What is a paradox? Simply stated a paradox is a statement or a situation that is self contradictory. Some apparent paradoxes are just a clever play with words and I would argue that they are not paradoxes at all but represent mere semantic baubles or playthings. In this category I would place the following statement: All Cretans are liars. I am a Cretan. The first sentence is fine however, the second sentence introduces a logical inconsistency which forces the argument into a never ending loop of doom.
Other paradoxes are more perplexing and can be very profound. Some are not paradoxes at all and are the consequence of faulty reasoning. In logic, a conclusion is valid if each premise in the argument is true. If in a chain of reasoning an error is introduced then the conclusion must be false. However, if every link in the reasoning is correct then the conclusion, no matter how implausible it may seem, must be true. Sherlock Holmes based his whole career upon this self evident truth.
Consider this situation: Three friends dine out in a restaurant. The total bill comes to $30. Each of the friends chip in $10 a piece. The waiter, Mr Mugumbo (who else), collects the bill but is informed by the cashier that the diners have been overcharged by $5 and this amount should be returned with aplomb and alacrity. However, the waiter is dishonest and decides to return only a $1 to each of the customers, thus pocketing the residual $2. As each diner receives a $1 rebate, they each pay $9 each for the meal. This equals 3 x $9, or $27. Now we know the waiter, Mr Mugumbo, has pocketed the $2 and that the customers have paid $27 for the meal: $2 + $27 = $29- but what has happened to the other $1? Remember the diners originally handed over $30!
Logic dictates that the single $ cannot magically disappear, but where has it gone? Therefore, we conclude that the reasoning must be faulty. But it is not a problem that can be readily solved, superficially at least. It can be solved with a little simple, but elegant mathematics. I won’t provide a solution- try and work it out for yourselves without going mad.
How would you classify this puzzle, also known as Theseus’s ship? Imagine a ship constructed entirely of wood. Every day a plank is removed and replaced with a new plank. This continues every day for three years until the whole of the ship is replaced with new planks. I forgot to mention that all the old planks are used as fuel for Mr Mugumbo’s furnace in his new restaurant which he managed to buy with money he defrauded from customers. My question is this: Is the ship the same ship which left the shipyard, admittedly on brief day trips, three years ago? To help you along, think about this: The human body replaces all its cells over a period of seven years. Even so, you still remain ‘you’, whatever that means. Just to be cruel, I would like to change the scenario. In this second instance, the old wood from the ship is used to construct a second ship (in your face Mugumbo!). After three years, there are therefore, two ships. Which of these ships is the original ship?
My final contribution is called the grandfather paradox. This is a problem well loved by science fiction writers. I will conclude with an answer which I think addresses the problem in a satisfactory manner. I attempt this because I really can’t help myself as it appeals to my mischievous nature. I’ll leave it up to you to decide whether I have been successful, or not.
The Grandfather Paradox
What if you could build a time machine and return to a time when your grandfather was but a little boy. You then kill your grandfather. Therefore, you are never born. But if you are never born how can you return to kill your grandfather? Therefore, you are born, which means etc…….
Many physicists would argue that this really isn’t a paradox at all because time travel is impossible. Mathematicians disagree and state that time travel can be modelled with equations. Of course, everything is possible on a piece of paper. There are some physicists who contend that a massive black hole could bend space-time into a loop and therefore travelling back in time is theoretically plausible, but practically impossible (why?).
I would argue thusly: There is a fundamental law of nature: Energy and matter cannot be destroyed or created although they can be interchanged (Conservation of Matter/Energy). Therefore, the total energy and matter, at any one time, in the universe is a constant. If you suddenly disappeared from the present, you would be removing matter and energy from now ie there would be less mass/energy in the universe, at that instant. When you arrive in the past you would add extra mass and energy at that instant. And assuming you still exist in the future, you would be adding mass and energy into the universe which is clearly impossible and absurd. Before I go too deep, ponder this: My argument would not apply to an infinite universe, or would it?
But what if our decisions in the present cause the proliferation of alternative outcomes in the universe? If today I decide to stay in bed rather than go to work would two divergent realities spring into existence? One dependent on staying in bed, the other generated because I went to work. This line of reasoning opens up the possibility of contemplating an infinite number of scenarios (universes?). I don’t like this possibility, although some philosophers are drawn to it.
The great English physicist, Stephen Hawking, thinks he has proved that time travel is impossible. Some time ago he planned and advertised a party for ‘Time Travellers only’. At the appointed date and time he was the only person to turn up. You may well ask why did Professor Hawking go to the party in the first place- another paradox, mayhap?
Finally, just a few words about the importance of paradox and knowledge formulation. Clearly, many so-called paradoxes are simple mind games which provide pleasure and stimulation of thought. There are some paradoxes, so deep, so intellectually intense that they provide a mental goad for further intellectual discovery. A good example is ‘Russell’s Paradox ‘and the Russell-Myhill Paradox. The initial and simple and easy to state Russell paradox was formulated in 1902. The Russell-Myhill version was a logical outgrowth of the original paradox. Bertrand Russell was one of the greatest minds of the 20th century and made significant inroads in mathematics, formal logic and philosophy. I recommend his book: ‘A History of Western Philosophy’; a masterful exposition of the subject, written in beautiful, crystal clear, prose. I can honestly say it is one of the few books I have read that have changed the way I view the world. I won’t get mired in these paradoxes however, for those who would like to drink deep into the wells of knowledge, I’ve provided a link. Be warned, it is not for everyone’s taste and a facility with formal logic notation is required. Arse.