paradox /‘paeredoks/ n. seemingly absurd though perhaps actually well-founded statement; self-contradictory or essentially absurd statement; person or thing conflicting with preconceived notion of what is reasonable or possible; paradoxical nature; paradoxical /-’doksikel/ a. [L f. Gk (para-, doxa opinion)]
From ancient folklore came puzzles; little games using objects, then drawings, then words. From these arose the paradox. These were puzzles with no apparent solution, questions with no answer or questions with too many correct answers. These little puzzles have kept philosophers busy for centuries and the sleep lost over them can be counted in decades!
Even the definition of the word paradox has been argued to death.
I’ll start with a few examples of well-known paradoxes. Then I will try to illustrate the usefulness of these little puzzles, their pervasiveness in life and their profound implication about the limits of our perception and knowledge.
For my purposes, I’ll define paradox as two apparently contradictory statements (or images, ideas, things) that both seem to be true. The two statements are usually joined by the conjunction, “and yet”.
“Everything is amazing and (yet) nobody is happy”
The paradox may present itself in question form, “Which came first, the chicken or the egg?” The choice offers two seemingly correct but contradictory answers. The chicken had to come from an egg and yet the egg had to come from a chicken. (Note: I think this one has been solved, in the egg’s favor).
A paradox can inspire merely a grin or a shrug. It can also make your head spin and make you start questioning everything. The puzzle can sometimes be “solved” in one way or another, but the hard-core paradoxes are the insolubles, which have been argued for millennia. Examples of these will be presented after we look at a few visual paradoxes.
Here is a drawing of a Necker Cube, first published by the Swiss crystallographer Louis Albert Necker in 1832.
Look at the cube for a few moments. Do you notice it jumping from one cube facing downward left to another cube facing upward right? After a few moments you can actually choose one cube or the other. You’ll notice though, that you can only see one cube at a time. Each view of the cube is a sort of tunnel vision that takes one point of view and excludes the other.
The cube presents a paradox in that the same drawing can be seen as one cube and yet it can also be seen as another cube.
The Necker Cube can also be seen as a gem or chocolate shape…
This gem shape can be seen with the centre rectangle in the foreground and the sides sloping away or with the rectangle at the back with the sides sloping in towards it.
Of course the Necker Cube can be just seen as a two-dimensional wire structure, like the frame of a stain-glass window…
Whether these optical illusions can be solved or not is not the point here. What the Necker Cube shows us is that one drawing (or idea, thing) can be viewed in several different ways. We can actually choose the way we look at it and we can only see one interpretation at a time.
Here’s another take on the Necker Cube…
This drawing incorporates elements of both cubes, becoming an impossible cube. Although the drawing is right there in front of us, it represents something that can’t (and doesn’t) exist.
Here are a few more impossible objects.
A triangular shape that seems to be made up of three 90o angles…
A pitch pipe with two or three pipes?…
A staircase that continues up (or down) forever…
And, the elephant at the beginning with four legs (from the top) and yet five legs (from the bottom).
Of course the trick in these can be explained but the point is that there are things that can be named and drawn and yet don’t actually exist.
Some of these impossible things are used to great effect for very practical, everyday purposes.
In perspective drawing, for example, a road leading off to the distance would show the two sides of the road and the horizon all meeting at the “vanishing point” (X). And yet this point doesn’t actually exist. The lines don’t actually meet in reality.
And So On…
The number (pi) in mathematics is an irrational number which cannot be accurately written. It can be named and used but can only be approximated by 3.1416… (The dots meaning “and so on”). Similarly with (root 2). It is very useful but cannot be written out. Neither number can be accurately expressed as a ratio of whole numbers (hence irrational).
The number (root -1) is an imaginary number that doesn’t exist at all and yet it is used in equations as if it was a real number. (What number times itself equals -1?)
Other examples from mathematics…
The set of integers is written like this: (… -3, -2, -1, 0, 1, 2, 3 …) with the ellipsis (…) at both ends. It means “and so on” to infinity in both directions and yet there are no infinite numbers. Every number is finite. “Infinity” doesn’t, in itself, exist.
Space and time are like the set of integers. Space seems to go on and on in all directions (inward and outward). Time is equally precarious… consider that the past is not now (anymore), the future is not now (yet) and we live squarely in the “present” which is a nonexistent, durationless instant!
All things (that exist) are “coming to be” or “passing away”. Since the absolute, eternal and infinite do not come to be or pass away, they do not exist.
The number Googol (not Google) is 10100 (10 to the hundredth power or 1 followed by a hundred zeroes). This number is already higher than the estimated number of particles in the universe (1070 to 1080).
Googolplex is GoogolGoogol ( Googol to the Goolgolth power, or 1 followed by a Googol zeroes). This number is unimaginably immense but obviously doesn’t get us any closer to infinity because there is no infinity to get close to.
Suffice to say there are several things in everyday life that are simply taken for granted and yet either don’t exist or are beyond our understanding.
Lies and Other Truths
This sentence is false.
This is a simple version of the classic Liar Paradox. If the sentence is true then it is false. If it is false, then it is true. The jury is still out on this one and volumes have been written about it. The sentence has been analyzed down to the last detail… IE What does “is” mean? or “What is true?” or “What is false?” or “Can a sentence speak for itself?” etc.
Here are a few more to ponder…
Our ancestors were the survivors… the killers vs. the killed, the stuffed vs. the hungry, the lucky vs. the unlucky. The aggressive fools won over the wise. The most ruthless, violent, unsustainable groups won over the peaceful and harmonious.
This is as counter-intuitive as parasites that kill their host. But there are forms of bacteria that kill their hosts. Bacteria DO die and many die with the host, after they drop the kids off at another host.
“It’s the space within the vessel, where nothing exists, that makes the vessel useful.”
Paleolithic man created ever more efficient ways of killing buffalo. The better he got at it, the faster the buffalo went extinct.
The Fate/Free Will paradox: If fate exists, then free will does not. If free will exists, then fate does not. A good argument for the existence of each one can be made, but no one can prove either.
An Infinite Regression Paradox: It seems that everything comes from something but what was the first cause? If it was a “creator”, then who created the creator? (Ad infinitum).
Ancestor paradox: As we go back in history we have exponentially more ancestors (2 parents, 4 grandparents, 8 great grandparents etc). If we go back far enough, we have trillions of ancestors and yet there were fewer people then than now.
(I’ve actually read a “solution” to this one and found it more confusing than the paradox itself!).
“Something unknown is doing we don't know what.”
(Sir Arthur Eddington)
Most of the “big ticket” paradoxes are concerned with the mysteries that we live with everyday, also known as the “visible unknown”. These are the things we experience and can talk about but which we know nothing about.
These include space and time (as above) as well as concepts such as gravity, light and death.
When we watch astronauts space-walking, we are amazed at the concept of zero gravity. But is the presence of gravity any less mysterious? Isn’t it odd that we stick to the earth without flying away?
Light can be perceived as either particles or waves but not both. What it is depends on how we look at it.
Everything is absolutely cosmic (and) yet absolutely ordinary.
We spend our entire lives acquiring material goods and yet everything turns to dust, including us.
We seem to be endowed with great notions of purpose, meaning, truth and justice and yet we live in a universe in which none of these actually exist.
We use the word “silence” all the time and yet it doesn’t actually exist.
Death comes as a shock every time and yet it happens to every living thing.
We witness death everywhere and everyday, and yet we don’t fully understand it.
We seek standard healthcare when we are sick and yet iatrogenics (harm done by the healer) kill more people every year than almost any other disease!
“That’s the thing about life; everything feels so permanent, but you can disappear in an instant”
(Jonathan Tropper, This is Where I Leave You)
The greatest paradox is that there are even paradoxes in the first place! We see contradiction in opposites and yet the entire universe is made up of opposites. We see things as unusual simply because of our own limitations.
Perception, reason, logic and language (including mathematics) are technologies. They may be vast in scope but they are limited.
You can’t use technologies if you don’t know their limits.
Paradoxes simply show us those limits.
It seems as if our logical, common sense perception can only see part of the picture (a very small part) at any given time.
It also seems that we can be ultra-skeptical within our limits of reason, but when we reach our limits we tend to leap into total fantasy. We jump to ridiculous conclusions. These become beliefs with no foundation. Then we make really bad assumptions and predictions based on these beliefs.
We simply have to surrender to the fact that there are things outside of the limits of our knowledge. These things are considered odd or unusual because they’re behind our blind spots, outside of our reach.
From ancient folklore came the hare and the tortoise, the chicken and the egg. It appears man knew very early on the limits to his own knowledge.
Although a paradox can take us right to the limits of our knowledge, it can also show us the vast array of possibilities within those limits. This is where paralogical or paradoxical thinking comes in.
“The opposite of a true statement is a false statement,
but the opposite of a profound truth is usually another profound truth”
Paralogical or paradoxical thinking involves looking at opposite, illogical tangents from an original idea or strategy. This ensures that otherwise overlooked ideas will be considered.
In his book, The Paradox Process, Derm Barrett does a thorough investigation of paradoxical thinking. Although the book is designed for creative business solutions, it can easily be generalized to all creative endeavors and problem solving.
Barrett defines three types of paradoxical thinking.
1. Contrary Thinking: This is simply adopting, thinking and acting from an opposite or contrary perspective. It involves considering the opposite view to what you believe or to what is commonly believed.
2. Janusian Thinking: (based on the Greek god Janus, usually depicted with two heads facing in opposite directions) This involves holding contradictory perspectives in your mind simultaneously.
3. Hegelian Thinking; (after German philosopher G.W. F. Hegel (1770 - 1831) Involves combining or synthesizing two or more contradictory ideas to create a totally new entity.
These three can be considered individually or as a continuous process… Think the opposite of a premise, hold both in mind together, then fuse the two into a new idea. In this way, every idea can produce a multitude of new ideas.
A simple example…
You set out to paint your room and think white is a logical color. Consider painting it black! Even if the idea is repulsive, suspend judgement while you hold up both colors in your mind. Combining the two ideas gives you a new shade, grey. From grey you can then consider a multitude of colors and finally settle on, say, a pastel, coffee color.
The important thing here is not so much the solution but the fact that many ideas, including some contradictory and bizarre, were considered before reaching the final decision. Paradoxical thinking leads us into un-chartered territory, into realms that we might not have considered using our standard decision making practices.
Formulate an idea or strategy, then look for ways to refute it. Like a chess master, who looks at “candidate” moves, then tries to find the strength AND weakness in those moves. This is harder to do than you might think.
Often, when trying to solve a tough problem, we start daydreaming or even fall asleep. Instead of thinking this is counter-productive, consider that daydreaming and sleep may provide the actual solution to the problem.
Try looking carefully at some of the clichés that affect your day-to-day decision making…
“If at first you don’t succeed, try, try again” … consider quitting.
“Everything in moderation” … consider wild extremes in all directions. There is good evidence that our early ancestors did NOTHING in moderation!
“Never say never”… consider saying never now and again.
If you think the world is orderly then consider randomness. It’s fascinating, humbling and puts us very quickly out of our league. It forces us to see our limits.
“We don’t know very much, we overestimate what we know and we underestimate uncertainty.”
(Nassim Nicholas Taleb, The Black Swan)
If you think the universe has meaning, consider meaninglessness.
Of course, anyone who even considers that there is no meaning or purpose in the universe is bound to be labeled a nihilist and a pessimist. The first people who were labeled nihilists were skeptics who believed that nothing existed and therefore all knowledge was impossible.
Nihilists were accused of trying to destroy the universe. But why would someone who believes nothing exists be interested in destroying anything?
I’ll admit that the idea of meaninglessness does lead one in a certain direction. A nihilist cannot know or believe anything, However I don’t see how this would lead to more anxiety than a life filled with knowledge and belief (and arrogance).
A nihilist by definition would be a total atheist in that he/she would not believe in ANYTHING.
A nihilist could not simply contrive some fairytale to “get them through the night”. But then the nihilist probably wouldn’t see time (night or otherwise) as something to be “got through”.
Nihilists would not have the luxury of dismissing “this life” in favor of a better life in the future (after death).
Finally a nihilist would not be disappointed in anything, as that would amount to asserting some sort of meaning.
But I digress…
Consider all sides of a problem or situation. Even if you are certain that something is true, consider its opposite. Then expand to holding two or more contradictory ideas at once. Then synthesize the ideas into a new whole.
The Turkey Problem
Let us now look at the problem of inference.
We see events follow one another and we infer that one caused the other. And yet many causes can go into one effect. and one cause can have a multitude of effects. We assume that what happened today will probably happen tomorrow. Like the contented turkey that gets food and water every day and starts assuming this will always be his lot, then two days before Thanksgiving… SURPRISE!
The tendency for humans to make inferences quickly is ingrained in us from the earliest times, when induction was probably more accurate IE “I hear a tiger over there (danger) so it’s not over here” (safety, escape route).
Similarly, belief was probably more effective in simpler times and with simpler circumstances.
The Paradox of Our Age by Dr. Bob Moorehead (http://garywolff.com/paradox.html) points out a host of ironies about our modern world. It also shows that many of our strategies for a better life have actually had the opposite effect! I think most of these strategies failed because they were based on false premises and beliefs, un-examined traditions and faulty inferences. Perhaps a little paradoxical thinking could have shown us some unlikely alternatives.
We don’t ever really get the big picture because of the limitations of our perception and our technologies (including knowledge).
Keep in mind that “limitations” may have a negative connotation but they are also the highest positive we can achieve.
Try not to believe everything you see and hear. Question everything. If you must believe something, don’t invest a lot in it and don’t base any important decisions on it. Definitely don’t impose your beliefs on anyone else.
Belief may be an indication that you’ve gone beyond your limits.
Don’t be a sucker (n. a gullible person that is easily deceived). Things are seldom what they appear to be but they are almost never what they are believed to be.
Don’t buy into every new belief or strategy that comes along. The belief of course indicates that someone else has gone beyond their limits. The strategy is probably based on that belief.
A strategy can be followed for years and yet have little or no effect toward the original intended goal.
Look back at predictions you have made and strategies you have adopted in the past. See how many were wrong. What were the consequences?
Look at the strategies you are using right now, purposefully looking for weaknesses and limitations. Adjust the strategies accordingly (from a small tweak to total abandonment). Note: Traditions are loaded with un-examined strategies.
Find time every day to do nothing… consider nihilism.
Be at least pessimistic enough to see the downside of a strategy. Blind optimism is never a good strategy.
Be aware of your own and other’s use of impossible words… “Ultimate”, “Infinite”, “Forever”, “Always”.
Also watch for certainty clauses...“There must be…”, “There can’t be…”.
We are forced to explore concepts like “infinity” with finite tools.
We base strategies and predictions on beliefs and on the erroneous assumption (inference) that the future will be just like the past.
“Anything can happen”
(Annie Dillard, Pilgrim at Tinker Creek)