Mathematics - The Language of Science
“Like the number, the history of mathematics has a beginning but it has no end,” is an accurate reflection of most disciplines in history. This thought was expressed by David Berlinski in his excellent short history of mathematics, Infinite Ascent (2005). For those that have not yet read Berlinski’s historical summary, or have never thought about the evolution of mathematics, this article gives you the highlights of humanity’s “language of science.”
Scott Kim produces ambigrams. This 1997 word design features the word which can be read from left to right ot right to left.
The Babylonians, Egyptians, and Greeks developed knowledge of geometric forms of various shapes and sizes from building construction, agriculture, and constructing tools for their livelihood. Pythagoreans of ancient Greece from the sixth century before the common era believed in the mystical qualities of the numbers. Numbers were masculine (1 and 3) or feminine (2), prime (5 and 7) or perfect (10), and the harmony among numbers reflected the harmony between things in nature. There was some debate among followers about perfection of the heavens and why there wasn’t ten heavenly bodies. As it turns out, the Sun and the nine planets of the twentieth century, once Pluto was discovered in 1930, provided this view of numerical perfection of the heavens.
While numbers garnered significant attention from the Pythagoreans, it was two centuries later when Euclid wrote his seminal work Elements and introduced the notion of precision in mathematics. Euclid was a geometer who introduced certainty with the method of the proof. Black and white absolutes were introduced, all based on the power of logic. His work in geometry remains in force to this very day with his 23 definitions, five common notions, and five axioms. The word geometry comes from the Greek geos meaning earth and metron meaning measure. In ancient times, geometry was used in many practical applications such as astronomy, surveying, and navigation to name a few. The Egyptians, beholden to the regular flooding of the Nile, were most interested in land surveying because of the swollen river taking out landmarks and reshaping fields.
Euclid’s geometry text, Elements, is reincarnated in this Canadian text. The contributions of Thales, Pythagoras, Plato, and Aristotle all found life in Euclid’s work that persists for more than 2,000 years.
After the great contributions of these intellectually inclined Greeks, mathematical advances in the West would have to wait for another thousand years. This is because of the mathematical incompetence of the Romans, even though this empire absorbed the Greeks as they conquered the world. The Romans were great military men. Consequently, their contributions were directed at engineering, medicine, and law, pursuits that helped the people rather than mathematical exercises that challenged the intellectual elite. Instead, the elite focused on political posturing, oral discourse, and propaganda. Mathematics was set aside.
Mathematical Newcomer - Terence Tao
Terence Tao (b. 1975) and prime numbers have a special relationship. Prime numbers on a number line include 2, 3, 5, 7, 11, 13,... and so on, any number that can only be divided by itself and the number 1. Rather than picturing them as points on a line consider visualizing them as star bursts on a canvas. Tao, a UCLA mathematician, recently partnered with Cambridge University mathematician Ben Green, to prove that this canvas can produce every possible shape. This discovery, the Green-Tao theorem, resolved a major question in number theory and revealed a suspected and new property of prime numbers. It was the mathematical event of 2004.
Tao was awarded the Fields Medal in 2006 as a result of this contribution and many of his other works. Such recognition to mathematicians is the equivalent of the Nobel Prize of their discipline.
Prime numbers become fewer and far between as one makes his way up the number line. However, ancient Greek mathematicians knew that they never run out because there are an infinite number of them. Appearances can be deceiving too. Prime numbers seem to occur in a random fashion, however, Jacques Hadamard of France and Charles de la Vallée Poussin of Belgium proved at the close of the nineteenth century that a hidden pattern can be discerned from the way they peter out. They do so in a way described by the natural logarithm function, revealing that there is a method to the prime madness.
Long, finite arithmetic progressions of primes was suspected to be one such pattern. No one had been able to prove it until Green and Tao showed up. Some progress was made in 1939 when Johannes van der Corput of the Netherlands demonstrated that three-term arithmetic progressions of primes are an infinite pattern. Then Green and Tao announced that they had solved the problem: arithmetic progressions of primes of any finite length exist.
As his web site reveals, Tao is a Professor at the Department of Mathematics, UCLA. He works in a number of mathematical areas, but primarily in harmonic analysis, partial differential equations (PDE), arithmetic and geometric combinatorics, analytic number theory, compressed sensing - a field he modestly created through his musings - and algebraic combinatorics. He was a finalist to become Australian of the Year in 2007.
Tao is a busy guy, but sticks to what he loves and does best. He will tell you that subjects, such as creative writing, are some of his worst. Like many other mathematicians and engineers, Tao prefers situations in which there are black and white guidelines and procedures. Math and science are such subjects that create these fun situations where an answer can be found or derived. Tao also recently relayed that he liked physics but encountered difficulty with chemistry, organic chemistry, in particular. Tao says of chemistry that there are “too many little facts to memorize.” Although there are many facts to memorize in math, Tao says that he can derive many of them from first principles, something that can’t be done to obtain the chemical formula for butane in organic chemistry, for example.
Newsletter Perspectives on Mathematics
The table below highlights stories about mathematics published in the Newsletter, 'A World Perspective.'| Topic | Issue of AWP | Pages |
|---|---|---|
| Kenneth Arrow | | 2 |
| Marquis de Condorcet | | 2 |
| Game Theory | | 2 |
| James Clerk Maxwell | | 3 |
| Speed of Light | October, 2112-1 | 5 5 |
| Sir Isaac Newton | 3 |
Everyday Math
In American English, 40 or 'forty' is the only number when spelled out that shows an ascending letter sequence.Meteor showers drop 45,360 kilograms of fresh space dust each year. This is the equivalent of almost 15 Cadillac Escalades falling from the sky.
The date November 19, 1999 or 11/19/99 was the last date where all the digits in the date are odd. The next date like this is January 1, 3111 or 1/1/3111.
The average U.S. worker contributes $63,885 to the nation's GDP. This makes Americans the most productive employees in the world, according to the International Labor Organization. Furthermore, the average number of hours worked were 1,804 compared to employess in France and Norway, who worked 1,564 hours and 1,407 hours, respectively.
Mathematician of the Month
Each month a mathematician is recognized for his or her outstanding achievement or contribution to the field. Their impact on society is also broached.Kenneth Arrow and Mathematics Applied to Politics
We make decisions by voting most of the time, thriving or fearing the choices we eventually make. The voting process is supposed to be fair because all voters get a say in the outcome. However, mathematicians have a lot to say about voting fairness, rooted in the period of the French Revolution and the thoughts of the philosopher Marquis de Condorcet (1743-1794).
Building upon these early ideas, Kenneth Arrow (b.1921) proposed the impossibility theorem that revolutionized welfare economics, decision-making processes, and voting systems. The theorem, developed in 1951, proves mathematically that a perfect form of government can never be achieved. This is because voting systems, which are mechanisms for obtaining a decision from a wide range of voter's preferences, cannot be aggregated without running into some kind of unfairness. In fact, from another perspective, you could say that Arrow determined that no electoral system could address the needs of a democratic system.
Kenneth Arrow at the 2004 National Medal of Science award ceremony held in his honor
One example of a poor electoral system was the ‘out of date’ British system of 1830. At that time, few men had the right to vote since a hodge podge of local customs, rather than national law, determined who had voting rights. Representation was both uneven and unfair as some members of Parliament (MPs) represented boroughs (towns) of shrinking population, in some cases of only a few households. Other MPs represented boroughs where one landowner controlled the votes while some large industrial towns in the north had no MP at all. Many of these towns had grown quickly since the start of the Industrial Revolution, yet lacked representation. Manchester, for example, had no MP because it was not designated as a borough. England’s Reform Act of 1832 brought greater fairness and representation to the people. The Act enabled many more of the middle-class to vote and it abolished many of the small boroughs and eliminated those controlled by special interests. It also reduced the number of members for other boroughs and established burroughs for the northern townships. Little did the writers of the Reform Act know that they could never achieve complete democracy.
Arrow’s attempt to apply mathematical rigor to society and its social conventions earned him a Nobel Prize in 1972. With regards to electoral systems, his theorem establishes that no electoral system meets all of the following principles of democracy:
- the individual freedom to vote for a candidate of one’s choosing, without any restrictions,
- an election result that depends only on candidate votes received,
- complete agreement where the winning candidate receives all of the votes, and
- a non-dictatorship, which means voting does not simply follow the preferences of a special individual while ignoring all others.
The very premise supporting Arrow’s theorem are the key factors driving any democratic culture. Since Arrow proved all of these principles cannot be simultaneously in force, it would appear that no democracy could exist. What I find exciting is the fact that Arrow’s argument is mathematical in nature and can best be explained by Condorcet’s paradox.* Although Arrow's theorem is mathematical, it is often expressed in statements such as "No voting method is fair” or "Every ranked voting method is flawed,” or "The only voting method that isn't flawed is a dictatorship.” These statements are not completely true as they over simplify Arrow's result. However, Arrow’s theorem shows that a voting mechanism cannot comply with all of the conditions listed above. The theorem Arrow introduced reinforced the usefulness of applying mathematics to the humanities, an area heretofore believed to be unsuitable for rigorous analysis.
* Condorcet’s paradox may best be understood by considering Arrow’s thesis itself, where he shows that it is impossible to find a voting rule under which one option emerges as the most preferred. Condorcet's paradox furnishes the simplest example of this idea and it goes as follows: There are three candidates running for office: Reagan (R), Carter (C), and Thatcher (T). One-third of the voters rank them R, C, T. One-third ranks them C, T, R. The final third rank them T, R, C. Then a majority will prefer Reagan to Carter, and a majority will prefer Carter to Thatcher. It would seem, therefore, that a majority would prefer Reagan to Thatcher. But in fact a majority prefers Thatcher to Reagan. Arrow's general proof is more complicated but holds for an infinite number of voters. Consequently, no one voting rule is the most preferred. Why? Another way of expressing this is if Thatcher is chosen as the winner, it can be argued that Carter should win instead, since two voters (Reagan and Carter) prefer Carter to Thatcher and only one voter (Thatcher) prefers Thatcher to Carter. However, by the same argument Reagan is preferred to Carter, and Thatcher is preferred to Reagan, by a margin of two to one on each occasion.
Arrow's Views on Capitalism - A Complex Mathematical System for Business
In the early 1970s Kenneth Arrow wrote an essay called “Capitalism, for Better or Worse.” At the time he was a Professor of Economics at Harvard and had just become a Nobel laureate in economics in 1972. Arrow’s article examines six fundamental contradictions of capitalism and he suggests these very contradictions, if resolved, may actually strengthen capitalism. First, it is well known that the capitalist system excels at productive efficiency. This is good for business. However, Arrow points out that “inequalities in the distribution of material wealth and...the power and control over the activities by which it is created constitute a steady indictment” against capitalism, but one that has not proven fatal to this complex system.
The six indictments examined include capitalism’s ideological weakness, the alienation it exhibits, the increasing concentration of wealth, the destructive effects of unemployment, and the economic dangers of inflation. In terms of ideological weakness, capitalism is tied to the selfish motives of humanity. It thrives on the greed of owners or a company’s managers. The common driver is the desire for increasing profits with the success of the capitalistic enterprise depending on a fine balancing act between costs and revenues. Its weakness is its very strength, ignorance of any idealistic commitment to support the public good.
The second weakness is the alienation that capitalism engenders. Personal relationships and support for the community are eroded with the continued capitalistic emphasis on impersonal exchanges and business transactions. Arrow points out that this aspect was reinforced by Karl Marx who said the worker was alienated from the product made by the fruits of his or her labor.
The third weakness addressed was the increasing concentration of wealth and power placed into the hands of fewer and fewer of the elite of industry. Both competition in the marketplace and advances in technology make this possible.
The fourth indictment against capitalism has been the emergence of working-class solidarity. The solidarity of workers gave rise to the union movement in North America and to the rise of socialism and syndicates in Europe.
The fifth problem with capitalism is the unemployment it produces. Since its emergence 150 years ago, capitalism has experienced recurring cycles of unemployment. However, Arrow points out that the rise of real income for people in society has more than offset the cyclical economic disasters that occur over time.
The last indictment against capitalism Arrow discussed was inflation. The theory that full-employment will lead to inflation seems to have been mitigated by the fact that a level of price stability can occur from market-force equilibrium or be enforced by government intervention.
Although capitalism is a complex system, Arrow believes these six contradictions suggest some social action is in order as follows:
- achieve macroeconomic stability,
- redistribute income and power to improve fairness in society, which in turn, will ensure the liberty and equality of individuals, and
- increase individual and local control over one’s destiny.
Please find below a table highlighting a number of the great mathematicians that have been featured in Rob's Newsletter and have contributed to advancing the language of science:
| Month | Mathematician | Century |
|---|---|---|
| February 2008 | 17th | |
| January 2008 | 19th | |
| September 2007 | 19th | |
| June 2007 | 13th | |
| May 2007 | 5th AD | |
| April 2007 | 18th | |
| September 2006 | 3rd B.C. | |
| April 2006 | | 20th |
Please see the article “Mathematical Rigor Applied to Society” found in the March 2006 issue of “A World Perspective” newsletter. Thomas Hobbes set the wheels in motion for the acceptance of mathematics used to study social behavior in society. The article addresses the fact that the current voting systems we use in Canada, England, and the United States are unfair - at least from the mathematical point of view.
Featured Book on the History of Mathematics
I recently finished reading Eric Temple Bell’s “Men of Mathematics,” which looked at the history of mathematics from the days of the Babylons through to the dawn of the twentieth century. As the subtitle of his quasi-historical work states, Bell looks at “the lives and achievements of the great mathematicians from Zeno to Poincare.”
He wrote the book in 1937 and it was posthumously reprinted in 1965, the version I enjoyed reading. Although Bell’s book of biographical sketches inspired many people to study mathematics, historians of mathematics do not consider it as completely accurate. Bell in his preface said he was not writing history, but a summary of the mathematicians’ humanity and their seminal achievements. It is an excellent read and I heartily recommend it.
E.T. Bell (1883-1960) image courtesy of JOC/EFR
The world’s great mathematicians have played a major part in the evolution of scientific and philosophic thought. Many believe thre three greatest mathematicians of all time include Archimedes, Newton, and Gauss. In fact, some historians postulate that if the Greeks took their cue from Archimedes instead of Euclid, Plato, and Aristotle, they might have advanced the era of modern mathematics of Newton and physical science of Galileo during the 17th century by 2,000 years.
Modern mathematics began with two great advances - analytic geometry in 1637 with Rene Descartes (1596 - 1650) and calculus with Sir Isaac Newton (1642 - 1727) and Gottfried Leibniz (1646 - 1716) around 1666. Descartes took that final step with his Cartesian coordinate system in building out analytic geometry “with its structure of geometrical proof, discovery, and invention.” Descartes can be said to have invented geometry and he documented his mathematical findings in an appendix to his great work “Method” (1637). Newton and Leibniz independently developed calculus in order to explain the physics of nature. Both built upon the incredible insight of Archimedes regarding the idea of limiting sums from which the integral calculus emerges.
The entire development of mathematics owes its progress to the ongoing battle between the notion of the discrete and the idea of the continuous. The use of the discrete, captured by the numbers 1,2,3,..., attempts to define the natural world atomistically, as individual elements put together to describe the whole. The world of the discrete belongs to algebra, symbolic logic, and the theory of numbers. The continuous tries to describe nature as undulating, coursing, and flowing phenomena, such as the rise and fall of the tides, the orbits of the planets, and the movement of electricity. Such continuity takes us into the mathematical world of the calculus and to a huge array of applications common to the fields of science and technology - the world of mathematical analysis. Engineers make much use of the tools of mathematics to solve practical problems.
The appearance of continuous mathematics can be attributed to Pythagoras’s failed attempt to describe the world solely with discrete mathematics. What we call the Pythagorean Theorem proves the point. If two sides of a triangle have a length of one unit, the diagonal produced with geometry yields an answer of the square root of 2. This number cannot be derived with a finite number of meaurements simply because the square root of 2 is an irrational number, a series of never-ending numbers needed to describe the length. Some of the Greeks must have been less tolerant, as the young mathematician who raised the issue of irrational numbers was thrown overboard and killed. The followers of the discrete did not want to be derailed by the ideas of the continuous.
Other biographical sketches Bell shares with his readers include:
- Fermat, who founded the theory of numbers, shared in the creation of the theory of probability, and developed both Fermat’s Theorem and Fermat’s Last Theorem,
- Pascal, who cofounded the theory of probability, invented the first calculating machine, carried on Toricelli’s work on atmospheric pressure, and solved many aspects of the cycloid,
- the Bernoulli family, who produced eight mathematicians over three generations, the greatest of which was Daniel who developed principles leading to the conservation of energy postulate and is best known for his work in pure and applied fluid motion,
- Euler, who was brillaint in both discrete and continuous mathematics, has never been surpassed as an algorist. He developed a solution to the three-body problem useful for navigation and made mechanics and analytical science,
- Lagrange, who was close friends with the great chemist Lavoisier, used methods of approximation to develop six-body solutions in celestial mathematics, - Laplace, who was a mathematical astronomer, proved the stability of the Solar System in his era and developed the theory of the potential, key to understanding the basics of electromagnetism,
- Monge, the inventor of descriptive geometry used for all mechanical drawings and graphical methods that helped make mechanical engineering a reality,
- Fourier developed the theory of heat conduction, which led to useful ideas on boundary-value problems, concepts critical to the development of electrical engineering and accoustical engineering,
- Poncelet developed projective geometry and introduced the principles of continuity and duality,
- Gauss, considered the Prince of Mathematicians, applied rigor to mathematical analysis, inventing the law of reciprocity, the method of “least squares” - the Gaussian law of normal distribution of errors in statistics and the associated bell-shaped curve that is familiar to many, developed the laws of biquadratic and cubic reciprocity, discovered the double periodicity of certain elliptical functions, unified cartesian and polar coordinates by noting that multiplication by i has the effect of rotation through a right angle, laid a mathematical theory of electromagnetism and coinvented an electric telegraph, developed differential geometry, and built up conformal mapping,
- Cauchy introduced rigor into mathematical analysis and the combinatorial, which led to the theory of groups
- Abel and Jacobi jointly developed the theory of elliptic functions and the Hamilton-Jacobi equations contributed to quantum mechanics,
- Cayley developed the theory of invariants of great importance to the theory of relativity, the notion of geometry of “higher space” (or of n dimensions), and the theory of matrices, which proved most useful for Heisenberg when he used matrix multiplcation for his work in quantum theory 67 years later,
- Weierstrass created a theory of irrational numbers to address the concepts of limits, continuity, and covergence,
- Boole’s original work added mathematical logic to the domain of algebra,
- Hermite solved the general equation of the fifth degree using elliptical functions instead of radicals, which proved to be impossible as a solution set, and developed the concept of transcendance of the number e (2.71828...),
- Kronecker combined the the theory of numbers, the theory of equations, and elliptic functions into one pattern, and
- Reimann, one of the most original mathematicians of modern times, defined curvature and recognized its invariance, and devised processes for the investigation of quadratic differential forms, both which found their physical interpretation in Einstein’s theory of relativity.
Read Bell’s summary of these great mathematicians and learn about their humanity, appreciate their strengths and frailties, and admire their seminal achievements.
Good reading!
Books of Fiction Using Mathematics
1. In every crime there is an element of truth, a single true explanation among all the possible explanations,
2. Material clues are akin to axioms in mathematics and logic,
3. Theories of the crime are derived from “thinking up conjectures, possible explanations that fit the facts [the material clues] and attempting to prove them correct,” and
4. The “police adhere to Ockham’s Razor, as long as there is no physical evidence to the contrary they always prefer a simple hypothesis to a more complicated one.”
If this is truly the case for all murder investigations, it seems the priority is the need for speed to close a case versus seeking the truth. This mind set will more often than not sway who is charged with a crime. In murder mysteries like this, many twists result when the killer believes he is smarter than the police, an idea reminiscent of the great works of Agatha Christie.
The challenge for the reader is to see if this desire by the police to close the case and find “any” killer or killers wins out over getting at the truth. In this fictional world of shady characters, detectives, and mathematicians who believe, “Because [mathematics] harms no one. Because it’s a world that has nothing to do with reality,” the solution to the case is like solving a puzzle.
The story includes delightful references to great mathematicians and logicians of the past such as Turing, Godel, Witt, Fermat, Mandelbrot, and Pythagoras, to name a few. As the author writes, “Man is no more than a series of his actions.” And with any mathematical series, the reader is challenged to see if the sum of the series leads to a correct solution.
Good reading!
Quotations by Mathematicians
“Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics.”
- Karl Friedrich Gauss (1777 - 1855)
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