"I know one thing," Socrates famously said. "That I know nothing."

It's a crucial insight from one of the founders of Western philosophy: You should question everything you think you know.

Indeed, the closer you look, the more you'll start to recognize paradoxes all around you.

The dichotomy paradox has been attributed to ancient Greek philosopher Zeno, and it was supposedly created as a proof that the universe is singular and that change, including motion, is impossible (as posited by Zeno's teacher, Parmenides).

People have intuitively rejected this paradox for years.

From a mathematical perspective, the solution -- formalized in the 19th century -- is to accept that one-half plus one-quarter plus one-eighth plus one-sixteenth and so on ... adds up to one. This is similar to saying that 0.999... equals 1.

But this theoretical solution doesn't actually answer how an object can reach its destination. The solution to that question is more complex and still murky, relying on 20th-century theories about matter, time, and space not being infinitely divisible.

This is called the arrow paradox, and it's another of Zeno's arguments against motion. The issue here is that in a single instant of time, zero seconds pass, and so zero motion happens. Zeno argued that if time were made up of instants, the fact that motion doesn't happen in any particular instant would mean motion doesn't happen.

As with the dichotomy paradox, the arrow paradox actually hints at modern understandings of quantum mechanics. In his book "Reflections on Relativity," Kevin Brown notes that in the context of special relativity, an object in motion is different from an object at rest. Relativity requires that objects moving at different speeds will appear different to outside observers and will themselves have different perceptions of the world around them.

Another classic from ancient Greece, the Ship of Theseus paradox gets at the contradictions of identity. It was famously described by Plutarch:

"The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same."

While we're at it, how can evil exist if God is omnipotent? And how can free will exist if God is omniscient?

These are a few of the many paradoxes that exist when you try to apply logic to definitions of God.

Some people might cite these paradoxes as reasons not to believe in a supreme being; however, others would say they are inconsequential or invalid.

Moving ahead to a problem posed in the 17th century, we've got one of many paradoxes related to infinity and geometry.

"Gabriel's Horn" is formed by taking the curve y = 1/x and rotating it around the horizontal axis, as shown in the picture. Using techniques from calculus that make it possible to calculate areas and volumes of shapes constructed this way, it's possible to see that the infinitely long horn actually has a finite volume equal to π, but an infinite surface area.

As stated in the MathWorld article on the horn, this means that the horn could hold a finite volume of paint but would require an infinite amount of paint to cover its entire surface.

Here is one of many self-referential paradoxes that kept modern mathematicians and logicians up at night.

An example of a heterological word is "verb," which is not a verb (as opposed to "noun," which is itself a noun). Another example is "long," which is not a long word (as opposed to "short," which is a short word).

So is "heterological" a heterological word? If it were a word that didn't describe itself, then it would describe itself; but if it did describe itself, then it would not be a word that described itself.

This is related to Russell's Paradox, which asked if the set of things that don't contain themselves contained itself. By creating self-destructing sets like these, Bertrand Russell and others showed the importance of establishing careful rules when creating sets, which would lay the groundwork for 20th-century mathematics.

"Catch-22," a satirical World War II novel by Joseph Heller, named the situation where someone is in need of something that can only be had by not being in need of it -- which is a kind of self-referential paradox.

Protagonist Yossarian is introduced to the paradox with regard to pilot evaluation but eventually sees paradoxical (and oppressive) rules everywhere he looks.

After all, 1 is the first nonzero natural number; 2 is the smallest prime number; 3 is the first odd prime number; 4 is the smallest composite number; etc. And when you finally reach a number that seems not to have anything interesting about it, then that number is interesting by virtue of being the first number that is not interesting.

The Interesting Number Paradox relies on an imprecise definition of "interesting," making this a somewhat sillier version of some of the other paradoxes, like the heterological paradox, that rely on contradictory self-references.

Quantum-computing researcher Nathaniel Johnston came up with a clever resolution of the paradox: Instead of relying on an intuitive notion of "interesting" as in the original paradox, he defined an interesting whole number as one appearing somewhere in the Online Encyclopedia of Integer Sequences, a collection of tens of thousands of mathematical sequences like the prime numbers, the Fibonacci numbers, or the Pythagorean triples.

Based on this definition, as of Johnston's initial blog post in June 2009, the first uninteresting number -- the smallest whole number that didn't show up in any of the sequences -- was 11,630. Since new sequences are added to the encyclopedia all the time, some of which include previously uninteresting numbers, as of Johnston's most recent update in November 2013, the current smallest uninteresting number is 14,228.

Conditional statements in formal logic sometimes have counterintuitive interpretations, and the drinking paradox is a great example.

At first glance, the paradox suggests that one person is causing the rest of the bar to drink.

In fact, all it's saying is that it would be impossible for everyone in the bar to be drinking unless every single customer were drinking. Therefore, there is at least one customer there (i.e., the last customer not drinking) who by drinking could make it so that everyone in the bar was drinking.

The Banach-Tarski paradox relies on a lot of the strange and counterintuitive properties of infinite sets and geometric rotations.

The pieces that the ball gets cut into are very strange-looking, and the paradox only works for an abstract, mathematical sphere: As nice as it would be to take an apple, cut it up, and reassemble the pieces so you have an extra apple for your friend, physical balls made of matter can't be disassembled like a purely mathematical sphere.

Even when working with old-fashioned finite quantities, math can lead to strange results.

The key to the potato paradox is to closely look at the math behind the nonwater content of the potato. Since the potato is 99% water, the dry components are 1% of its mass. The potato starts at 100 grams, so that means that it contains 1 gram of dry material. When the dried-out potato is 98% water, that 1 gram of dry material now needs to account for 2% of the potato's weight. One gram is 2% of 50 grams, so this must be the new weight of the potato.

Another surprising math result, the birthday paradox comes from a careful analysis of the probabilities involved. If two people are in a room together, then there's a 364/365 chance they do not have the same birthday (if we ignore leap years and assume that all birthdays are equally likely), since there are 364 days that are different from the first person's birthday that can then be the second person's birthday.

If there are three people in the room, then the probability that they all have different birthdays is 364/365 x 363/365: As above, once we know the first person's birthday, there are 364 choices of a different birthday for the second person, and this leaves 363 choices for the third person's birthday that are different from those two.

Continuing in this fashion, once you hit 23 people, the probability that all 23 have different birthdays drops below 50%, and so the probability that at least two have the same birthday is better than even.

This seems impossible but is true when you consider the math.

The friendship paradox is caused by how, in most social networks, most people have a few friends, while a handful of people have a large number of friends. Those social butterflies in the second group disproportionately show up as friends of people with smaller numbers of friends, and drag up the average number of friends-of-friends accordingly.

Time travel, if possible, could result in some extremely strange situations.

The bootstrap paradox is the opposite of the classic grandfather paradox: Rather than going back in time and preventing oneself from going back in time, some information or object is brought back in time, becoming a "younger" version of itself, and enabling itself later to travel back in time. One then has to ask: How did that information or object come into being in the first place?

The bootstrap paradox is common in science fiction and takes its name from a short story by Robert Heinlein.

Finally, some see the silence of our universe as a paradox.

One of the underlying assumptions in astronomy is that Earth is a pretty common planet in a pretty common solar system in a pretty common galaxy, and that there is nothing cosmically unique about us. NASA's Kepler satellite has found evidence that there are probably 11 billion Earth-like planets in our galaxy. Given this, life somewhat like us should have evolved somewhere not overly far away from us (at least on a cosmic scale).

But despite developing ever-more-powerful telescopes, we have had no evidence of technological civilizations anywhere else in the universe. Civilizations are noisy: Humanity broadcasts TV and radio signals that are unmistakably artificial. A civilization like ours should leave evidence that we would find.

Furthermore, a civilization that evolved millions of years ago (pretty recent from a cosmic perspective) would have had plenty of time to at least begin colonizing the galaxy, meaning there should be even more evidence of their existence. Indeed, given enough time, a colonizing civilization would be able to colonize the entire galaxy over the course of millions of years.

The physicist Enrico Fermi, for whom this paradox was named, simply asked, "Where are they?" in the middle of a lunchtime discussion with his colleagues. One resolution of the paradox challenges the above idea that Earth is common and posits instead that complex life is extremely rare in the universe. Another posits that technological civilizations inevitably wipe themselves out through nuclear war or ecological devastation.

A more optimistic solution is the idea that the aliens are intentionally hiding themselves from us until we become more socially and technologically mature. Yet another idea is that alien technology is so advanced that we wouldn't even be able to recognize it.

It's a crucial insight from one of the founders of Western philosophy: You should question everything you think you know.

Indeed, the closer you look, the more you'll start to recognize paradoxes all around you.

Even Socrates is baffled.The Death of Socrates by Jacques-Louis David (1787) |

**To go anywhere, you must go halfway first, and then you must go half of the remaining distance, and half of the remaining distance, and so forth to infinity: Thus, motion is impossible.**The dichotomy paradox has been attributed to ancient Greek philosopher Zeno, and it was supposedly created as a proof that the universe is singular and that change, including motion, is impossible (as posited by Zeno's teacher, Parmenides).

People have intuitively rejected this paradox for years.

From a mathematical perspective, the solution -- formalized in the 19th century -- is to accept that one-half plus one-quarter plus one-eighth plus one-sixteenth and so on ... adds up to one. This is similar to saying that 0.999... equals 1.

But this theoretical solution doesn't actually answer how an object can reach its destination. The solution to that question is more complex and still murky, relying on 20th-century theories about matter, time, and space not being infinitely divisible.

**In any instant, a moving object is indistinguishable from a nonmoving object: Thus motion is impossible.**This is called the arrow paradox, and it's another of Zeno's arguments against motion. The issue here is that in a single instant of time, zero seconds pass, and so zero motion happens. Zeno argued that if time were made up of instants, the fact that motion doesn't happen in any particular instant would mean motion doesn't happen.

As with the dichotomy paradox, the arrow paradox actually hints at modern understandings of quantum mechanics. In his book "Reflections on Relativity," Kevin Brown notes that in the context of special relativity, an object in motion is different from an object at rest. Relativity requires that objects moving at different speeds will appear different to outside observers and will themselves have different perceptions of the world around them.

**If you restored a ship by replacing each of its wooden parts, would it remain the same ship?**Another classic from ancient Greece, the Ship of Theseus paradox gets at the contradictions of identity. It was famously described by Plutarch:

"The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same."

**Can an omnipotent being create a rock too heavy for itself to lift?**While we're at it, how can evil exist if God is omnipotent? And how can free will exist if God is omniscient?

These are a few of the many paradoxes that exist when you try to apply logic to definitions of God.

Some people might cite these paradoxes as reasons not to believe in a supreme being; however, others would say they are inconsequential or invalid.

**There's an infinitely long "horn" that has a finite volume but an infinite surface area.**Moving ahead to a problem posed in the 17th century, we've got one of many paradoxes related to infinity and geometry.

"Gabriel's Horn" is formed by taking the curve y = 1/x and rotating it around the horizontal axis, as shown in the picture. Using techniques from calculus that make it possible to calculate areas and volumes of shapes constructed this way, it's possible to see that the infinitely long horn actually has a finite volume equal to π, but an infinite surface area.

As stated in the MathWorld article on the horn, this means that the horn could hold a finite volume of paint but would require an infinite amount of paint to cover its entire surface.

**A heterological word is one that does not describe itself. Does "heterological" describe itself?**Here is one of many self-referential paradoxes that kept modern mathematicians and logicians up at night.

An example of a heterological word is "verb," which is not a verb (as opposed to "noun," which is itself a noun). Another example is "long," which is not a long word (as opposed to "short," which is a short word).

So is "heterological" a heterological word? If it were a word that didn't describe itself, then it would describe itself; but if it did describe itself, then it would not be a word that described itself.

This is related to Russell's Paradox, which asked if the set of things that don't contain themselves contained itself. By creating self-destructing sets like these, Bertrand Russell and others showed the importance of establishing careful rules when creating sets, which would lay the groundwork for 20th-century mathematics.

**Pilots can get out of combat duty if they are psychologically unfit, but anyone who tries to get out of combat duty proves he is sane.**"Catch-22," a satirical World War II novel by Joseph Heller, named the situation where someone is in need of something that can only be had by not being in need of it -- which is a kind of self-referential paradox.

Protagonist Yossarian is introduced to the paradox with regard to pilot evaluation but eventually sees paradoxical (and oppressive) rules everywhere he looks.

**There is something interesting about every number.**After all, 1 is the first nonzero natural number; 2 is the smallest prime number; 3 is the first odd prime number; 4 is the smallest composite number; etc. And when you finally reach a number that seems not to have anything interesting about it, then that number is interesting by virtue of being the first number that is not interesting.

The Interesting Number Paradox relies on an imprecise definition of "interesting," making this a somewhat sillier version of some of the other paradoxes, like the heterological paradox, that rely on contradictory self-references.

Quantum-computing researcher Nathaniel Johnston came up with a clever resolution of the paradox: Instead of relying on an intuitive notion of "interesting" as in the original paradox, he defined an interesting whole number as one appearing somewhere in the Online Encyclopedia of Integer Sequences, a collection of tens of thousands of mathematical sequences like the prime numbers, the Fibonacci numbers, or the Pythagorean triples.

Based on this definition, as of Johnston's initial blog post in June 2009, the first uninteresting number -- the smallest whole number that didn't show up in any of the sequences -- was 11,630. Since new sequences are added to the encyclopedia all the time, some of which include previously uninteresting numbers, as of Johnston's most recent update in November 2013, the current smallest uninteresting number is 14,228.

**In a bar, there is always at least one customer for whom it is true that if he is drinking, everyone is drinking.**Conditional statements in formal logic sometimes have counterintuitive interpretations, and the drinking paradox is a great example.

At first glance, the paradox suggests that one person is causing the rest of the bar to drink.

In fact, all it's saying is that it would be impossible for everyone in the bar to be drinking unless every single customer were drinking. Therefore, there is at least one customer there (i.e., the last customer not drinking) who by drinking could make it so that everyone in the bar was drinking.

**A ball that can be cut into a finite number of pieces can be reassembled into two balls of the same size.**The Banach-Tarski paradox relies on a lot of the strange and counterintuitive properties of infinite sets and geometric rotations.

The pieces that the ball gets cut into are very strange-looking, and the paradox only works for an abstract, mathematical sphere: As nice as it would be to take an apple, cut it up, and reassemble the pieces so you have an extra apple for your friend, physical balls made of matter can't be disassembled like a purely mathematical sphere.

**A 100-gram potato is 99% water. If it dries to become 98% water, it will weigh only 50 grams.**Even when working with old-fashioned finite quantities, math can lead to strange results.

The key to the potato paradox is to closely look at the math behind the nonwater content of the potato. Since the potato is 99% water, the dry components are 1% of its mass. The potato starts at 100 grams, so that means that it contains 1 gram of dry material. When the dried-out potato is 98% water, that 1 gram of dry material now needs to account for 2% of the potato's weight. One gram is 2% of 50 grams, so this must be the new weight of the potato.

**If just 23 people are in a room, there's a better-than-even chance at least two of them have the same birthday.**Another surprising math result, the birthday paradox comes from a careful analysis of the probabilities involved. If two people are in a room together, then there's a 364/365 chance they do not have the same birthday (if we ignore leap years and assume that all birthdays are equally likely), since there are 364 days that are different from the first person's birthday that can then be the second person's birthday.

If there are three people in the room, then the probability that they all have different birthdays is 364/365 x 363/365: As above, once we know the first person's birthday, there are 364 choices of a different birthday for the second person, and this leaves 363 choices for the third person's birthday that are different from those two.

Continuing in this fashion, once you hit 23 people, the probability that all 23 have different birthdays drops below 50%, and so the probability that at least two have the same birthday is better than even.

**Most people's friends have more friends than they do.**This seems impossible but is true when you consider the math.

The friendship paradox is caused by how, in most social networks, most people have a few friends, while a handful of people have a large number of friends. Those social butterflies in the second group disproportionately show up as friends of people with smaller numbers of friends, and drag up the average number of friends-of-friends accordingly.

**A physicist working on inventing a time machine is visited by an older version of himself. The older version gives him the plans for a time machine, and the younger version uses those plans to build the time machine, eventually going back in time as the older version of himself.**Time travel, if possible, could result in some extremely strange situations.

The bootstrap paradox is the opposite of the classic grandfather paradox: Rather than going back in time and preventing oneself from going back in time, some information or object is brought back in time, becoming a "younger" version of itself, and enabling itself later to travel back in time. One then has to ask: How did that information or object come into being in the first place?

The bootstrap paradox is common in science fiction and takes its name from a short story by Robert Heinlein.

**If there's nothing particularly unique about Earth, then there should be lots of alien civilizations in our galaxy. However, we've found no evidence of other intelligent life in the universe.**Finally, some see the silence of our universe as a paradox.

One of the underlying assumptions in astronomy is that Earth is a pretty common planet in a pretty common solar system in a pretty common galaxy, and that there is nothing cosmically unique about us. NASA's Kepler satellite has found evidence that there are probably 11 billion Earth-like planets in our galaxy. Given this, life somewhat like us should have evolved somewhere not overly far away from us (at least on a cosmic scale).

But despite developing ever-more-powerful telescopes, we have had no evidence of technological civilizations anywhere else in the universe. Civilizations are noisy: Humanity broadcasts TV and radio signals that are unmistakably artificial. A civilization like ours should leave evidence that we would find.

Furthermore, a civilization that evolved millions of years ago (pretty recent from a cosmic perspective) would have had plenty of time to at least begin colonizing the galaxy, meaning there should be even more evidence of their existence. Indeed, given enough time, a colonizing civilization would be able to colonize the entire galaxy over the course of millions of years.

The physicist Enrico Fermi, for whom this paradox was named, simply asked, "Where are they?" in the middle of a lunchtime discussion with his colleagues. One resolution of the paradox challenges the above idea that Earth is common and posits instead that complex life is extremely rare in the universe. Another posits that technological civilizations inevitably wipe themselves out through nuclear war or ecological devastation.

A more optimistic solution is the idea that the aliens are intentionally hiding themselves from us until we become more socially and technologically mature. Yet another idea is that alien technology is so advanced that we wouldn't even be able to recognize it.

**(Source: BI UK)**
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