain’t convinced, Rationally Speaking, 11 December 2013. / My philosophy, so far — part I, Scientia Salon, 19 May 2014.

[4] Nominalism in the Philosophy of Mathematics, by O. Bueno, Stanford Encyclopedia of Philosophy.

[5] The Unreasonable Effectiveness of Mathematics in the Natural Sciences, by E. Wigner, Communications in Pure and Applied Mathematics, 1960.

[6] Indispensability Arguments in the Philosophy of Mathematics, by M. Colyvan, Stanford Encyclopedia of Philosophy.

[7] Yes, one could go with the Max Tegmark’s mathematical universe hypothesis, or with David Lewis’s modal realism, but that’s just crazy talk. (Yeah, I know, this is going to be controversial, so bring it on!)