Sunday, October 18, 2015

From Primitive to Modern: The Evolution of Mathematics

Book Review: Clifford A. Pickover's The Math Book

From the very beginning in the Introduction, Pickover makes it clear that this book is not the complete, definitive list or the final word on what the milestones of math should be. As he mentions, these are his selections that cover a wide range of mathematical topics, from games to important theorems, and are written in such a way that an average person could understand, in a general sense, what the topic is about. And to his credit, Pickover does just that, and does it well.

The list is in chronological order and the topics have one-page summaries with an accompanying picture that can help the reader visualize the math concept or milestone featured. The breadth of time Pickover covers is from BC to the early 2000s. The list starts, surprisingly, not with a human mathematical milestone, but with an animal one. In fact, the first two milestones describe how animals such as ants and monkeys have mathematical capabilities. And the next milestone describes a mathematical pattern in Cicada (insect) behavior. The first human mathematical milestone, which is the fourth milestone in the book, is the presence of knots. Then the milestones become more historical with the mathematical advancements from ancient civilizations, like the ancient greeks. Famous games, equations, numbers, and texts are featured in this book, as well as technological advancements, like computers that can help prove conjectures.

The reason Pickover showcased the milestones chronologically, as he states in his introduction, is to have the reader see the evolution of mathematical thought, and how one milestone is connected and/or spurred another milestone. Pickover delivers on this goal, since I saw how mathematics is ultimately a compounding network of many topics, each connected - influencing, advancing and creating others. He not only points out these connections, he even lists the related topics at the bottom of the page. Furthermore, Pickover also points out how a milestone is important or useful outside the field of mathematics like in economics or the sciences. The best example of this are knots. Pickover mentions knots as the first primitive human milestone, and the topic continues to progress in complexity with advances in knot theory like Jones Polynomial, which has applications regarding the DNA molecule and proteins (pg. 24, 478, 490).

But why start the book, which is about mathematical milestones, with a mathematical capability of an animal, instead of a human? Why would it even be considered a milestone? Pickover discusses how mathematics is not solely human, and it implies how human mathematical capability has evolved from our animal ancestors.

All in all, this book is a great read for those who want a condensed history of mathematical milestones. Since this is my first major exposure to the history of mathematics, it's hard for me to compare this list to other milestones and say "Why isn't such-and-such in this book?!?" However, 3 out of my 5 milestones that I could barely come up with in this blogpost were featured in Pickover's book (Zero, Calculus, and Alan Turing). The first one that didn't show up was Albert Einstein's E=mc2, even though Pickover mentions Einstein in the book, and specifically that equation in the Introduction. The second was a milestone that happened after the publication of the book, which was the first woman to win the Fields Medal (1).

None of the people or organizations below have endorsed my work, nor am I affiliated with any organization or any person. I am a college student.

Pickover, Clifford A. The Math Book. Sterling Publishing Co., Inc. New York, NY. 2009.

(1) Ball, Phillip. "Maryam Mirzakhani Becomes First Woman To Win Prestigious Fields Medal." The Huffington Post.

Sunday, October 11, 2015

Math: Making Science Possible

Nature of Mathematics

Math has always been taught as a separate subject from science throughout my education. Furthermore, math was taught first. Addition and subtraction were 1st through 3rd grade concepts. Multiplication and division were in 4th and 5th grade. And that's when science was introduced as a subject - not until the 4th and 5th grade. Based on the structure of my education alone, it seems that the concepts of math (numbers, operations etc.) had to be taught first before the concepts of science could be learned, since science uses a lot of mathematics. But could math be considered a science field because of this connection? 

In my class the other day, we conducted an experiment where we dropped objects from a bridge that were attached to a certain number of rubber bands. We were supposed to determine the right number of bands that would make our object come closest to the ground without touching it. For me, I felt like I was doing both science AND math while doing this experiment. The scientific component: forming a hypothesis, factoring gravity, weight, and elasticity to make that hypothesis, and testing that hypothesis with an experiment. The mathematical component: the acceleration of gravity is a mathematical constant, the mass of the object, and the force of the weight, which is described in a mathematical equation. Even though they overlap, I still see them as two separate fields.

To help illustrate this point further, I've made a Venn Diagram showing how I see math and science as two separate fields that do overlap. 

Math/Science Venn Diagram

So why did I organize the Venn Diagram the way I did? I won't be able to explain my decision-making process for each subtopic (my blogposts are long enough), but I will try with a few examples. I put geometry squarely in Math, and Astronomy in Science, even though I can think of ways they overlap. For example, orbits around the sun in astronomy can be described as ellipses and math was applied to the data to discover this. But shapes, like ellipses, exist and have mathematical properties in geometry independent of astronomy. Furthermore, there are concepts in astronomy that are independent of math, like which planets have or don't have atmospheres and how that impacts that planet's environment. So, that's why geometry and astronomy, and other subtopics, are not in the overlap part of the diagram.  

So, what connects the two? What is in the overlap? Data most importantly. Science collects data during experiments. Math and Statistics - a field I consider separate from Math, but is included as a subfield of math for simplicity - are used to make sense of the data. Data and math were used to discover that planets orbit the sun in an ellipses shape. Physics, even though it's usually referred to as a science, is so dependent on math, I put it in the overlap. I can't think of physics (gravity, force, acceleration etc) without thinking of math, like trigonometry and quadratics, which are all in the overlap.

There are two concepts in Math and Science that are similar but have different processes and functions that help separate Math and Science from each other, and they are Proofs and Experiments, respectively (look at the bottom of the circles). Scientists conduct experiments to test hypotheses, which are predictions based on observations of the natural world. Mathematicians, when they observe a pattern with numbers or shapes, make a conjecture and then they try to prove it. When they do, the conjecture becomes a theorem and it is considered fact in the mathematical community. This is where the line between math and science is most visible for me. In science, hypotheses are never considered fact. When the data is inline with the hypothesis, scientist say the hypothesis is supported, never proven. It takes multiple experiments done repeatedly before a hypothesis or hypotheses are considered to have enough supporting evidence to be a theory, which is still not considered a fact.

And there are other fundamental differences that separate math and science. Science is about understanding how things in the physical world/universe work. As my class discussed, math is about quantifying the world/universe and understanding numerical patterns, which are often found in nature, like the Fibonacci Spiral in pine cones (1). But finding patterns in the natural world that can be described using numbers doesn't make mathematics a science. Math exists without science, as illustrated with subtopics like geometry. In geometry, there are abstract concepts that don't exist in nature, like a one-dimensional line that expands to infinity. Also, discoveries or advancements in math are concerned with patterns in numbers, not physical processes in the world/universe, like in science. Furthermore, it seems that math makes science possible since math is used to interpret scientific data. And even though, math describes gravity with quadratics, science would not perish if the connection between science and math was less strong. For example, what if gravity was random, and couldn't be represented by a quadratic? (Let's not think about the fate of humanity and all life for the moment). Scientists would still be trying to find out how things work, but it would have less of a mathematical connection.

None of the people or organizations below have endorsed my work, nor am I affiliated with any organization or any person. I am a college student.


(1) Vi Hart. "Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3]." YouTube Video. Date of Access: October 12, 2015.

Figure created using GeoGebra.