Author Archives: clarkclan2014

Math: Is It All In Our Head?

After years of math classes, the crazy truth is finally coming out.  It is all just in our heads.  No way! How can that be? There’s an interesting debate in the world of math.  Are math principles the creation of humanity, or are they universal truths that humans discovered? There are compelling arguments on both sides of the debate and both sides have several different sub-levels of thought.  In this article, I will discuss them both generally.

The realists maintain that mathematical principles would exist even without people.  Humans discovered the principles and brought them into practical use and any intelligent human being could also discover the same principles.  This argument is supported by the fact that many cultures have discovered mathematical principles independent of one another.  Also, mathematical concepts, such as the Fibonacci sequence and some fractals, occur in nature which would suggest that they exist even without people.   Some realists, like the Pythagoreans, believe that the world was created by numbers.  The realist point of view can lead to an almost supernatural view of mathematics.

The challenge with mathematical realism is that there is no physical domain where math entities exist.  We cannot draw a perfect circle or even a line.  We can conceptualize these things in our mind and we can prove them in theory; however, we cannot actually manipulate math entities in the physical world.  Many math concepts exist only in the context of our understanding about them and conceptualizing them.

Another view is the anti-realists.  They maintain that math is the creation of humans in order to make sense of the world.  They recognize that math is an amazing, complex system and that it works as modeled by science.  However, some argue that scientific principles could be explained without math.  One anti-realist, Hartry Field, demonstrated this by explaining Newton Mechanics without referencing numbers or functions.  He explained that, in his opinion, math is fictional and is true only in the context of the story in which it is being told.

So, is it all in our heads?  A fiction that was created to explain properties in our world?  In reality we may never be able to settle the debate and it may not matter.  Math works.  That is the beauty of it.    In his article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, Eugene Wigner observes that

the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.  This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.

Albert Einstein, 1921. Image: Public domain, via Wikimedia Commons.

Albert Einstein, 1921. Image: Public domain, via Wikimedia Commons.

Perhaps the best thing to commend mathematics as being real, is that it works.  Time and time again, it works.  Its principles, laws and theorems, applied over and over, in different settings produce accurate results and predictions.  Einstein commented in a 1921 address titled Geometry and Experience, “It is mathematics which affords the exact natural sciences a certain measure of security, to which without mathematics they could not attain.”  He explored the question of how math, a product of our mind can be so applicable to the concrete world.  He asked, “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirable appropriate to the object of reality?”  Einstein answers this question with the statement, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”  He looks specifically at the field of geometry and the need humans have to learn about the relationships of real things to one another.  Even though the axioms of geometry are based on “free creations of the human mind”,  he says, “Solid bodies are related, with respect to their possible dispositions, as are bodies on Euclidean geometry of three dimensions.  Then the propositions of Euclid contain affirmations as to the relations of practically-rigid bodies.”  The abstract principles, when applied to “real” world situations prove to be accurate.  Einstein continues to explain how the theory of relativity rests on the concepts of Euclidian and non-Euclidian geometry.  He challenges the mind to conceptualize a universe which is “finite, yet unbounded”.  In the end, it is this ability to use conceptualized principles and apply them to our world that makes mathematics work.  So yes, mathematics may be all in our head and it may be a huge puzzle created by humanity, but it is effective, useful, and even beautiful.

Sources

Einstein, Albert. Geometry and Experience. http://www-history.mcs.st-andrews.ac.uk/Extras/Einstein_geometry.html

Wigner, Eugene. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Wikipedia. Philosophy of Mathematics. http://en.wikipedia.org/wiki/Philosophy_of_mathematics

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Women, Math, and the Gender Gap

As a female math student, I often find myself in the minority in my classes. In fact, in one class, I am one of only three girls. Now, if I were young and single, I might really appreciate these odds, however, as a mother of four daughters, I find it rather concerning. Recently in class, we studied the work of Sophie Germain, her contributions to mathematics, and the study of Fermat’s Last Theorem. We also discussed some of the challenges she faced as a woman scholar in the early 1800’s. I have been thinking about the roles of women in math, and I have wondered–even with so many programs to encourage women in the fields of math and science, why do we still see such a large gender gap?

Christine Ladd-Franklin By Smithsonian Institution from United States [see page for license], via Wikimedia Commons.

Christine Ladd-Franklin
By Smithsonian Institution from United States [see page for license], via Wikimedia Commons.

Women have truly struggled over the years to have equal opportunities for education, and, while in many subjects women have equal footing, they have been slower to catch up in math and science. Historically, women who have contributed to math have had a difficult time pursuing higher education. Sophie Germain had to use the pseudonym of a male student to submit papers to the university. Almost 60 years later, another promising mathematician, Christine Ladd, also had a difficult time obtaining a fellowship at John Hopkins University because of her gender. Her experience is described as follows:

[T]he university first announced its fellowship program in 1876, and one of the first applications to arrive was one signed “C. Ladd.” The credentials accompanying the application indicated such outstanding ability that a fellowship in mathematics was awarded to the applicant, site [sic] unseen, and was accepted. When it was discovered that the “C.” stood for Christine, several embarrassed trustees argued she had used trickery to gain admission, and the board immediately moved to revoke the offer. They failed to reckon, however, with the irascible Professor James J. Sylvester, stellar member of the first faculty. In 1870 Sylvester had been named the world’s greatest living mathematician by the Encyclopedia Britannica, and his presence at Hopkins was a real coup for the struggling university. He was indispensable and knew it, in an ideal position to insist on virtually anything he wanted; in this case, he had read Christine Ladd’s articles in English mathematical journals, and he insisted upon receiving the obviously gifted young woman as his student. Miss Ladd was admitted as a full-time graduate student in the fall of 1878. Though she held a fellowship for three years, the trustees forbade that her name be printed in circulars with those of other fellows, for fear of setting a precedent. Dissension over her continued presence forced one of the original trustees to resign (Riddle, 2014).

Despite such difficulties, woman continued to push forward in mathematics. Now, women don’t face the same discrimination in education; in fact, there are many programs designed to encourage girls to pursue their interests in math, and research shows that the gender gap may be narrowing in education. An article from Time Magazine, titled “The Myth about the Math Gender Gap”, reports about a study by researchers at University of Wisconsin and University of California, Berkeley. The study found that there was very little difference between the scores of girls and boys on federally mandated tests. They also found that equal numbers of boys and girls were taking advanced math classes in high school. This, researchers concluded, is why we are seeing a decrease in the gender gap. An earlier study from 1990 found that although test scores were equal in the elementary years, boys outpaced the girls in high school years. This coincides with the fact that fewer girls at the time were continuing in higher level math classes (Park, 2008).

However, there does continue to be a gap in higher education. Although equal numbers of male and female students are graduating with bachelor degrees in math, fewer women continue to the graduate level and even fewer to the associate professor level. There are several theories about what may be responsible for this gap. Another article in Time titled, “Explaining the Complicated Women + Math Formula”, explores different thoughts on this subject. The article specifically looks at four different theories—ability, prejudice, interest or choice, and the affect of family roles (Luscombe, 2010).

Ability
“As far as ability goes, studies are pretty clear that on average women and men are about the same at math” (Ibid). While an equal number of male and female students graduate with bachelor degrees in math related fields, only 25% of PhDs are given to females and “at the next level, tenure-track associate professor, the proportion of females shrinks to single digits” (Ibid). With this information, we can conclude that a large number of women who have the ability to continue at the higher level of math related fields are not continuing.

Prejudice
Ceci explored the possibility that women were experiencing prejudice from male college faculty in the hiring process, however, this actually showed to be the opposite. “We found that if candidates of matched ability are applying for a position, women are slightly more likely to get the job,” says Ceci.

Interest and Choice
Williams and Ceci found that the interests of girls may be another factor. While some boys may be very good at math, the girls who excel at math also excel at other subjects. It has also been found that girls tend to be more interested in working with living things rather than inanimate objects. Therefore while capable of excelling at math careers, girls may be choosing to follow other interests.

Family and Social Roles
The final factor they found was that, when given the choice, women would choose to follow family interests over their careers. The typical track to a tenure position came at a time when many women were interested in starting families and opportunities were no longer available if they wanted to come back to their career at a later time. “Ceci and Williams believe that the solution lies in changing the way tenure is attained. ‘The tenure structure in academe demands that women having children make their greatest intellectual contributions contemporaneously with their greatest physical and emotional achievements,’ they write, ‘a feat not expected of men.’ The process could be spaced out, candidates given more time. ‘There’s no reason to do it the way we do it, except tradition,’ says Williams (Luscombe, 2010).”

Although women are making huge strides in narrowing the gender gap in mathematics and I have a lot of reasons to be optimistic that my daughters will have opportunities that women throughout history did not, there are definitely still issues to be addressed. Some feel that women are socialized away from math and science, which may be true, but there are definitely other issues at play. The important thing to recognize as we look to the future is that there are no simple solutions and no one single factor at play. Ultimately, we want to ensure that those women who want to pursue careers in math have every opportunity to do so and that the field is free of the prejudice and stereotypes of the past.

Sources:

Luscombe, Belinda; October 28, 2010 “Explaining the Complicated Women + Math Formula.” Time. http://healthland.time.com/2010/10/28/the-complicated-women-math-formula/

Park, Alice; July 24, 2008, “The Myth of the Math Gender Gap.” Time. July 24, 2008. http://content.time.com/time/health/article/0,8599,1826399,00.html

Riddle, Larry; January 10, 2014, Biographies of Women Mathematicians—Christine Ladd-Franklin. http://www.agnesscott.edu/lriddle/women/ladd.htm

Exploring Limits

In calculus, a limit is defined as the value of a function as it approaches some point. Sometimes, a function has no finite limit at a point because it just keeps growing, and we say the limit is infinite. In this case, the function never reaches the limit but the value grows arbitrarily large as it gets nearer and nearer to the limit. In our reading, I have been considering limits in a different light. I have been thinking about the limits of civilizations as they progress in their development of mathematics. Some civilizations seem to reach a limit of understanding and because of cultural restraints, their limited number systems, or even because they outwardly reject an idea, they stop progressing. Fortunately, sometimes their discoveries shape and influence other cultures and, as a whole, progression continues. I would like to explore different limits in the progress and development of mathematics and consider what limits us today.

Plimpton 322. Image: Public domain, via Wikimedia Commons.

Plimpton 322. Image: Public domain, via Wikimedia Commons.

In ancient Mesopotamia, more than 4000 years ago, the Babylonians used the base of 60 to develop a high level mathematical system. They developed positional notation and could use fractions as well as whole numbers. They developed systems to figure square roots. Clay tablets from that time show tables with logarithms, multiplication facts and reciprocal pairs. There is information about calculating compound interest and solving quadratic equations. Writings on the tablets suggest that math was a subject that was taught and studied. In many ways, they seem to have exceeded the capabilities of other civilizations that came much later in history. No one can question that their accomplishments were amazing, to say the least, and perhaps influenced other cultures. However, because most of their mathematics were only for very practical purposes like conducting business, surveying land and constructing buildings, they stopped short of exploring some of the deeper meanings of things. For example, our text points out, “In the Babylonian square-root algorithm, one finds an iterative procedure that could have put the mathematicians of the time in touch with infinite processes, but scholars of that era did not pursue the implications of such problems.” (Merzbach and Boyer, pg. 26) What might have been the implications if they had? As they approached the limit, they stopped rather than exploring the infinite possibilities. They stood on the brink of even greater discovery, but did not pursue it.

Pope Sylvester II. Image: Public domain, via Wikimedia Commons

Pope Sylvester II. Image: Public domain, via Wikimedia Commons

One of the most dramatic examples of cultural influences limiting the progress of mathematics is the example of the progression of Indian positional decimal arithmetic to Europe. Mathematicians in India had developed a number system with ten digits, including zero, and used it to develop methods of computing fractions, square roots and π. In the tenth century, Gerbert of Aurillac attempted to introduce the system to Europe. He had learned the system first hand from Arab scholars in Spain.   However, he was rejected and during this time of the Crusades in Europe, he was rumored to be sorcerer. He died after a short reign as Pope Sylvester II. “It is worth speculating how history would have been different had this remarkable scientist-Pope lived longer” (Bailey and Borwein, 6).” The Indian system was reintroduced 200 years later by Leonard of Pisa, but was rejected again and considered “diabolical”. It wasn’t until the beginning of the 1400’s that scientists began using the system. “It was not universally used in European commerce until 1800, at least 1300 years after its discovery” (Bailey and Borwein, pg. 6). While many other areas of the world were able to do complicated computations using the Indian system, Europe, because of its cultural restraints, was still laboring with Roman numerals. Imagine what the brilliant minds of the Europeans might have discovered or developed if they had the ease of the Indian number system? In this case their culture may have created a limit that kept them from infinite discoveries.

Today in our world we have amazing tools to help us progress. Not only do we have the combination of a well-developed number system, thousands of theorems and laws and the knowledge of centuries of learning, we also have technology that assists in remarkable ways. Indeed we have all the tools of the past plus the technology of our day. However, are there things yet to be discovered, or have we reached a limit? Are there obstacles in our society or ways of thinking that limit us? As recently as the early 1900, women had a difficult time pursuing their mathematical interests. Even today, women and minorities continue to be underrepresented in the math and science fields. What might have been the result if woman had been afforded the same educational opportunities as men over the years? Do we limit ourselves by the way we approach math? Are there different number systems or “languages of math”? In recent years, computer scientists have given us other “languages” for coding. Are there similar languages for math? The challenge for our day is to not be content and accept that what has been learned is all there is.

In our reading for class I have been amazed at how often a group or civilization is on the brink of great mathematical discovery, but because of varying reasons they stop short of the mark. Sometimes cultural influences limit the progress and other times it seems individuals do not look far enough to find deeper meaning or answers. It is true that hindsight may be twenty/twenty, but I can’t help wondering what future civilizations may look back on and see that we barely missed. What are we on the brink of discovering if only we would look forward and push closer and closer to the undefined limits?

Sources:

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Bailey, David H., and Johnathan M. Borwein, “The Greatest Mathematical Discovery?,” 2011.