Saturday, August 31, 2013

The Three Wise Men

Two things I love to read about are math and history. Both subjects have intrigued me for as long as I can remember. Not only were those topics I was drawn to in school, taking extra classes in both that weren’t required for my degree, but I also searched the bookstores for good books on both areas of study. I was particularly keen on the combination of the two, the history of mathematics. Like regular history, the history of mathematics was a history of the men (and sometimes the women) who made the discoveries and how these ideas built on each other. Regular history is also about people and discoveries and great events and battles. History of math held all of that too, even the battles … on occasion.

I remember when I first found Bell’s “Men of Mathematics” in a college bookstore. I took it home and stayed up most of the night reading since I just couldn’t put it down. I’ve filled my bookshelves and bookcases with stories about mathematics and how it was developed and the personalities involved. It isn’t just math. Other technical topics are interwoven in the advancement as these great mathematicians were also great physicists and great astronomers and great engineers, but mostly great physicists.

Since counting is a natural part of mathematics, and putting things in order is the essence of much math, it is only fitting that I discuss the three greatest mathematicians of all time. This is a pretty settled list. There is no real controversy about which three are in the list I’m about to describe. Oh, some would disagree that an individual is the greatest mathematician because they consider him the greatest physicist. But that’s about the only disagreement.


We will start back in ancient Greece, the cradle of mathematics. Other ancient cultures developed advanced mathematics, at least to a degree, but it is the Greeks and their math that has had the greatest influence on modern mathematics and the development of that math through the twenty plus centuries following the ancients. This is largely due to the development of the axiomatic method by these historical thinkers. Names like Euclid and Pythagorus and many others are at the basic foundation of mathematics and, for over a thousand years, their writings were used as text books by “modern” students up until the sixteen or seventeen hundreds. In fact, you will find Euclid’s Elements still being used at the turn of the twentieth century.

But the first wise man that I will describe in this trio of greatness is Archimedes. He lived in the late part of the Greek empire, around the time that the Romans were conquering. In fact, he was killed by a Roman soldier. If you look him up you will find out that he is listed as a mathematician, physicist, engineer, inventor, and astronomer as he made significant discoveries in all of these fields. Among his advances in physics are the foundations of hydrostatics, statics [mechanics], and an explanation of the principle of the lever. He is credited with designing innovative machines, including siege engines and the screw pump that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors.

Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi. In so doing, he came within a hairs breadth of inventing The Calculus. He also developed a form of numeric expression that we now know of as “scientific notation.” That is where you give a set of numbers and a power of ten to show the overall magnitude. This method was used by Aristotle to estimate the number of grains of sand on a beach. This was a tremendous advancement since simple number notation of that time was very clumsy and held back advanced numeric thinking. He also defined the spiral bearing his name, formulae for the volumes of solids of revolution, as well as his ingenious system for expressing very large numbers.

Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere inscribed within a cylinder. Archimedes had proven that the sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest of his mathematical achievements.

Some of his writings have survived to this day, but his fame was also spoken of in other ancient documents and we have a pretty good picture of his work even though he lived a couple hundred years before Christ. In The Sand Reckoner, Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos, as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8×1063 in modern notation.

In his Methods of Mechanical Theorems, Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, another short work by Archimedes, The Method of Mechanical Theorems was written as a letter to Eratosthenes in Alexandria.

There are many reasons that Archimedes is held as the greatest of all of the ancient Greek mathematicians and natural philosophers. His work covered many different areas of study and he came so close to inventing calculus, which is what our next great mathematician is remembered for. He came at the end of a long line of great Greek thinkers and one wonders if the Romans, great engineers but not good scientists, had not concurred Greece, what more inventions would these natural thinkers have discovered. Would there come a Greek even greater than Archimedes?

We can’t change history, so we’ll never know the answer to that question. Instead, following the rule of the Romans, the western world fell into a period of darkness in which little scientific progress was made. For over fifteen hundred years Archimedes was a shining star of invention. Then, in the sixteen hundreds, new lights began to appear. On Christmas day, 1642, this gift to the world of mathematics (and physics and astronomy and …) was born.

Isaac Newton

In June 1661, Newton was admitted to Trinity College, Cambridge as a “sizar” — a sort of work-study role. At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers, such as Descartes, and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalized binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the university temporarily closed as a precaution against the Great Plague.

Although he had been undistinguished as a Cambridge student, Newton's private studies at his home in Woolsthorpe over the subsequent two years saw the development of his theories on calculus, optics, and the law of gravitation. In 1667, he returned to Cambridge as a fellow of Trinity.

It was during these few months at home that Newton had his eureka when, as the tale goes, he saw an apple fall to the ground and realized that the moon was constantly falling around the earth, which led to his development of the theory of gravity and his three rules for motion. He perfected his mathematical method we now call calculus in order to solve the equations that he created.

When Newton published his ideas about light and color in Opticks, Robert Hooke, head of the British Royal Society, criticized some of his conclusions. Newton was so offended that he withdrew from public debate. The ensuing controversy turned the shy Newton off to the process of publication. His conclusions about light were eventually shown to be true, as Newton knew they were since he had proven it with mathematics and experiment, but the experience led him to keep the results of his work on gravity and calculus secret for nearly twenty years. At issue was whether light was a particle or a wave. Oddly, both men were right as it was later shown at the advent of quantum physics that light displays both qualities depending on the experiment. However, most of the controversy back in the 1600's had more to do with the poor quality of prisms and optics available to perform experiments, which made it difficult for others to reproduce the results that Newton reported.

Later, Newton’s interest in astronomical matters received stimulus by the appearance of a comet in the winter of 1680–1681, on which he corresponded with John Flamsteed and Edmund Halley, both Royal Astronomers. After the exchanges with Flamsteed and Halley, Newton wrote out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector.

As the story goes, Halley inquired of Newton about a particular mathematical relationship that would result in the elliptical orbits described by Kepler, and Newton instantly responded. When asked how he knew, he said he had worked it out years before. Halley then encouraged his publication of these ideas and even financed the publishing.

Newton communicated his results to Robert Hooke and the Royal Society in De motu corporum in gyrum, a tract written on about 9 sheets which was copied into the Royal Society's Register Book in December 1684. This tract contained the nucleus that Newton developed and expanded to form the Principia, one of the most influential scientific texts ever written. The Principia was published in July 1687 with encouragement and financial help from Edmond Halley. In this work, Newton stated the three universal laws of motion that enabled many of the advances of the Industrial Revolution which soon followed and were not to be improved upon for more than two hundred years, and are still the underpinnings of the non-relativistic technologies of the modern world. He used the Latin word gravitas [weight] for the effect that would become known as gravity, and defined the law of universal gravitation.

In the same work, Newton presented a calculus-like method of geometrical analysis by “first and last ratios,” gave the first analytical determination of the speed of sound in air, inferred the oblateness of the spheroidal figure of the Earth, accounted for the precession of the equinoxes as a result of the Moon's gravitational attraction on the Earth's oblateness, initiated the gravitational study of the irregularities in the motion of the moon, provided a theory for the determination of the orbits of comets, and much more.

Newton made clear his heliocentric view of the solar system — developed in a somewhat modern way, because already, in the mid-1680s, he recognized the "deviation of the Sun" from the center of gravity of the solar system. For Newton, it was not precisely the center of the Sun or any other body that could be considered at rest, but rather "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World", and this center of gravity "either is at rest or moves uniformly forward in a right [or straight] line."

Certainly his work in physics would classify him as the one of the greatest of that branch of science, but pure mathematicians memorialize him for this invention of calculus, although his delay in publication allowed another contemporary, Gottfried Wilhelm von Leibniz, to independently discover the important mathematical method. This led to a long fight over priority which actually set British mathematics back since Newton’s notation was not as clear as the notation invented by Leibniz, and the British used Newton’s notation as an act of support.

Even the history of something as dry as mathematics has its controversy and national pride and prejudice.

Perhaps more important than all his fabulous discoveries was his impact as a sort of mentor to the enlightenment. It was Newton's conception of the Universe based upon Natural and rationally understandable laws that became one of the seeds for Enlightenment ideology. He received fortune and acclaim and even a civil service job and a knighting in response to his great work.

In his own words, he said, “I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”

Carl Freidrich Gauss

Gauss, often called the “Prince of Mathematics” was also busy in other disciplines including physics and astronomy. Born at the time of the American Revolution, Gauss anchored the first half of the Nineteenth Century, a century that formed the foundation for all the wonderful discoveries to come in the Twentieth. He contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics. He is honored in physics by a naming a unit of magnetism, the “gauss,” after him.

Although the mathematics discovered by the other two great mathematicians may be familiar to most high school graduates, even if they never took a course in calculus, Gauss’ work is more advanced and so average students not involved in advanced math may not have be experienced with the areas that he advanced.

Sadly one of Gauss’ principles was “few but ripe.” Therefore he only published ideas that he had fully developed leaving us to wonder at further ground breaking and original ideas he hinted at in letters, but never published. Mathematics would have gained greatly from even the random thoughts of this mathematical prince. He was so significant in the work that he did publish, one can only ponder what other gems of mathematics were not yet ripe in his genius, so he withheld any hint of his insights.

For the benefit of the non mathematical reader, I won’t go into the details of his discoveries except to note the number of modern day prizes, buildings, and astronomical and geological objects that carry his name and reputation. For the mathematical reader, there is no mystery why this modern mathematician is listed with the other two greats.

He supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician. I’ve written about that identity which I call the "most beautiful equation in the world," a view often expressed by other mathematicians. It is most interesting that Gauss thought of that equation as a touchstone for measuring math prowess.

I have to admit that I had to study the identity, and I’m still amazed how it shows that an exponential with an imaginary number is somehow equivalent to rotation or how transcendental and irrational numbers can combine with imaginary numbers to produce the most basic integer. I now understand how, but — believe me — it wasn’t, nor is it now, intuitive. I still struggle to understand that damn and wonderful equation, and I run through the transformation in my head from exponential to trigonometric identities anytime I consider it. It is always something I have to go through step by step, rather than leap to the conclusion.

I think Gauss was correct. I will never, ever, ever be a world-class mathematician. All I ever will be is someone scratching the surface of the concepts invented and conquered by these three wise men and all the other wise men and women of science that have come before or since. In all reality, I don’t think I’ll ever get my Ph.D., nor would I have gotten it when in my twenties. I just don’t think I’m good enough. That realization is not going to stop me. I’ll give it my best, using every trick of learning and understanding that I’ve got stuffed up my sleeve. I’ll give it the “old college try.”

These guys are my heroes and my mentors. I will try … try my best … but I don’t guarantee the results. After all, as these three men demonstrate, my sights are set very high. I may miss the target all-together. Yet, I too, feel like a small boy, a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. I may not comprehend that great ocean, but I see it out there … and it excites me to just look out at it. I don’t have what these great, or most great, scientists have, but I do match them in desire. I get it! … I may not understand it, … but I get it!!

No comments:

Post a Comment