Infinity is a puzzling concept in itself and also in the way it has shaped the development of mathematics. Take for example the number line, the infinity of the infinity of the integers and the larger infinity of the real numbers. Cantor asked: is there also an infinity of intermediate size? This is called the Continuum Problem. Since it is independent of ZFC, there is a worry that it does not have a determinate solution. This raises a difficult question that I’ve been wrestling with for some time now: What is the provenance of this concern? Mathematicians worry about it, certainly. But is the worry itself mathematical? Or is it philosophical or perhaps a question ultimately of cognitive science or physics? And anyway, who has the right to pose questions in this fashion?
These are hard questions, so before I go further let me put two more examples on the table:
- The development of Non-Euclidean geometry from the work of Bolyai and Lobachevsky in the first half of the 19th century. They now co-exist side by side but neither can lay claim to the old mantle of synthetic a priori insight into the geometry of physical space.
- Intuitionism. Brouwer declared that the Law of Excluded Middle has the same status as the view that the Earth is the centre of the Universe. For him, the Copernican Revolution in mathematics was wedded to the view that only denumerable infinites exist. And his opinion was taken seriously. But unlike the case of Ptolemaic astronomy, it did not yield revisions in the practice of mathematics — or rather, its revisions were additions rather than the sort of thing Hilbert had in mind when he referred to Kronecker as the Verbotsdiktator. The answer everyone could agree on turned out to be ecumenical: classical and intuitionist mathematicians are free to proceed as they please. Let a thousand flowers bloom.
Question: when a set of axioms is demonstrably independent of some further hypothesis, is there a determinate answer regarding the truth or falsity of that hypothesis?
This question is explored by Penelope Maddy in her book Defending the Axioms. Her main focus is the addition of new axioms to Zermelo-Fraenkel set theory in order to settle the question of whether Cantor’s Continuum Hypothesis is true or false. She quotes Gödel on this (2011: 56): set theory “…must describe some well-determined reality, in which Cantor’s conjecture must be either true or false. Hence its undecidability from the axioms being assumed today can only mean that these axioms do not contain a complete description of that reality.” The key assumption here is that the reality in question is determinately there to be described.
What sort of process is involved in deciding the undecidable? Shapiro describes it as follows in his book Thinking About Mathematics (2000: 208): a proposition or formula may be undecidable in arithmetic or in some framework, so we decide it by embedding it into another more expressive framework. As we just saw, Gödel takes this process — undecidability + embedding in a richer framework — to constitute an argument for Platonism. But given that more than one embedding is available, what rule prescribes that we go in one direction rather than the other? Hilbert eventually declared Cantor’s conjecture to be meaningless on this basis. Interestingly, Shapiro also echoes this sentiment in his paper in the collection Meaning in Mathematics (p. 104): disagreements over set-theoretical axioms, he says, are in a certain sense not real disagreements but just talking-past-one-another phenomena. What is needed is not fruitless argument but what he calls an “Euclidean rescue,” wherein we declare the competitors equally valid in their own domains, which are then strictly segregated. (At the level of logic he repeats this gesture.)
All of this brings us back to the questions raised above: when mathematicians worry about Cantor’s Paradise, is the worry itself mathematical? Maddy mentions Dedekind’s belief, that the natural numbers are a free creation of the human mind, in order to illustrate the class of irrelevant conceptual accretions attendant upon but irrelevant to mathematical practice (2011: 53). On this point, I can’t resist sharing the following humorous story recounted by Dirk van Dalen in his biography of Brouwer (2013: 92). Brouwer was trying to assuage the doubts of his advisor Korteweg about the contents of his dissertation thesis. He wrote about Russell: “To put something higher than mathematics, you must feel a non-mathematical intuition; Russell has none of that, and yet he starts to bullshit.” Korteweg’s response included the following: “A kind of pessimistic and mystic philosophy of life has been woven into [the dissertation] that is no longer mathematics, and has nothing to do with the foundations of mathematics. It may here and there have coalesced in your mind with mathematics, but that is wholly subjective.”
Here is the point. You may agree that Brouwer operated on the basis of a fair amount of irrelevant philosophy when working out his alternative to classical mathematics. And although I am leery of this attitude, few would deny that there is some distinction to be had here between what is merely subjective and what Maddy describes as “shared convictions that actually drive the practice” of mathematics. The question is whether and to what extent this distinction supports the weight Maddy wants to place on it. To give just one example, it seems to rule out by fiat the entire project of cognitive science of mathematics proposed by Lakoff and Núñez in Where Mathematics Comes From. And that is a strange thing for an avowed naturalist to want to do.
Here I just want to flag the issue and mention what Mary Tiles says about this aspect of Maddy’s position in her review of Defending the Axioms. She leans on Bourdieu to make the point that there is a danger here of telling an epistemological or methodological story that simply “reproduces the dominant discourse of a sector of pure mathematics about itself while denying legitimate access to non-dominant discourses. While not denying that set theorists have internalised a sense of objectivity, there are grounds for thinking that this sense cannot be the sole arbiter of reliability when it comes to non-mathematical uses of their products.”
Of course, there are more and less extreme ways of fleshing out the complaint about “legitimate access to non-dominant discourses.” For example, it is clear that whatever exactly it is, the debate about Woodin’s program in contemporary set theory is not a debate about whether physics tells us that set theory has to be one way or another. The debate over whether we should we force mathematics to pass through the eye of the physicalist needle — is conducted on the basis of relatively lesser common ground. And this may be seen to count against it to some degree.
At any rate, there is no narrowly mathematical procedure for answering this latter question — except in the sense of a very narrow internal-to-mathematics perspective that makes the answer trivially affirmative — so it makes sense to view it as a question imposed on mathematics from the outside. Given this, are we right to explore such a question, or is this just wrong-headed and confused? Are we allowed to ask?
The distinction between internal and external questions is often wielded with deflationary intent — this is how Maddy deploys it. Set theorists have the wherewithal to function independently of external oversight, relying on an institutionally shared sense of mathematical depth and objectivity that does not even require that sets exist. A familiar way to make sense of the significance of this is as follows: we can view as ipso facto external to mathematics any global question — such as whether abstract objects exist as such — so long as none of the possible answers to this question are conceived of as capable of affecting mathematical practice. This may be part of the reason why it is felt reasonable to dismiss such external questions — they define themselves into irrelevance by requiring that mathematics be potentially based on an error undetectable not only to mathematicians themselves, but to us looking in from the outside.
If this is what the strategy is, then it’s a nice trick. External criticisms that propose to modify mathematical practice are dismissed on that basis, whereas criticisms that leave practice as it is are declared irrelevant. As for internal criticisms, one simply parameterises the disputants and returns to work. I have to admit that, in my more cynical moments, this strikes me as a Potemkin imitation of objectivity!
Regardless, there is a more specific issue here pertaining to the foundational aspirations of set theory in particular. To see what this is, begin by noting that there is a class of beliefs pertaining to what is invariant across any possible mathematics or logic, a class whose boundary shifts with time. The belief that classical mathematics is the only legitimate mathematics, and more generally the belief that a particular mathematical theory is in the intersection of ‘universal’ mathematics (in the sense, for example, that there is a universal geometry that is equivalent to Euclidean geometry minus the parallel postulate), illustrate what I am referring to here. When a hypothesis is formulated that is independent of a given set of axioms, and we relativise it to two different expanded sets of axioms, one yields that hypothesis as a formula and the other yields its negation. My point: there seems to be no universe, then, in which either the formula or its negation are mathematically necessary. And so the question becomes: what is mathematical necessity, ultimately speaking? What, for that matter, is mathematically universal? The idea of providing a foundation for all of mathematics seemed to require accounting for one or both of these notions (necessity, universality), and doing so in way that was internal to mathematical practice. But it seems that this cannot be done. So now we ask: are beliefs about mathematical necessity or universality ultimately mathematical or philosophical (or within the purview of empirical science)?
Being that metamathematics exists, I have to say I am wary of Maddy’s attempt to cast objections to set-theoretical methodology as both external to mathematical practice and inherently global (and by implication irrelevant). Regarding the latter, external questions are not always or even predominantly global in nature. Schematically, if I give you an argument for why there are no sets, but there are zets, where a ‘zet’ is defined such that at least some of the things you took to be existing sets do in fact exist, then I have not given you a global anti-realist argument against sets. (Compare ‘set’ here with ‘electron’.)
Let me try another tact and motivate arguendo the assertion that external questions are always global. Well, how many elves lived in Rivendell? Are we asking from within the perspective of Tolkien’s universe, or from an external perspective? Maddy’s complaint might be glossed like this: philosophers or physicists ruin the fun when they come along and ask which axiomatisation is ‘really’ correct, which abstract objects ‘really’ exist, and so on. The complainer might insist, in a Carnapian way, that this question should be taken only in an ‘internal’ manner, and that it is senseless to pose it externally, as to do so is to render it undecidable and theoretically inert. (But if it is undecidable why not transpose the mathematician’s own practice and embed it into another theory — this time perhaps that of cognitive science?) She might then observe, moreover, that the question of how many elves are in Rivendell is, necessarily, either an internal question to be answered as such, or senselessly narrow and specific. Narrow and specific in this sense: asking it from the external perspective is wholly unmotivated and weird — like a scientist who comes along claiming that although elves are complete myth and fantasy, the land where Rivendell was built was an actual geographical location in our planet’s prehistory. It is just weird. So how can it be less weird to say, for example, that although the rational numbers exist, the irrational numbers are just fictions that should never be reified into actual beings? This, perhaps, is the reasoning behind the conclusion that external questions are essentially global.
Yet, if this attitude is correct, why not apply it within mathematics as well? Let me try to articulate my discomfort here a little more clearly. Why not just dissolve foundational debates by allowing each otherwise competing set theory to bloom independently and freely? Each one will then have its own Rivendell, with as many or as few elves as you like. The existence or non-existence of a particular elf will be axiomatically decided one way or the other within these competing systems. If external critiques of mathematical ontology are senseless and undecidable, is it not the same way within mathematics if we frame things thus? “My own practice will not, after all, be affected by the correctness of whatever ontological revisions you wish to impose on me. So your objections are quite irrelevant! In fact, even if you could falsify my belief, I could just add a new axiom to my system to inoculate myself from your attack! Maybe, then, we should just get on with the business of developing our own systems, and not waste each other’s time…”
But from the viewpoint of mathematical necessity, this looks suspiciously like saying that we can project an indefinitely large number of incompatible patterns across the same data set, none superior to any other. The data set is random in this sense, and I find this hard to distinguish from the observation that a false belief ‘represents’ the empty set — an analogy that if accepted suggests that beliefs about mathematical necessity/universality are all false.
In general, the idea of arbitrarily adding axioms whenever we please should give us pause to consider what sort of constraint there is on set-theoretical activity. But this is where we seem to have ended up, with dismissing the validity of external questions making possible a corresponding internalising of the strategy underlying this dismissal, which in turn erodes the legitimacy of internal questions by rendering them all external in principle. In general, if a global challenge to a domain is simply the idea that nothing exists in that domain (but this fact is invisible from within), then any sub-domain could use this strategy to inoculate itself from any ‘outside’ criticism. But this is absurd. And this suggests that to avoid eroding the purchase of internal questions, we need to preserve the openness of mathematics to its outside, at least in principle — whether this be philosophy or science. (And this is the reason why I am leery of saying that e.g. Brouwer’s background philosophy is merely subjective/irrelevant to his mathematics.)
Set theorists do recognise a legitimate question about what unity, if any, underlies the plurality of competing axiom schemes — but for Maddy this question only has mathematical (i.e. legitimate) sense if taken as a question about which axiomatisation allows for the broadest and most complete reconstruction/accommodation of mathematical practice — not a question about whether mathematics as such models an ‘independent’ reality. Within the meta-axiomatic ‘game’ of choosing between rival set theories, selecting a different axiomatisation involves making a choice about what to accept as existing or not existing. A question of this sort is an ontological question, so there are in some sense ontological questions included in mathematical practice beyond those of axiomatic fiat (since there is no axiom system for making this choice). This is a step towards unity, but it does leave something to be desired, at least in my opinion. I want a mathematical problem to teach me something about reality — not simply about a ‘game’ we play or something which may turn out to be as illusory as Rivendell! (I’m also inclined to say that, to the extent that she relies on shared practice and perception of mathematical depth to guide axiom choice, Maddy ends up with a non-classical meta-theory, because facts about mathematical depth are presumably finite and physical.)
I would like to find a way, then, to render it plausible and right to continue asking unifying ontological questions at broader and broader scopes without succumbing to the charge of regressing into objectionable metaphysics. So here it is: we noted that the viewpoint of a greater unity is invoked when adjudicating issues within foundational mathematics such as which set theory best captures the mathematical hierarchy. This immediately generates a new line of questions: who decides how unified mathematics is? Who decides that pure mathematics is one unity whereas applied math is another, or more to the point, that the unity of mathematics is not itself merely a part of a greater unity, over which mathematics cannot have any sort of legislative fiat or veto? Tiles makes this point interestingly in her review: pure and applied mathematics appear to be merging once again, but Maddy’s strategy requires them to remain firmly separated. I think we can see a way towards formulating an external but non-global challenge to mathematics by focusing on this dividing line between pure and applied mathematics. This line is what filters the purely abstract and brings it back down to Earth.
But hang on a minute! Imagine someone saying: “Mathematics methodologically differs from natural science; this is enough to account for the unity of each and their robust difference from one another. So that’s why it is illegitimate for physicists to pose external questions to mathematicians.” It is worthwhile to draw attention to this line of response. We can even add to it a bit: Gideon Rosen, for example, in his contribution to the book Meaning in Mathematics (p. 116) reasons as follows: mathematics is unassailable by all pertinent mathematical, scientific and common-sense standards — so to assail it necessarily requires dubious philosophical tools. In other words, isn’t mathematics the best method we have for evaluating the ontological status of mathematics? By contrast, armchair philosophers would only be bringing their own methodologically obscure techniques to bear in trying to adopt an external perspective.
In response, consider this: I would not want to be told that science cannot adjudicate the question of whether elves exist because Tolkien’s universe is unified by the methodology of creative writing and so can only be internally adjudicated. Elves exist in the story — but if we wish to say that they also exist tout court, then science gets to have an opinion on whether this is a wise thing to want to say, or not. Anyone — even a philosopher — can adopt a perspective methodologically internal to science in order to externally evaluate the claim of a particular discipline or field of inquiry. This is simply not the same as relying on dubious a priori methods, whatever these might be. And it means that people outside of mathematics get to have a say on the ontological import of mathematics — in particular, over what ‘exists’ means. My conclusion: it is subtly illegitimate to use the word ‘exist’ in speaking of mathematical objects unless one is prepared to allow this point to be adjudicated by a perspective external to mathematics itself.
Which brings me to my competing desire: suppose we want to develop a single or univocal ontological viewpoint. How should we proceed? One suggestion, which I would just like to float for now, is that everything must bend to the perspective of empirical science broadly conceived. This yields the following response to the charge of weirdness articulated above: the scientist is not going in willy nilly and arbitrarily declaring classes of similar objects either non-existent or existent without further ado. There is a demarcation corresponding, for example, to what could be applied in principle to the physical universe or some possible universe, and what cannot. The former are taken as potentially descriptive or representational, the latter as fictive, expressive or broadly epistemic.
Let me summarise briefly. For two supposedly rival hypotheses x and y, we can treat them in two different ways. First, we can treat them as belonging to the same domain of inquiry, in which case the question of which is correct is an internal question of that domain. Second, we can also treat them as belonging to different domains, such that y is external to x, and in this way the Carnapian manoeuvre seems to trivially solve any dispute or problem. This seems to be a reason for looking suspiciously upon that manoeuvre. I have suggested that we must pay due respect to this suspicion in order to properly follow the imperative of developing a unified — univocal — ontological perspective.
In this respect I am a little less sanguine than Maddy. Her strategy is to try to have internal realism and external anti-realism behave like two sides of a Möbius strip, such that it is indeterminate and undecidable whether mathematical objects exist or not. External evaluations of mathematics are like party-crashers that rip the cloth off of the dinner table so perfectly that nothing is disturbed or altered at all. But I’ve identified weaknesses in this approach. It is stronger when able to equate external challenges with global challenges — and correspondingly weaker at coping with non-global but still external challenges. The external perspective can, fairly obviously on reflection, be as fine-grained and as discriminating as you like; it can even, for example, choose one particular side in a debate otherwise internal to mathematics — e.g. the debate about what constitutes real as opposed to merely apparent mathematical depth, to use Maddy’s vocabulary — and attempt to buttress that side by embedding it in a broader ontological perspective.