Paradoxes: Aiding in scientific advancements

|Difficulty level: Medium|

"How quaint the way of paradox –
At common sense she gaily mocks.”
–  W. S. Gilbert

 A paradox is a statement that is a contradiction in itself, apparently devoid of logic. Consider the declaration – “This statement is false”; if it is a lie, then the assertion is true. However, if it is true, then it is indeed a lie. Isn’t this seemingly innocent statement, an instance of the Liar paradox, enough to baffle the logician in you? That’s what paradoxes are all about!

Since the beginning of rational thinking, these have played a valuable role in philosophical discussions. The first paradox in recorded history can be credited to the Greek thinker Anaximander of Miletus (610BCE – 858BCE) [1]. In order to address the question of whether everything has an origin, he surmised that everything owes its existence to something that existed before, thereby giving birth to the concept of an infinite past. Furthermore, since we always contemplate about the future, that too is infinite. This school of thought is somewhat similar to what we now know as the Theory of Evolution attributed to Charles Darwin. So, what do you think – did the hen come first or the egg😉?

Renowned 20th-century philosopher Williard Van Orman Quine divided the paradoxes into three classes. A veridical paradox often leads to conclusions that seem nonsensical. Even though incomprehensible at first, careful reconsideration often leads us to simple and logical answers that are true – much like solving a puzzle. A popular example is the Birthday paradox, which states that the probability of two people having the same birthday is 0.5 for 23 people and 9.99 for 75 people. A falsical paradox is the outcome of an underlying error in a step leading up to the conclusion. Absurd algebraic proofs (like, 1=2) that involve a division by 0 fall in this category. Any paradox that is neither veridical nor falsical is an antimony.

Paradoxes frequently arise out of a flawed notion about a theory, or a gap in it. Therefore, a resolution often strengthens our understanding of the theory or paves way for a more advanced and complete one. Take the example of the Dichotomy paradox proposed by the celebrated 5th century BCE philosopher Zeno of Elea (495BCE – 430BCE). Suppose to reach from point A to B that is separated by 1 km, you walk a certain distance, say 500m and then walk half of it, i.e., 250m, followed by 125m, and so on. If you continue this way, you will get a series with an infinite number of terms –

Zeno argued that it should be impossible to reach point B as it would take an infinite amount of time, owing to the number of terms that are added in the sum. His argument was based on pure logic. In the present day for anyone with basic knowledge about infinite series, it is not difficult to fathom that the paradox is a consequence of the assumption that an infinite number of terms can’t add up to a finite number. The same pattern occurs when a square is divided into smaller areas by halving the existing part. The sum of these areas is always equal to that of the larger square. Nonetheless, it took centuries of brainstorming to arrive at this explanation.

“Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.” 
– Blaise Pascal

There is no dearth of splendid paradoxes in Physics either. Be it the Schrödinger’s cat or Maxwell’s demon, some  paradoxes have baffled scientists for generations. The hidden brilliance and elegance of a simple question posed by Heinrich Wilhelm Matthias Olbers – “why is the night sky dark when there are so many stars” is manifested in the fact that it acts as a proof of the Big Bang theory [2]. This is the Olbers’ paradox. We know that the speed of light is finite; so, the visible universe includes only the parts where light has reached. Thus, the radius of our visible universe is much smaller than the entire universe. This also implies that the universe is of finite age, thereby providing evidence in favor of the Big Bang.

The Wheeler paradox is an interesting problem that exposed a gap in the physical theories [3]. John Wheeler was haunted by the predicament that since a black hole at rest had only charge, mass and angular momentum and not much entropy, thus the entropy of a complex system that is sucked into a black hole would vanish; this was in clear violation of the second law of thermodynamics. The inconsistency was a result of not knowing how to apply the concept of entropy to blackholes. His student Jacob D. Bekenstein realized that the entropy of a black hole is proportional to the square of its mass and any system falling into the black hole will increase its mass and therefore also its entropy. This resolved the paradox. Throughout the history of philosophy, mathematics and sciences, several such paradoxes have played a crucial role in broadening our knowledge.

Apart from this, puzzles and riddles based on certain paradoxes are quite engaging and fun. One such group of logic puzzles is the Knights and Knaves riddles, popularized by Raymond Smullyan. These are based on the liar paradox. Two tribes inhabit an island – the Knights who always speak the truth and the Knaves who always lie. The puzzles involve a series of interactions between the inhabitants and an outsider, in which the latter has to conclude which clan the former belongs to.  Try answering a classic Knights and Knaves puzzle taken from Smullyan’s book What is the name of this book: The riddle of Dracula and other logical puzzles: There are three natives A, B and C. A says, “All of us are Knights”; B says, “Exactly one of us is a Knight”. What are A, B and C? (Answer: A – Knave, B – Knight, C – Knave)

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