In order to find clues about how to face the challenges that the new century present us, we can take a look at the man’s mental development, as well as the revolutionary moments in the history of humankind, in particular the history of mathematics. We can examine more closely the different mental stages we all go through (according to Piaget), in order to have a better sense of human potential. Two of the innovative moments in the history of mathematics are the creation of non-Euclidean geometry by Nikolai I. Lobachevsky (1793-1856) and the formalization of the concept of infinity and the transfinite numbers by Georg F. Cantor (1845-1918). These achievements were the result of efforts performed by minds working against traditional ways of thinking, freed from the concrete reality where so many mathematicians before them had been stuck. As mathematics teachers at the beginning of the most demanding century ever, we ought to better know our students’ potential, and grant those students who think differently all the attention and support that likely creators of changes in history command.

According to Jean Piaget (1896-1980), in the same way our body evolves until it reaches a relatively stable state with the maturity of our organs, so our mental life can be thought of as an evolution toward a form of equilibrium characterized by the adult spirit. Thus, since birth, human beings (the subject) tend to reach and maintain a state of equilibrium with their environment, that is, with the world (the object). When this equilibrium is broken (by a change of position, a question, or an emotional experience, for example), an action or response (a movement, a thought, or a feeling) is needed to re-establish the broken equilibrium. These interactions between subject and object continue throughout the rest of life. According to Piaget, there are two ways human beings respond in order to re-establish a broken equilibrium: assimilation (the use of perceptual or mental schemata already existing in the individual to interpret new experiences) and accommodation (the addition of new schemata to those already existing to be able to interpret new experiences). In terms of the individual’s mental development, Piaget constructed a model which identifies four stages (see previous blog On Algebra and Logic): a) *Sensorimotor stage (birth – 2 year-old)*. b) *Preoperational stage (ages 2-7)*. c) *Concrete operations (ages 7-11)*. d) *Formal operations (beginning at ages 11-15)*. For Piaget, the actual mental growth occurring in the individual can be described as the gradual construction of a map of reality. This construction steadily shows connections between any two points—by means of short and direct routes first—which become greater in space and time and take progressively more complex paths. Thus, the first interactions—perceptual assimilations and accommodations—are direct and simple interactions. In the sensorimotor stage (the first in Piaget’s model) interactions include reversals and detours. In the preoperational stage (the second in Piaget’s model), the child acquires representative thought and thus is able to invoke absent objects. This possibility of mental representation makes the child capable of reaching pieces of reality that are not visible, since they lie either in the past or partially in the future. But these representations are more or less static figures, pictures of a mobile reality, which represent only some states or paths out of a great number of possible states or paths. In this stage, the ongoing construction of the map of reality still leaves a lot of blank spaces and not enough conditions to go from one point to another. In the concrete operation stage (the third in Piaget’s model), these static images become mobile, and are able to embrace the dynamic of reality. Also the child’s responses (to regain a state of equilibrium with the environment) follow greater distances in space and time and take more complex routes. The map of reality is in this stage becomes progressively more complete. However, this map is a representation of reality only. In the formal operation stage (the fourth in Piaget’s model) more than just reality is included: The world of the possible becomes reachable. The individual’s mind is freed from concrete reality, and mathematical creation can flourish powerful and unbounded.

One of the most withstanding mathematics works has been the *Elements* of Euclid (about 325 BC- 265 BC). This work has been become a model for the organization of knowledge. As a matter of fact, Newton (1643-1727) used the structure of Euclid’s Elements to write his new physics *Principia*. In addition to the structure of this work of Euclid’s, the achievement of the *Elements* had remained unchallenged for more than twenty centuries, due to the feeling of absolute certainty that it had brought about. The structure of this work is based on the formulation of definitions, axioms or common notions (these are general statements, not specific to geometry and whose truth is obvious or self-evident), postulates (these are the basic suppositions of geometry), and theorems or propositions (which are the consequences deduced logically from the definitions, axioms and postulates). Euclid proposed five postulates, each of which he believed to be independent—not a logical consequence of the other postulates of the set. The following statement is equivalent to the fifth of these postulates: Through a given point not on a given straight line can be drawn only one straight line parallel to the given line. For centuries, mathematicians had difficulty in regarding the parallel postulate as independent of Euclid’s other postulates and made repeated and failed attempts to show that it is a consequence of the other ones. These attempts date back to Claudius Ptolemy (AD 90—AD 168) followed by other mathematicians after him, and more recently Girolamo Sacheri (1667-1733), Johann Lambert (1728—1777), and Adrien Legendre (1752—1833) made new attempts. These last three mathematicians tried to prove the dependence of the parallel postulate by reductio ad absurdum. In fact, they assumed that “through a given point not on a given straight line, not one, but at least two lines parallel to the given line can be drawn,” and expected to find a contradiction from this assumption. However, for about 150 years Saccheri, Lambert, and Legendre failed to find such a contradiction, and the problem of the dependency of Euclid’s fifth postulate remained unsolved. The first to suspect the independence of the parallel postulate were Carl F. Gauss (1777—1855) from Germany, Janós Bolyai (1802—1860) from Hungary, and Nicolai I. Lobachevsky (1793—1856) from Russia. However, it was Lobachevsky the first mathematician to publicly express—in an article titled *On the Principles of Geometry*—that Euclid’s parallel postulate could not be proved on the basis of the other postulates; therefore, an opposite statement to this postulate, that is, “through a given point not on a given straight line, not one, but at least two lines parallel to the given line can be drawn,” could be accepted as a valid and independent statement instead of Euclid’s fifth postulate. As contradictory to common sense as it can appear, it is as consistent—not generating contradiction when put together with the other four postulates—as that of Euclid’s fifth postulate. This was the origin of what is known as non-Euclidean geometry. Lobachevsky’s geometry revolutionized not only geometry, but the entire fields of mathematics and philosophy. It took him a lot of courage to share this idea with the world. As a matter of fact, Karl F. Gauss—by many considered the greatest mathematician of all time— remarked that he had thought of this possibility before, but didn’t make any public statement about his idea opposing that of Euclid’s parallel postulate, because he was too concerned about mathematicians’ (and non-mathematicians’) reactions. Lobachevskian geometry showed that Euclidean geometry was not the absolute truth it had been considered to be for more than twenty centuries. Even more, it freed geometrical reasoning from spatial intuition. The concept of space also had to be revised as a consequence of this new geometry. Although Lobachevsky became the rector of a university in the city where he grew up—Kazan University—almost until the end of his life, he faced economic difficulties. Even more, during his lifetime his work was not given the consideration that his revolutionary ideas deserved. Only after his death was he given general acknowledgement for such a creative work. Indeed, his perceptions changed our comprehension of the universe.

The concept of infinity has been a struggle for philosophers, mathematicians, and even theologians since the days of the ancient Greeks. Infinity has been always a mysterious or confusing idea, even feared by the greatest minds in history. As a matter of fact, Aristotle (384 BC—322 BC) thought that being infinite was a poor quality, rather than perfection. Also, Galileo Galilei (1564-1642) affirmed that “by its very nature, infinity is incomprehensible to us,” and Gauss was opposed to the use of an infinite quantity as a real object, which, in his opinion, should never be allowed in mathematics. For Gauss, the infinite was only a way of speaking. Galileo did observe that there are as many squares, as there are natural numbers , but he didn’t go further with this observation. In 1872, Richard Dedekind (1831-1916) defined a set as infinite if it could be put into one-to-one correspondence with one of its subsets. This was Cantor’s starting point. According to Dedekind’s definition, both the set of natural integers and the subset are infinite, but Cantor called them “countably infinite.” For him, countably infinite sets are those that can be put into one-to-one correspondence with the set of natural numbers. Cantor went even further and showed that there are infinite sets that cannot be put into one-to-one correspondence with the natural numbers. In fact, he proved in 1973 that the set of real numbers is one of them, and called these sets “uncountably infinite.” Three years later, Cantor proved what neither Euclid nor anybody else had ever imagined to be possible: that the points of a surface, for example a square that includes its boundary, could be put into one-to-one correspondence with the points of a straight-line segment that includes its end points. Thus, one of Euclid’s common notions, that the whole is greater than the part—which stood without question for more than 2,000 years—had been disarmed! But this was only the beginning of Cantor’s unconventional way of thinking. In order to distinguish the countably infinite sets—the sets of integers, rational numbers, and roots of polynomial equations with rational coefficients, to take some examples—from the uncountaby infinite ones—as the real numbers, or planes, or spaces of *n* dimensions, for example—he established that all countably infinite sets have the same power (or cardinality). He denoted this power by using the first letter in the Hebrew alphabet, the aleph, and the subscript zero, , calling it **aleph-null**. Then conjecturing that there were no sets with intermediate levels of infinity between the countably infinite sets—like the set of rational numbers , for example—and the set of real numbers , Cantor assigned to the latter the next power order or level of cardinality. This conjecture of Cantor’s is known as the continuum hypothesis. Although he struggled the rest of his life to prove this conjecture, his daring way of thinking about a topic that had been so elusive as that of infinity, gave rise to a new (aleph-based) number system: The transfinite numbers, where and are the first two transfinite numbers. Cantor also proved that the set of subsets of a given set is always of a higher power than that of the given set—the set of subsets of a given set cannot be put into one-to-one correspondence with the original set. Therefore, the power of the set of subsets of the real numbers is the third transfinite number, the power of the set of subsets of the previous set is the fourth transfinite number, and so forth indefinitely. His work was so original that Cantor himself acknowledged that his ideas about this new system of (nonfinite) numbers were in opposition to widely spread beliefs about infinity in mathematics. He was not only not understood by many of his contemporaries, but also underestimated. As a matter of fact, Cantor could never join the faculty of what he considered a high-caliber school. He remained at the University of Halle, a rather small school, for his entire career. However, as the great German mathematician David Hilbert (1862—1943) said, “Cantor’s work was the most astonishing product of mathematical thought, one of the most beautiful realizations of human activity in the domain of the purely intelligible.”

Piaget’s interpretation of his mental development model as a gradual construction of a map of reality, up to the point where the subject’s mind frees from this concrete reality and makes reachable the world of the possible, is wonderfully illustrated by Lobachevsky’s and Cantor’s mathematical achievements. Not only did they liberate themselves from the constraints of a petty reality that became short to the greatness of their thoughts, but also had the courage to share their ideas with the world, despite the hard opposition they had to face for going against “common sense.” The history of these achievements also shows us, mathematics teachers, that among those students who think differently, those who don’t follow conventional ways of thinking and surprise us with estrange ideas—which may even seem like silly ideas—there might be the ones who will transform the world. The requirements of this extraordinary 21^{st} century demand extraordinary thoughts, capable of reaching what could be seen as impossible achievements. The future achievers of this kind of accomplishments, exponents of the new generation of Lobachevskys and Cantors ready to reach those heights, may now be in your classroom. Don’t ignore them.

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