Today is February 29th, which only occurs once every four years! Of course, when I think of February 29th, I think of the Gilbert and Sullivan opera "The Pirates of Penzance" (not really, but maybe I should!). At the beginning of the opera, the character Frederic believes that he is going to be released from his pirate apprenticeship, since he has reached his 21st birthday. Unfortunately for Frederic, he learns that he was born on February 29th ("Leap Day"), and so he technically has had only five birthdays and is therefore still bound to the Pirate King as his apprentice. Frederic's dilemma (which is described in the song, "Paradox") is an example of what is called a veridical paradox, i.e. one where the result is absurd but nevertheless true. There's another well-known example of a veridical paradox called the "Potato Paradox". If you look this paradox up on the Internet, you will find a reference to the 2015 movie "The Martian", starring Matt Damon.
I don't think that the "Potato Paradox" actually came up in "The Martian" (or the book by Andy Weir on which the movie was based), but it certainly seems like it could have been mentioned. If you've read the book, watched the movie, or both, you will remember that Mark Watney grows potatoes to try to survive until his rescue from Mars (he actually uses his own feces as fertilizer). In one memorable line, Watney says, "They say once you grow crops somewhere you officially colonized it. So technically, I colonized Mars. In your face Neil Armstrong!"
So, what exactly is the "Potato Paradox"? It goes something like this:
Mark grows 100 kg of potatoes, which consist of 99% water. He then leaves them outside overnight so that they consist of 98% water. What is their new weight?
The answer may surprise you - it's 50 kg! Allow me to explain. If the potatoes are 99% water, that means that their dry mass is 1%. In other words, 100 kg of potatoes contains 1 kg dry mass, which does not change. In order for the potatoes to change to 98% water, the dry mass has to equal 2% of the overall weight of the potatoes. If the dry mass cannot change, it has to follow that the total mass of the potatoes has to decrease. Since the proportion of dry mass has doubled, the total mass must be halved to 50 kg!
The American philosopher and logician W.V.O. Quine distinguished between three different classes of paradoxes - the aforementioned veridical paradox, the falsidical paradox, and the antimony. As I mentioned above, the "The Pirates of Penzance" paradox and the "Potato Paradox" are two examples of a veridical paradox, i.e. one where the result is absurd but nevertheless true. The Monty Hall problem, Arrow's impossibility theorem, and the birthday problem are other examples of a veridical paradox, all of which I've posted about in the past.
A falsidical paradox is one that not only appears to be false but actually is false. Zeno's paradoxes are classic examples here. Recall the race between Achilles and a tortoise - "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead." As the story goes, Zeno imagined that the Greek hero, Achilles was challenged to a footrace by a tortoise. Achilles was a gracious and fair fellow, so he gave the tortoise a head start of 100 paces. If both Achilles and the tortoise start at exactly the same time ("On your mark, get set, go!"), and if both Achilles and the tortoise run at constant speeds (Achilles being very fast and the tortoise being very slow), then after a finite period of time, Achilles will reach the point where the tortoise started (100 paces away). Importantly, the tortoise will no longer be there! He has moved, albeit slowly, to a new place, just a little farther away. Now, it will take Achilles a finite period of time to cover the new distance that separates him from the tortoise, and he will eventually reach where the tortoise was after the race started. Again, the tortoise will not be there, as he has moved to a new position, just a little farther away. In this manner, whenever Achilles arrives at the point where the tortoise has been, he will still have some distance to go before he can reach the tortoise.
An antimony cannot be classified as either veridical or falsidical and reaches a self-contradictory result by applying properly accepted methods of logic and reasoning. I'll talk about the classic Liar's paradox, one famous antimony, next time. But for now, Happy Leap Day!
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