Paradoxes in the History of Mathematics

Greg Stiffler's Online Teaching Portfolio

Zeno's Paradox Cantor's Paradox Russell's Paradox Einstein and Twin's Paradox Works Cited

History’s mathematicians have always faced problems as they expanded their knowledge of their field. Most of the predicaments that mathematicians come across can be solved eventually. However, some seem to have no solution and can even challenge the way mathematics has been interpreted, which is why they have posed such a problem to mathematicians. These are known as paradoxes, which are statements that seem to contradict themselves or appear illogical, but nonetheless could be true. An example is saying, “I always lie.” If you are lying, you are telling the truth, but if you are telling the truth, you are lying. The paradoxes of Zeno with infinity, of Cantor and Russell with set theory, and the twin paradox within the physics of relativity have created problems and arguments for mathematicians, as well as forcing them to think about the subject of mathematics in different ways than before.

Zeno, the Greek philosopher who lived in the fifth century B.C.E., created several paradoxes to show the idea of space and time being separate, and that by dividing them one comes to many contradictions. Two of the several paradoxes he presented exemplify such contradictions. The first stated that a tortoise and the fast runner Achilles were going to race, and that the tortoise would be given a head start. Zeno stated that if Achilles wanted to beat the tortoise, he would first have to catch up with it, but to do that he would first have to cover half the distance between them. Zeno continued on, stating that after Achilles did make up half of the original distance between he and the tortoise, the tortoise would have moved forward, creating a new gap between the two. Then Achilles would have to cover half of this new gap before catching the tortoise. However, once he covered half of this new gap, the tortoise would have moved again and created another new gap. This meant that Achilles would continually be covering half the distance of a gap, only to discover that he had to cover half the distance of a new gap. Zeno concluded that as long as the tortoise had a head start, Achilles would never be able to catch him because he would always be covering a finite distance in an infinite sequence of time intervals. The second paradox examines an arrow in flight. Zeno said that if you start to break down the time of flight to smaller and smaller increments, then you can examine an arrow at a given instant, and at that time the arrow will be motionless. He went on to say that if time is composed of instants, then the arrow is never moving because at any specific instant the arrow is at a point in space but not in motion (Katz 57).

Zeno’s paradoxes created a problem for mathematicians because they examined the idea of infinity and infinitesimals in finite space. Aristotle was the first to try to refute these statements, claiming that in the Achilles example, “a finite object cannot come in contact with things quantitatively infinite,” meaning the infinite divisibility of time would not affect the runner. In the arrow problem Aristotle said that time is not made up of indivisible instants, which was Zeno’s assumption, and that although the arrow may not be moving at an instant, motion is not defined at instants but over a period of time (Katz 56-7). Nonetheless, because infinity has no real value and is not mathematically tangible, there has always been much controversy around it. Zeno’s paradoxes caused mathematicians to think carefully about the concepts of infinity and infinitesimals and not to make assumptions about them. In a lecture on Pythagoras and Pythagoreans with Dr. Shirley we learned that infinitesimals created problems for the Greeks. Pythagoreans encountered the first major crisis in mathematics when they came across the square root of 2 when working with triangles. They assumed all right triangles would have finite lengths, and were shocked when they discovered a 45-45-90 triangle, which has the square root of 2 as the length of the hypotenuse.

Zeno’s examination of infinitesimals was important to mathematics as it helped lead to great developments in calculus. Limits find the approximation of a function as it approaches infinity, and in Dr. Shirley’s lecture on calculus we learned it was a limit that solved the second crisis in mathematics about how to interpret an extra “dx” in a derivative problem. Furthermore, in the late 1600’s Leibniz became bothered by his use of infinitesimals in differentiation, and decided to justify their use. Although to Leibniz it never really mattered whether or not infinitesimals existed, he found that if a certain ratio is true when quantities are finite, then the same ratio will be true when dealing with limits and infinite values. This manipulation technique became very useful to Johann and Jakob Bernoulli who accepted infinitesimals as mathematical entities and used them to make important discoveries in calculus and its applications (Katz 530-1).

The paradox created by Cantor in the latter half of the 19^th century involves the concept of cardinality and its relation to set theory (Katz 734). Cardinality basically describes how many numbers are in a set; for finite sets it is as simple as counting, but infinite sets cannot have a cardinality that can be represented by a whole number. He found that if the members of an infinite set can be put into one-to-one correspondence with one another, leaving no extra numbers in either set, then the two sets have the same cardinality. One-to-one correspondence means that for each member in one set, there is a corresponding member in a second set. For example, in an e-mail with my professor, Dr. Shirley notes that the set of positive integers and the set of perfect squares are both infinite and have the relationship of nón² for each member of the set, meaning they have one-to-one correspondence. Cantor proved that the set of real numbers has a larger cardinality than the set of integers, paradoxically meaning that the infinite set of real numbers is “bigger” than the infinite set of integers. More generally, Cantor’s paradox begins by stating that the set of all sets (call it set B) is its own power set, where a power set is the set of all subsets of a given set A. Power sets are always bigger than the sets associated with them (Weisstein, “Power Set” 1). The paradox concludes that given set B, the cardinality of set B must be bigger than itself. To understand the paradox, one must consider Cantor’s Theorem, which states that the cardinality of any set is lower than the cardinality of all of its subsets (Weisstein, “Cantor’s Theorem 1). The paradox is that if the set B is the set of all sets, then the cardinality of the subset of B would be bigger than set B; however, the cardinality of set B should be the same since set B and the subset of B are the same (Weisstein, “Cantor’s Paradox 1).

Russell’s paradox, discovered at the beginning of the 20^th century, gives an even more generalized view of the set theory paradox discovered by Cantor. It states that R is the set of all sets which are not members of themselves, meaning that all of the sets in R do not contain themselves as elements. The question then becomes, does R contain itself as an element? If one assumes that R does contain itself, then by definition R cannot contain itself and vice versa. The problem is most often given as the barber paradox. Suppose in a small town there is only one barber who is defined as the one who shaves all the men who do not shave themselves. Then the question is “who shaves the barber?” If the barber does shave himself, then he does not by definition. If the barber does not shave himself, then by definition he does (Russell’s Paradox 3).

Cantor and Russell’s paradoxes were important to the field of set theory because they caused mathematicians to examine assumptions they had previously made. These paradoxes showed that the set theory at that time (much of which was devised by Cantor) had many inconsistencies because much of it was purely intuitive and not based on any type of axiom or proof. This forced mathematicians to formulate a way to make set theory more consistent and to give it clearly defined restrictions. In the early 1900’s Ernst Zermelo devised seven axioms which gave clear rules for set theory (Katz 809-11). One of them, the axiom of separation (or regularity) avoided Cantor and Russell’s paradoxes by prohibiting self-swallowing sets (“Russell’s Paradox” 1). These paradoxes were important to the development of set theory because they expressed the need for rules, just as in algebra or geometry.

The third paradox of this paper, the twin paradox, has its roots in Einstein’s theory of relativity. His theory was revolutionary in its perception of physics as velocity approached the speed of light. The special theory of relativity was proposed by Einstein in 1905, and had two important postulates. The first is that the physics of relativity are true for all phenomena; the second states that nothing can exceed the speed of light, and that the speed of light is the same for all observers. Ten years later in 1915, Einstein published the second part of his original theory, the theory of general relativity, which describes gravity as a form of geometry that distorts time and space (Case 213).

Einstein’s theories of relativity have expanded on the physics of Newton by relating space and time, but have also created paradoxes and complications, the most famous being the twin paradox, that have confused mathematicians and scientists alike for years. The twin paradox begins by assuming we have a pair of fraternal (usually identical twins are used but fraternal and different gender make it easier) twins who are astronauts, and one (the male for example) decides to take a trip to Alpha Centauri, four light years away. His spaceship travels at 6/10 the speed of light, or .6c. When the brother gets to Alpha Centauri, he turns around and comes straight back to earth. From his sister’s perspective, his clocks ran slow by the time dilation factor, sqrt(1-(v²/c²)), meaning that although on earth 160 months (or 8/.6 years) have passed, the brother has only aged by 4/5 of that, or 128 months. However, from his perspective the earth is moving at .6c, which would mean that time for his sister has slowed down, thus making her 32 months younger after the trip (Fowler 2-3).

The famous twin paradox was developed by Einstein himself in 1905 after having created his special theory of relativity, and it exemplifies some of the problems scientists had in understanding the theory. The paradox grew out of the postulate to the special theory of relativity that the speed of light is always constant, despite one’s frame of reference (Crotti 1). The importance of the paradox is that it exposed a problem with the special theory of relativity: frame of reference. Einstein’s first theory does not specify which twin’s frame of reference would be correct, meaning either could say he or she was older (Curiosa 1). The existence of the paradox forced Einstein and other physicists to develop an explanation.

Although the twin paradox was eventually resolved after a number of years, even today it still creates controversy on how the problem is viewed. It is accepted among physicists that the twin who speeds off to Alpha Centauri will ultimately end up younger than the twin on Earth. Some solutions state that the resolution comes from the fact that the twin in the spaceship is in motion, under the force of acceleration, and therefore aging less. This explanation leads into the general relativity theory and its treatment of acceleration (Curiosa 1). The resolution that seems to be more commonly accepted involves the use of frames of reference. The argument simply states that there are three frames of reference in the paradox: that of the sister on earth, the brother in his trip to Alpha Centauri, and the brother in his trip back from Alpha Centauri (Curiosa 1). As the brother speeds away to the star, he is in a frame of reference where he is older than his sister, and when he comes back, the brother shifts to a frame where his sister is older. This changing of frames is why the brother ends up younger than his sister (Bridgman 1). The twin paradox continues to encourage mathematicians and physicists to formulate new ideas for its resolution and for the theory of relativity.

The paradoxes of Zeno, Cantor and Russell, and Einstein’s twin paradox have caused much confusion throughout history, but have all been important in advancing mathematics. Zeno’s paradoxes about the divisibility of space and time and infinitesimals influenced mathematicians to think more about infinity and the assumptions that had been previously made. Leibniz’s work with infinitesimals led to great discoveries in calculus and influenced the work of the Bernoullis. The paradoxes of Russell and Cantor led mathematicians to formulate axioms and guidelines for set theory, and showed how it could not be based on intuitive logic. Finally, the twin paradox showed both Einstein and the physics community that although the special theory of relativity was brilliant, it still had some inconsistencies that needed to be worked out. Although paradoxes are troubling and confusing by nature, they have nonetheless been important to mathematicians at identifying problems and inconsistencies in mathematics throughout history. Furthermore, by challenging the thinking of the time, paradoxes can lead to even more brilliant discoveries in math. Clearly, paradoxes have been important to mathematics, and the discipline might not be where it is today without them.

Works Cited

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Case, William B. “Relativity.” The World Book Encyclopedia. 1988 ed.

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Twins, Doppler. 1996. University of Virginia. 7 Nov. 2002

<http://www.phys.virginia.edu/CLASSES/252/srel_twins.html>.

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---. “RE: cardinality” E-mail to the author. 12 Nov 2002.

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<http://mathworld.wolfram.com/PowerSet.html>.