Thin Realism, Arealism, and other Big Ideas

Penelope Maddy’s recent Defending the Axioms is my sort of book. It is short (150 pages), beautifully clearly written (if there are obscurities, they are in the philosophy, not the prose), and I’m in fact rather sympathetic to her overall approach (I share her doubts about ‘First Philosophy’, I share in particular her doubts about the force of supposedly a priori arguments for nominalism and against abstracta, and I rather like her post-Quinean conception of the task for the ‘Second Philosopher’).

I’m not sure, however, that there is much in the book that will be a major surprise to the reader of Maddy’s previous two books. Still, the brevity and the tight focus ‘On the Philosophical Foundations of Set Theory’, to quote the subtitle, might help to make her ideas available to new readers daunted by the length and sweep of Second Philosophy. How well does Maddy succeed in doing that?

I’ve a lot of questions. The book is dedicated to ‘The Cabal’ of Californian set theorists. And — talking things over with Luca Incurvati — one interesting issue that came up is whether Maddy’s vision of the working methods of set theorists isn’t rather skewed by a restricted diet of examples from her local practitioners. But I’ll leave that issue for Luca to pursue, even though it is more interesting than my topic here: for he knows a heck of a lot more about the ins and outs of recent set theoretic practice than I do. Here, rather, I’m going muse about the central metaphysical theme in the book: does the book succeed in selling Maddy’s story about that?

Maddy discusses two positions that she suggests are prima facie open to those with no first-philosophical axes to grind. One she dubs ‘Thin Realism’. Starting from the mathematized science of the early nineteenth century, mathematical ideas came to be pursued more for their own sake, leading inter alia to the great unifying endeavour which is set theory. This is hugely productive of ideas and results and is mathematically deep. If you are not already in the grip of some extra-mathematical prejudices, what’s not to like? So as good second philosophers, who don’t pretend to have special extra-mathematical reasons to criticize successful mathematical practice,  we should say that there are sets, and that set theory tells us about them — and also that there is no more to be said about them than what set theory says (other, perhaps, than negative things such as that they aren’t already-familiar bits of the causal, spatiotemporal, order). A gung-ho ‘Robust Realist’ is tempted to say more: she is gripped by an extra-mathematical picture of the sets as genuinely ‘out there’ quite independently of the natural world, forming a parallel world of entities sitting in a platonic heaven, with a great gulf fixed between the mathematical abstracta and the sublunary world. The harder she pushes that picture, the tougher it is for the Robust Realist to account for why we should think that the methods we sublunary mathematicians use should be a reliable guide to the lie of the land beyond the great gulf. It can then seems that our methods need backing up by some kind of certification that they will deliver the epistemic goods (and what could that look like?). Maddy, by contrast, thinks that the demand for such a certification is misplaced: why — as a second philosopher, working away in the thick of our best practices in science and maths — suppose that perfectly standard mathematical reasoning should stand in need of the sort of external supplementation that a Robust Realism seems to require?

Now, this might make it sound as if Maddy’s Thin Realist is going to end up with the sort of thin line about truth that we find e.g. in Wright. Then the thought would be: here is a discourse in good order, with appropriate disciplines and standards for making moves within it, so we can construct a minimalist truth-predicate for it. So we can not only say e.g. that the set of natural numbers has a powerset, but that it is true that there exists such a powerset, etc. However, this quick route to a thin realism isn’t Maddy’s line. Indeed, she explicitly contrast her Thin Realism with Wright’s minimalism:

Wright’s minimalist takes set theory to be a body of truths because it enjoys certain syntactic resources and displays well-established standards of assertion that our set-theoretic claims can be seen to meet: the idea is that a minimalist truth predicate can be defined for any such discourse in such a way that statements assertable by its standards come out true. In contrast, the Thin Realist takes set theory to be a body of truths, not because of some general syntactic and semantic features it shares with other discourses, but because of its particular relations with the defining empirical inquiry from which she begins.

It is important for Maddy’s Thin Realist, then, that our set theory — however wildly abstract it seems — has its connections to less abstract mathematics, which in turn has its connections … ultimately to the messy business of engineering-level science. Set theory, the rather Quinean thought goes, is an outlying but not entirely disconnected part of a network of enquiry with empirical anchors.

But put like that, we might wonder whether the Thin Realist protests too much. To be sure, looking at the historical emergence of modern mathematics, we can trace the slow emergence from roots in mathematized science of purely abstract studies driven increasingly by a purely mathematical curiosity, and can see the (albeit very stretched) lines of connection. Starting from where we now are, however, the picture changes quite sharply: here we are with feet-on-the-ground physicists doing their thing using one bunch of methods (ok, let’s not worry about string-theory fantasists!) and over there are the modern set theorists doing their improvisatory thing with a quite different bunch of rules of play. They are, it now seems, playing an entirely different game. We might ask: whatever the historical route by which we got here, is there really still a sense, however stretched, in which the physicists and the set theorists remain in the same business, so that we can sensibly talk of them both as trying to ‘uncover truths’?

The new suggestion, then, is that mathematicians have such very different fish to fry that it serves no good purpose for the second philosopher to say that the mathematicians too are talking about things that ‘exist’ (sets), or that set-theoretic claims are ‘true’. And note that it isn’t that the mathematicians should now be thought of as trying to talk about existents but failing: to repeat, the idea is that they just aren’t in the same world-tracking game. No wonder, then, that — as Maddy puts it on behalf of such a difference-emphasizing philosopher — our ‘well-developed methods of confirming existence and truth aren’t even in play here’. Call this second line, according to which set theory isn’t in the truth game, ‘Arealism’.

So what’s it to be for the second philosopher, Thin Realism or Arealism? What’s to choose? In the end, nothing says Maddy. Here’s modern science and its methods; here’s modern maths and its methods; here’s the developmental story; here’s a chain of connections; here are the radical differences between the far end points. Squint at it one way, and a sort of unity can be seen: and then we’ll incline to be Realists across the board — while, of course, eschewing over-Robust mythologies. Squint at it all another way, and the modern disunity will be foregrounded, and (so the story goes) Arealism becomes more attractive. There’s no right answer (rather, what this all goes to show is that the very notions of ‘truth’ and ‘existence’ are more malleable that we sometimes like to think).

Thus, roughly put, goes a central line in Maddy’s thought, if I’m following aright. But I wonder what underpins Maddy’s hankering here after a more-than-logical conception of truth? The Thin Realist, recall, thinks that Wrightian minimalism about truth isn’t enough: she wants to talk of set-theoretic truth so as to point up the (albeit distant) links from the maths to good old empirical enquiry. The Arealist doesn’t want to talk about set-theoretic truth because she wants to point up the differences between maths and good old empirical enquiry. But why this shared basic anchoring of the idea of truth not to the cross-discourse formal role of the notion but (so it seems) to the idea of correct representation of the empirical world?

What we need here, if we are going to make progress, is more reflection on the very concepts of truth and existence. And I don’t mean that we need an unwanted injection of first philosophy! Grant that our malleable inchoate ideas can indeed be pressed in different directions. Still, the naturalistic second philosopher can take a view about the best way to go. She will want to look at our practices of talking and thinking and inferring, and she will want to have a theory about what is going on in various areas of discourse (empirical chat, moral chat, pure-mathematical chat, etc.). Her preferred developed notion of truth should then be the one that does the best theoretical work in her story about those discourses. And it certainly isn’t obvious at the outset how things should go. Will she end up more like a Blackburnian projectivist, privileging a class of representational discourse as the home territory for a basic notion of truth (so that other discourses are playing a different game, and have to earn their right to borrow the clothes that are cut to fit the representational case)? Or will she end up like a more thorough-going pragmatist (there is no privileged core), or — in a different key — like Wright’s minimalist? Certainly, the reader of Defending the Axioms isn’t given any reason to suppose that things must fall out anything like the first way, keeping room for a substantial notion of truth.

And that gap in the end makes the current book rather unsatisfactory as a stand-alone affair (which isn’t to say it is not a fun read).

Of course Maddy herself has written extensively about truth elsewhere so as to fill in something of what’s missing here. But this means that, after all, you really will have to go back to read the weighty Second Philosophy to get the whole story, and hence the full defence of the line Maddy wants to take about sets.

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