Resolution of Paradox: a Gateway to Mathematical Progress

Weyl-49-Cover

Hermann Weyl (1885-1955)

A paradox is a statement that appears to contradict itself, or that is counter-intuitive. The analysis of paradoxes has led to profound developments in mathematics and logic. One of the richest sources of paradox is the concept of infinity. Hermann Weyl, one of the most brilliant mathematicians of the twentieth century, defined mathematics as “the science of the infinite”  [TM192 or search for “thatsmaths” at irishtimes.com].

Ever since there has been time to wonder, humankind has been intrigued by the enigma of infinity. The idea has perplexed philosophers and mystified mathematicians for millennia. The Greeks were contemplating infinity from the time of Pythagoras. Head-on confrontation of infinity led to contradictory conclusions; the paradoxes of Zeno are amongst the most noted of these.

“Immeasurably Subtle and Profound”

In one of Zeno’s paradoxes, Achilles must complete a race of a fixed distance. To do this, he must first reach the half-way point. Before this, he must reach the midpoint of the first half, and so on without limit. Since he must arrive at an infinite number of midpoints, Zeno concludes that Achilles cannot even begin to move: motion is impossible! Bertrand Russell described Zeno’s ideas as “immeasurably subtle and profound”.

Zeno-Dichotomy-Paradox

The Greeks could not imagine that the sum of an infinite number of distances could be finite. It was not until 1821 that French mathematician Augustin Cauchy defined mathematical limits in a rigorous way, and showed what it means to say that the sum of an infinite number of terms is finite.

Cantor Counts On and On

Although we have been aware for millennia that it is possible to continue counting without limit, everyone until Georg Cantor stopped before reaching infinity. Cantor got to this point and continued far, far beyond. He created a theory of infinite quantities. He was the first person ever to see beyond the limit of the counting numbers.

Cantor showed that there is not one infinity, but an unlimited hierarchy of them. Infinite sets do not all have the same size: some are more infinite than others. Cantor’s discovery of a whole galaxy of infinities had an enormous impact on mathematics and on philosophy.

Paradoxes of Probability

MontyHall-1

If player has selected Door 1, should he or she switch to Door 2?

Probability theory is a rich source of counter-intuitive results. Well-known is the birthday paradox: to have a better than 50% chance that two people in a room have the same birthday, it is sufficient to have just 23 people. Another familiar example is the Monte Hall paradox: a game-show host invites the participant to choose one of three doors: behind one is a sports car, behind the other two are goats.

When the player has chosen, the host opens another door to reveal a goat, and then allows the player to change to the third door. It is hard to imagine any advantage in this, but analysis shows that the odds of winning are substantially increased by changing.

An even more surprising example is the two-child paradox. “In a two-child family, given that there is at least one boy, what is the probability that there are two?” We assume that boys and girls are born with equal frequency and that the sex of each child is independent of that of the other. These assumptions can be challenged, but we are not concerned here with genetic niceties or calendrical quirks.

The intuitive answer is 50%. More careful investigation leads us to a 1-in-3 chance of a second boy. The mystery is resolved by considering the “sample space”, the set of all possibilities.

While many of these paradoxes appear to be curiosities, their clarification has enhanced the preciseness of human thought.

 


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