As part of a current project to revisit the Personal Development aspect of the Henley DBA (Doctor of Business Administration) programme, I’ve been forced to start to think about the concept of paradox. Let me tell you – it’s painful, mind-twisting stuff. And I blame my mentor, Professor Jane McKenzie for everything!
So, what is a paradox, and why does it matter for the DBAs? Good question. Answers on a postcard please. In the meantime, here is what I think.
First of all, a paradox must be both self-contradictory and not self-contradictory and it must be (or appear to be) both at the same time. When it is, it isn’t. And when it’s not, it is. Commonly cited examples of this include the Liar Paradox. A logic paradox sets up a kind of warp-speed oscillation, whereby you have to jump instantly from one side to the other as soon as you have comprehended either.
It’s rather like in the diagram below (maybe you see a smaller box in the corner of a room, then you see one large cube with a cube-shaped chunk missing (or vice versa), but you cannot see both simultaneously):
The cube illusion doesn’t usually occur to us as a paradox, though, because there isn’t a pressing need for us to understand one way over the other. We can grasp that it’s both, and neither. The sky does not fall on our heads. In fact, without a conflictual aspect or consequence, most paradoxes – like sleeping dogs – should be let to lie. We’re often quite unaware of the paradoxical nature of much of our perception and sense-making. The sorting out and sifting of all the possible double and contrary meanings happens mostly at an unconscious or habitual level – leaving us free to get on with the business of whatever we think our business ought to be. Paradox matters only when we are involved in some kind of change or learning process.
In the pure sciences such as mathematics paradoxes have been seen as non-axiomatic and can exist only in theory (in other words, in the imagination) and not in reality. That doesn’t prevent paradox being talked about a lot by mathematicians. Paradoxes appear a lot in philosophy, too. The eye, our organ of sight, can never see itself. In fact, the one thing none of our senses can do is sense themselves. And a statement such as “today is the only day that is not different” is self-referentially impossible because if it’s true that today is the only day which is the same as the others, then it instantly becomes different, which instantly makes it like all the others, so not different, which…
Near the beginning of the 20th century Bertrand Russell infamously dealt with paradox in mathematics by means of a the deus ex machina of the hierarchy of logical typing. Is the set of things that are non-cats itself a member of the set of things that are non-cats (i.e. a member of itself)? No, was the answer, because a set is always of a higher logical type than its members. This is very useful, as it turns out that this is why it is logical not to eat the packaging of your pizza but the contents inside, even though the package says “pizza”.
And yet, psychologically, socially, zoologically and aesthetically there are some nice paradoxes of identity and we do seem able to bend, twist and break Russell’s rule when it comes to social interaction. In fact, it may be necessary for us to do so. One famous example is the Ship of Theseus, which is the ancient question of whether a wooden sailing ship which over time, piece by piece, has every bit of wood, every rope and every scrap of sail replaced is still the same ship? A more modern and terribly funny equivalent is “Trigger’s Broom” from the British sitcom “Only Fools and Horses”:
I suppose one of the tensions present in the identity paradox is that between permanence and change, and this seems one of the interesting aspects for the DBAs as they are there on the course precisely because they wish to attain both, and this is surely contradictory. There are others they will find – the dichotomy of perceived gaps between “research” and “practice”, or “rigour” and “relevance” (and so on… and on) which social identity via membership of practitioner or academic communities prizes and demands. These sorts of paradox are experienced as real mainly because the oscillation between one side and the other is made possible by the passage of time.
No doubt it’s a paradox that will resolve itself – sooner or later. In the short run, a person is free to enlighten themselves and shake off the need to resolve a paradox at all (better to dissolve it through awareness), while in the long run – as Keynes reminds us – we’re all dead. This last view I take to mean that we should relish living in the present, not that we should feel helpless or gloomy about it.
“If the fool would persist in his folly he would become wise.” William Blake
Chris Dalton's Personal Development Blog 
OK Chris you started 🙂 . Would you agree that the quickest/easiest way to switch (toggle) between the 2 views of the box paradox is to focus on the intersection line right in the middle…Just because of human eye/brain coordination is not fast enough doesn’t mean that it is not possible to see it both ways at the same time, same as dogs can sniff cancer while the best scientific detection method today has a high degree of error.
So can one useful (you need to check with your professor of course) practical benefit of paradox mindset (thinking) [btw a proposal for your new HBR article title, also nicely reminding ‘paranoid mindset’ :)] is to be able to think your are right and dead wrong at the same time, if nothing else it brings humility, listening, learning, empathy into DBA/MBA whatever…
as you always say ‘it depends’ …whatever it is (or it isn’t), keep the smile…
Hakan Birincioglu
This is not difficult, Chris. Mathematics defines what is required from an “equivalence relation”. You’re toying with relations that do not meet the requirements.
For example the image of the cube in the room; it’s an image on top of another; the relationship “on top of” does not meet the mathematical requirements for being an equivalence relation.
@ Martin, thanks for the reply. I certainly don’t have the training, or the cajones, to argue a mathematical case with a mathematician, and I hope that my post conveys my respect to the treatment of paradox in mathematical terms. I’m guessing that by equivalence relation you’re referring to propositions in math that are reflexive, symmetrical and transitive? If I got it right, mathematical paradox disappears the instant it is identified.
I’m not toying with that stuff at all. It may be, for clarification, that we need to say that paradox in the social or living sciences is not the same as paradox in mathematics. Anyway, instead I’m arguing that the stuff of paradox in the social or the biological world is relational and does not necessarily dissolve upon detection. It may even be that something like a paradoxical relationship is part of the evolutionary process.
The cube is not the best illustration, I’ll admit. But following your logic, which image is on top of which? In reality, there are no cubes and the image is not in 3D, but rather the idea of either perspective exists in the relationship between the viewer (and the boundaries of circuitry of perception in our vision/brain etc.) and the 2D print on the paper/screen. It’s a bit like asking how many bats there are in a Rorschach ink blot.
Sorry, I just saw this now, which explains and, I hope, excuses why I didn’t respond earlier.
So “paradox” means something else in social sciences than in mathematics? Well, yeah, OK, sort of – I’ve met ambiguities like that before; for example psychologists talk about something called “energy”, which is not the same as the physics “energy”.
But, in that case, the meaning of the word “paradox” becomes a function of the genre or context in which it is used; and there is nothing in the image of the boxes to identify genre; and if there was, it would be more physics than sociology.
In terms of genre, the liar paradox belongs neither in sociology or physics, but in philosophy. However, I would propose that your DBAs’ difficulties with integration of academic and commercial paradigms is an example of, not a genuine philosophical paradox, but rather a psycological mechanism called “splitting”; and splitting works; but it leaves a wake of cognitive dissonance.
To resolve the unpleasantness, I would propose that a better understanding of social constructs is a good place to look for answers – philosophy never really cut it with that.
For example, have a go at Christianity’s creation story: science sees it as an incorrect hypothesis, sociology sees at as a socially constructed illusion, politics treats it as a group of voters with a legitimate claim to representation, phycology sees it as a response to deep-rooted fears and desires, and Christianity itself sees it as something that happened when Eve bit in an apple.
The trick, in my modest opinion, is to start with a solid social construct, and then, when your DBAs are ready, generalize it to the more flimsy constructs of “academia” and “commerce”.
Just my two cents. Have a wonderful day.