Article by Sanderson M. Smith, Mathematics Instructor, Cate School, Carpinteria, California
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The title presents a fascinating question. I am a mathematics teacher. I should be able to readily respond to the question. After all, I do teach the subject. OK, my somewhat overly-simplified answer is that mathematics is the language of our universe. Be you evolutionist, creationist, or holder of some other belief, empirical evidence overwhelmingly suggests that our universe is mathematically designed. Still, if you are an intelligent reader (and I assume you are), my response to the question is clearly unsatisfactory. Among other things, it does not address questions such as "why does mathematics work?" and "where does it come from?"
Despite my love and appreciation for the magnificent academic discipline of mathematics, I do not have a good answer to the question "what is mathematics?" Since the time of Plato, mathematicians and philosophers have addressed this question. Answers, many of them contradictory, have been produced for over two thousand years, but none have survived the test of time and critical analysis.
Do we, as teachers of mathematics, simply present the academic discipline of mathematics as an established, unchangeable, and irrefutable block of knowledge that everyone should learn? Do we accept this interpretation of mathematics? Do our students view mathematics this way? Regretfully, I think the answer to these questions is all-too-frequently "yes."
Contrary to this static viewpoint, I believe it is possible that students can gain a tremendous appreciation for mathematics if they understand that the question "what is mathematics?" has been analyzed and debated since the time of the Pythagoreans, a mystical cult that surfaced around 550 B.C. I also believe that humans love and appreciate a good mystery. If this is so, why not present the truth in a mathematics classroom? After all, the truth is very simple: The very nature of mathematics is a mystery!
The purpose of this writing is to present a brief summary of types of mathematical thought that have surfaced since the ancient Greeks realized that humans have the mental capacity to reason. Prior to doing this, it is important to note three discoveries that shocked the mathematical world. These, and a few other discoveries, shattered the historical beliefs of many intellectuals who thought that their version of mathematics had a firm foundation. While a perfect analogy is not possible, imagine yourself having purchased, and living in, a luxurious tenth floor condominium in a very desirable location... and then finding out the foundation of the building was unstable. The shockers:
The discovery (invention?) of non-Euclidean geometries in the 19th century. Prior to this time, many schools of mathematical thought had accepted the laws of Euclidean geometry (studied in modern schools) as indubitable truths about the universe. Meaningful geometries that did not conform to the laws of Euclid's famous historical work, the Elements, were discovered by Hungarian Janos Bolyai (1802-1860) and Russian Nikolai Lobachevsky (1792-1856). For example, in some geometries, Euclid's famous parallel postulate ("Through a point external to a line, there exists, in a plane, exactly one line parallel to the given line") is false.
The discovery (invention?) of quaternions by Irish mathematician William Rowan Hamilton (1805-1865). A quaternion is a meaningful type of number that does not "behave" like the numbers we commonly use. Among other things, in quaternion arithmetic, x times y does not equal y times x. Mathematicians learned that the basic laws of common algebra, accepted as truth for twenty centuries, are not universal truths.
Godel's Incompleteness Theorem: In the 20th century, German Kurt Godel (1907-1978) proved that consistency can never be established by methods of mathematical proof. Every logical system must contain statements that can't be proved. In other words, a formal mathematical system could never prove its own consistency. Something must be accepted on pure faith. (In modern day geometry, the terms point, line, and plane are never formally defined. And, a postulate such as "In a plane, two non-parallel lines intersect at unique point" is accepted as true without proof. However, there are simple non-Euclidean geometry models where this postulate is not true.)
These discoveries, and some others that have not been mentioned, are important in understanding why many theories about the nature of mathematics have failed to pass the test of time. What follows is a very brief summary of some historical schools of thought which made attempts to answer the question "what is mathematics?"
Greatly influenced by the earlier Pythagoreans, Plato (c.427-c.347) asserted that mathematics represents a separate universe of abstract objects existing outside of what we know as time and space. Mathematical objects (such as numbers) aren't created by humans. They always existed. (An analogy might be represented by a great piece of sculpture. The end result was already there. The sculptor simply removed the excess marble.) Platonism asserts that a mathematician is an empirical scientist who can only discover what is already there. He or she can't invent new mathematics. Mathematical truth possesses absolute certainty.
[PROBLEMS WITH THIS VIEW: Platonists never really explain how flesh-and-blood mathematicians come to interact with the external universe of mathematics. The discovery of non-Euclidean geometries and quaternions contradicted the "absolute truth" view of the Platonists. Other historical interpretations reject the mysticism surrounding an external world of mathematics.]
German mathematician David Hilbert (1862-1943) headed this group. Formalists assert that mathematics must be developed through axiomatic systems. Formalist and Platonists agree on the principles of mathematical proof, but Hilbert's followers don't recognize an external world of mathematics. Formalists argue that are no mathematical objects until we create them. Humans create the real number system by establishing axioms to describe it. All mathematics needs is inference rules to progress from one step to the next. The Formalists tried to prove that within the framework of established axioms, theorems, and definitions, a mathematical system is consistent. In the mid-twentieth century, formalism became the predominant philosophical attitude in math textbooks.
[PROBLEMS WITH THIS VIEW: Godel's Incompleteness Theorem contradicts the consistency philosophy of formalism. It has been pointed out that accepted results from theorems were used before axioms were created to establish the theorems. The modern emphasis on the concrete and the applicable is not consistent with the formalist philosophy that you don't really do mathematics until you state a hypothesis and begin a proof.]
English mathematicians and philosophers Bertrand Russell (1872-1970) and Alfred North Whitehead (1861-1947) cofounded this school of thought. This school claims that mathematics is a vast tautology. All of mathematics is derivable from principles of logic. Many of the logistic ideas are similar to those of the formalists, but the latter group does not believe that mathematics can be deduced from logic alone. Among other things, the Logicists attempted a logical construction of the real number system, whereas the Formalists constructed it axiomatically. Logicism also uses mathematical sets in its logical development.
[PROBLEMS WITH THIS VIEW: Logicism, despite many attempts, could not successfully resolve paradoxes that arose in set theory. Godel's Incompleteness Theorem was a death blow to the "math is a tautology" philosophy expounded in Principia Mathematica, a monumental work constructed by Russell and Whitehead.]
Intuitionism (sometimes called Constructivism):
Building on the philosophies of Immanuel Kant (1724-1804), Dutch mathematician Luitzen Brouwer (1881-1966) emerged as the leader of this school of thought, which differs considerably from those previously discussed. Intuitionists claim that mathematics originates and thrives within the mind. Human minds intuitively possess the forms of space and time. The natural numbers are given intuitively, and they represent the fundamental datum of mathematics from which springs all meaningful mathematics. Mathematical laws are not discovered by studying nature; rather, they are found in the recesses of the human mind.
[PROBLEMS WITH THIS VIEW: The intuitionist view doesn't give any insight as to why mathematics works. We don't know how intuitive knowledge is held in the brain. Mental representations of concepts such as love, hate, etc. differ considerably from human to human. Is it realistic to assume humans share the same intuitive view of mathematics? Why do we teach mathematics if it is all intuitive?]
I suspect that most folks would not associate mathematics with the word fiction, which represents that which is feigned or imagined. Yet, consider facts such as the following:
* The square root of 2 is a symbol that appears frequently in algebra and geometry texts. The symbol represents an irrational number (non-repeating decimal) that no human has ever written out in complete decimal form. In fact, humans have never viewed any irrational number, including pi and e. For each irrational number, we have accepted a representation without ever having seen what is represented. Does that mean irrational numbers are fictional?
* Has anyone ever seen a segment of length 1 inch actually divided into infinitely many parts? The wonderful branch of mathematics we know as calculus accepts representations of this without any human observing what is actually represented. Again, are we talking fiction?
Here's an interesting question about the amazing number, pi, which makes numerous appearances in math textbooks: Is there a sequence of 100 consecutive zeros in the decimal representation of pi? Well, we can note that modern computers have calculated pi to billions of decimal places, and no such sequence has been found. But what about the next billion places, and the next ten billion after that? The point is, we simply don't know, although probability certainly suggests that we would eventually find such a sequence.
In conclusion, despite the fact that I have taught mathematics for many years, I really cannot explain what it is, where it comes from, why it works, or how we can make such amazing use of things that could be classified as fictional. Mathematics is, to a great extent, a mystery. As a thinker, I can only say that my personal philosophy of mathematics takes bits and pieces from each of the historical schools of mathematical thought. As previously mentioned, I see mathematics as a language. As we become more and more proficient with this language, we will better understand the universe that we inhabit. I believe the Creator (and you may define Creator however you wish) put mathematics out there for us to discover, but I don't believe humans will ever discover all of the mathematics that exists. The mystery of our existence in a mathematically designed universe is what makes living interesting and exciting. As teachers, shouldn't we let our students know that the question "what is mathematics?" has yet to be answered?
"Mathematics is not a perfect gem, and continued polishing will probably not remove all flaws. Still, it is the most precious jewel of the human mind and must be treasured and husbanded."
-Morris Kline, New York University
Eves, Howard. An Introduction to the History of Mathematics. New York: Holt, Rinehart and Winston, 1990.
Guillen, Michael. Bridges to Infinity: The Human Side of Mathematics. New York: St. Martin's Press, 1983.
Hersh, Reuben. What is Mathematics, Really? New York: Oxford University Press, 1997.
Hollingdale, Stuart: Makers of Mathematics. New York: Penguin Books, 1989.
Klein, Morris. Mathematics: The Loss of Certainty. New York: Oxford University Press, 1980.
McLeish, John. The Story of Numbers: How Mathematics Has Shaped Civilization. New York: Fawcet Columbine, 1991.
Smith, Sanderson. Agnesi to Zeno: Over 100 Vignettes from the History of Mathematics. Berkeley, CA: Key Curriculum Press, 1996.
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