(Fun!) Brain Teasers

Some accident in an online editorial committee allowed the publication of an article with…yes…Philosophy in the headline. What was neat about this piece is that it contained a few “brain-teasers” that are usually kept within the confines of boring dinner parties or law school library coffeeshops. Included in this Top 10 list are some old-school paradoxes from Zeno, and also some more modern mind benders. I thought it would be fun to briefly cover each one since these types of questions tend to come up in conversations with high-school and college students.

In what follows I have edited the stated paradox for relevancy. If you read the original article, you can pick up links to pages with possible solutions and explanations.

#10 – The “Heap”

If a man has zero hairs on his head, we say he’s bald. However, a man who has 10,000 hairs on his head is not considered to be bald. But what if we add a single hair to the head of the man with zero hairs? He would still clearly be bald.

Now let’s say that a man has 1,000 hairs only. But the strands are evenly spaced and really thin. Would this man be bald or not bald? Would you consider a single grain of wheat a “heap of wheat?” Definitely not. How about two grains? Still, probably not. So, when do a few grains or a few hairs end and a whole heap or baldness actually begin? The problem is one of vagueness. Where does one description end and another begin?

This one is more tied up with language and definition than any philosophical principle. We could just say that a man is not technically bald unless he has zero hairs on his head. “Bald” means not one hair. Calling a man bald with any hail would technically be a false statement, or perhaps an inaccurate judgment. We could say the man is “balding” in the sense that he is in the process of becoming bald, and that is probably how the conventional language means it.

In terms of what constitutes a heap, we could also place a rigid definition on this. We could say that 10 or more grains is a heap. The question then becomes why 10 grains and not 2? And this gets at the issue of an arbitrary notion of abstract objects, like “heap,” or even “baldness.” However, baldness is accidental to the man and a heap (or any kind of collection of natural substances) is a technically an artifact. Neither is what an Aristotelian-Thomist would deem a substantial form. Therefore, no anti-realist problem arises. We can make sense out of heap in various cases by being clear what in a given instance is what we mean by ‘heap’.

 #9 – The Liar Paradox

The first sentence of this paragraph is a lie. Stop and think about that sentence for a second. Is it true? Or a lie? A true lie? This is called The Liar Paradox, and it’s also from the time of Eubulides. It’s straightforward and fun and takes the form of one short statement: “This sentence is a lie.” Another incarnation of the paradox is: “Everything I say is false.”        

       

How this one made the list is puzzling. Nothing is being said when a contradiction is uttered. “X is a square circle” does not tell us anything. We do not have any referent in the first sentence of this paradox, so there is nothing to think about. The last statement “everything I say is false” is nonsensical. The examples in this paradox are like saying “it is true that there is no truth.” Such a statement collapses upon itself. Often these “philosophical” paradoxes seem to twist our mind because they violate the principle of non-contradiction somewhere. We know something is amiss even though we cannot always articulate it immediately.

# 8 – Limited and Unlimited

Zeno wanted to show that the idea of a plurality of things (which all exist side by side in time and space) brought with it some serious logical inconsistencies. The Limited And Unlimited Paradox displayed this. Does one thing exist or many? What separates one thing from the next? Where is the line?

This is also called The Paradox of Density, and let’s put it a little differently. This works with multiple objects, but we’ll start with just two. If there are two things, what separates them? You need a third thing to separate the two.

The Paradox of Density takes place on many different scales, but you get the basic idea. So, is there just one massive entity called the universe that contains indistinguishable matter of varying densities (air, the floor, a tree, etc.)? Is all matter perpetually divisible? Or if we divide matter into objects small enough, will we eventually reach the object so small that it cannot be divided?

Zeno was the master of paradox. I think many of his antinomies are good thinking exercises. Before this list, I had not previously seen the paradox of limited and unlimited written in this way. To cash this one out per above, we will first need to unpack it a little more. Take the issue of two things needing something to separate them. First of all, if there are actually two things, then they are distinguishable in some way. In this case, say they have different material constituents. It seems Zeno wants to know what is this “space” thing that we might evoke to explain our separation of them. The “space” between object A and object B cannot be nothing, because then we could not invoke it to explain anything. So, the space between A and B becomes thing C. At least that’s what Zeno thinks. But then he will ask us, what separates A from B and B from C? It seems that a fourth thing must be posited, and so on ad infinitum.

There are probably different ways to address this paradox. On an Aristotelian-Thomist view (A/T), the principles of nature are matter, form, and privation. Thing A is a composite of matter and form. Privation might be thought of as the lack of form that A has. A is what it is, and it is not B because A has the substantial form of A. The same thing for anything B, C, and so on. A and B each can be said to have form, matter, and privation. We predicate these principles of each thing, keeping in mind privation is lack or absence. Zeno’s demand for us to tell him what separates A and B therefore betrays an incorrect view of substances. He wants us to admit that privation is a substance, which it is not.

Still, someone might protest that the “space” between A and B is comprised of various subatomic particles. Space really is “something” if we utilize modern physics. On this view, Zeno’s demand seems to gain traction. First, let us remember what Zeno is trying to establish. He wants to show that if there are many things there must be infinitely many things. But if Zeno just wants to prove that there is an infinite number of things, I do not really see what real problem there is. It seems the specter is predicated on an assumption that if we say there are many things we mean that there is a finite quantity. To me the actual problem with this is unclear. An SEP entry breaks Zeno’s actual paradox down as follows “If there are many things, then there must be finitely many things; and if there are many things, then there must be infinitely many things.” I do not see any reason to think the first conditional has any force such that we need accept it. We could just go this route and let Zeno hit us with another zinger. 

There is still might be a physicist demanding an answer to this quandary without “appealing to the outdated physics of Aristotle.” The A/T view will still look at substances A and B, where the privation in substances could simply be termed matter in potency to something we know not what. Or, if we broke things down to the subatomic level, we would just identify other distinct form/matter composites filling the “space.” After all, “space” on this analysis is something of a placeholder for substances not observable by the senses. If the subatomic particles exist in the way we think they do, they would be form/matter composites with essences/ natures. Responding to Zeno, we would not necessarily need to say that what exists within what we call ‘space” is infinite, though I cannot see the problem if were (and we would need to know a little more about how ‘infinite’ is being used – e.g. potential or actual, etc.).

The second part of the paradox above seems odd to me. It makes little sense to say the universe could just be one thing with indistinguishable matter of various densities because densities and other properties are just what distinguishes matter. And there is the example of trees and air, which have distinct material constituents. Here it seems “matter” is used in something of an equivocal sense, which we need not accept. It seems to be used in both generically and specifically. To say, “everything that exists is material” is not the same thing as saying, “everything is ultimately the same thing.” To ask if all matter is perpetually divisible is a benign question because even if matter is infinitely divisible it does not follow that substances, comprised of matter and form, are.

#7 – The Dichotomy Paradox

Let’s say that you decide to walk to the store and buy a soda. For you to get there, you’ll have to cross the halfway point. No problem, this makes sense. But from the halfway point, you’ll have to next cross the halfway point of the halfway point (three-quarters of the way from your house to the store). Then you’ll have to cross the halfway point of that distance and the halfway point of the next smaller distance.

So wait a minute. If you keep dividing your trip into halfway points, you’ll never actually be across the halfway point . . . ever. How is this possible? You know that you can go to the store and get a soda. But when do you actually cross the last halfway point (where there are no more halfway points)

This one has gotten a lot of attention. Most of the time people try to solve this mathematically, which is fine as far as math is concerned. I will not attempt nor really address that type of resolution. For what it’s worth, the disputations in higher level math about infinite sets as they apply to this paradox are just doing metaphysics in a different way. In any event, I think Zeno’s false assumption is that extrinsically denominated measurements like “halfway” have substantial reality instead of having accidental being and leading to mental abstractions and subsequent judgments which start from reality and then fold into other concepts which do not have extramental existence. [yes, that last sentence was long]

Zeno’s paradox, in this case, rests upon substantive predication of place and time, which on the A/T view are only accidental to substances. A substance undergoes accidental change by going from place A to place B. Zeno is ultimately saying that it takes an infinite time to get from A to B. This is based on an infinite divisibility of the walking distance. But, as Aristotle pointed out, the infinite divisibility of the sidewalk only exists for the one measuring, not necessarily the one traversing. This sounds like a cop-out, but if we think about it, the sidewalk is not actually divided as such, but only so divided by abstraction/calculation of the one measuring it (or the one creating the paradox!). There is no actual problem for the unreflective guy just walking to get a soda. Zeno treats “halfway” and other measurements as though they are real things, which they technically are but only in a derivative and/or abstracted way. Time itself is an extrinsically denominated accident. What is time if not that which tracks/measures change? The very notion of there being any time measurement on the paradoxical journey in question presupposes that change is real. Zeno needs to assume change is real to disprove change, so his argument does not work.

I am not an anti-realist about time or measurement, they are not mere constructs. But they are of and about substances. And we must agree that substance undergoes change, or the paradox does not get off the ground.

Consider what happens to the paradox if there is an earthquake and ground collapse into a black hole oblivion behind Zeno as he travels to get a soda. On his finite journey, Zeno has gone 1 mile out of the 5 it is supposed to take him infinity to traverse. The 1 mile behind him is gone. His “line” of travel has shrunk by 20%. By the rationale in the paradox we can then have an earthquake happen on all ground in front of him as well. Zeno is just left standing on one plot of soil, say it is 2 feet across. Still, poor Zeno is apparently looking at an infinite amount of time to even kneel in prayer for rescue. For him to kneel, he has to move just 1 inch.  But the space cutting infinity still must happen. The result of Zeno’s paradox is that change, or motion, just cannot happen at all.

Another way to look at this is to borrow something from the mathematical side. Zeno is admittedly traveling between two finite points. He is not traveling an infinite distance. How does it even make sense in principle that it would take infinite time to cross a finite distance? The distance is fixed, and not expanding. Zeno seems to be predicating a finite and an infinite accident of substances which are at the same time and same relationship. But this can only occur by an error of our judgment or the pressing of a mental abstraction onto reality.

Further, even if we take Zeno’s bait and start “slicing” the line, why should we think that it is infinitely divisible? The infinite time traversing the line seems to be predicated on no stopping point to the slicing. But if there is a stopping point, a measurement or fundamental block of reality by which the line could be sliced no more, then traversing it is no longer logically problematic. The line folds or is halved into infinity only if we agree with Zeno that this is possible in principle. Yet, Zeno’s position then seems to imply an infinite regress of material constituents; that we can just keep going forever and never get at the ultimate “stuff” of physical reality. If there is nothing foundationally physical, then it presents some significant problems for how we would have anything physical at all to start slicing. To take another tack, it seems that for Zeno the house of quarks, bosons, leptons, etc. would just be based on a sinking foundation of nothingness unless we posit that the physical gives way to the metaphysical and ultimately just lands on pure Being itself. And if Zeno wants to land here, great. I think that would relieve the tension in his paradox.

Much more could be said on this. And much brighter minds than me have presented ways to escape from Zeno that probably make this short attempt seem inconsequential. But the whole point is to try and have some fun, right?

#6 – Achilles and the Tortoise

In this puzzle, Achilles races a tortoise. To be a nice guy (demigod), Achilles gives the tortoise a 100-meter (328 ft) head start because Achilles is an extremely fast runner and the tortoise is . . . well . . . a tortoise. As soon as the gun fires and the race begins, Achilles quickly closes in on the slow-moving tortoise. In no time, Achilles has crossed the 100 meters (328 ft) of the head start that he gave the tortoise. Simultaneously, the tortoise has traveled 10 meters (33 ft). So Achilles still hasn’t caught the tortoise. But again, Achilles will quickly close in, crossing the additional 10 meters (33 ft). During this time, however, the tortoise has traveled another 1 meter (3 ft). By this logic, Achilles can never truly catch the tortoise, can he? How can this be possible? Every time he gets closer, the tortoise goes further. Does this mean that motion itself is impossible even though we experience it daily?

The resolution to #7 (above) applies equally to this one. Zeno is denying that change/motion is possible, but the same flawed assumptions and problematic conclusions await. Achilles should feel good about his speed.

#5 – The Paradox of Inquiry

As Meno said, “And how will you inquire into a thing when you are wholly ignorant of what it is? Even if you happen to bump right into it, how will you know it is the thing you didn’t know?” Socrates rephrased the paradox this way: “A man cannot search either for what he knows or for what he does not know. He cannot search for what he knows—since he knows it, there is no need to search—nor for what he does not know, for he does not know what to look for.”

This paradox seems to betray a fundamental misunderstanding of knowledge. It treats knowledge as if it is this “thing” that we look for like lost car keys or El Dorado. How do we even look for something if we do not even know what “it” is?

One way to address this is to take the empiricist (in the classic sense) tack. What man knows are things in the world, composites of form and matter. The intellect apprehends the form of the object and an identity/relation obtains between the knower and object known. The intellect can form abstractions from the things it encounters, and can organize, arrange, and make judgments about them. To borrow from Aristotle, nothing is in the intellect that was not first in the senses. We can understand this as the starting point for knowledge. Man encounters, learns, and interacts with the world. This is a progressive process. We did not start out asking “where can I find an automobile?” Instead, by living and operating in the world of things, man begins to ask questions and seeks to solve problems. Still Plato would persist “how do you know what questions to even ask?” To which we might reply that such a question presupposes an understanding of human nature that we need not accept, an understanding that goes against reason. Man can “know” what to look for in many instances because He has a true grasp of the world as it is. We can understand by the very nature of what we are, what natures are, and that something is out of order and seek a way to properly order things. To argue in the way of the Meno paradox is to simply assume a certain philosophical anthropology that is unnecessary.

Of additional note is that the Meno paradox is predicated on the idea that we do not really come to know things, only to remove the scales covering what we already possess in some way. Via the philosophical dialectic, we recall the really real things that we grasped before our soul was imprisoned in our body. Plato’s theory of reminiscence is, I think, implicit in Meno and need not be accepted.

#4 – The Double Liar Paradox

Let’s move up to more modern times and toy with a fun extension of The Liar Paradox called The Double Liar Paradox. First dreamed up by mathematician P.E.B. Jourdain, this brainteaser goes as follows: Take a flash card or a piece of paper. On one side, write: “The sentence on the other side of this card is true.” Now flip it over and write on the other side: “The sentence on the other side of this card is false.” If the second sentence is true, then the first sentence is false. (Flip the card.) Here, you end up moving into an indefinite changing of sides—side A to side B on the card. But if the sentence you first wrote is false, as the second sentence claims, then the second sentence would also be false. Thus, both sentences are right and wrong at the same time. Have fun with that one.

This one is just like #9. There is nothing to it because there is no actual referent to the propositions. The words on the paper are not conveying anything about reality.  It is like saying “married bachelor.” A “sentence” that has no content is essentially vacuous, the words are said but have no signification. I could just write something like “there is no meaning except that there is no meaning about meaning…” and it would be non-sensical. Contradictions are not things. We can talk about them abstractly only because we know the non-contradictory nature of reality.

#3 – The Monty Hall Problem

This one can be seen on game shows everywhere. Let’s say there are three doors. Behind each of two doors is a brick, but one door masks $1 million. You get to pick a door and see if you win the million. Let’s suppose you choose Door A and hope for the million. Then the game show host opens another door at random to see if you won or lost. The host chooses Door B, and it reveals a brick. With Door B out of the way, the one-third odds just got a lot better. You’re left to choose between Door A and Door C. You can even switch to Door C now if you want. Since you don’t know what is actually behind your door, you’re still picking between two doors. So your odds are 50/50, right? Door A, Door C . . . it’s one out of two . . . can’t get any simpler than this. Wrong. At this point, it sounds counterintuitive to say that you have a two-thirds chance of getting the $1 million if you switch doors and a one-third chance if you stay put. But it’s true. Can you figure out why?

Since this is more of a probability related problem, I will leave it to those more inclined in that direction. But this is certain a fun question to think about in case you ever need it.

#2 – The Barber Paradox

The puzzle is simple: A barber says he’ll shave any man who does not shave himself and all men who do not shave themselves if they come to be shaved. The question is: Does the barber shave himself? If he does, then he no longer shaves all men who do not shave themselves because he shaves himself. If he does not shave himself, then he does not shave all men who do not shave themselves.

By including himself in the “bargain,” the statement of the ambitious barber results in a contradiction. If he maintains his deal, the falls on his own sword (or really sharp scissors). We have no reason to accept the bargain as having any meaning as it is presented.

#1 - Schrodinger’s Cat

Moving on to the best brainteaser, which is arguably not a paradox, let’s talk about Schrodinger’s cat. It begins with the idea that we take a cat and place it in a soundproof box. Now, without lifting the lid to observe the cat, how do we know whether the cat is alive or dead?

In a nutshell, every time you look at something (a chair, for instance), you get a definite answer as to its state. (It is there.) When you turn your head, you can only get probable chances of whether it is still there or not. Yes, it’s safe to say that the chair didn’t get up and walk away. But without observation, you’ll never really know. So, at what point can the things we observe be certain to exist (or exist in the state we observe them)?

In the early 1700’s, Bishop George Berkeley is credited with arguing “esse est percipi” (“to be is to be perceived”). Schrodinger’s Cat has always seemed to me a lot like Berkeley’s position. Berkeley solved this problem by positing that things existed when we were not thinking about them or perceiving them because God was thinking about them. The modern science from which Schrodinger’s Cat springs would not offer such rationale.

To think that nothing exists until we perceive it is of course an absurd claim. Science could not work if this was true, because science takes for granted in one observation all kinds of unseen/unperceived things. Further, this way of understanding reality is impossible because it presupposes a knowing subject in order for a known object to exist. But our actual interaction with the world is the exact opposite. We only know ourselves because the external world first impinges upon us. To ask, “how do I really know that the chair exists when I am not in the room” betrays a skepticism rooted in a false understanding of what knowledge is and what things are (e.g. having natures, substantial forms, etc.). The idealist undertones of this paradox/dilemma need not be accepted. If knowledge is of things and not ideas or beliefs about things, and if things exist as substantial forms having certain natural operations, etc. then Schrodinger’s Cat does not present any real problem. There may be other ways around this as well, such as the fact that we simply have no good reason or evidence to think that the chair should have moved, so we are rational or warranted in thinking that the chair is still there. In any event, I hope no animals were harmed in the formulation of this paradox.