A paradox is a statement which appears to contradict itself and yet might be true (or even false). Paradoxes have interested philosophers, mathematicians, logicians and scientists for thousands of years and have also promoted the value of critical thinking and analysis. Paradoxes have arisen in many branches science, mathematics and philosophy and there are hundreds of them. A few paradoxes like the Grandfather paradox is very well known. Here, I have decided to share a few of them, mostly logical in nature, which I find rather interesting.
The barber paradox or the Russell’s paradox (named after English philosopher Bertrand Russell) refers to a hypothetical town with just one barber. In this town, every man keeps himself clean-shaven which is done so in two ways: he either shaves himself or he is shaved by the barber. Also, the barber is a man in town who shaves all those, and only those, men in town who do not shave themselves. Here starts the question: “Who shaves the barber?” Clearly, he can either shave himself or go to the barber (which happens to be himself). The paradox arises because the barber is the man in town who shaves those who do not shave themselves, rendering both of the options invalid (since, ultimately, he has to shave himself).
This paradox poses the question of whether something remains fundamentally the same if all of its old components have been replaced by new ones. Suppose we have a ship and we decide to remove all the wooden pieces of it, one by one, replacing them with new ones. After the process has been completed, are we left with the old ship? Or is it something entirely new? It essentially asks the question of identity. Resolutions to this paradox have been proposed since the times that date back to the classical Greek era. It has a more modern resolution in the form of four-dimensionalism.
Much like the Ship of Theseus, the Sorites paradox is one that arises from vague predicates. In a typical formulation, one starts out with a heap of sand and proceed to remove single grains from the heap. Any deduction using common logic would suggest that the removal of a single grain of sand would not turn the heap into a non-heap. The same considerations apply after removing two, three, four etc. grains. The paradox arises when, after removing an increasingly large number of grains, only a single grain of sand remains. The question posed is whether this grain is a heap or a non-heap? If no, when did the heap to non-heap transition occur?
There are many proposed resolutions to this paradox. One is setting a fixed boundary: a limit which divides heaps from non-heaps. One might claim that fewer than 10,000 grains of sand will render it a non-heap. However, this resolution suffers greatly from the arbitrary nature of the limiting number, thus lacking precision.
This falls under a list of paradoxes of motion first proposed by Zeno of Elea. This particular one is mind boggling and yet, it is a part of our everyday life.
In a race between Achilles and tortoise, the latter is given a headstart- say 100 m. It is obvious that Achilles is faster, and he quickly covers up the 100 m deficit. However, within the time that Achilles took to cover the distance, the tortoise further moves a finite distance ahead of Achilles. Our Greek hero takes a smaller period of time to cover this new deficit but, within this short interval, the tortoise treads along. It seems that no matter what Achilles does, he can’t overtake the tortoise, despite being the faster one.
The resolution of this paradox is mathematical, but easy to grasp. One has to imagine that the deficits created between Achilles and the tortoise form an infinite series: 1/2 + 1/4 + 1/8 + 1/16 +… which adds up to 1 (convergent series for those who are familiar with the term). As long as Achilles moves at a sufficiently fast pace, he can cover the distance in a measurable amount of time and catch up with the tortoise. There is one construct where Achilles can never catch up with the tortoise, even though he is faster (in that case the deficits form a divergent series).