2011

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!

In defining a first order syntax, there’s a choice-point at which we can go two ways.

Option (A): we introduce a class of sentence letters (as it might be, ) together with a class of predicate letters for different arities (as it might be , , ). The rule for atomic wffs is then that any sentence letter is a wff, as also is an -ary predicate letter followed by terms.

Option (B): we just have a class of predicate letters for each arity (as it might be ). The rule for atomic wffs is then that any -ary predicate letter followed by terms is a wff.

What’s to choose? In terms of resulting syntax, next to nothing. On option (B) the expressions which serve as unstructured atomic sentences are decorated with subscripted zeros, on option (A) they aren’t. Big deal. But option (B) is otherwise that bit tidier. One syntactic category, predicate letters, rather than two categories, sentence letters and predicate letters: one simpler rule. So if we have a penchant for mathematical neatness, that will encourage us to take option (B).

However, philosophically (or, if you like, conceptually) option (B) might well be thought to be unwelcome. At least by the many of us who follow Great-uncle Frege. For us, there is a very deep difference between sentences, which express complete thoughts, and sub-sentential expressions which get their content from the way they contribute to fix the content of the sentences in which they feature. Wittgenstein’s Tractatus 3.3 makes the Fregean point in characteristically gnomic form: ‘Only the proposition has sense; only in the context of a proposition has a name [or predicate] meaning’.

Now, in building the artificial languages of logic, we are aiming for ‘logically perfect’ languages which mark deep semantic differences in their syntax. Thus, in a first-order language we most certainly think we should mark in our syntax the deep semantic difference between quantifiers (playing the role of e.g. “no one” in the vernacular) and terms (playing the role of “Nemo”, which in the vernacular can usually be substituted for “no one” salve congruitate, even if not always so as myth would have it). Likewise, we should mark in syntax the difference between a sentence (apt to express a stand-alone Gedanke) and a predicate (which taken alone expresses no complete thought, but whose sense is fixed in fixing how it contributes to the sense of the complete sentences in which it appears). Option (B) doesn’t quite gloss over the distinction — after all, there’s still the difference between having subscript zero and having some other subscript. However, this doesn’t exactly point up the key distinction, but rather minimises it, and for that reason taking option (B) is arguably to be deprecated.

It is pretty common though to officially set up first-order syntax without primitive sentence letters at all, so the choice of options doesn’t arise. Look for example at Mendelson or Enderton for classic examples. (I wonder if they ever asked their students to formalise an argument involving e.g. ‘If it is raining, then everyone will go home’?). Still, there’s another analogous issue on which a choice is made in all the textbooks. For in an analogous way, in defining a first order syntax, there’s another forking path.

Option (C): we introduce a class of constants (as it might be, ); we also have a class containing function letters for each arity (as it might be , ). The rule for terms is then that any constant is a term, as also is an -ary function letter followed by terms for .

Option (D): we only have a class of function letters for each arity (as it might be ). The rule for terms is then that any -ary function letter followed by terms is a term for .

What’s to choose? In terms of resulting syntax, again next to nothing. On option (D) the expressions which serve as unstructured terms are decorated with subscripted zeros, on option (C) they aren’t. Big deal. But option (D) is otherwise that bit tidier. One syntactic category, function letters, rather than two categories, constants and function letters: one simpler rule. So mathematical neatness encourages many authors to take option (D).

But again, we might wonder about the conceptual attractiveness of this option: does it really chime with the aim of constructing a logically perfect language where deep semantic differences are reflected in syntax? Arguably not. Isn’t there, as Great-uncle Frege would insist, a very deep difference between directly referring to an object and calling a function (whose application to one or more objects then takes us to some object as value). Again, so shouldn’t a logically perfect notation sharply mark the difference in the the devices it introduces for referring to objects and calling functions respectively? Option (D), however, downplays the very distinction we should want to highlight. True, there’s still the difference between having subscript zero and having some other subscript. However, this again surely minimises a distinction that a logically perfect language should aim to highlight. That seems a good enough reason to me for deprecating option (D).

Here’s a small niggle, that’s arisen rewriting a very early chapter of my Gödel book, and also in reading a couple of terrific blog posts by Tim Gowers (here and here).

We can explicitly indicate that we are dealing with e.g. a one-place total function from natural numbers to natural numbers by using the standard notation for giving domain and codomain thus: $latex f\colon\mathbb{N}\to\mathbb{N}$. What about two-place total functions from numbers to numbers, like addition or multiplication?

“Easy-peasy, we indicate them thus: $latex f\colon\mathbb{N}^2\to\mathbb{N}$.”

But hold on! $latex \mathbb{N}^2$ is standard shorthand for $latex \mathbb{N}\times \mathbb{N}$, the cartesian product of $latex \mathbb{N}$ with itself, i.e. the set of ordered pairs of numbers: and an ordered pair is standardly regarded as one thing with two members, not two things. So a function from $latex \mathbb{N}^2$ to $latex \mathbb{N}$ is in fact a one-place function that maps one argument, an ordered pair object, to a value, not (as we wanted) a two-place function mapping two arguments to a value.

“Ah, don’t be so pernickety! Given two objects, we can find a pair-object that codes for them, and we can without loss trade in a function from two objects to a value to a related function from the corresponding pair-object to the same value.”

Yes, sure, we can eventually do that. And standard notational choices can make the trade invisible. For suppose we use $latex `(m, n)’$ as our notation for the ordered pair of $latex m$ with $latex n$, then $latex `f(m, n)’$ can be parsed either way, as representing a two-place function with arguments $latex m$ and $latex n$ or as a corresponding one-place function with the single argument $latex (m, n)$. But the fact that trade between the two-place and the one-place function is glossed over doesn’t mean that it isn’t being made. And the fact that the trade can be made (even staying within arithmetic, using a pairing function) is a result and not quite a triviality. So if we are doing things from scratch — including proving that there is a pairing function that matches two things with one thing in such a way that we can then extract the two objects we started with — then we do need to talk about two-place functions, no? For example, in arithmetic, we show how to construct a pairing function from the ordinary school-room two-place addition and multiplication functions, not some surrogate one-place functions!

So what should be our canonical way of indicating the domains (plural) and codomain of e.g. a two-place numerical function? An obvious candidate notation is $latex f\colon\mathbb{N}, \mathbb{N} \to\mathbb{N}$. But I haven’t found this used, nor anything else.

Assuming it’s not the case that I (and one or two mathmos I’ve asked) have just missed a widespread usage, this raises the question: why is there this notational gap?