This is a very belated follow-up to an earlier post on Penelope Maddy’s short but intriguing Defending the Axioms.
In my previous comments I was talking about Maddy’s discussion of Thin Realism vs Arealism, and her claim that there in the end — for the Second Philosopher — there is nothing to choose between these positions (even though one line talks of mathematical truth and the other eschews the notion). What we are supposed to take away from that is the rather large claim that the very notions of truth and existence are not as central to our account of mathematics as philosophers like to suppose.
The danger in downplaying ideas of truth and existence is, of course, that mathematics might come to be seen as a game without any objective anchoring at all. But surely there is something more to it than that. Maddy doesn’t disagree. Rather, she suggests that it isn’t ontology that underpins the objectivity of mathematics and provides a check on our practice (it is not ‘a remote metaphysics that we access through some rational faculty’), but instead what does the anchoring are ‘the entirely palpable facts of mathematical depth’ (p. 137). So ‘[t]he objective ‘something more’ our set-theoretic methods track is the underlying contours of mathematical depth’ (p. 82).
This, perhaps, is the key novel turn in Maddy’s thought in this book. The obvious question it raises is whether the notion of mathematical depth is robust and settled enough really to carry the weight she now gives it. She avers that ‘[a] mathematician may blanch and stammer, unsure of himself, when confronted with questions of truth and existence, but on judgements of mathematical importance and depth he brims with conviction’ (p. 117). Really? Do we in fact have a single, unified phenomenon here, and shared confident judgements about it? I wonder.
Maddy herself writes: ‘A generous variety of expressions is typically used to pick out the phenomenon I’m after here: mathematical depth, mathematical fruitfulness, mathematical effectiveness, mathematical importance, mathematical productivity, and so on.’ (p. 81) We might well pause to ask, though, whether there is one phenomenon with many names here, or in fact a family of phenomena. It becomes clear that for Maddy seeking depth/fruitfulness/productivity also goes with valuing richness or breadth in the mathematical world that emerges under the mathematicians’ probings. But does it have to be like that?
In a not very remote country, Fefermania let’s say (here I’m picking up some ideas that emerged talking to Luca Incurvati), most working mathematicians—the topologists, the algebraists, the combinatorialists and the like—carry on in very much the same way as here; it’s just that the mathematicians with ‘foundational’ interests are a pretty austere lot, who are driven to try to make do with as little as they really need (after all, that too is a very recognizable mathematical goal). Mathematicians there still value making the unexpected connections we call ‘deep’, they distinguish important mathematical results from mere ‘brilliancies’, they explore fruitful new concepts, just like us. But when they turn to questions of ‘foundations’ they find it naturally compelling to seek minimal solutions, and look for just enough to suitably unify the rest of their practice, putting a very high premium on e.g. low-cost predicative regimentations. Overall, their mathematical culture keeps free invention remote from applicable maths on a somewhat tighter rein than here, and the old hands dismiss the baroquely extravagant set theories playfully dreamt up by their graduate students as unserious recreational games. Can’t we rather easily imagine that mind-set being the locally default one? And yet their local Second Philosopher, surveying the scene without first-philosophical prejudices, reflecting on the mathematical methods deployed, may surely still see her local mathematical practice as being in intellectual good order by her lights. Why not?
Supposing that story makes sense so far (I’m certainly open to argument here, but I can’t at the moment see what’s wrong with it) let’s imagine that Maddy and the Fefermanian Second Philosopher get to meet and compare notes. Will the latter be very impressed by the former’s efforts to ‘defend the axioms’ and thereby lure her into the wilder reaches of Cantor’s paradise? I really doubt it, at least if Maddy in the end has to rely on her appeal to mathematical depth. For her Fefermanian counterpart will riposte that her local mathematicians also value real depth (and fruitfulness when that is distinguished from profligacy): it is just that they also strongly value cleaving more tightly to what is really needed by way of rounding the mainstream mathematics they share with us. Who is to say which practice is ‘right’ or even the more mathematically compelling?
Musings such as these lead me to suspect that if there is objectivity to be had in settling on our set-theoretic axioms, it will arguably need to be rooted in something less malleable, less contestable than Maddy’s frankly rather arm-waving appeals to ‘depth’.
Which isn’t to deny that may be some depth to the phenomenon of mathematical depth: all credit to Maddy for inviting philosophers to think hard about its role in our mathematical practice. Still, I suspect she overdoes her confidence about what such reflections might deliver. But dissenting comments are most welcome!