the stuff of proof

Interview by Richard Marshall.

Penelope Maddyis the candy-store kid of metaphilosophical logic and maths. She's stocked up with groovy thoughts about the axioms of mathematics, about what might count as a good reason to adopt one, about mathematical realism, about Gödel's intuitions, naturalism, second philosophy, Hume and Quine, world-word connections, about where mathematical objectivity comes from, about the limitations of drawing analogies, about depth, about Wittgenstein and the logical must, about the Kantianism of the Tractatusand about the relationship between science and philosophy. Suck it and see, this one has a fizz ...

3:AM:What made you become a philosopher? Are you a lone brooder or prefer to think and argue aloud with others?

Penelope Maddy:I started out in mathematics and was moved from there to philosophy by others, oddly enough, without really understanding what was going on. Foundational questions captured my interest early on: one of my most cherished memories is the sudden realization that the number 1 could be defined in naive set theory! Poking around in my great high school math teacher’s secret book closet, I soon came to understand that 2+2=4 and the rest of classical mathematics could be proved from the standard assumptions of axiomatic set theory, but that one of the first and most natural questions about infinite sets, the Continuum Hypothesis (CH), couldn't be settled one way or the other on the basis of those same axioms. What could a solution to such an open question even look like?!

At the time, UC Berkeley was the place to go to study set theory: forcing was new, and larger and larger large cardinal axioms were being proposed in turn. Another vivid memory is watching in awe as one of my professors showed us the proof that if there's a measurable cardinal, then one of the open questions (not CH, alas) has an answer (there are sets outside Gödel's minimal universe). This was just the answer one would want and expect, but why in the world would one think that this candidate for a new axiom -- 'there are measurable cardinals' -- is true?! Perhaps there could be new axioms even to settle CH, but what counts as a proper argument for or against a proposed axiom?

Without realizing it, I'd slipped into philosophy. When I applied to the Princeton math department for graduate school, they admitted me instead into the program in history and philosophy of science on the basis of my statement of interests. Being from Berkeley, I figured this must be a program like their Logic and Methodology, but when I arrived, it turned out I was pretty much just in the philosophy department. The transition took some fierce adjustments and teetered on disaster at times, but I eventually came to see the wisdom of those admissions officers.

Given your two choices, I guess I have to go with 'lone brooder', though ‘brooder’ doesn’t quite fit -- often these days I feel more like that kid in the well-stocked candy store.

3:AM:You’re interested in metaphilosophical issues of logic and maths. So let's start with the status of mathematical proof. It was once thought that maths rested on logic – ‘the stuff of proof’ as you put it, and axioms of proof - but now we don’t think there are such axioms. Is that the issue, and what happened that meant the desired axioms went missing? Was it Godel?

PM:I think what happened was the rise of pure mathematics in the 19th century. Before that, mathematics was done in tandem with science; often enough the two weren't even distinguished. But when mathematicians gradually started thinking about mathematical structures with no apparent connection to applications, for their purely mathematical interest, a need arose for a precise way of specifying those structures and making sure they were coherent. The eventual solution was to specify them by precise 'axioms' -- definitions, really -- and to establish their coherence by showing that they could modeled by sets, that is, that the existence of a set with the desired structure could be proved from the axioms of set theory. So the axioms of set theory became the fundamental assumptions of mathematics.

As you say, this wouldn't have been unsettling if the set-theoretic axioms could have been thought of as purely logical; it wouldn't be a big deal to assume them if they were really just obvious or intuitive or self-evident or whatever (these are often termed ‘intrinsic’ considerations). But that hope collapsed with Russell's discovery of a paradox in Frege's system. Some of the standard axioms of contemporary set might be 'obvious' -- 'two sets are the same if they have the same elements' might even be taken for a definition -- but many observers don't think this can be said of all the standard axioms, and the new axiom candidates that we need to be evaluating in hopes of settling the open questions leave this sort of justification far behind. So there’s that question again: what could count as a good reason to adopt an axiom if it's not obvious or intuitive or self-evident?

3:AM:Now your idea of 'mathematical realism' was thought to be an account that answered the problem – can you say what this position claims about the status of maths?

PM:The main problem my early realism was intended to solve arises before you even start to worry about justifying axioms. When faced with the sad fact that CH can't be settled from the standard axioms, some reacted by denying that there’s any further question to be raised: the only question that makes sense, this line of thought goes, is whether CH or not-CH follows from the axioms; once that's settled, as it has been, there's nothing more to ask -- case closed.

Gödel thought CH was still in some sense a real mathematical question, and he espoused a strong sort of realism about sets to defend that view: there is an objective world of sets where CH is either true or false; the axioms we now have aren't enough to fully describe that world; we need new axioms to settle the question. CH also seemed to me to be a real question that I wanted answered, and as a novice philosopher, I figured I could do worse than following Gödel's lead. Unfortunately, Gödel's realism apparently relies on a kind of 'mathematical intuition' that strikes most observers, including me, as disconnected from anything we know about human cognition; one of Benacerraf's seminal challenges is aimed at just this point. So I tried to replace Gödel's intuition with something based on ordinary perception, with appeals to experimental psychology and neuroscience.

Not everyone was convinced, to put it mildly. But the resulting account, if it worked, would not only certify CH as a real question, but also provide a framework for approaching the problem of justifying axioms: as Gödel suggested, we would think of set theory and theoretical natural science as roughly analogous; perception is to science as intuition is to set theory; and theoretical confirmation is to science as a corresponding sort of argument (often termed ‘extrinsic’ considerations) is to set theory. I hoped this set-theoretic counterpart to scientific theorizing would be the necessary key to assessing axiom candidates by something other than their 'obviousness', etc.

3:AM:You find it unsatisfactory don’t you? What’s the problem with it?

PM:Alas there were three problems with this hopeful position. Much as I'd helped myself to Gödel's view as a starting point, I'd also taken the Quine/Putman indispensability argument off the shelf: the view, widely accepted at the time, that the existence of mathematical abstracta is confirmed along with the scientific theories in which it plays an indispensable role. When I got down to thinking about these matters more carefully, it didn't seem to me that science and mathematics are interrelated in the way it describes. So that was one problem.

Another came when I got serious about trying to understand extrinsic justifications as analogous to theory confirmation in natural science. On closer inspection, arguments that seemed to me to present a good mathematical case for or against some axiom candidate looked like just so much wishful thinking when compared with scientific reasoning. To put the point roughly, the fact that the existence of measurable cardinals deliver a desired conclusion (mentioned in (1)) would count as extrinsic evidence in its favor, but if we’re really out to describe an objectively existing world of sets, why should we expect it to conform to our desires? Faced with this extrinsic evidence, my realist could reasonably respond: yes, I see that the axiom of measurable cardinals would lead to lots of welcome mathematical results, would generate a lot of deep and important mathematics, but what if -- sadly, sadly! -- there just aren’t any measurables?! This seemed like a bad response to me -- the mathematical virtues should carry the day -- but if welcome theoretical structure were a criterion for acceptance in physics, we’d never have ended up with quantum mechanics. So the strong analogy between theoretical natural science and higher set theory began to lose its charm.

The third reason is the one that moved me into philosophical methodology. If you're a 'naturalist', you think that science shouldn't be held to extra-scientific standards, that it doesn't require extra-scientific ratification. I thought I was a naturalist because I was out to make Gödel's intuition scientifically respectable, and because I didn't subject mathematics to extra-mathematical criticism, but I was still trying to base a mathematical decision -- is CH a question worth pursuing? -- on extra-mathematical grounds. In a slogan: if extra-mathematical considerations can’t count against mathematical moves, they can’t count for them, either. Realizing this made me slap my forehead for having missed the obvious for so many years.

3:AM:Your response to rejecting mathematical realismis to turn away from metaphysics and towards mathematics and you call this ‘naturalism.’ What is this position and why is it better than realism?

PM:By ‘naturalism’ in this context, I mean the idea that mathematical decisions -- is CH worth pursuing?, is the axiom of measurable cardinals a good axiom? -- should be taken on mathematical grounds, without interference one way or the other from extra-mathematical considerations of metaphysics or epistemology. This may sound like a truism, but it actually has bite: it not only rules out my realist’s complaints about extrinsic justifications, but also rejects metaphysical arguments against the Axiom of Choice, arguments from philosophical semantics over classical logic, and so on.

3:AM:Is this an example of what you mean by ‘second philosophy’? I think you root its rather austere approach to the Wittgenstein of the ‘Tractatus’ where he says that the right philosophy would be to say nothing except what can be said and that what can be said is exhausted by propositions of natural science. This seems odd in that isn’t mathematics not a proposition of natural science so presumably you’re not drawing the same conclusion from Wittgenstein’s idea that he drew but are looking to extend his suggested methodology?

PM:Yes, a broader version of this ‘naturalism’ is what I came to call ‘Second Philosophy’. Early on, I’d assumed that everyone knew what it was to be a naturalist, that all I had to do was explain how to extend this to mathematics, but over the years -- when the first question at so many of my talks began ‘but that can’t be naturalism, because naturalism says … ‘ -- I came to understand that a bewildering array of views are called ‘naturalism’. So I set out to elaborate what I meant by ‘naturalism’ and coined a new term for it to avoid squabbles over the word. My goal was to lay out this metaphilosophical backdrop and to take a position on the ground of logical truth, so that I could then swing back and address the metaphysical and epistemological questions about set theory. Though I’d argued that these considerations are irrelevant to methodological decisions in mathematics, I certainly didn’t conclude that they were unimportant or pseudo-questions (though I was interpreted to have said that more than once).

As to the roots to Second Philosophy, it’s true that I do use that remark from the Tractatus as a rhetorical device in the introduction to the book, but the real forebears are Hume and Quine, and as I now understand matters, Thomas Reid. Wittgenstein actually had a decided anti-scientific bent, but we’ll get to that in a moment.

3:AM:Guys that seem obvious naturalists – like Quine and Hume say – don’t really make it as second philosophers do they? Why not?

PM:Hume takes a huge step forward in ‘introducing the experimental method of reasoning into moral subjects’, but pioneer that he is, he understandably trips up in the execution. One oddity that I’ve so far been unable to fathom is his assertion that since mathematics and natural science ‘lie under the cognizance of men, and are judg’d by their powers and faculties’, it follows that they are ‘in some sense dependent on the science of Man’ (Hume’s investigation), that ‘the science of man is the only solid foundation for the other sciences’ (from the introduction to the Treatise). The Second Philosopher would happily grant that the study of human perceptual and cognitive mechanisms could lead us to re-examine the evidence we’ve accepted for other physical or mathematical theories: e.g., Kepler was worried about the distortions introduced into astronomical observations by the fact the human eye, like the telescope, involves optical transmission through a small aperture. (This is how he came to discover the retinal image and revolutionize vision science.) But this hardly means that psychology trumps all!

I don’t know how this oddity connects to the more conspicuous problem: though Hume announces that his experiments will involve ‘a cautious observation of human life … in the common course of the world .. men’s behavior in company, in affairs, and in their pleasures’ (again the introduction), what he actually does is introspect into the nature of impressions and ideas. This approach reaches its apex in part IV, where a quick run through the argument from illusion leads him to the theory of ideas, and from there to radical skepticism. As Reid observed, when your train of thought ‘ends in a coal-pit’, you ought to acknowledge that you’ve gone wrong somewhere and try to figure out what misled you!

As for Quine, I don’t think his holism captures the structure of scientific inquiry, and despite his insistence on ‘naturalized epistemology’, he often falls into thinking of some pure sensory input or surface stimulation as the ‘data’ for our ‘theory’ of ordinary physical objects and the rest. But there is no such data from which we infer the existence of objects; there’s just a long causal process from object to eyes to brain to belief. (Barry Stroud has made this point against Quine.) In the philosophy of mathematics, I’ve already mentioned that I don’t think the indispensability arguments properly capture scientific methods, mathematical methods, or the relationship between the two -- and I depart from him on the ground of logic as well.

But all these are quibbles. Though Second Philosophy differs in these ways from the ‘naturalisms’ of Hume and Quine, they are both towering figures in the history of naturalistic thinking. (For what it’s worth, in the historical pantheon, it seems to me that Reid actually comes closest to what I have in mind.)

3:AM:How does your examination of the contemporary debate over the nature of world-word connections help you show what second philosophy can do? I guess it’s through showing what it contrasts with that helps.

PM:That was an important motivation for including the part on word-world connections in the book: showing how the Second Philosopher would wind her way through a familiar debate, so as to help characterize her point of view. I also wanted to sketch in Mark Wilson’s position on this, both for its inherent interest and importance, and as an example of a contemporary philosopher who (I would claim) is functioning as a Second Philosopher would. (Mark might dispute this!) Finally, I wanted to have a developed take on the debate between the correspondence theorist and the disquotationalist on the table so that I could make a point in the later discussion of mathematics: people often think that disagreements about metaphysics in mathematics are closely linked to disagreements between correspondence and disquotation; I wanted to be able to argue that this isn’t right.

3:AM:There’s a puzzle that seems to arise from your approach isn’t there. If maths isn’t to be grounded on metaphysical truths then how come it’s so fruitful in the concrete world? How do you approach this so-called miraculous aspect of maths? Is yours a sort of Yabloistfictionalism?

PM:To be honest, I’m not sure how grounding math in a world of abstracta, causally isolated from us, would help explain why it works so well in applications. In any case, my take on the so-called ‘miracle of applied mathematics’ is that it’s not really so miraculous.Years ago, when I was thinking hard about this problem, I happened to hear a lecture by Persi Diaconison coincidence. Diaconis is a statistician by trade, but also a magician and a well-known debunker of psychic phenomena. Some of that debunking involves psychological and statistical observations about what appear to be striking coincidences. One is the ‘new word’ phenomena: you learn a new word, then suddenly, by absurd coincidence, you hear it three times in the next 24 hours! But of course it isn’t an absurd coincidence; now that you know the word, you notice it. Similarly, when you discover a mathematical tool that can solve a certain kind of problem, you tend to notice when problems with that structure turn up -- what a striking coincidence that so many situations can be described mathematically! There’s also ‘selective memory’: you notice the case where it works and forget the cases where it didn’t; how amazing that it works so often!

Think of all the worldly situations that can’t be effectively modeled mathematically. Or there’s the ‘law of large numbers’: in a population of 250 million, a ‘million to one chance’ happens 250 times; with the huge range of well-studied pure mathematical structures, it’s not surprising that some of them find application. As I listened to Diaconis’s lecture, I realized that each one of the errors that lead us so naturally to think there’s a coincidence demanding explanation (e.g., this person must be reading my mind!), could also lead us to think that the applicability of mathematics is an amazing coincidence.

3:AM:And where does mathematical objectivity come from – are you sympathetic to someone like Stillman who argues that elegance and aesthetic justifications for theories are not subjective – he draws on Nietzsche who contrasts Kant’s super-subjective view of beauty with Stendhal’s more active view – and does this help answer the question what pure mathematicians and set theorists are doing?

PM:The idea that mathematical decisions of the sort I’ve been talking about -- evaluating axioms, for example -- are really aesthetic judgments is a popular one, but I confess I find it unhelpful. First off, we don’t understand aesthetic judgments very well, so it wouldn’t be a clear gain even if it were true. But more importantly, it doesn’t seem to me that the mathematical judgments we’re considering are of the same sort, or of a sort analogous to, aesthetic judgments. Something makes the axiom of measurable cardinals a good candidate, but I don’t think it has much to do with what makes a Cezanne’s landscape a great painting, or Beckett’s ‘Happy days’ a great play, or Trollope’s Last Chronicle of Barseta great novel, or Cunningham’s ‘Points in Space’ a great dance. (For that matter, I’m not even sure how much there is in common between the painting, the play, the novel and the dance.)

So, for example, one of the extrinsic considerations in favor of measurable cardinals is that their existence implies the existence of a very special set of natural numbers called 0#, and this set of natural numbers helps us see in great detail how it is that Gödel’s minimal universe goes wrong. You might try to see this as something that could also apply to paintings and novels, but I’d wager than in doing so, you’d have to loosen it up so much that you’d lose the mathematical texture that gives this extrinsic justification its force. Wittgenstein once said that if you wrap tables, chairs, and cupboards in enough paper, they’ll all look spherical -- the point being that you can make them all seem the same, but you lose what’s important about them in the process.

I think what’s happening here is that it’s so hard to understand what’s going on in mathematics that we grope around for an analogy: math is like science, math is like a game, math is like fiction, math is like art. One conviction I’ve developed over the years is that in the end, these analogies aren’t going to help: the only way to understand what’s going on in mathematics is to buckle down and look at the mathematics itself, at how it’s done and why.

My own view is that the objective ground of these mathematical decisions is in a quality of the mathematics itself, something mathematicians call depth. Measurable cardinals are good because (at least as far as we can tell at this point) they facilitate deep mathematics and don’t block any alternative deep mathematics. I’ve claimed that depth is an objective feature, but in fact this is a notion that hasn’t received much philosophical attention, so it’s a little early to say.

3:AM:Your latest book is about the problem of the logical-must and through it you argue that Wittgenstein in both his Tractatusphase and then his Philosophical Investigationphase he’s grappling with a way towards your idea of second philosophy. Is that right? If you’re right then there’s a link between Kant’s approach to logic and Wittgenstein’s. So can you sketch Kant’s thinking about logic first before we look at Wittgenstein so we get to see what he’s trying to explain?

PM:Well, I don’t think it would quite do to say that Wittgenstein was grappling toward Second Philosophy; as noted earlier, he severely disapproved of scientific thinking (‘the poverty and darkness of this time’) and firmly distinguished philosophy from science, in both his early and his late work. What I do try to argue is that what he says about the ground of logic in the Tractatus -- that it lies in the structure of the world -- and about our inferential practices in the Philosophical Investigations-- that they rest on our interests and abilities, and very general facts about the world -- that these views are entirely consistent with the Second Philosopher’s view of logical truth. There’s an extra assumption about the nature of meaning that separates the Tractarian view from Second Philosophy, but that assumption is rejected in the Investigations as well. All that’s left separating them in the late work, then, is Wittgenstein’s rejection of all things scientific, and I argue that this opinion of his isn’t essential intertwined with his core philosophical views. In other words, it can be removed without affecting the rest, at which point the second philosophical position can be seen as an empirical elaboration of that shared core.

3:AM:So what is Kantian and Second Philosophical about the ‘Tractatus’?

PM:I first came to the second philosophical view of logic by thinking about Kant’s position, so I use it as a point of entry. This then helps with the interpretation of the Tractatus, not because the Tractarian position is Kantian, but because it can seem Kantian, and seeing why it nevertheless isn’t Kantian is revealing. Or so I claim!

3:AM:Several scientists seem dismissive of philosophy. What do you say to such skepticism about the value of philosophy? Do you think the criticism is symptomatic of a deeper philistinism in society general?

PM:I don’t think Robert Boyle would have dismissed Locke when Locke was working in his lab, and later, when Locke was devising his metaphysical and epistemological framework in close engagement with Boyle’s corpuscular science. For that matter, Descartes wrote the Meditationsas part of a project to replace Aristotelian scholasticism with the new mechanistic physics. Berkeley’s New Theory of Visionremains a landmark in vision science, turning the field from optics to the psychology of perception, not to mention his cogent and enduring critique of contemporary calculus and his sensitive grappling with the new action-at-a-distance in Newtonian science. Reid was also instrumental in the development of vision science, with his fundamental distinction between sensation and perception.

This kind of fundamental interconnection between philosophy and science continues today. Philosophers of physics contribute to the foundational debates in quantum mechanics, general relativity, thermodynamics, continuum mechanics and more. Philosophers of psychology work hand-in-hand with cognitive scientists, neuroscientists and experimental psychology in areas from the theory of vision to the theory of consciousness. Austin predicted and contributed to the emerging science of linguistics, for example, with his theory of speech acts. Wittgenstein’s family resemblances provide a basic theoretical tool in the psychology of concepts. The philosophy of set theory described above is explicitly motivated by a real problem in the practice : what are we to do about CH and why?

There’s much more of this, of course, but my point is that many scientists welcome the contributions of philosophers, for good reason. Many others, in this age of specialization, simply don’t appreciate the role philosophers have played and still play in their disciplines. But it’s also true that philosophy itself sometimes loses touch with the project it shares with the sciences, the project of investigating the world and our place in it. When philosophy becomes too detached, even willfully detached, from that broader project, I’m sometimes tempted to side with its scientific detractors.

3:AM:And for those of us here at 3:AM are there five books you could recommend to us to take us further into your philosophical world?

PM:It’s a pleasure to recommend a few of my current favorites! --

Margaret Wilson, Ideas and Mechanism.

George Berkeley, New Theory of Vision.

Thomas Reid, Inquiry into the Human Mind on the Principles of Common Sense.

J. L. Austin, Sense and Sensibilia, and Philosophical Papers (especially, ‘A plea for excuses’ and ‘Other minds’).

Gary Hatfield, Perception and Cognition.

(I’ve cheated a bit by adding a couple of essays from Austin’s Philosophical Papers, but Sense and Sensibiliais short.)

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