Sunday, December 5, 2010

Pi In The Sky

Pi In The Sky:
A Review of "Where Mathematics Comes From"

I found this book to be quite interesting because it attempts to answer the question: where does our knowledge of abstract concepts in mathematics come from? Their quite unique answer is that abstract concepts in mathematics (i.e. a priori knowledge) can not be learned without a significant amount of sense data (i.e. a posteriori knowledge). I suggest reading the book whenever you have the time; it's well worth your time.

Summary of the Book: Eminent psychologists challenge the long-held conviction that mathematics exists on a transcendent plane above humans



If a tree falls in the woods and nobody is there, does it make a sound? Did the number pi exist before humans studied circles? The question of whether mathematicians discover or create is still a heated topic of debate. Mathematics, in all its transcendent beauty, is assumed to have an objective existence external to human beings. This isn’t the case, according to cognitive psychologists George Lakoff and Rafael Núñez. They believe that mathematics, like all human endeavors, “must be biologically based.” In the book, Where Mathematics Comes From, the authors have the goal of convincing both lay readers and mathematicians alike that cognitive psychology will explain the ways in which we learn mathematics. For them, mathematics exists only as a tool for the human mind to use when perceiving the world.

As simple as this sounds, we should consider ourselves lucky to have such a gift. Their research suggests that only a few animals have the ability to count. Our mind formulates integrals, parabolas and rational numbers by expanding on our innate ability to count. Lakeoff and Núñez argue that mathematics is fundamentally a human enterprise, arising from basic human activities. We can only comprehend mathematical concepts, such as imaginary numbers, through metaphors that compare these concepts to activities in real life.

Lakeoff and Núñez take us on a journey through mathematics’ past and cognitive psychology’s future in an attempt to bring mathematics down to earth. We start in a laboratory watching rats and children in counting experiments and we eventually work our way through the minds of great mathematicians, such as George Boole and Leonard Euler.

In their study of the history of mathematics, the authors observed that each layer of mathematics built upon the ones below. Like the iron and steel that holds together modern skyscrapers, a series of grounding and linking metaphors holds mathematics together. The authors uncovered a few grounding metaphors from which we would not be able to visualize or understand concepts like addition or multiplication. These metaphors are the concrete foundation for mathematics. Linking metaphors connect the different fields of mathematics, such as arithmetic, algebra, geometry, trigonometry, calculus, and complex analysis, just as elevators connect the different stories of a skyscraper.

As cognitive scientists go, Lakeoff and Núñez are as different as they get. George Lakeoff has been studying linguistics and the neural theory of language since the 60s. Rafael Núñez is just beginning a promising career in the field, having devoted the last decade studying the origin of mathematical concepts. What they share in common is a Berkeley Psychology department and an interest in learning where mathematics originates in our mind.

Here is an example of the beauty of mathematics, which happens to be the climax of their book.
e to the i time pi is equal to negative one
e^i*pi = -1

The Euler Equation has remained to this date mysterious, and even somewhat mystical, because it is by no means instinctive to believe that raising an irrational number to an imaginary, irrational power could possible yield negative one! One would expect the result to be imaginary, or at least beyond our comprehension. Lakeoff and Núñez argue that pi, i, and e are more than just numbers. They represent ideas. These ideas can only be grasped by a human mind that understands concepts such as periodicity and rotation.

The authors devote such a large section of their book to the Euler equation because this equation combines many branches of mathematics and therefore is a good place to locate linking metaphors between these branches. To give a flavor for the Euler equation, a few of the metaphors used to understand the numbers i and pi will be presented. The beautiful proof of the equation is left for the authors.

The concept of pi from trigonometry is not the same as the concept of pi that comes from geometry. Pi is no longer just the ratio of the circumference to diameter on a circle nor does it just take on the value of 3.14.159…. It is also a measure of periodicity for recurrent phenomena. To understand pi’s importance to periodicity, image yourself on a circle. Any point will do. Image traveling around the circle. How far do you travel before you return to the same point on the circle? It depends on the radius of the circle, so let’s just say that it’s a circle of length of one unit. You must travel through a distance of two pi before reaching one’s starting point on any circle. In the same way, one must travel an octave to reach the same note on a musical scale. To reach the point opposite from you on the circle, you must travel a distance of pi. Outside of the realm of mathematics, we are left with a language that describes recurrence through the circular metaphor, as can be seen by the line, “I can’t wait for the holiday season to come around again.”

The number i is altogether a mysterious quantity. It is by definition the square root of negative one. If you think about this, i can’t be a positive number. A positive number times another positive number is still positive. However, i times i is negative. By the same logic, i can’t be negative because a negative number times a negative number is positive. So, if it can’t be positive or negative, and it definitely isn’t zero, then it can’t be a real number.

Instead of focusing on the number i, it will be more important to understand multiplication by i. We begin by visualizing the number line. This requires having learned the grounding metaphor that numbers can be represented as points on a line. A real number is one that can be visualized on a number line. From there, we need to add a second dimension, which can be done by drawing an axis perpendicular to the first. For real numbers we have the intuitive understanding that multiplication by negative one means finding a number symmetric with the respect to the origin, which is the same as saying multiplication by negative one equals a rotation by 180 degrees in these 2-D plane. Multiplication by i times i is the same as multiplying by negative one. Two rotations of 90 degrees equal to 180 degrees. So, multiplication by i is the same as rotation by 90 degrees. This means that real number on the first number line move to the second axis when they are multiplied by i.

This explanation of i and pi provides enough background to understand Lakeoff and Núñez’s proof of the Euler Equation using the metaphors found earlier in their book. What they show is that Euler and all mathematicians since who have proved this equation could only have done so by learning important metaphors relating mathematics to physical concepts such as lines, rotation and periodicity.

It is fascinating to wonder whether every theorem and proof in mathematics is written down in one Book, drawn from by a few inspired mathematicians. This seems unlikely though because this would mean that the metaphors we use are universal. A species living on a distinct planet probably does not visualize numbers as points on a line. Perhaps for them numbers are represented as stars in the sky. It could ever be possible for them to have a high level of mathematical sophistication without the use of pi.

2 comments:

  1. Its interesting. I think of mathematics as a human concept, but one which is useful to explain real physical phenomena in the universe. Hence there is math as a distinct human cognitive concept but also maths as some kind of underlaying law of the universe. What is interesting is that things like i are numbers we discovered using mathematics that are also useful at describing the universe. My history of maths is not that great, but my understanding of i is that it was not developed from empirical observation (unlike pi) but from a purely theoretical stance. The fact that it is useful to describe the universe is very interesting...

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  2. I wrote this article 12 years ago. It's strange to see how much I've changed in the last twelve years. At the time, I was much more likely to agree with philosophers who stated or who tried to prove that mathematics was purely a human phenomena (i.e. that mathematics is a creation of the human mind.)
    Today, I tend to agree with philosophers who state or who try to prove that mathematics existed before there was a universe. For example, the fact that there are only four division fields (real, complex, quaternions, and octonions) is likely to have been true longer before the Big Bang, and long after any heat death to the universe. The same is likely true for the fact that there are only a finite number of simple groups (the largest of which we call The Monster Group.) The Monster group of discrete group theory (likely) existed before the Big Bang, and will (likely) exist after the Big Bang. The Monster (or the Baby Monster) are not (in my opinion) inventions of the human mind based off of experience. Having an experience requires certain hardware and software before the experience can be registered. The experience, the software, and the hardware are but features of the underlying self-replicating, self-reflective structure.
    My guess is that the Monster is inside of us (i.e. that the group of relations exists between particles in our bodies), and that there is something in the mathematics of the Monster that it is capable of self-recognition. What I'm trying to say is "to be a living, thinking human" means "to be a self-replicating, self-recognizing mathematical structure."
    In other words, it's not humans who invent mathematics, but the other way round. It is mathematics that allows the invention of humans, and allows humans to eventually realize that they are, like everything, are mathematical.

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