Divisions & Paradoxes



For your continued comfort, please note the safety announcement: Dear reader, this article, as is in so much of value in the Art & science, is full of paradoxes. Of course, some may be apparent paradoxes; not just food for thought then, but a lifetime’s feast. Some paradox alerts will appear in this article, some won’t be sign-posted. Naturally this is on purpose to maintain your interest, not because I missed them. Don’t mock kind reader; it is healthy to hold on to some harmless delusions.


The more closely the scrutiny of the Art, the Sciences and their foundations, the more likely the chance of finding so many similar paradoxes that they may be the meeting places of ideas that link the two.

To try to put paradoxes into some sort of context, how does this sound? If learning is how adults play, paradoxes are the bits in the playground designed to maintain our interest, intrigue and amuse us. Before our thoughts float away like a balloon forgotten by a small child distracted by an ice cream, let’s look at a definition (taken and adapted from Wikipedia and others) as the string and possibly the sticky hand to anchor us. A paradox (Greek paradoxos –opposed to existing notions) can be:


* An apparently true statement or group of statements that lead to a contradiction or a situation which defies intuition, or it can be

* A seemingly opposite or apparent contradiction that actually expresses a non-dual truth. It may be a bit like a Kaon, containing aspects that are inaccessible to rational understanding but accessible to intuition.

Paradoxes can reside in a person, assertion or thing, giving each an element of incongruity, self-contradiction and ambiguity. Each based on a valid deduction from acceptable premises.


A famous statement with all the required apparently inexplicable and contradictory aspects was written by Mary Shelley, the author of the gothic novel “Frankenstein” and husband of the wonderfully named Percy Bysshe (a man whose parents clearly had a sense of humour). She wrote “The silence of midnight, to speak truly, though apparently a paradox rang in my ears.” I am sure I have experienced that myself.



Some paradoxes can seem self-contradictory but may be true, like “Standing is more tiring than walking” or “A musician must practise improvisation to become a good improviser”. Some are more challenging like “I always lie”; if true, it must be false. Some are both daft and amusing. Have you seen a form with “This page has been left blank for you to write notes” at the top of one page?



Some of my favourite are visual and it is hard to find better than the Dutch Meister himself. Here is “Waterfall” by M. C. Escher (1961). He helpfully advised that water should be added occasionally to replace that lost by evaporation. The basic structure of the drawing is two linked and impossible triangles.










Cunningly, I have chosen the Penrose triangle or tribar, the mascot of visual paradox, as the paradox alert for this article. Some that are incredibly obscure to me may well be blindingly obvious to you, dear reader. If so my admiration is unbounded.









Independently devised and popularised by the mathematician Roger Penrose in the 1950s, “impossibility in its purest form” was first created by the Swedish artist Oscar Reutersvard in 1934. It so commonly found in Escher’s work some people think it is his brain child. This is fair enough as Escher’s illustrations switched on the light of inspiration for others. Reutersvard seems to have lost out a bit, perhaps because his name is so difficult to pronounce for those outside Scandinavia.



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