## the metaphysics of logic

Interview by Richard Marshall.

Penny Rush is a hard core straight scotch who never stops thinking about the logic of logic, its metaphysics and other crazy depths. She thinks about the relation between logic and reason, about the idea of the one true logic, about classical vs non-classical logics, about the four basic logical issues, about the metaphysics of logic, about Brady’s meaning containment and its implications, about mathematical realism, about Derridas’ analysis of metaphysics as theology and Husserl’s phenomenology, about meaning, objectivity, Barad’s model, Crispin Wright’s, Quentin Meillassoux and the paradox of independent reality and whether logic is in the same state of ‘otherness’ as mathematics. Drink each dram with care as dark nights fall in on us and things freeze…

**3:AM:** What made you become a philosopher? Is your preferred style lone brooding or dialogic argument or something completely different from these?

**Penny Rush:** I used to love argumentative discussions: talking way into the night about all sorts of things, especially so when it felt as though what was happening was a combined effort to understand or to get closer to some truth. But I love actual argument/dialogic discussion less now. I’m not sure why this is exactly, perhaps it’s to do with a new appreciation for space and quiet (which perhaps is itself something the responsibilities of work and motherhood taught me, or maybe it’s simply the result of an increasing awareness of the vastness of our ignorance). So I suppose lone brooding is preferable to me now, though I would never knock back the chance to question or listen to a genuinely thoughtful mind.

I’m not sure I did become a philosopher – I think it was probably more a process of realising that is what I am. As soon as I was aware enough to start trying to decipher my self, I knew what I most wanted to do was to think, especially about open questions: things most people do not yet suppose settled or understood. So I guess unanswered questions, but also revisiting interesting questions presumed answered, have always attracted me. I mean attracted in the sense of being pulled toward or entirely engrossed; there’s a strange element of compulsion in it for me: as soon as I hear phrases like ‘we can’t account for …’ or ‘this seems right except …’ I’m lost in a wondering that never really stops.

**3:AM:** You’ve addressed the question about whether logic is a model of reason, correct thought, laws of thought and such like. So why are logic and reason completely different animals?

**PR:** I’m not sure that I think they’re completely different animals, although there are plenty of reasons to think so, most of which have to do with the weird ways we actually use logic when we reason or the very different outcomes each can deliver in various scenarios – but I’ll talk more about those soon. What I’ve been interested in is exploring the idea that we may be better off if we draw back from the notion that logic is (or has to be) a model of correct reason. But this idea does not by itself mean that logic and reason must be completely different. For all the differences, there are a lot of similarities and a lot of shared features between the two. For example, correct reason (at least at times and in some contexts) shares with logic the goal to discover what follows from what or, at times, how and when the truth of certain ideas might entail the truth of others.

Conversely, a drive toward clarity and rigor is something logic can share with reason.

So, I’ve tried to challenge the idea that if logic does not describe or model the way we ought to reason, then it is in someway not doing its job or not being what it is supposed to be. This is in part because if we withdraw that idea we uncover a host of interesting questions which ought not, I think, be covered. Dislodging the underlying assumption that we know the relationship between reason and logic reveals the nature of that relationship, as well as of logic and reason themselves, as the open problems I suspect they ought to be. And, by withdrawing that assumption, we also further enable logic to freely develop as it will – perhaps in ways quite other to those we might anticipate while holding fast to any preconception of its proper role. For an example, the study of logic can include the exploration of the wealth of pure logical structures for their own sake, their fascinating interrelationships, and the many and various ways they relate to mathematical structures.

I’m thinking here of work like that of Walter Carnielli and Marcelo Coniglio, who seem to me to explore abstract logical structures in much the same way pure mathematicians explore abstract mathematical structures. Carnielli and Coniglio look at translations between logics, and resultant new logics, using mathematical theories (particularly sheaf theory and category theory). It seems to me that this sort of work follows the formalisms for their own sake in much the same way mathematics can – and, as has been demonstrated in mathematics, we can follow more freely if we don’t always have an eye on application or potentially suppressive canonical constraints. Some of higher order mathematics has no (known) application in the (known) physical world, but this does not stop it being interesting and important – nor does it stop mathematicians studying it. Logicians’ study of logical structures should, I think, be similarly unrestrained.

**3:AM:** Would this perspective problematise the view that logics are cognitive tools, prosthetics for the mind as someone like Dutilh Novaes might argue?

**PR:** Not really, because even if it turns out that the best way to think of logic and correct reason is as completely different things, they can still share important relationships, one of which may be the one Novaes proposes.

Stewart Shapiro characterizes the question of the relationship between logic and reason as “an instance of the philosophical problem of explaining how mathematics applies to non-mathematical reality”. It is, of course, a special instance due to the long history of the conception of logic as a model of correct reason, but I like this characterisation of the situation very much. It invites reflection on the similarities between logic and mathematics and suggests ways we might make some headway in understanding both.

Shapiro also offers a suggestion similar to Novaes’: that we could characterise logic as an addendum to natural language and human reason – something we use, for instance to more rigorously elucidate concepts in philosophy and mathematics. This is a nice, relatively neutral, alternative to the image of logic as prosthetic (which can carry the implication it is somehow human-purpose built).

**3:AM:** When we ask if there is one true logic do we raise the issue of how we could decide without already presupposing one on the grounds that we’d choose the one that is most logical to choose? Is there a vicious circle in this? Is it the same sort of question about how we might revise logic?

**PR:** Yes, it does seem as though we use logic to help us decide which logic to use (or which logic is best), even in more mundane contexts than asking whether there is one true logic, and this does present a problem to the extent that we might want to justify our use of the original logic in selecting any preferred logic, but also to the extent that we’d like to have any justification per se of any logical laws that we then find need to be utilized in any such justification. This sort of circularity has been much discussed in the literature, notable examples include Paul Boghossian’s chapter in his and Peacocke’s book *New Essays on the A Priori* and Crispin Wright’s paper: *Intuition, entitlement and the epistemology of logical laws*. In his *New Essay’s* chapter, Boghossian attempts to justify our use of the classical rule modus ponens (A, A->B, therefore B). He puts forward the notion (there and elsewhere) that some basic logical rules are themselves an inherent part of what their logical concepts actually mean (and that in this case, the otherwise unjustified or blind use of these rules in an argument attempting to justify them is warranted).

Crispin Wright takes a somewhat different view – he argues that, at base, there are some rules we just have to take on trust although, because it is rational to do so, we are entitled. So basic logical laws, for Wright, may be ‘beneath the scope of cognitive enquiry’. So, our use/faith in modus ponens may not be something which we can or need justify in the same way we’d need (or might want to) justify a choice of one logical system over another – the former may be subconscious, automatic, default reasonable, or a case of blind reasoning, and the latter conscious and deliberate, so it seems quite sensible to at least entertain the possibility that different standards apply in each case, hence mitigating the apparent circularity.

I like the idea of blind reasoning particularly – it’s a nice way to understand, not just the way infants and animals seem to utilize logical rules, but also the way the brilliant arguments of some astonishing individuals can illuminate, if only in flashes, a complex logical structure they themselves may never have encountered. The unfortunate inverse of this is perhaps less rare, though more difficult to explain!

The idea that their inference rules constitute the meaning of logical concepts actually has quite a long tradition, taking in Gentzen’s original system which introduced rules intended to entirely capture the meaning of each logical symbol, and Brandom’s inferentialism – a general theory of meaning that extends this idea to all concepts. For Brandom, the meaning of a concept just is the set of inferences involved (or that we ought to draw, depending on the context) in its application. His, though, is a very broad notion of inference, incorporating such things as simple observation and physical responsiveness to our surroundings.

But I think it’s important to note here that there’s quite a lot we really don’t know before we even begin any of these types of inquiries. For instance, it may look very clearly as though modus ponens is one of the rules that ‘the best’ logic would include (it seems obviously correct), but the exact nature of both our use of that rule and the rule we use is open to question. JC Beall points out that the rule we use when we think we’re using modus ponens might actually best be understood as a related multi-conclusion rule: a rule with the same premise set, but with a conclusion set encompassing A and ~A, or B. Or (as Jc more precisely summarized in a recent conversation) the rule that says: there’s no way to make everything in the premise set {A, A->B} true while making everything in the conclusion set {B, A&~A} untrue. It just so happens we often reject the conclusion set A and ~A, and choose B as the conclusion, just as in the original rule. But (as Jc puts it), what we learn from Gil Harman is that this choice is a sort of ‘rational inference’ (generally accompanied by a host of background rational rejection principles): it is not ‘logic’ (Beall’s paper on all of this is reprinted at *Philosopher’s Annual*)

So, perhaps, when we use modus ponens, our faith in that rule (which, by the light of reasons he offers in the same paper, Jc argues is not valid) is less a faith in the certainty of a basic logical law than in our own particular use (which may be largely subconscious) of the related multi-premise rule, which is valid. That is, our use may be less in the spirit of certainty and more a sort of stop gap: a rough and ready assumption that the context in which we are reasoning is consistent. But of course this assumption is also, as Jc puts it, extra-logical: part of our rational behavior, in this case, our tendency to reject contradiction (A and ~A) for most subject matters (Jc’s argument is a good example of what I talked about earlier – of the way in which holding logic and correct reason apart can better illuminate the nature of both).

So the questions of whether and how to revise logic and how to justify logical rules are separate, but both are involved in the question as to whether there is one true logic, and if there were, how we could know this (as well as which it was) along with the question of how we could justify such knowledge.

**3:AM:** Is the answer to the question ‘is there a one true logic?’ about deciding between classical logic vs non-classical logic or is it about showing how we can have both? And what are the four basic logical issues?

**PR:** I think the kind of research inspired by both sorts of answer is important, so I guess I’d not like to categorically rule one out! It’s important to think about whether we can (or whether it is useful to) judge the correctness of both or either against some overarching set of standards – which may be the usual sort of criteria by which we judge any theory (fruitfulness, simplicity, etc.) or may be the extent that they describe or discover some ideal or realist notion of the way logic itself is (say, independently of the way we think it is, or the way it can be). This sort of context motivates a search for a ‘universal logic’: a logic that is as widely applicable as possible, or the most widely applicable in comparison to other logics. And such a search, in turn, may help with questions like where and how to draw the line between theories – e.g. between the quantum (which is where quantum physics applies) and the middle-sized (which is where classical physics and classical logic seem to apply – though, perhaps other logical systems equally apply), and then again between the middle sized and the absolutely huge (which is where Einstein’s relativity theory comes into its own). Paul Francis offers just one reason this is an important question: namely that we don’t yet know how to theorize about something very big and very small at the same time (like black holes). Relatedly, asking after ‘one true logic’ in this sense also motivates ‘refinements’ of logic – driven by the notion that we can improve our grasp on the way logic (or validity or consequence or whatever) itself actually is.

Jc also had something interesting to say here (also in recent conversation): that, if we grant that logic is the universal closure for our theories, then there is exactly one logic: the “weakest (topic-neutral, universal, etc.) entailment relation governing all our theories”. This, intuitively, is an appealing way to characterize one sort of motivation underpinning a search for that logic. It is also one way to show up a lot of what logic won’t/can’t tell you – it can’t and shouldn’t, for instance, provide topic-sensitive information such as whether or not a particular subject is consistent or inconsistent, or, say, the nature of belief or knowledge.

Then again, it’s also important (and fruitful) to consider ways in which classical and non-classical logic (or logics) may be equally good or justified, and so we may in fact have a plurality of logics (having a look at Beall and Restall’s very readable Logical Pluralism should convince readers of the fruitfulness and inherent value of thinking this way).

In his chapter in my new book *The Metaphysics of Logic*, Stewart Shaprio characterizes the idea that there is one true logic as the idea that there is such a thing as being ‘simply valid’ (regardless of frames of reference). So the natural counter-point of such a view is logical relativism: the idea that logical validity depends on the wider structure or frame of reference in which it occurs. For me, both of these perspectives are interesting and fruitful in their own right.

Ross Brady’s four basic logical issues can be understood as issues for a non-classical logician working in the current time – that is, given the relatively recent proliferation of non-classical logics and questions arising out of that: about the status of non-classical and classical logic, the best logic or formal system in which we can or should talk about a logic itself, and the relationships between all of these.

So Brady identifies the first issue as choosing between classical and non-classical logic – of course this presumes a choice should be made. Brady and I make the case that his logic is to be preferred over classical logic, so at least in this respect, we work on the basis that there is (or may be, given the information to hand) ‘one true logic’. The second issue also works on that basis and is the issue (if the outcome of the first choice is pro non-classical logic) which of the non-classical logics to choose. Brady’s logic, MC (which I’ll talk more about soon) is a good candidate, given that it can easily accommodate ‘classical’ scenarios (where key classical rules can be proved, or where ‘ideal conditions’ hold sway – i.e. where we know, for a given sentence ‘p’ whether it or ~p is the case). So the third issue is whether and when, within our chosen non-classical logic, we can or ought to justify this sort of use of certain aspects or rules of classical logic; and the fourth is the related issue of whether classical logic should be used to prove results about our non-classical logic: i.e. it is about the use of classical meta-logic for non-classical systems. So all of these issues build on each another, and arise in the course of considering a non-classical logic.

**3:AM:** What are the powerful options in the non-classical logic camp – and can you suggest what they can do that you can’t do with classical logic? I ask this because as you know there are some who think classical logic can handle everything the alternatives can handle.

**PR:** One important thing that non-classical logics have done that classical logic has not (although, who knows, it may have, had Frege lived longer) is, after stepping carefully in problematic domains, to revise or rebuild completely in the light of suspicious results: classical paradoxes or limitations in areas like quantum physics, the foundations of mathematics, and plain old everyday reasoning in inconsistent or even just possibly inconsistent situations – have all inspired non-classical logics, and as a result we now have logics offering more nuanced and accurate models of deduction across at least some contexts and at most, more contexts than those classical logic can handle.

The advent of such logics means there are not many who would make the claim that classical logic can handle everything the alternatives can – it is widely acknowledged, for instance, that paraconsistent logics can handle inconsistent situations, and classical logic cannot (it explodes). But some may still claim that classical logic is to be preferred nonetheless. Even that sort of claim, though, has now to accommodate the fact of other logics and the apparent failures of classical logic (often quite glaring): so, even if it is to be preferred, the role and nature of classical logic are no longer the straightforward matters they were once considered to be.

More specifically, different non-classical logics provide various ways we can retain some of the intuitively appealing axioms of mathematics and still either avoid or make sense of a number of important paradoxes these axioms generate, whereas classical logic can really only try not to run into them (say, by withdrawing the axioms). For readers interested in following this up, I’m thinking specifically here of the set-theoretic paradoxes for naïve set theory (for more on this, see Mares and Paoli).

Another thing non-classical logic can do (in this case quantum logic) is model some of what goes on at the quantum level, something classical logic struggles to do (as Bueno and Colyvan point out, when applied to electron spin and direction, classical logic delivers two determinate possibilities for both possible directions, which is simply wrong or at best meaningless since, due to Heisenberg’s Uncertainty Principle, it is not possible to measure the spin of an electron, given its momentum or position, let alone in two different directions).

And non-classical logics can help us reason within our specific epistemic predicament – namely, we often get things wrong. Our knowledge is not complete or consistent and so relying on a logic which rules out both incompleteness and inconsistency could be dangerous. Jay Garfield adds that our predicament also includes the ‘epistemic hostility’ of our environment – that often our environment is actually hostile to our acquiring knowledge of it. And this seems to me quite obviously right, though it’s not quite as obvious where the boundaries of such hostility might lie. Non-classical logics that allow the possibility of inconsistent and incomplete states of affairs are just better equipped to help us reason in epistemically hostile environments.

And this is just scratching the surface of the list of things non-classical logic can do that classical logic cannot!

Perhaps the right way to look at it is that classical logic (possibly or by and large) applies to the middle sized world – and is sketchy at its edges (the very small and the very big) – although even this is an idea we should be cautious of: what if it turns out that middle-sized stuff behaves just like quantum stuff? Where do we draw the line? Brady points out that ‘middle-sized’ could be thought of as a rough guide, with the concepts of consistency and finitude providing more accurate boundaries.

Either way, though, it seems preferable to choose a logic that can accommodate complete and consistent domains as well as go beyond them, over one whose reliability beyond such domains is dubious.

I’m not sure I’d like to list powerful options – there are too many and too many contexts in which that question might be asked. The ones I guess I’m most familiar with are intuitionistic logic, Priest’s LP and Brady’s relevant logic, MC. I very much like the motivation and overall power of Brady’s logic – and, as Brady points out (in conversation) “there are a number of non-classical logics which might best be characterized as ‘experimental’, given by a technically interesting set of axioms and rules, but without a clear concept”. That MC has a clear concept – meaning containment – sets Brady’s logic apart in this respect.

**3:AM:** The entailment logic of ‘Meaning containment’ is one of these non-classical logics isn’t it – it doesn’t have the law of the excluded middle and it doesn’t have the disjunctive syllogism? Why does this show that there are problems with Cantors diagonal argument?

**PR:** Yes, meaning containment (MC) is Brady’s logic: it is characterized by content semantics – a semantics which could be viewed as a generalization or abstraction of the way meaning (content) behaves. All of the notions normally utilised in spelling out logical concepts or logical consequence (e.g. ‘sentences’, ‘formal’ and ‘follows from’) rely on some implicit notion of content. So it is a straightforward, though not usually explicit, formal constraint on logic that it ranges over contents. But what if content itself has a structure: what if there are some basic, intuitive axioms for contents analogous to the basic axioms for the Natural Numbers? If there are, it is as important to preserve these basic ‘laws of content’ in any extension of the concept as it is to preserve the basic laws of arithmetic in any extension of the concept ‘number’. And I think that the semantics for MC can be justified in just this way: i.e. by an appeal to step carefully and stay within the constraints of our original grasp on just this concept – the concept of meaning as content (compare Brandom’s inferentialism with the content-semantics idea that meanings are contents within which sets of other meanings are contained – or, as Brady more precisely puts it, with the content semantics idea of ‘intensional set-theoretic containment’: contents are intensional sets of formulae which are analytically closed). But showing this – that the content structure of MC can naturally be distilled from this original grasp (i.e. can indeed be seen as the ‘bare bones’ of meaning) through a process of abstraction or formalisation – is a work in progress.

MC is a relevant logic, one of the logics motivated by the basic idea that the premises of an argument should be relevant to its conclusion. And the law of excluded middle (~A v A) and the disjunctive syllogism (AvB, ~A therefore B), both of which are valid in classical logic, do indeed make logical arguments valid which seem quite convincingly to be not valid, e.g. the argument ‘I am wearing jeans, so either evil exists or it does not’. Two things to note about this argument – there are some cases where it seems wrong to hold that things must be simply either true or false: maybe we just don’t know, or maybe they’re best characterized as somehow being both. Just noting this much means we ought to doubt the legitimacy of the LEM (which states that everything must be true or false, or that either a claim or its negation hold). The other is that it seems a clear case against the use of irrelevant reasons to justify conclusions (but this is exactly what classical logic allows). But as Brady points out, the question of the status of the law of excluded middle and the disjunctive syllogism is not definitively tied to considerations of relevance: Ackerman’s original relevant system had both, as an axiom and a rule, respectively.

To see why we should doubt the disjunctive syllogism, suppose I believe A. But then I can also, by the rules of classical logic, believe A v B for any B whatsoever, regardless of whether or not I believe B, or even whether or not I know what B is. So, crucially, B can be entirely irrelevant to A. The problem with this is, if I subsequently come to believe that A, after all, is not true, the disjunctive syllogism says that from this information, and utilizing my original A v B I can now infer B. But what if B had nothing to do with A? It doesn’t seem right that I could infer there are no whales in the ocean (call this claim B) from first of all believing that it’s sunny outside, and then coming to believe it is not raining outside. This would actually be irrational! (Jay Garfield’s dog provides a great concrete example of this sort of reasoning in action).

This does not mean that we can never reason rationally using rules like these – but it does mean that it’s worth asking whether that use is actually reasonable in a given context or whether its use there may be over-reaching. The principal problem with Cantor’s diagonal argument is just this: that it overreaches – it utilizes the LEM in an area where that utilization can’t (upon inspection, and given the sorts of considerations I’ve just very briefly touched on) be reasonably defended.

Brady and I argue in our paper on the subject that the LEM for the natural numbers can be justified only when we have good rational or technical reasons to assume it, or when it can be proved (or perhaps by some combination of proof and rational assumption). A proof could take a number of forms: we could show (for some statement or function ‘A’ over the natural numbers) that rejection of A entails acceptance of ~A; or we could prove A or ~A by mathematical induction for an arbitrary n; or we could directly prove either A for all n, or ~A for all n. In the paper, Brady shows that a proof for this sort of general form of the LEM (for any A) over the natural numbers should not go through, so its legitimacy needs to be considered case-by-case (via a close look at the particular A proposed). And in the case of Cantor’s argument we’re relying on a function (the infinite array of 1’s and 0’s) and a set (down the diagonal of this array), for neither of which we can justify or prove the LEM (the function is an arbitrary 1-1 correspondence so there’s no principled way to establish that the LEM holds for it, but even if there were, we can’t assume or show the LEM for the specific extra element of the diagonal set – i.e. there’s no foundation for supposing that it either is or is not an element of the diagonal set (maybe it’s neither) and no way of proving this either). So, Cantor’s Theorem is actually an open question.

**3:AM:** Does this logic have an impact on other branches of math?

**PR:** It has an impact on our understanding of the deductive processes underlying other branches of mathematics, and offers a new way of understanding the key theorems, so yes! One very definite impact is the retention of the appealing axioms of naïve set theory without engendering the paradoxes (see *The Simple Consistency of Naïve Set Theory*). This sort of result in turn gives us further reason to doubt those classical rules which do engender paradox in mathematics. There’s a discussion on this in the last section of the paper on Cantor’s proof, for anyone wanting to know more.

**3:AM:** You’ve wondered whether mathematical realism isn’t too much like anti-realism to be anything more than an ad hoc stipulation. So what is realism in maths and why do you think it is pretty lame as it stands – and are you thinking this is important because without realism a claim of objectivity and certainty of mathematical truth is threatened?

**PR:** ‘Realism’ has become an over appropriated term, to the point that now it’s almost meaningless. But it is probably safe to say that it is (or was) usually intended to pick out a position that involves the belief that mathematical reality is somehow independent of us – that it is, and is the way it is, regardless of whether or how we think of or construct its objects.

Certain types of realism and anti-realism look compellingly like two sides of the same coin, but particularly so in mathematics, where the objects are abstract and, in a sense, fully given by the formalisms. Physical reality is somewhat different – there we have a clear sense that there’s more to an object than our impression of it, whereas it’s hard to see how there could be any more to mathematics than the formal construction or description of its objects (where’s the rest of it supposed to be?). So one central problem for a mathematical realist who wants to say that the objects of mathematics exist independently of the way we conceptualize or describe them, is to introduce at least something like that sort of gap – between our impression and reality itself – for mathematical objects.

So, I think a mathematical realist needs this gap in their account: a gap between the way mathematics is, in reality, and the way we take it to be. But I also think we can’t simply stipulate it (tagged onto an account of mathematics which could just as easily – or in effect – be anti-realist). If a realist account of how we know mathematics delivers structures whose existence could be dependent on, or nothing more than, the end result of that process, but then simply adds ‘and they’re independent’, it’s not convincing (or, better, I personally feel as though something crucial has been left out).

So yes, that does seem a bit ad hoc to me. I think mathematical realists need to grapple with the gap or at least to articulate it in such a way as to make the notion of an independent mathematical reality convincingly different from that of a constructed or even imagined one. Or, again, to put it more accurately, I think it’s worth wondering about what such a mathematical realist position would involve, and how it might be defended. And it’s worth it, not only for the reasons you mention, but also because some sort of realism seems most apt to describe the actual workings of mathematicians, whose motivation seems best captured by the notion of ‘discovery’ than ‘construction’. And finally, it’s worth it because when we don’t have the answers, we should be careful to leave all of the options on the table, especially the interesting ones!

**3:AM:** How have Derridas’ analysis of metaphysics as theology and Husserl’s phenomenology helped you think about the idea of what mathematics is – and is it your claim that maths is radically ‘other’, uncircumscribable by any account and literally beyond comprehension?

**PR:** Well, anything that’s not us is, in a very literal sense, beyond comprehension: if something would be (and be as it is) regardless of whether or not we humans were around to think about it, then it’s beyond comprehension.

So I’m not sure how radical this sort of ‘otherness’ is. Lately, I have started to wonder whether it is simply a part of our ordinary or common sense. But, analyzing the content of the idea, we get something like this: that what we know and perceive (and comprehend) is also not what we know and perceive (and comprehend). We tend to try soften this apparent contradiction when we come to articulate it (maybe by saying something like: ‘there’s an aspect of reality which is outside of what we comprehend’, or ‘there’s a sense in which reality lies beyond what we can know about it’, etc.) but my suspicion is that these sorts of ideas hide the deeper tension of paradox, or at least of contradiction.

So I’m very interested in thinking about this idea: the idea that (independent) reality is essentially both within and without (or better, outwith) our comprehension. And yes, there are ways and ways of cashing it out, but on the whole, I feel a niggling dissatisfaction with the ways I’ve seen that try to say that reality is in some way independent and to avoid any sort of truck with it being ‘beyond comprehension’. It seems to me that these sorts of accounts are, in the end, unable to offer any clear principled or non-ad-hoc way in which they are different from an anti-realist account where what we call independent reality just is (or is easily identifiable with) what we comprehend, or know or understand (or, as I said in answer to question 9, is the end point of this sort of process – of comprehension, construction, etc. – what‘s ‘given’ to comprehension). And this is particularly so for mathematical realist accounts. So, I think the mathematical realist just has to grasp this contradictory nettle, or risk that their account will (at best) be inter-translatable with, or (at worst) effectively indistinguishable from, an anti-realist account.

The nub of it is that I suspect that the whole idea of an independently real domain (particularly a mathematical one, but quite possibly any such domain) is able to avoid this sort of collapse into anti-realism just if it is, in this sort of sense, also ‘radically’ other.

Derrida and Husserl both take seriously the idea that an independent reality is entirely ‘other’ – but they do so in quite different ways. Derrida’s is negative – he shows just how tricky it is to posit a reality which is essentially different from us (or, to draw any sort of line between ‘us’ and ‘reality’) without somehow coming unstuck. He gives lots of good reasons to suspect that something strange is going on whenever we try to articulate any sort of external ‘ground’ in philosophy at all – which he thinks means that traditional, foundational philosophy is in a quagmire, and I guess I agree, but I like the quagmire – we are there because there’s a very interesting fracture that we just keep falling into: one that no bridge (built after scrabbling up one side or the other) can span.

On the other hand, Husserl’s phenomenology grapples more directly with the fracture itself and so the resultant picture, messy and confusing as it is, portrays our situation more faithfully than one which tidies things away too neatly, or is apparently without holes. The point, to reiterate, is that I like that it’s a strange situation we’re in: for one thing, I think allowing it to be strange (rather than trying to resolve it) casts light on a number of other such fractures through philosophical enquiry, but also it just seems right to me that what goes on at the everyday level when we simply discover our world is in fact not simple in the sense of able to be captured in any account reducing the phenomenon: be it to an epistemological process, ontological division, or scientific method. Any such reductions are our own – and that what is not our own (not us) is unsayable needs, I think, saying!

**3:AM:** You’ve an original account of where meaning is that is ontological, epistemological and hermeneutical. Firstly, can you say what puzzles you were attempting to answer with this so readers can orientate themselves?

**PR:** It’s been a while since I’ve looked at this, but I think I wanted to focus particularly on the ways we might think of meaning at an ontological level. I was trying to pin down what might be going on when we want to say (and again, think especially of mathematics here) not only does something mean x, but it also is x (e.g. the sense in which, when we ask what ‘2’ means, we are really asking what ‘2’ is).

So, I was trying to look at all of the ways meaning might be construed, but particularly as somehow objective. Meaning is often understood as social, or tied in other ways to human activity and language. We perhaps think less often of the ways certain sorts of meaning can also be tied to the world, or the reality it describes. But I think the relationship between meaning and world is exactly where everything gets most interesting – for all the same reasons that I like our quagmire.

**3:AM:** So where is meaning and how does your answer help sort out the puzzles?

**PR:** If meaning is, or can be, objective: something that is as well as something we somehow make, then the puzzle is sorted out in much the same way as the realists’: by allowing it to remain puzzling.

The relationship between word and world would then be one of difference and identity: we just have to allow (at least in some cases) that the content of each can be the same, and yet they are also irreducibly different.

Karen Barad has a promising model of this relationship, inspired by Bohr’s interpretation of quantum mechanics. In Barad’s model, the observer and the observed or measured, act together to draw their own boundaries within the moment they encounter one another. I like her thought that our accessing ‘external’ reality is no more problematic an idea than our accessing its meaning or ‘representational content’. I think any such dividing lines between us and pre-existing objects are by their very nature paradoxical. But I think what Barad’s model and my own share is a fundamental optimism: they both allow for the possibility that what we know or understand, while it does not infallibly capture, nonetheless somehow does essentially involve, reality itself.

**3:AM:** It’s considered an intellectual virtue to be objective in many circles. But objectivity seems difficult to capture. You’ve examined efforts to do so – what are the problems we have in trying to capture what this virtue is?

**PR:** Wright’s *Truth and Objectivity* does a wonderful job of considering what might be involved in pinning down objectivity – and his attempts to find an expression of objectivity that captures the realist concept without the typical realist problems are, I think, some of the most fascinating and challenging pieces of philosophy I’ve read. But ultimately, I think that so long as we grant the ‘transcendent’ version of realism as at all relevant or important to think about, then there will always be an important version of objectivity that resists any such attempts. This is one problem.

Other problems here include deciding what sort of a concept we want or is best suited to a given subject matter. In some cases, an attempt to be ‘objective’ could suitably be characterized as an appeal to what our collective best guess might be (say, were we all in an ideal state for knowledge acquisition) but in others, we might want to appeal to the way an ‘objective fact’ has nothing to do with us or our opinion.

And yet another problem is whether or not the question of whether or not a given subject matter is objective in this second sense. I would like to think it is, but of course this invites the problem of just how to find that objective framework in which such decisions might be made without first having to make a decision about that framework itself! (Shapiro’s defense of the objectivity of logic in Boghossian and Peacocke’s *New Essays* includes an interesting footnote on just how we could formulate this problem for logic).

Perhaps an appeal to objectivity traditionally (or before Einstein and quantum physics) meant to be an appeal to an at most defeasible knowledge of a fixed independent fact or reality. In this sense, neither quantum nor plain old everyday entangled reality is objective at all (and we can argue that ‘entangled’ encompasses what is ‘given’ in construction, comprehension etc.) But if, as realists, we try to un-entangle ourselves – to draw a (non-ad-hoc) line between ‘us’ and ‘it’ – we encounter the effects of the fracture again: the ‘objectivity’ we recover seems always to turn out to be either unknowable, or entangled after all.

**3:AM:** What is the paradox of independent reality and how does it link up with realism and tolerance? And are you arguing that rather than dissolving the paradox we should rather get to grips with it and acknowledge it? Wouldn’t that mean we had to think an impossibility?

**PR:** To take the last part of the question first: maybe not – maybe it means a logical impossibility (or contradiction) is the best way (that we currently have on hand) to describe the situation we’re in when we’re encountering a reality entirely independent of us, but also accessible by us. This does not necessarily mean that when we’re in that situation we think an impossibility – we don’t often describe our own knowledge process and its relationship to reality to ourselves, mostly we just know and relate – it’s when we come to thinking about what we’re thinking about, or asking how we know what we know, that the paradox comes in.

As for an articulation of the paradox itself and how it links up with realism, I can’t do much better than quote Quentin Meillassoux: “A realist, from the perspective of modern philosophy, is basically someone who claims to think that which is where there is no thought. That is to say, a realist is someone who keeps doing the opposite of what he says he’s doing: he speaks of thinking a world in itself and independent of thought. But in saying this, does he not precisely speak of a world to which thought is given, and thus of a world dependent on our relation-to-the-world?”

Then again, Husserl puts it beautifully as well: “[the problem, for anyone inclined to suppose a ‘transcendent’ or ‘independent’ realm, is] how cognition can reach that which is transcendent … [i.e.] the correlation between cognition as mental process, its referent and what objectively is … the source of the deepest and most difficult problems. Taken collectively, they are the problem of the possibility of cognition”.

Regarding tolerance, well, I think the paradox offers an as yet largely unexplored way to both acknowledge and respect ‘otherness’. Most people are probably familiar with attempts to engender tolerance via an appeal to some sort of relativity of truth, or some ‘internal’ account of what we can know: e.g. appeals to the idea that ‘reality’ depends on our perception, background etc., and so there’s no privileged perspective from which we can claim True knowledge (and so on). But, while such ideas may be helpful in some contexts (and I’d generally recommend them over their extreme opposite if those are the only two options on the table), I don’t think we always need some version of this sort of thing in order to grant that another point of view is as deserving of respect as our own.

If we allow for the paradox (or fracture), then we allow that even when we do know, we also do not. And this, I think, invites humility and the tolerance subsequent to humility, without the damage to optimism that internalist or relativist accounts can bring in their wake: that is, without sacrificing our faith in the possibility of genuine insight and interaction with what may lie entirely outside our ken (but, again, only hopelessly so if our ‘ken’ were indeed isolated, entirely internal or unable to so genuinely interact).

**3:AM:** Is logic in the same paradoxical state of ‘otherness’ as maths on your account? Are you committed to some idea of logical realism?

**PR:** I think what I’m trying to do is explore the possibilities: and that includes the possibility that logic and mathematics are independent and real in some essential way. I think it’s a bit dangerous to be committed to one stance as a philosopher – in my chapter in The Metaphysics of Logic, I explore how we could argue that logic, similarly to the way in which mathematics can, could be considered to be a real, independent structure (or structures) and there I also revisit the contradiction I’ve just discussed, but try to characterize it in a number of different ways, including logically.

**3:AM:** And for the intrigued readers here at 3:AM, are here five books you could recommend to take us further into your philosophical world?

**PR:** That’s very difficult! Rather than try to figure out which criteria I could possibly measure any such set against, I think I’ll just try to picture which of the books on my shelf are the most well- thumbed. By this criterion, I’d recommend:

Mark Steiner’s *The Applicability of Mathematics as a Philosophical Problem*; Feferman et al.’s

*Kurt Gödel: Collected Works*; Stewart Shapiro’s *Philosophy of Mathematics: Structure and Ontology *

Lévinas’ *Discovering Existence with Husserl*; and Derrida’s Edmund Husserl’s ‘Origin of Geometry’ An Introduction.

**ABOUT THE INTERVIEWER**

**Richard Marshall** is still biding his time.

Buy his book here to keep him biding!

First published in 3:AM Magazine: Friday, December 5th, 2014.