Paradoxes

Introduction

In this article, I want to discuss paradoxes. Paradoxes can be useful because they can thrill us and teach us unexpected things. To demonstrate, I will briefly analyze a random survey of paradoxes and pseudo-paradoxes. I did not select these for any particular reason, except that they came to mind.

 

Liar Paradox

In college, a professor presented this paradox:

     All Cretans are liars. I know, because I am a Cretan.

If a Cretan tells us all Cretans are liars, then that can only be true if he is telling the truth; but if he’s telling the truth and if he is a Cretan, then the statement is a lie because there is at least one Cretan who is telling the truth.

This paradox exists only because of a distortion of the colloquial meaning of the word All. While All technicaly means 100%, in actual usage, it’s frequently employed as hyperbole and really means a large number. I think this is common knowledge. With the obvious use of All, the paradox vanishes.

Another paradox seems to lurk in the second sentence where the speaker states he knows because he’s a Cretan. Is he saying he’s a liar? Not necessarily; he could just be saying that he is a Cretan and therefore knows his people. Again, there is nothing novel about this interpretation, although one could make a case for the alternative interpretation as well (certainly a better case than for the distorted interpretation of All).

If interpreted that way, the second sentence is basically a stronger of this paradox: I am lying now. It’s a purely linguistic construct, but it may be thrilling to feel the infinite regress caused by such a simple statement and contemplate how our language and concepts can multiply without end.

A more realistic version of this paradox occurs on some Jury summons forms. On those, the potential juror can opt out of jury duty due to “bad morals”. What if a person with good morals does not want to do jury duty, and so opts out by claiming “bad morals”? Would s/he then have bad morals for lying about having bad morals? It would only be a lie if s/he had good morals, but by lying, s/he would have bad morals and so on. Again, we’re back to the paradox and again it’s easily resolved with language and common sense. The opt-out is about the person’s morals in general.

You can find variations of the Liar Paradox at:

http://en.wikipedia.org/wiki/Liar_paradox

 

The Pop Quiz Paradox

http://en.wikipedia.org/wiki/Unexpected_hanging_paradox

A teacher tells the class there will be a surprise quiz sometime during the next 7 days. A student thinks about this, reasoning backwards. The quiz can’t be on day 7, because going into day 7, the student would know there was a quiz (since there wasn’t one on days 1-6). Likewise, it couldn’t be on day 6 since day 7 was already eliminated, and a similar argument discounts day 6 (since no quiz is on days 1-5). The student follows this line of reasoning to its logical conclusion: there is no quiz and doesn’t study. Then on day 3, the quiz is given and the student fails.

There is no paradox here. The example was contrived via a distortion of the word “surprise”. It’s obvious by “surprise” the teacher meant that the day of the test would not be specified in advance, which means it can be on any day. If a student manages to infer the day of the quiz, that does not contradict anything the teacher said, for the only promise was that the teacher would not announce the day of the quiz in advance.

However, in all fairness the real point of this “paradox” was to illustrate the inadequacy of this sort of reasoning in some circumstances. This reasoning – Backwards Induction – is one of the tools of Game Theory. When applied to circumstances like the above, it fails catastrophically. Not only does it produce wrong results, but it leaves the user worse off than if s/he had not used any reasoning at all. Indeed, it fails so badly, that it produces results contradictory to the very premises given in the problem it was meant to analyze!

This paradox is restated in another form as the Unexpected Execution.

 

Russel’s Paradox

http://www.scientificamerican.com/article.cfm?id=what-is-russells-paradox

Years ago, Bertrand Russel and Alfred North Whitehead started Principia Mathematica – an ambitious work to derive all of mathematics from logic. During this work, they discovered a paradox in set theory: the set of all sets that do not include themselves. This was a huge blow to mathematicians. Russel eventually introduced the theory of types to prevent this paradox, but many people had their faith in set theory shaken as a result.

This paradox can be stated in another way, with an example of the unshaved barber: A group of barbers shave only those who do not shave themselves. If a barber in this group does not shave himself, then by the definition he must shave himself. But no barber in this group can shave himself because if he did, he would be a man who does shave men who shave themselves.

Restated in this way, it’s another language issue: language is treated with mathematical precision rather than with common sense. Of course the barber would not shave himself, and the arguments are similar to those discounting the Liar Paradox.

However, since this occurred in math, which is treated with *ahem* mathematical precision, the issue caused more distress. Set theory was considered a very good candidate for the foundation of math, and finding a paradox in it – even one as seemingly contrived as the above – was enough to worry a lot of people. Mathematicians worry about things and are explicit about things that others dismiss as common sense.

On the other hand, mathematics worked perfectly well without foundations. Mathematics developed over time as a set of practices (rooted in real life) and from there abstraction upon abstraction was built until we have what we see today. Trying to add foundations to a body of practice that developed without them them is like drawing a blueprint for a house after it was built.

 

The Omnipotence Paradox

If God is omnipotent, then God can create anything. God can also lift anything. Therefore, God should be able to create a rock so heavy, God cannot lift it. But if God cannot lift it, then there is something God cannot lift and therefore God is not all powerful. If God can lift it, then God cannot create such a rock and therefore God is also not all powerful.

This is another case of definitions, but unlike the examples above, the definition it uses is the widely held one and not a contrived or distorted one. This invites us to study Omnipotence, something many take for granted. The paradox vanishes if we weaken or strengthen the definition of omnipotence, but this may leave many with even deeper questions and implications.

For instance, if we weaken the definition to say that omnipotence does not mean doing contradictory things, we have set bounds on the power of God. Does it matter? It would mean God is not supreme as God is subject to something, even if that something is a rule. On the other hand, if interest in God’s power is only to ensure salvation, does it matter? If one’s interest is solely in salvation, yet is still troubled, why is that? Just what does one expect of God?

On the other hand, we can strengthen the definition of omnipotence and say that God can do this and yet not be subject to a paradox because God also created logic and is thus not subject to it. We of course cannot conceive of how this would be possible because unlike God, our minds are limited by logic. For the faithful, this may be an even more inspiring taste of God’s might. To others, it may seem like dodging the question. But does it? God is often said to be beyond conception and in phrasing this paradox, the existence of God was assumed for the sake of the argument, which implies accepting the implications that go with definitions of God.

 

The Mind Reader Paradox

Two beings (X & Y) are playing a game of chicken. The rules are simple. They get in a car an drive towards each other at high speed. The first one to turn away loses. If neither turns, the cars crash and both die. X can read Y’s mind. Y has no powers, but knows X is a mind-reader. Who wins?

X seems like the sure bet because X can read Y’s mind, but not only can Y force a win, but Y can do so BECAUSE X is a mind-reader. Since Y knows that X knows what Y will do, all Y has to do is resolve not to swerve. Y knows X will veer so there is no doubt in Y’s mind. What’s more, X cannot bluff because X knows of Y’s resolve, so X must veer. Y can even break his/her steering column to ensure that swerving is impossible. X must move or die.

Again, X did not lose despite being able to read Y’s mind, X lost BECAUSE s/he could read Y’s mind.

This (admittedly) far fetched scenario is not a paradox because mind-reading doesn’t imply power and as such, one could argue that it’s another language issue. However, we often equate knowledge and power, so this scenario can encourage us to revisit that assumption.

A little more thought may even reveal the curious nature of power, and how the supposedly powerful may be pawns of those they control. Along those lines, Orwell wrote a short story called “Killing an Elephant”. You can read it here:

http://www.online-literature.com/orwell/887/

 

The Motion Paradox

http://philsci-archive.pitt.edu/1197/1/Zeno_s_Paradoxes_-_A_Timely_Solution.pdf

A person shoots an arrow at a target. It takes some time to cover 1/2 the distance to the target. It takes some more time to cover ½ of the remaining distance. And so on. Here’s the thing; since dividing a number repeatedly by 2 never brings you to 0 (it just gets you smaller and smaller numbers), the arrow should never reach it’s target.

This one’s due to a misconception of how motion works, and it’s been resolved in a variety of ways. One way is via calculus and infinite series. Another involves relativity (space/time are one concept). Yet the model is so intuitive (at some point 1/2 the distance IS covered), that it invites thoughts about other solutions.

What does the paradox assume? Well, it assumes that space and time are continuous. But are they? What if space and time are discrete, so that you can’t divide 1/2 infinitely? If this seems odd, ask yourself if it seems more logical that space and time should be infinitely divisible, with no smallest unit possible. There are some tantalizing hints that this may be so: while the Planck length and Planck time don’t say this, they seem to imply some limits, although just what those limits are is not clear. More to the point, some phycisists (most notably Ed Fredkin) believe that reality is discrete:

http://en.wikipedia.org/wiki/Digital_philosophy

Zeno had a whole host of such paradoxes:

http://en.wikipedia.org/wiki/Zeno’s_paradoxes

 

Banach Tarski Paradox

http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

You have a sphere which is made up of points. As geometry tells us, points are infinitely small and there are infinitely many of them in any surface. Because of this, you can construct any number of spheres of any size out of the original sphere and still have the original sphere left over, unchanged in size. You could take a sphere the size of a pea, construct a trillion spheres each the size of the sun and still have a sphere the size of a pea left over.

This is due to a misuse of infinity. Infinity is one of those troublesome concepts that leads to a lot of paradoxes and plain nonsense if misconceived. How do we misconceive infinity? By considering it a number. Infinity is not a number, it’s simply short-hand for being able to continue doing something. So when we say the set of integers is infinite, we are not saying there are infinitely many integers (although we frequently see statements like that), but simply that one can always construct a larger integer. This is important, because the moment we talk about infinite sizes, we’re talking nonsense; if infinity is never ending, how can you construct anything of infinite size, since you’d never finish the construction!

To see how the notion of infinity is problematic, let’s look at this sphere paradox again and realize that it contradicts a fundamental rule of mathematics, one with a lot more grounding and basis in concrete experience than infinity. This is the basic rule of subtraction. Subtraction states that the only way A – B = A is if B = 0. Yet the sphere example claims A,B > 0 AND A – B = A.

If a mathematical concept violates the laws of basic arithmetic, that concept is no longer mathematical and it has to go.

Again, this happens is because infinity is not a number, so you can’t subtract it or do anything numerical with it, including using it as the size of anything.

 

Conclusion

All but one of the paradoxes above were resolved by clarifying our use of language. This should come as no surprise because language is not precise; listeners do a lot of correction, disambiguation and completion on what they hear. Therefore, when ambiguous utterances are taken as precise, and and the role of active interpreters is ignored, we end up with problems. Think about anything you say and hear on a typical day and think of how many concepts, assumptions and processes you bring to bear to massage that into a coherent concept.

In fact, Wittgenstein argued that all philosophical problems were language problems:

http://en.wikipedia.org/wiki/Philosophical_Investigations

Yet even if some paradoxes were solely due to language, they are no less important, for language is a tool of the mind and we often mistake our words for the thing itself and suffer as a result. To look into how we use language can give us insight into our mind and the things we hold dear. For the religious, it’s not a linguistic game to precisely define Omnipotence. For many, it’s not unimportant to understand what knowledge does and does not imply.

Not only do these paradoxes invite us to improve our thinking, but they can also increase our happiness! We often mistake our mental models of reality for reality, so demonstrating a failure of our thinking (courtesy of the paradox) can make us realize the two are not equal. Maybe we go further; if I couldn’t even conceptualize something as trivial as motion correctly, how else is my mind confused? Maybe that anger, fear, or stressful goal that’s been bugging me was really unimportant to begin with? Maybe I should ignore it when my mind brings up thoughts of fear in particular situations?

In this sense, some paradoxes can be seen as Zen koans that more palatable to the logically oriented. While the Zen Koan also invites one to get out of one’s mind, it requires the attempt to solve intentionally nonsensical riddles. The problem – at least for me – is that I won’t try to solve a riddle I think is nonsense. Ask me what my face was before I was born and I will say I had none and walk away. Ask me what the sound of one hand clapping is and I’ll show you (yes, I know how to clap with one hand, thanks to a friend who taught me this invaluable skill). On the other hand, discuss something logical with me and I’m in. Lure me in with logic, let the paradox smack me upside the head and I will get that aha! As such, the more logically oriented mind is invited to contemplate and consider.

The mind is a blessing and a curse. It can solve problems and provide us with endless hours of amusement. Likewise, it can create problems and provide us with endless hours of torment. We have to know how to use it when appropriate and ignore it when not, and a healthy mistrust of it is can help us, and such mistrust can be fostered by an illustration of the many exciting ways it fails us.

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11 thoughts on “Paradoxes

  1. Another great post full of fascinating thoughts. There may be an interesting connection between your comments on Zen koans and on the dispensability of variables in logic. Two hands would be two variables. One variable does not allow a relationship. No relationship means no variable. One hand cannot clap. As a clapping thing one hand cannot exist. A phenomenon cannot exist except as one of a pair of relative opposites. Ergo all existent phenomena are dependently originated. Yin and Yang and all that. George Spencer Brown wrote a book of poetry about all this called (if I remember right) ‘It Takes Two to Play”. I’ve never been able to find it, but he states, rather unusually for a book of poems, that it is rigorous.

    1. Very nice connection!

      Actually, the only way one hand can clap is if part of the hand makes contact with part of the other, so you’re still dealing with two variables and thus a relationship. So the ability to clap with one hand actually strengthens your point. If there was no way that part of the hand could come in contact with another, we’d end up with a unity and hence no room for the relationship. The clapping emerges as the “between” of the two hands or parts of the hand.

  2. Yep. It seems relevant that Spencer Brown had to introduce imaginary values into Boolean logic in order to model the Zen worldview. But then I must admit I’m a bit hazy when it comes to mathematics.

      1. Yep, you turned me on to GSB, and I did a bit of reading on him. His foundation for math is interesting, and the “visual math” that arose in response to his work is pretty interesting stuff.

        I wonder if he’s neglected because he’s so hard to classify. Is his work mystical? Mathematical? Lyrical? It’s hard to read something like “Laws of Form” and know where to place it.

  3. Yes, that may be exactly it. A bit like trying to sell genre-less music. It just misses every market. He is often dismissed as a crank and equally often called a genius. Even people who have met him sometimes come away confused about which it is. I spoke to him once and he mentioned quite casually that he is a buddha. His book is mystical, metaphysical, mathematical, psycho-physical, psychological and metaphorical. Russell saw the mathematics but seems to have missed the rest. He did not understand metaphysics and so did not see that axiomatising set-theory would be equivalent to axiomatising Reality. It was hardly likely to be easy. Any such attempt, as we know from Kant, would require a phenomenon that is not an instance of a category. Brown introduces this phenomenon, and so does away with all paradox and any need for a Theory of Types.

    It seems obvious to me that this is the solution for the problem of origins, since it is the only one that works. Quite why so few people see this is a mystery to me, and I’m not being disingenuous. It really is a mystery. I don’t know why it isn’t taught in schools. yet I can tell you that you are one of a very small group of people if you can see the metaphysical significance of Brown’s calculus, even though it’s hardly rocket-science.

  4. It occurred to me come back and add, since what I said above might seem arrogant, that while it would be very difficult to understand the full implications of Brown’s calculus and might take a lifetime of study, it would not be at all difficult to see that his axiomatic ultimate phenomenon is the same as Kant’s, and will therefore have the same philosophical and scientific ramifications. .

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