Abstract
Morris Kline’s achievement in his field of mathematics did not come from the discovery of some new theorem applied to a relatively obscure analytic problem. Instead, Kline’s contribution to the field was his ability to surmise the health of the overall discipline of mathematics; and, like a doctor, attempt to prescribe a cure. Above all else, Kline pursued perfection for his field of mathematics. His philosophy regarding his chosen field was one of practicalism. To this end, his work, rather than being theoretical based, reflected a genuine desire to improve the field. As a noted historian and philosopher, he sought this through several different approaches throughout his career. Most notably Kline was a frequent critic of the way in which mathematics was taught. From elementary school through the college level Kline believed that students did not typically embrace mathematics purely for the sake of doing the problems, but instead only enjoyed the subject when it was being used to solve other problems. Thus Kline encouraged teaching methods that would not only make mathematics more enjoyable for students, but would show them the relevant usefulness of mathematics in other fields. Writing seventeen books over the course of his career Kline also sought to create a stronger sense of truth in mathematics that he surmised had fallen into decline over the last century. With a sound understanding of the basis for Greek thinking regarding the underlying truths of mathematical reasoning, Kline sought to encourage the use of these truths to pursue solutions in other fields. From encouraging students to understand the mathematics in the use of baseball statistics, to the use of mathematics in solving advanced physics problems, Kline argued that mathematics was made more relevant for individuals when performed in the context of another discipline.
Paper
A professor of Mathematics, Morris Kline wrote extensively on the history, philosophy, and teaching of mathematics. He taught mathematics at New York University and was a strong critic of the way in which mathematics was taught. Widely recognized for his critical analysis on mathematical curriculum, he voiced his oppositions to the protocol of mathematical instruction on a variety of occasions and sought to correct instructional methods on teaching techniques from elementary all the way through post-secondary education. His arguments were so frequent and impassioned that editors of magazines typically posted a counterpoint from another author just keep the appearance of balance in their publications. About teaching mathematics, Kline wrote;
I would urge every teacher to become an actor. His classroom technique must be enlivened by every device used in theatre. He can be and should be dramatic where appropriate. He must not only have facts but fire. He can utilize even eccentricities of behavior to stir up human interest. He should not be afraid of humor and should use it freely. Even an irrelevant joke or story perks up the class enormously (Kline, 1956, p. 11).
Thus Kline’s contributions to the field were not in a vast portfolio of new mathematical theorems, but rather an ongoing analysis and critique of the development of the field of mathematics itself. Kline wrote widely on the history of the field and his philosophy centered significantly on a strong understanding of the works of Socrates, Plato, Aristotle and other Greeks including Euclid and the growth of the field since that time (Kline, 1972, p. 52).
Morris Kline’s work reflected both a deep understanding of the history and foundations of mathematical reasoning, as well as the field’s development as an attempt to explain problems in other fields. Thus there can be said to be two significant truths regarding his personal philosophy. One was his appreciation of the certainty that mathematics intersected with science on a working level. This certainty, which Kline found in the natural world, informed his strong appreciation of mathematics ability to solve problems in other fields.
One significant area Kline focused on was urging mathematical research to go about solving problems posed in other disciplines besides mathematics. To this extent, Kline has been viewed as seeking to take mathematics beyond the world of solving math problems for math’s sake, rather he saw math as the starting point of certainty with which to view other problems in the natural world. In his book, Mathematics: the loss of certainty, he focuses on the decline of absoluteness in the field that had emerged over the last century. In the book, Kline begins with an explanation of how the Greeks came about with the first truths in the field. Truths that are understood in nature are called axioms and are the founding building blocks for much of mathematics. This reliance on deductive reasoning based on self-evident principles called axioms guaranteed the truth of what is deduced if the axioms are themselves truths. So by incorporating this apparently clear, reliable, and unsullied logic, mathematicians formed unquestionable and indisputable conclusions. These axioms of truth went beyond a mere statement of facts, but were observable in nature. Euclidian geometry is based in large part on these universal axioms of truth. Kline saw the decline of this reliance on axioms as a hindrance to the usefulness of mathematics in assisting the sciences. According to Kline;
“The loss of truth, the constantly increasing complexity of mathematics and science, and the uncertainty about which approach to mathematics is secure have caused most mathematicians to abandon science.” Instead, “they have retreated to specialties in areas of mathematics where the methods of proof seem to be safe. The also find problems concocted by humans more appealing and manageable that those posed by nature” ((Kline, 1980, p. 34).
Furthermore, this crisis over the certainty of mathematics discouraged the use of mathematical methods in other aspects of our society like aesthetics and philosophy.
Kline’s philosophy was also thoroughly influenced by the Greeks. Kline’s bases of certainty derived as much from these classic notions of truth. However, his own personal philosophy contradicted with some early mathematicians. Kline noted that Plato’s disposition toward astronomy succinctly illustrates Plato’s position on the important knowledge to be sought. This science, Plato said, is not disturbed with the activities of the noticeable heavenly bodies since mere observations and explanation of the motions fall significantly shy of actual astronomy. Instead, Plato noted that the true science of astronomy does not come from the observations of the stars but from an analytical deduction of the laws of motion that power the universe. Thus Plato held little interest in dealing at all with the natural universe, but rather the mathematical interpretation of the heavens that dealt only with a “theoretical astronomy”. So the heavenly bodies in the night sky are only to be referenced as markers to assist in the quest of the ultimate mathematical truth. “We must treat astronomy, like geometry, as a series of problems merely suggested by visible things” (Plato, 2004). Thus the uses of astronomy in practical matters like navigation, calendar reckoning, and the measurement of time were of no interest to Plato.
Aristole’s view of mathematics however, emphasized a stronger relation to the physical environment. Unlike Plato, Aristotle did not view the world in mathematical terms. Instead, he believed that the primary source of our reality lay in material things that are part of our physical world. He was a critic of Plato’s reduction of all science to merely mathematics and was likely much more a physicist than Plato. Since Aristotle believed that objects and material things were the foundation of our source of reality, he surmised that it was through the study of these physical attributes of the world that we would obtain truths from it. Mathematics, was a tool that would help lead to those truths. Thus true knowledge, according to Aristotle, is obtained only through experience and intuition gained from that experience. To this end, Aristotle sought to highlight universal qualities by abstracting them from real things. These universals were obtained by starting “with things which are knowable and observable to us and proceed toward those things which are clearer and more knowable by nature” (Kline, 1980, p. 72).
In many ways Kline’s view of mathematics was similar to how he saw Aristotle’s view of the mathematical world. Like Aristotle, Kline viewed mathematics in relationship to the natural world around him and as a tool with which to view other how the world works. Kline found that mathematics ability to make predictions in fields such as mechanics, optics, astronomy, physics, and hydrodynamics was the true brilliance of the discipline. To this end Kline frequently noted the need to reveal the applications and use of mathematics in other disciplines rather than expecting students to simply enjoy doing mathematical problems. Likewise, he encouraged that mathematical investigation focus on solving problems posited in other disciplines rather than creating structures of significance only to other mathematicians.
On all levels primary, and secondary and undergraduate - mathematics is taught as an isolated subject with few, if any, ties to the real world. To students, mathematics appears to deal almost entirely with things which are of no concern at all to man"….yet mathematics is the key to understanding and mastering our physical, social and biological worlds (Kline, 1986, p. 1).
This advocacy for students to use mathematics to pursue answers to problems in other disciplines even went so far as to have some arguing whether Professor Kline really wanted to be a mathematician. With his constant discussion and propensity towards mathematics ability to assist in other disciplines, Kline often found himself under attack. In deducting what was the primary motivation for Morris Kline to be such a proverbial agitator, one contemporary wrote.
I am wondering whether in point of fact, Professor Kline really likes mathematics”. [...] I think that he is at heart a physicist, or perhaps a ‘natural philosopher’, not a mathematician, and that the reason he does not like the proposals for orienting the secondary school college preparatory mathematics curriculum to the diverse needs of the twentieth century by making use of some concepts developed in mathematics in the last hundred years or so is not that this is bad mathematics, but that it minimizes the importance of physics (Meder, 1956, p.5).
So Kline’s passioned avocation for both the instinctive advance to learning, and for the full integration of science and mathematics were the legacy of his personal philosophy. Ultimately, Kline’s dogged criticism of the mathematical education system and his firm belief on the necessity to use problems in other disciplines within the field of mathematics shows that Kline’s philosophy towards the field was one of practicalism and relevance. He was an ardent advocate of these pursuits for the mathematical field throughout his career. He treasured the certainty that could be found in the deductive reasoning of mathematics, but found that true blessing of truths that were discovered through mathematics came from the application of these truths to other disciplines. He encouraged mathematicians to be bold in talking the problems of other disciplines and ultimately to savor and appreciate the ability of mathematics to inform the lives we lead everyday.
Works Cited
Kline, M. (1986). Focus, a Journal of the Mathematical
Association of America
Kline, M. (1956). State of Mathematical Education. Mathematics
Teacher, 49:171.
Kline, M. (1980). Mathematics: the loss of certainty. New York:
Oxford University Press.
Kline, M. (1972). Mathematical thought from ancient to modern
times. New York: Oxford University Press.
Plato. (2004). The Republic. Barnes & Noble Inc.
Meder, A. (1956). Rebuttal to Kline, M. Mathematics Teacher,
51:433.
