Inconsistent Mathematics, Reutersvärd, and Buddhism: An Interview with Chris Mortensen

by Michael Lopresto

IA_23_Bike Rack with Shadows

Chris Mortensen is Emeritus Professor of Philosophy at the University of Adelaide. He thinks that the inconsistent hasn't been taken seriously enough in Western philosophy, that the masterpieces of Reutersvärd rub our noses in the inconsistent, and that Western philosophy and Buddhism are complementary. He's the author of Inconsistent Mathematics (1995) and Inconsistent Geometry (2010).

Firstly, what made you get into philosophy?

I think I was always interested in it, really—since high school, anyway. I was diverted for while into maths and physics during my first couple of years at university, before coming back to philosophy. I realised that if what you want to do is what you like doing, then philosophy is the thing to do. I still kept up with the maths subjects, but philosophy was more fun, and I was better at it.


Had there always been a lot of overlap between your interest in philosophy and your interest in maths?

There was, but one thing I noticed was that my logic lecturers would always motivate what they were doing. They would tell you why this was interesting, why there was a debate here. Whereas my maths lecturers on the other hand tended to be very pure and syntactical, leaving aside motivation much of the time. Some logicians are very pure – some of my best friends are very pure. But perhaps it is possible to be a bit too pure and syntactical in philosophy, it depends on what you are trying to achieve I suppose. Just pop down to the library and have a look at Russell and Whitehead's Principia Mathematica. It doesn't contain too much English (even though Russell excelled as a philosopher, as opposed to a logician).

You argue that mathematics, including geometry, contain propositions that are true contradictions. Can you say something about how you arrived at this astounding result, and why you think it should be welcomed?

Well, I'm not sure that's the way I'd put it. What I'm really interested in is inconsistent content. I profoundly disagree with those who say that there's no such thing as inconsistent content, that the inconsistent has no structure, and that all inconsistent theories are trivial and therefore the same.

You might want to say that all inconsistent content in mathematics is false, but we can nonetheless make distinctions between the falsehoods. How do we make distinctions between the inconsistencies without using an inconsistency-tolerant logic? I don't think you can do it.

The same is true of inconsistent images. You can make a distinction between the Schuster Fork and the Ernst Stairs. They each have different 3D content. Note that it's 3D content, as opposed to just 2D content, which is just marks on a page. 3D content is where things get interesting, and that's where you get the inconsistencies.

3QD images

Do I think it represents some truths about geometry? Well look, it's got a 3D content of a set of stairs, and if you were to climb to the top one way, it would take two steps. If you were to climb up another way, it would take three steps. So we have this image that with a bit of argument we can see has as part of its content both an A and a not-A.

If you want to say that such is outright true, somehow external to geometry, that’s hard to agree. But we can certainly experience and think about these things, like we can think about unicorns. If I can think about unicorns, does that mean that unicorns have some sort of being? Maybe somebody like Heidegger might say that they do, but I don’t think that I’d want to say so.

I suppose you could say that we're thinking about unicorns that exist in other possible worlds?

I wish you wouldn’t say other possible worlds! Where are these other possible worlds? I don’t see them! No, I think possible worlds have their uses, independently of whether you see them as concrete or abstract, as long as you don’t put any explanatory weight on them for metaphysics.


Inconsistent Stairs

What do you make of attempts, like chunk and permeate—breaking inconsistent premisses up into consistent chunks to derive appropriate consequences—to consistentise various inconsistent phenomena?

It’s interesting that you should say that chunk and permeate is a method of consistentising an inconsistent set of premisses. In the original Brown and Priest paper, they call it an inconsistency-tolerant form of reasoning. Then they suggest that Newton’s reasoning about derivatives in the calculus took this form: in one chunk Newton reasons with the premiss that in the derivative function ∆x ≠ 0, and then in another chunk he assumes that ∆x = 0, and this gets the appropriate result at the end.

I think it’s a worthy project, and Newton does seem to have argued like that, but I think attempts like chunk and permeate and others are crypto-consistent. They belong to the context of discovery, and not to the context of justification. To be justified in believing in the conclusion of an argument, you have to believe each of the argument’s premisses equally, and that’s not what the chunk and permeate people are saying.

You're a champion of nonclassical logic. What do you think of resistance to making revisions to classical logic? For example, Tim Williamson (1992) says, “Classical logic and semantics are vastly superior to the alternatives in simplicity, power, past success and integration with theories in other domains.” I take it that you disagree with Williamson, or do you just think that nonclassical logic is valuable in other areas?

Richard Routley nailed this one nicely. He said that the issue of simplicity is not how simple the semantics is. Of course a two-valued Boolean algebra is going to be simpler in terms of semantics. But what matters is the issue of the applications of the logic across all areas of philosophy. Boolean logic struggles in its applications. Even so, it’s hardly to be dismissed: we understand the limits of Boolean logic, where it can be used, which explains its usefulness.


Inconsistent Columns

Are you a dialetheist about the semantic paradoxes, like the liar—”this sentence is not true”—or are you just a dialetheist about maths and geometry?

When the moon is full. Basically, I’m agnostic about the semantic paradoxes. What Graham Priest does in arguing for the dialetheist position is that he shows that you don’t need very much at all to get there. Just self-reference and a truth predicate—things that you and I, and nearly everyone else believes in. Then the challenge for the person who wants to resist the dialetheist conclusion is to say which one of the two they’re going to give up. Surely not the truth predicate. And surely not self-reference, since it’s completely unproblematic in most other cases: this sentence has five words. And furthermore, Gödel showed us that you can have self-reference in a mathematically respectable way. So I’m not convinced about the dialetheist conclusion, but I don’t have an alternative solution of my own. It’s a deep problem, but you can’t ignore the superb simplicity of the dialetheist position. Something really nice about the impossible pictures is that you have clear cases of contradictions without self-reference.

You study the masterpieces of Reutersvärd and Escher. Do these artists give us examples of true geometrical contradictions? Or do you make a somewhat lighter claim that their work only gives us experience with inconsistent content, and that we can only study this content with inconsistent methods?

Truth in geometry is a very complicated issue, so it's definitely the weaker claim that we're studying inconsistent content. And when you've got inconsistent-tolerant methods at your disposal, why not put them to use to do explanatory work. For instance, how do you explain the human ability to experience inconsistency? I can show you the formalisms, but it's so much more convincing when you've got your nose rubbed in it. Take Reutersvärd's original picture, Opus 1, which is very striking:


Opus 1

What do you think of the philosophical motivations of Reutersvärd and Escher?

They were both outstandingly creative. Oscar Reutersvärd made Opus 1 when he was 18 years old when he was bored at his Latin high school class, in 1934. He was twenty years ahead of the field! I don’t think that he knew what he had at first, it was more like an inspired accident, though he could see that he produced something uniquely creative. He worked on and off on these images, but basically pursued a career as a professor of art history at Lund university. There was really nothing like the kind of thing that Reutersvärd was doing until the Penroses, Lionel and Roger, published a short paper in 1958, and Escher started his impossible pictures around the same time. Reutervard only started again around that time. Then he was very prolific, some estimates say that he produced over 4000 impossible images. I was just at Indiana University, where they had acquired a collection of well over 350 of Reutersvärd’s works.

What do I think Reutersvärd was doing? He uses the word “paradox”, but a paradox is not the same as the proof of a contradiction. The Penroses wrote their paper in 1956, and it got published in 1958. It had a picture of the triangle, but not chopped into blocks like Oscar's original. They also had a picture of the stairs. But Oscar did a picture of the stairs in 1937—and it was better!

Escher on the other hand was quite a bit older than Reutersvärd, and started on the impossible pictures much later. Escher was trying hard to work through paradoxical content. I wouldn’t say that he did anything genuinely inconsistent until the mid-1950s, although he was working on things during the ‘30s and ‘40s. He did High and Low, where he tried to show the same scene from two different perspectives. It’s almost inconsistent, but not quite. Even though Oscar got there first, I have the highest regard for Escher. Artists got really keen on the work by Reutersvärd and Escher, and started a school called “Impossibilism”, whose purpose was to depict impossible objects and impossible situations.

Anglophone philosophy is often claimed to be very narrow in what it considers to be “actual” philosophy, and exclusionary to other traditions. Is this what led you to study Buddhism? How do you see the relation between Buddhism and traditional Anglophone philosophy?

I’ve been interested in Buddhism ever since I was in high school, when I first heard the Zen koans. Maybe you’ve had this experience too, but Zen stories have a curious rightness to them, which seemingly defies expression in language. It’s the Zen stories that can really hold your interest. But Buddhism is a much broader church than just Zen Buddhism. Ancient (Theravada) Buddhism, became rather monastic, with people just meditating until they reach personal enlightenment. Until the young Turks, the Mahayana Buddhists, came along and said, “well hang on, what about our obligations to our fellow humans? What about them?” This is 400 years after the Buddha now, and the word compassion starts appearing. Zen is later again, roughly what happened when Mahayana Buddhism met Chinese Taoism.

An excellent body of literature has cropped up in the last 20 or 30 years, incorporating both traditional Anglophone philosophy and Buddhist philosophy. One focus has been on the nature of the self. There’s a long tradition in Buddhism denying the existence of the self. In turn this is supported by another tradition in Buddhism denying the existence of composite objects, denying the existence of things with parts. In the famous text Questions of Milinda, the line is that nothing with parts exists, so not only do you and I not exist, but neither the chariot, the gatepost nor this table exists! A draconian solution!

I’m tempted to say that the self is just the body, the “is” of contingent identity. The skin is a natural boundary. Having a pin in your skin, or two millimetres away from your skin makes a world of difference, phenomenologically. Now, I think there’s a lot of similarity between Hume’s theory that the self is just a bundle of sensations, and the Buddhist denial of the self. I think what the Buddha really wanted to deny, and what Hume really wanted to deny, was the soul, not the self. For Buddhism, the doctrine of impermanence says that everything is impermanent, including you and me, so there’s no immutable, immortal soul. So denial of the soul is core Buddhism, but I don’t see the denial of the self being any more part of core Buddhism than I see reincarnation as being core Buddhism—although I know that the Buddha himself believed in reincarnation.

So I take core Buddhism to be the Four Noble Truths: there is suffering; the cause of suffering is attachment; relief from suffering is possible; follow the Eightfold Way. It’s an arresting view of the sources of psychological suffering. I think the doctrine of impermanence has got to be core Buddhism. All things are impermanent. That’s just common experience! (“Well,” says the clever young philosophy student at this point, “what about the universe itself?” But of course the Buddha wasn’t thinking that far away from human experience.)

There was a gentleman giving a paper here a few years ago saying quite strongly, “you do not have Buddhism without reincarnation.” But I’d rather save what’s saveable about Buddhism. So core Buddhism for me is what you can save as true, especially its arresting views on the sources and alleviation of psychological suffering. What’s non-core Buddhism is what you can’t rescue, such as reincarnation and the denial of the self. Does that make me a Buddhist? Well something like a Buddhist, I suppose. I don’t know if I’m all the way there.

You've written about the relationship between Buddhism and psychotherapy. Can you say a bit about this?

They're both in the business of alleviating psychological suffering. I don't think the Buddhist can do anything about it when you stub your toe. Well, as a matter of fact, meditation actually helps if you've stubbed your toe. But I don't think Buddhists say that they can magic away disease or anything like that.

So the story I gave in that paper was that the two are complementary. Buddhism addresses itself to the person who's suffering psychologically, from attachments and the like. Psychotherapy is much the same, but for those who are much more dysfunctional or dissociated. So first go to the psychotherapist, then go to the Buddhist teacher. It might be that they will operate in tandem—I've come across a number of traditionally trained psychotherapists who are flexible and open enough to have those sorts of arrangements, or to train in Buddhist teaching themselves.

Would you say that Buddhist philosophy often has a mystical component?

Just to interject, I thought you were going to ask me how well Buddhism and scientific realism fit together, to which I want to say, very nicely!

So is Buddhism mystical? I don’t think it’s mystical at all. I think it’s grounded in human nature. Take the first of the Four Noble Truths: there is suffering. Well that’s just common experience! Same for the doctrine of impermanence. So much of Buddhist philosophy is arguing from premisses based on experience, which makes it natural, not mystical. In Tibetan Buddhist schools, children are taught argument theory from the age of six or so. I saw some Tibetan debates in India when I was over there. Very lively! The Adelaide Philosophy Club has nothing on them.


This conversation has been edited for length and clarity.