Paradox is such a great looking word that I couldn’t resist it. I must be a little odd or should I say weird but I just love the X at the end and the way it sounds. Just don’t get me going about paradoxical! So what is a paradox?
A paradox is a statement that contradicts itself. It is so much easier to give an example rather than a load of words. A classic is ” I always lie” if it is true then it must be false and is therefore a paradox. Also, which came first, the chicken
or the egg? Space, time and the universe are full of paradoxes and just go to show how weird the place that we live in is.
It has been said that there are no true paradoxes, only apparent ones. It is certainly the case that many paradoxes come about when language is badly applied to experience. These paradoxes show that our language must be looked at more carefully. Finding the “resolution” of a paradox means finding a way of talking about it so that it’s paradoxical nature vanishes.
Here are a few paradoxes that I found particularly interesting.
Can Things Be Impossible For A God?
These paradoxes illustrate problems with omnipotence. Can God create an object too heavy for him to lift? Or, forgeting the physical problems entirely, can God ask a question too difficult for him to answer? And was that the question?
This paradox treads on theological grounds, which brings in special problems of heresy and damnation. Either God cannot in fact do everything, or he can, but he chooses not to thus avoiding the problem.
Lets Make 2=5 Well Perhaps
Here is an example of a paradoxical mathematical proof. Several such proofs exist but all contain faults. The first line is taken as a given:
- A = B
- A+A = A+B (adding A to both sides)
- 2A = A+B (simplifying)
- 2A-2B = A+B-2B (subtracting 2B from both sides)
- 2A-2B = A-B (simplifying)
- 2(A-B) = 1(A-B) (factoring)
- 2 = 1 (cancelling)
Once it is established that 2=1, it is easy to prove that any other pair of numbers are equal simply by multiplying and adding appropriate numbers to both sides of the equation.’
Proofs’ like this always have a fallacy. The fallacy in this one is in the last step, when the (A-B) on both sides is cancelled. This is division by zero, which is illegal in mathematics. That’s a pity isn’t it I could have been a rich and famous man.
Zeno’s most popular paradox runs something like this, suppose you want to cross the room, first you have to walk halfway across, then you have to cross half the remaining distance, then half the remaining distance, and so on. There are, eventually an infinite number of distances you have to cross to get across the room, and this is impossible to do in a finite time, therefore, all motion is impossible.
The resolution of this paradox does not require calculus, It is only necessary to point out that Zeno has assumed that the distance across the room is infinitely divisible, but not infinitely extended. Therefore the time required to cross it must also be infinitely divisible, and it is not required that the time be infinitely extended. Whether space or time are actually infinitely divisible may be a problem in modern physics, but that’s way over Poor old Zeno’s head.
The Lying Cretin
This and the following several paradoxes rely on headache inducing phenomenon known as self-reference. The Lying Cretin, like Zeno’s Paradoxes, comes to us from the Ancient Greeks. One version of the Lying Cretin paradox occurs in the Bible-
“One of themselves, even a prophet of their own, said, The Cretans are always liars, evil beasts, slow bellies. This witness is true. Titus 1:12-13″
The prophet in question was Epimenides, who said “all Cretins are liars” when he was being interrogated by Athenians. The problem here is that if what he says is true, then he, a Cretin, is always a liar, so what he says is therefore false. There must, in fact, be at least one Cretin who is not a liar. This is not quite a paradox, it is just an odd way to lie. The writer’s claim that “this witness is true”, cannot be correct. Lying Cretans: 1 – the Apostle Paul: 0.
The Barber Paradox
In a certain town lives a barber who has a peculiar rule by which he operates. The barber cuts the hair of everyone who doesn’t cut their own hair, and he doesn’t cut the hair of anyone who does cut their own hair. Who cuts the barber’s hair?
The resolution to this is that the barber is bald (Or he uses hair remover). Alternately, no barber can follow such a rule.
The Librarian Paradox
This one is a little tricky to follow; there is a chief librarian who is in charge of several libraries. He asks the librarians of each library to prepare for him, in book form, a catalogue of all of the books in their respective libraries. Each librarian is faced with a dilemma. Since the catalogue is a book, should it be included in the catalogue of books? Some decide yes, some decide no. When the chief librarian receives all of the catalogues, he divides them into two groups, those which list themselves, and those which don’t. There are many of each kind, and the chief librarian sets out to catalogue the catalogues.
He prepares one catalogue which lists all catalogues which list themselves. Another lists all catalogues which do not list themselves. Where is this last catalogue to be listed? If it does not list itself, then it really should, but if it does, then it really shouldn’t.
The resolution is simple; the librarian cannot make a catalogue according to the stated rule.
You are taking part in an experiment. In front of you on a table are two boxes, box A and box B. Box A is made of glass so you can see inside, and it contains £1000. Box B is opaque and contains either nothing or £1,000,000. You are
allowed to take either box B alone, or both box A and box B.
Here’s the catch, the people running the experiment claim to be able to predict with astounding accuracy which of the two options you will go for, and will have put the £1,000,000 in box B only if they predict that you will only take box B. If they think you will take both then they’ve left box B empty.
In the previous thousand experiments like this, their predictions about what the subject will do have been correct… so, what do you do? No, don’t run away.
One argument is that you’re just another subject, so if you take both boxes you get £1000, if you take just box B then you get £1,000,000. So take just box B.
On the other hand, either the money is sitting there in box B or it’s not, so whatever is in box B, taking both means you get £1,000 more than if you take just box B.
which box would you take?
This Is a Very Well Known and common Paradox around the Internet and amongst alien conspiracists and organisations like SETI.
Fermi realized that any civilization with a little rocket technology and a small amount of expansion ambition could rapidly colonize the entire Galaxy. Within ten million years, every star system could be colonised. Ten million years may sound long, but in fact it’s quite short compared with the age of the Galaxy, which is roughly ten thousand million years. This doesn’t seem to have happened.
Fermi’s paradox asks, where are they?
Drake’s equation shows that the galaxy should be colonised and makes us ask Fermi’s question once again. The equation is usually written:
N = R* • fp • ne • fl • fi • fc • L
Don’t let all those symbols confuse you, Drake’s equation is just a lot of different factors to do with the number of alien civilisations and the different things that would affect them multiplied together. These “things” are listed below.
- N = The number of civilisations in The Milky Way Galaxy whose electromagnetic emissions are detectable.
- R* =The rate of formation of stars suitable for the development of intelligent life.
- fp = The fraction of those stars with planetary systems.
- ne = The number of planets, per solar system, with an environment suitable for life.
- fl = The fraction of suitable planets on which life actually appears.
- fi = The fraction of life bearing planets on which intelligent life emerges.
- fc = The fraction of civilizations that develop a technology that releases detectable signs of their existence into space.
- L = The length of time such civilizations release detectable signals into space.
So Drake’s equation says there should be loads of aliens and Fermi’s paradox says well where are they Mr Drake??
Why is the night sky dark? The universe is rather big, well, we could call it infinite which means, we will say, that there are infinite stars. All those stars and all that light produced by those stars should fill up the night sky with light. But It doesn’t, does it?
This paradox has been around a long time and a few explanations have occurred. De Chesaux and Lord Kelvin (who gave his name to the temperature scale) suggested that there might be dust in between us and many of the stars, blocking out the light that we receive from them. However, absorbing dust would eventually cause an equilibrium, and emit as much radiation as it absorbed. Even if it was at a different wavelength, we would still receive the same amount of light as before.
Some astronomers thought that if the universe was expanding (as good old Hubble showed that it is) light from distant stars could be redshifted by the Doppler effect. While this effect provides a contribution, it doesn’t account for enough light to darken the entire sky, the left overs should still be detectable.
Benoit Mandelbrot suggested that there is fractal distribution of galaxies. This would mean that there was a lot of empty spaces between stars, and galaxies, so the light coming from each direction would not add up to infinity.
The real explanation to Ober’s paradox is that light takes time to travel and therefore it hasn’t reached us yet and by the time all the light does reach us, the stars will have died and others will have been born so the light can never make the nights day.
EPR Paradox And Schrodinger Cat
I have covered these both before in a quantum physics post.
The EPR Paradox is a paradox that was first suggested by Einstein who proposed a thought experiment that appeared to demonstrate quantum mechanics to be an incomplete theory.
Although a bit morbid, it demonstrates an interesting conundrum. If you had access to a time machine, what would happen if you traveled back in time and (let’s hope) accidentally killed Grandpa? What would happen to you? Your parents? Your kids?
This violates causality all over the place, it is an assertion of free will which transcends the functionality of time travel. It is best explained by example, a man builds a time machine from plans he received from a mysterious person years before. The man realizes that the stranger was himself, using the time machine to travel back and give the plans to his younger self.
A bilking paradox would occur if the man built the machine, tested its reliability, and then refused to give himself the plans.
Interesting and mind bending, paradoxes keep us thinking. From Catch-22 “a concern for one’s own safety in the face of dangers that were real and immediate was the process of a rational mind”.
In this novel, this meant that an airman might be crazy and could be grounded, all he had to do was ask. But if he did ask, he was deemed sane and so had to keep flying. He might be crazy to fly more missions, and sane if he didn’t – but if he was sane, he had to fly them. I expect this will not be airmen one day but spacemen.