IMPORTANT NOTE The original files I submitted to the MAA for publication in 1996 were all written in a word processing package that is long out of date (I don’t recall what I used), and when I managed to open them in a simple text editor it was hard to locate the original text among the acres of weird characters, and I had to re-construct the order in which the various clauses, sentences, paragraphs, and the like had actually appeared. (I was clearly getting the internal code that had been generated during my writing and self-editing. It seems I make many changes as I write.) The twelve posts published here are the best I could do to reconstruct the originals. I am confident they will differ only in minor ways from what was originally published. Reading them now, it is clear we were all learning to use this new online medium as we went along. By 1997, I was composing posts in HTML (which I learned for the purpose) in plain-text files, so the remainder of these archives will be as they originally appeared apart from (usually minor) editorial changes made by the MAA Online editor.
JANUARY 1996
Good Times
Welcome to the new MAA on line information service, and to my own particular corner—though the idea of a “corner” does not transfer well from the paper page to the scrolling screen. In fact, the vocabulary we use is just one of the things that has to change as we move to ever greater use of electronic media.
If you are reading these words, then you have already taken the plunge and made at least the first tentative steps into the strange new world of the World Wide Web. And it really is a different world, with different rules, different advantages, and different dangers—as the story below indicates.
The following electronic mail message arrived on my computer recently. The source of the message was a long established and highly regarded academic society in England. How would you respond if the same thing happened to you? (The reference to FCC in the message is to the United States Federal Communications Commission.)
BEGIN MESSAGE
The FCC reports a computer virus that is being sent across the Internet. If you receive an e-mail message with the subject line “Good Times”, DO NOT read the message, DELETE it immediately. If you get anything like this, DON’T DOWNLOAD THE FILE! It has a virus that rewrites your hard drive, obliterating anything on it.
What makes this virus so terrifying, says the FCC, is the fact that no program needs to be exchanged for a new computer to be infected. It can be spread through the existing e-mail systems of the Internet. Once a computer is infected, one of several things can happen. If the computer contains a hard drive, that will most likely be destroyed. If the program is not stopped, the computer’s processor will be placed in an nth-complexity infinite binary loop – which can severely damage the processor if left running that way too long. Unfortunately, most novice computer users will not realize what is happening until it is far too late.
Luckily, there is one sure means of detecting what is now known as the “Good Times” virus. It always travels to new computers the same way in a text e-mail message with the subject line reading simply “Good Times”. Avoiding infection is easy once the file has been received – not reading it. The act of loading the file into the mail server’s ASCII buffer causes the “Good Times” mainline program to initialize and execute. The program is highly intelligent – it will send copies of itself to everyone whose e-mail address is contained in a received-mail file or a sent-mail file, if it can find one. It will then proceed to trash the computer it is running on.
If you receive a file with the subject line “Good Times”, delete it immediately! Do not read it!
END OF MESSAGE
Pretty scary, huh? It scared me the first time I received it. That is well over a year ago now. Since then, I seem to have received the same message maybe half a dozen times, each time from a different source, sometimes from one individual, sometimes from an organization, as this last time. The actual text varies from message to message, but the gist is the same on each occasion.
It is an indication of the speed at which the Internet is growing that, as indicated by the latest distribution of the “Good Times” message, there are still plenty of folk on the net who are receiving this warning for the first time, and occasionally passing it on.
It’s time for some facts.
Fact 1. There is no “Good Times” virus. More precisely, there is no virus that goes about under that name and which trashes people’s hard drives, which is what the warning message says happens. In fact, though I have met dozens of people who have received a message warning them not to read any e-mail with the subject line “Good Times”, I have yet to meet a single person who has actually received a message having such a subject line, regardless of what was in that message, friend or foe.
Fact 2. Even if there were such a virus, the FCC would not get into the picture. They are the folk you go to if you want to get a piece of the airwaves to broadcast radio or television programmes in the United States. They have nothing to do with tracking or warning about computer viruses.
Fact 3. There is no such thing as an intelligent program. And claims made by the artificial intelligence community notwithstanding, there probably never will be. What there are, are programs that can reap the maximum advantage from human intelligence. The “Good Times” warning message is just such a program.
By most people’s definition, a virus is something that causes you harm, discomfort, or inconvenience, and which is capable of spreading through a community. For generations, human beings have had to suffer the consequences of viruses spread from person to person by touch and by air. Then came the AIDS virus, the world’s first major virus spread around the world by jet aircraft, as infected individuals unknowingly spread the virus from continent to continent. After AIDS, it was clear that in an age of world travel on a large and growing scale, geography was no longer a defense against the spread of a virus.
At about the same time that the world was becoming aware of AIDS, the growth of computer technology also gave rise to a new kind of virus, the computer virus. Spread at first by “touch”, namely the passing of computer disks from person to person, and then later by “jet aircraft”, the growing interconnections of computers around the world by the Internet, these new kinds of virus did not cause sickness or death in people, they infected computers. The more benign ones simply made a nuisance of themselves; the more virulent forms were able to destroy all the data stored in any computer they found their way into.
Computer users soon learned to fear these new electronic viruses as much as they did their biological cousins. Enter the “Good Times” virus. Make no mistake about it, there is a virus here, and it has spread like wildfire. To date, the only known defense to this virus – information – has failed to stem its spread. Fortunately, the virus does no major harm. It certainly does not destroy all the data on your hard drive. It is very much in the class of “minor nuisance” viruses, a bit like a minor cold. It spreads through humans. It depends for its continued life on human good will and a desire to help each other. In the virus’s favor is the fact that there is a lot of that good will about. People around the world receive a message warning them about this dangerous piece of software that is “somewhere out there” and, what do they do? They pass on that warning to everyone they know. Voila! The virus has spread further.
There is, you see, no data-destroying “Good Times” virus out there. The warning message is itself the virus. Its spread does not require any sophisticated software. It utilizes human beings in order to replicate and spread. And that makes it the closest thing yet in the computer world to a regular, human-infecting, biological virus
Where did this particular electronic-human virus originate? No one knows. At least, the originator has yet to declare his or her hand. It is, without doubt, a clever idea, and one that is at most an annoyance. At present, like the horrendous Ebola virus that seems to be contained in a single continent, the “Good Times” virus has not escaped the borders of Cyberspace. But one of these days, it is going to find its way into another world, perhaps the world of printed information exchange, such as newspapers or magazines. And then, perhaps in a slightly mutated form, it will have a whole new population to infect. All it will take is one careless newspaper article. I, for one, pray that day will never come.
But in case it does, if you ever come across an article on the Internet with the phrase “Good Times” in the title, don’t read it.
Keith Devlin is the Dean of Science at Saint Mary’s College of California, and the author of “Mathematics: The Science of Patterns”, published by W. H. Freeman in 1994.
FEBRUARY 1996
Base considerations
What is the difference between a modern electronic computer and a 13th century English wine merchant? The answer is “Not as great as you might think.” The major clue lies in the system of measurement used in the wine and brewing trade in England from the 13th century onwards, parts of which are still in use:
2 gills = 1 chopin, 2 chopins = 1 pint,
2 pints = 1 quart, 2 quarts = 1 pottle,
2 pottles = 1 gallon, 2 gallons = 1 peck,
2 pecks = 1 demibushel, 2 demibushels = 1 bushel or firkin,
2 firkins = 1 kilderkin, 2 kilderkins = 1 barrel,
2 barrels = 1 hogshead, 2 hogsheads = 1 pipe,
2 pipes = 1 tun.
As a child growing up in England in the 1950s, a large part of my early schooling was spent learning various archaic systems of measurement. Not content with a monetary system that mixed base 12 and base 20 arithmetic, my ancestors had bequeathed me and my compatriots all manner of number systems for measuring land area, liquids, grain, and whatever.
Among those number systems was the delightful one above. The only hard part about learning it was remembering the order of the words. The number system itself is good old binary arithmetic, these days found all over the world etched into wafers of silicon.
Leaving aside the wonderfully evocative vocabulary of the system, 13th century English wine merchants performed their arithmetic in the same way that a modern computer does.
We are so used to computers nowadays that it seems obvious that arithmetic should be performed in a binary fashion, this being the most natural form for a computer, which is ultimately, a “two-state” machine (the current in a circuit may be either on or off, an electrical “gate” may be either open or closed, etc). But this was not always the case.
When the first American high-speed (as they were then called) electric computers were developed in the early 1940s, they used decimal arithmetic, just as their inventors. But in 1946, the mathematicians John von Neumann (essentially the inventor of the “stored program” computer we use today) suggested that it would be better to use the binary system of arithmetic, since which time binary computers have been the norm. (Not that this was the first time that calculating machines made use of the binary system. Some French machines using binary arithmetic were developed during the early 1930s, as did some early electric computers designed in the United States by John Atanasoff and by George Stibitz, and in Germany by Konrad Zuse.)
There is, of course, nothing special about the decimal number system we use every day. Certainly it was convenient in the days when people performed calculations using their fingers. Assuming a full complement of same, it is essential that there is a “carry” when we get to ten. The number at which a “carry” occurs in any number system is called the “base” of that system. In base 10 arithmetic (decimal arithmetic), 10 entries in the units column are replaced by 1 entry in the 100s column, and so on. This means that we require ten “digits” in order to represent numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, all other numbers being composed of a string (or “word” if you like made up from these digits.
Computers (and electronic calculators) use the binary system to perform their arithmetic. Here there are only two digits (known as “bits” short for “binary digits”), 0 and 1. In binary arithmetic there is a “carry” whenever a multiple of 2 occurs. So, counting from one to ten in binary looks like this: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010.
Arithmetic in binary (addition, multiplication, etc) is performed just as in the decimal arithmetic we learn in primary school, except that we “carry” multiples of 2 into the next column rather than multiples of 10. (So instead of having a units column, a tens column, a hundreds column, and so on, we have a units column, a twos column, a fours column, an eights column, a sixteens column, and so on.)
It is the fact that in binary notation all numbers can be expressed using just two digits, 0 and 1, that makes the binary system particularly suited to electronic computers, since the ultimate construction element is the “gate,” an electrical switch that is either on or off (1 or 0).
Of course, we do not use binary notation when we communicate with a computer or a calculator. We feed numbers into the machine in the usual decimal form, and the answer comes out in this form as well. But the computer/calculator immediately converts the number into binary form before commencing any arithmetic and converts back into decimal form to give us the answer. All that is involved is a matter of notation (or language, if you like). The actual numbers are the same. 111 in binary means the same as 7 in decimal notation, just as das Auto in German means the same as the car in English.
All of this is a good excuse for bringing in the following teaser, one which can be used to demonstrate the absurdity of many of the questions beloved by testers of IQ in children. Fill in the next two members of the following sequence: 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, 31, 100, —, —.
Devlin’s Angle is updated at the start of each month. The above article is adapted from Devlin’s book “All The Math That’s Fit To Print”, published by the MAA in 1994.
MARCH 1996
Reflections on Deep Blue
“The . . . question, “Can machines think?” I believe to be too meaningless to deserve discussion. Nevertheless I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted.”
The above paragraph is taken from Alan Turing’s celebrated and oft-quoted paper Computing Machinery and Intelligence, written in 1950.
As Ivars Peterson describes in his recent column in MAA Online (February 26), the IBM chess-playing computer Deep Blue just gave World Chess Champion Garry Kasparov a good run for his money. Should we then credit Deep Blue with intelligence?
Certainly, we generally assume that the ability to play a good game of chess requires intelligence when it comes to people. Why not for chess-playing machines as well?
Well, here is one argument that shows that things are not so simple. One aspect of human intelligence, and a regular component of tests designed to measure intelligence in children and adults seeking employment, is skill at arithmetic. And yet a ten dollar calculator can outperform almost any human being when it comes to arithmetic. Does it follow that a calculator is intelligent? Most people would answer, “No.”
How about the ability to solve algebraic equations? Again, when the solution is found by a person, this task is generally regarded as requiring intelligence, but there are a number of computer algebra systems available that can solve algebra problems far quicker, and with far less chance of making an error, than can most people.
In both cases, arithmetic and algebra, the computer arrives at the answer in roughly the same way as a person, by following the appropriate mathematical rules. And yet we regard human proficiency at arithmetic and algebra as requiring intelligence but an even greater machine proficiency at the same tasks as being ‘merely mechanical rule following’.
The distinction between the ways mind and machine operate is clearer when it comes to playing chess. Considerable effort has been put into the development of chess-playing computer systems such as Deep Blue. However, they achieve their success not by adopting clever strategies, but by essentially brute force methods. Taking advantage of the immense computational power and speed of today’s high performance computers, they examine huge numbers of possible plays, choosing the one that offers the greatest chance of success. There is no ‘intelligence’ involved. (See Peterson’s article.)
In contrast, a good human chess player might consider as many as a hundred possible moves (generally far fewer), and follow the likely ensuing play for at most a dozen of those. The initial choice of moves to be considered in detail is one of the things that marks the good human chess player.
Moving away from chess, is it even theoretically possible for a computer system—that is to say, a computer program running on a conventional electronic digital computer—to be intelligent? If a computer program performs a task that requires intelligence if performed by a person, should we describe that program as behaving in an ‘intelligent fashion’?
On the basis of the above observations about arithmetic, algebra, and chess, the answer would seem to be “No,” but are we being guided by human pride rather than rationality—by a desire to be unique in our intelligence. After all, a jet airplane does not stay in the air by converting food to energy and flapping its wings the way a bird does, but we still say that the plane ‘flies’. It achieves the same end—flying—in a completely different way, a way more suited to its structure and design.
One difficulty in deciding whether or not to ascribe ‘intelligence’ to a computer system is that we must first be clear just what we mean by intelligence. Unfortunately, though most of us might feel that we know intelligence when we see it in other people, science has yet to come up with an acceptable definition.
It was precisely the difficulty of defining intelligence in humans that led Alan Turing himself to propose a definition of ‘intelligence’ that could reasonably be applied to computers. Writing in the same article quoted earlier, Turing formulated a simple test for machine intelligence, known nowadays as the Turing Test.
The Turing Test asks you to imagine you are sitting at a computer terminal through which you can carry out a conversation (typing at a keyboard and reading responses on a screen) with two partners, one a computer, the other a person. Your two partners are named A and B, but you do not know which one is the computer and which one the person. You cannot see either the person or the computer; your only communication with them is through the terminal. If you address a question to A, then A will always answer, and likewise B will always answer a question directed to B. Your task is to try to decide, on the basis of a conversation with A and B, which one is the computer and which one the person.
If you are not able to identify the computer reliably, then, says Turing, it is entirely reasonable to say that the computer is ‘intelligent’—it passes the Turing test for intelligence.
I should point out that, although the hidden person might be expected to answer truthfully, the computer is under no obligation to tell the truth, so in particular, questions such as “Tell me, A, are you a computer?” are unlikely to resolve the issue for you.
With the invention of the digital computer, it was only five years after the appearance of Turing’s 1950 paper—in which he specified programming a computer to play chess and to understand natural language as the two most obvious challenges to attempt first—that work began on both challenges.
In 1955, Allen Newell wrote a paper analyzing the problems facing anyone trying to program a computer to play chess, and by 1956, a group at Los Alamos National Laboratory had programmed a computer to play a poor but legal game of chess.
At about the same time, Anthony Oettinger began work on automated language translation by programming a Russian–English computer dictionary.
However, neither of these two projects offered anything that might be termed ‘intelligent behavior’ on the part of the computer; they were simply ‘automation’ processes whereby straightforward tasks were implemented on a computer. The first genuine attempt to create machine intelligence was made by Allen Newell, Clifford Shaw, and Herbert Simon of the RAND Corporation, who in 1956 produced a computer program called The Logic Theorist.
The aim of this program was to prove theorems in mathematical logic, in particular the theorems in the early part of Whitehead and Russell’s Principia Mathematica. In fact, The Logic Theorist proved 38 of the first 52 theorems in Principia Mathematica.
It has to be admitted that propositional logic is ideally suited to being performed on a computer, having an extremely simple, rigidly defined language and a small set of fixed, well-defined axioms and rules. Turing’s choice of human–machine conversation as an intelligence test for computers was a far more challenging task. Natural language communication is now known to be one of the hardest tasks that faces anyone trying to build ‘intelligent machines’. Forget all of those smooth talking computers and robots on television and in the movies, such as Kit, the talking automobile in the TV series Knight Rider, and HAL, the eventually malevolent on-board mission-control computer in Stanley Kubrick’s space-travel movie 2001: A Space Odyssey. In the real world, no computer system has come close to passing the Turing test, and, I claim, there is good reason to assume that none ever will.
What that good reason is will be the topic of future columns in this series. My claim—which many readers will regard as heretical coming from a mathematician—is closely bound up with what I see are significant limitations on what can be achieved by the methods of mathematics in the domain of human reasoning and communication. In this regard, I am at least in good mathematical company. Let me end with the following words of Blaise Pascal.
“The difference between the mathematical mind and the perceptive mind: the reason that mathematicians are not perceptive is that they do not see what is before them, and that, accustomed to the exact and plain principles of mathematics, and not reasoning till they have well inspected and arranged their principles, they are lost in matters of perception where the principles do not allow for such arrangement. . . . These principles are so fine and so numerous that a very delicate and very clear sense is needed to perceive them, and to judge rightly and justly when they are perceived, without for the most part being able to demonstrate them in order as in mathematics; because the principles are not known to us in the same way, and because it would be an endless matter to undertake it. We must see the matter at once, at one glance, and not by a process of reasoning, at least to a certain degree. . . . Mathematicians wish to treat matters of perception mathematically, and make themselves ridiculous . . . the mind . . . does it tacitly, naturally, and without technical rules.”
Devlin’s Angle is updated on the first of each month. The above article is adapted from his forthcoming book “Goodbye Descartes: The Quest for a Science of Reasoning and Communication”, to be published in the fall by Wiley.
APRIL 1996
Are Mathematicians Turning Soft?
“Soft Mathematics” is the title of an article I wrote for Mathematics Awareness Week 96, which takes place April 21-27. The theme for this year is “Mathematics and Decision Making.” The focus of my article is the blend of mathematics and techniques of the social sciences that is increasingly prevalent in economics, management science, psychology, cognitive science, sociology, linguistics, and the like. The article appears on the Mathematics Awareness Week Web entry, which can be accessed directly from MAA Online. Choose “Resources for Mathematics and Decision Making” and then pick “Original Articles”.
By “soft mathematics” I don’t mean “applied mathematics.” Nor do I mean “mathematical modeling.” And I certainly don’t mean the use of statistics in the social sciences. Social scientists do indeed make frequent use of statistical techniques to collect the data for their studies. There is no question that practically all sciences, natural or social, rely on mathematics in one way or another when it comes to data collection.
Rather, what I mean by the term is a genuine attempt to blend mathematics with other approaches in trying to analyze or describe some phenomenon. The result is certainly not mathematics in the sense I was brought up and trained to understand the word. Just as counterfeit money is not money, so too soft mathematics is not mathematics. Of course, counterfeit money can occasionally make people genuinely rich, and likewise soft mathematics can lead to real (hard) mathematics. (Printing your own money can also, of course, land you in prison, and a lot of what might seem to qualify as soft mathematics is pure junk that by rights ought to land the perpetrators in mathematical prison.)
As an example of soft mathematics, take the following formula from linguistics:
XP –> (SUB) X’ YP*
(I said “take it.” I did not ask you to understand it.) The symbol X’ should be rendered as X with an overbar, a typographical convention that is not universally available on WWW servers and readers. The formula reads: “An X-phrase consists of an optional subject, coupled with an X-bar, coupled with any number of Y-phrases.” It is a grammatical rule in what linguists call “X-bar theory”, and it is a candidate for one of the rules of language that may be “hard-wired” into the human mind as part of what linguist Noam Chomsky called Universal Grammar.
According to the algebraic rules that go along with this formula, you can replace the symbolic variables X and Y by any of N (for noun), V (for verb), P (for preposition), and A (for adjective) to give a particular rule of grammar. For example,
NP –> (SUB) N’ VP*
which says that a noun phrase consists of an optional subject coupled with a noun-bar (a noun is the simplest case) coupled with any number of verb phrases.
To anyone not familiar with contemporary linguistics, this doubtless seems pretty bizarre, and it would involve too great a digression to explain just what is going on. Those sufficiently intrigued should consult linguist Steven Pinker’s excellent best-selling book The Language Instinct, published in 1994. My point is simply to indicate that, in order to capture some of the abstract patterns and structures of mind and language, linguists sometimes find it convenient (perhaps necessary) to make use of mathematics, or at least mathematical notation or techniques.
The result is not that linguistics becomes part of mathematics, or even a “mathematical science” (in the sense of, say, physics). The aim is not to prove theorems. There might not be any relevant theorems to prove. Ever. There might not even be any deep or revealing definitions. The aim is to do linguistics, to understand language and investigate the way people use language to communicate.
To some mathematicians, the absence of theorems means that what is going on is not only not mathematics, but no self-respecting mathematician should become involved in such a dubious pursuit. I don’t agree. Besides the fact that such a view would damn any number of revered mathematicians (for example Leibniz, for his algebra of mental concepts), such an attitude strikes me as dangerously isolationist. There have been a number of infamous occasions where social scientists have made use of mathematical techniques in a highly inappropriate and misleading manner. (Chomsky was not one of them, I hasten to add. He was well-trained in mathematics. He even proved some theorems!) Far better for those of us properly trained in the discipline to offer our services when we are asked (or even if we are not, provided we do it in a collegial fashion, aware that as mathematicians we have only been handed one of many sets of stone tablets, all of which may be valid in their own domains).
To my mind, it is not a cause for dismay that, say, a linguist or a management scientist uses what for me as a professional mathematician is an entirely trivial piece of apparatus. I don’t see such use by others as besmirching my discipline. Rather, I view it as an affirmation of the incredible power of mathematics that even its simplest elements can be put to good use by others.
Devlin’s Angle is updated on the first of each month.
MAY 1996
The Five Percent Solution
As mathematicians, we are rarely satisfied with less than the complete solution. In fact, even that is not enough. Once a problem has been solved, we look for shorter, cleaner, more elegant solutions. That is the nature of mathematics. For the engineer, searching for the “perfect” solution is a luxury that can rarely be afforded. There are deadlines to meet, and delay increases the cost. Find a solution that works within a certain toleration—that’s the name of the engineer’s game.
For the business manager, things are far less clear-cut than in either mathematics or engineering. The world of people, business processes, economics, finance, and marketing is orders of magnitude more complex (“messy” might be more appropriate) than the world of mathematics. For the business manager, a “five percent solution” is often worth a considerable search. A small increase in efficiency can mean cost savings of millions of dollars. A five percent edge over the competition can mean the difference between domination of the market and commercial extinction.
Entire new ‘sciences’ have even grown up around mathematical models. Decision theory, for example, has grown up around the models of games studied in the mathematical discipline known as game theory. In its modern form, game theory is the product of the mathematician John von Neumann. Game theory models the results of actions of competing parties in competition, with each acting in its own self-interest. In 1994, the Nobel Prize in economics was shared by John Harsanyi and the mathematicians John Nash and Reinhard Selten for their introduction of several different concepts of market equilibria, situations in which each party is in an optimum position relative to its competitors.
Decisions affect our daily lives. Mathematics idealizes the process of evaluating information and balancing alternatives that is part and parcel of rational decision making. Mathematical models of decision strategies underlie computer programs that either make or support decisions in various contexts.
Mathematical subdisciplines such as statistics, optimization theory, probability theory, queuing theory, control theory, game theory, and operations research are all used routinely in making difficult choices in public policy, health, business, manufacturing, finance, law, and various other human endeavors. Mathematics plays a significant role in arriving at decisions in all walks of life—making policy decisions connected with global warming, deciding on the most economic way to generate electric power, making a profit in financial markets, approving new drugs, weighing legal evidence, flying aircraft safely, managing complex construction projects, choosing new business strategies, and so forth. Mathematics produces the daily weather forecast we see on our television screens each evening. And mathematics routes our telephone calls across the nation and around the world.
But whenever a mathematical model is employed in the course of making an analysis or informing a decision, the question arises: how reliable in the model? For no matter how sophisticated and precise the mathematics, if the model does not accurately reflect the reality it is supposed to represent, then results obtained from the model will be of little or no use.
For example, mutual funds can include investments in derivatives such as currency repurchase options, financial instruments whose prices are tied to prices of other commodities in the market. In today’s markets, both the value and the hedging structure of many derivatives are decided by mathematical models of economic behavior. Stochastic differential equations are used to express the market’s intrinsic uncertainty. They lead to valuation formulas that balance risk and expected return. But at the end of the day, wise investors would be better advised to put their faith (and their money) in the hand of an experiences (human) market analyst, not a fancy computer program bristling with the latest mathematical models. (The analyst may well use such a fancy computer program. But that is a different matter.)
Similarly, for all its mathematical interest, game theory remains a virtual irrelevancy when it comes to making many real, human decisions. Despite all the impressive mathematics of game theory, the model for most real-life conflicts is the age-old game of ‘chicken’. (Remember the James Dean movie where the two cars race for the cliff edge?) In that kind of game, human experience, courage (foolhardiness?), and instinct rule the roost.
And just recently, a political scientist and a mathematician extended an old technique for dividing a piece of cake between two individuals—one cuts, the other chooses—to yield a fair division when there are many parties and when economics and other complex forces are at work. In the real-world, such disputes arise when it comes to dividing land and natural resources at the close of a multi-nation war. The theoretical solution of the underlying mathematical problem, that of fair, envy-free division among many parties, should then be useful when heads of state try to negotiate an end to hostilities. But only the most naive mathematician would expect that the new mathematical theorem will lead to a division of territory in Bosnia that satisfies all parties.
There is, then, a wide gulf that separates the worlds of mathematics and business. At least in terms of what constitutes a “solution” to a problem. But that gulf does not prevent the world of business being able to profit from the odd bit of mathematics from time to time.
I was prompted to reflect on the role of mathematics in the business world by a conversation I had recently with a business manager. He was commenting on the use of the word “Quality” in the business world. “Before the introduction into business of tools like the Gaussian distribution, the fishbone diagram, the Pareto chart, and so on,” he observed, “the word ‘quality’ was soft.” Everyone felt they could recognize “quality” when they saw it, but no one could define or measure it. But a few fairly simple mathematical ideas made all the difference.
I was sufficiently intriguiged by all of this to check out an read a recent book on quality control and improvement in business, Accelerating Innovation, by Marvin Patterson, director of Corporate Engineering for Hewlett-Packard. The book is littered with graphs and charts and other mathematical ideas (including some ideas from decision theory). None of it would earn you a college credit from a mathematics department. But—and I have to take this on faith since I have no knowledge of the world of business management—this assortment of simple mathematical ideas can be worth millions of dollars to a company in the competitive marketplace.
At the very least, it suggests that a good education in mathematics could be a tremendous asset to the upwardly mobile young business executive. Providing, of course, that they can rid themselves of the overpowering urge to go for the 100 percent solution.
It also, dare I suggest, casts a more positive light on the business majors in our college and university.
Devlin’s Angle is updated on the first of each month.
JUNE 1996
Laws of Thought
By the time they graduate, most mathematics students have heard of Zeno, he of the paradoxes of motion: Achilles and the tortoise, the arrow paradox, and so forth. Less well known is the fact that there was not one but two highly influential Zenos in the world of ancient Greek mathematics.
The better known Zeno, of Zeno’s paradoxes, was Zeno of Elea, in Magna Graecia, who lived about 450 B.C. That Zeno was a student of Parmenides, the founder of the Eleatic school of philosophy.
The lesser known Zeno, Zeno of Citium, lived some 150 years later. The legacy of this Zeno is present-day computer science.
Well, that’s making a bit of a leap. What Zeno of Citium actually did was found the Stoic school of logic. Though modern mathematical logic is popularly credited as having its beginnings in the syllogistic logic of Aristotle, most of the fundamental notions of contemporary propositional logic began not with Aristotle but with Zeno and the Stoics.
As in modern propositional logic, the patterns of reasoning described by Stoic logic are the patterns of interconnection between propositions that are completely independent of what those propositions say. The Stoics examined a number of ways in which two propositions can be combined to give a third, more complicated proposition.
For instance, one, simple way that two propositions may be joined to form a single, new proposition is by means of the connecting word (or ‘connective’) “and”. Present-day logicians bring out the abstract pattern of connectives such as “and” by using algebraic notation. The letters p, q, r are generally used to denote arbitrary propositions, and a symbol such as & is used to abbreviate the word “and”.
Thus, [p&q] denotes the proposition [p and q].
It is of interest to note that at no time did the Stoics themselves hit upon the idea of using algebraic notation, with letters denoting arbitrary propositions and symbols denoting connectives. They wrote everything out in ordinary language. This often resulted in their having to write down long and complicated sentences that are difficult to follow, and that almost certainly hampered their possible progress in logic. The modern, algebraic way of expressing the notions of propositional logic is much better. The algebraic expressions are far shorter and much easier to read. (It is one of life’s many ironies that a linguistic device introduced to make things clearer and simpler, namely algebraic notation, should have precisely the opposite effect on so many people.)
Other means of combining propositions analyzed by the Stoics were “or” (abbreviated by the symbol v in modern logic), “not” (often denoted by the symbol ~), and “implies” (denoted by ->).
As alluded to above, the key idea behind the Stoics’ approach to logic was that you do not know what the constituent propositions are about, or even whether each constituent proposition is true or false. All that you know is that any proposition must be either true or false. When the Stoics came to analyze the combining of two propositions by one of the connectives, they did so by looking at the pattern of truth and falsity. For example, in the case of “and”, the pattern is straightforward: if both p and q are true, then the proposition [p&q] will be true; if one or both of p and q are false, then [p&q] will be false.
Modern logicians generally display such a ‘pattern of truth’ in a tabular form, using a truth table, a nineteenth century device not available to the Stoics. The Stoics had to express the truth pattern for [p&q] in the following cumbersome fashion:
“If the first and if the second, then the first and the second. If not the first, then not the first and the second. If not the second, then not the first and the second.”
If you replace “the first” by the letter p, “the second” by the letter q, and abbreviate “and” by &, this rather perplexing looking sentence becomes:
“If p and if q, then [p&q]. If not p, then not [p&q]. If not q, then not [p&q].”
From this version, it is but a small step to the truth pattern for “and” as originally expressed above (or to the truth table for “and”).
The Stoics realized that there is ambiguity in the meaning of “or”: does it mean the inclusive-or or the exclusive version? They tended to prefer the exclusive variant; present-day logicians plump for the inclusive version.
The Stoics had great trouble trying to understand the nature of “implies”. As any present-day student of logic will know, when you try to define the meaning of the symbol -> in terms of truth values alone, the result is a logical connective that only partially captures the notion of implication.
The Stoics formulated five rules of inference. Expressed in modern-day algebraic notation (but with the symbol v denoting exlusive-or), they are:
From [p -> q] and p deduce q.
From [p -> q] and ~q deduce ~p.
From ~[p&q] and p deduce ~q.
From [pvq] and p deduce ~q.
From [pvq] and ~q deduce p.
The first of these rules is the modern-day logical inference rule of ‘modus ponens’. Here is how the Stoics themselves expressed this rule:
“If the first then the second, and if the first, then the second.”
Starting with their five inference rules, the Stoics were able to deduce a number of other patterns of reasoning. For example, they showed that the following deduction is valid:
From [p -> (p -> q)] and p deduce q.
Using the Stoics’ own terminology, this was expressed like this:
“If the first then if the first then the second, and if the first, then the second.”
Given algebraic notation and the modern technique of truth tables, much of Stoic logic reduces to some simple algebraic manipulations together with the filling-in of truth-values in a table. However, it took over two thousand years for Mankind to reach that stage. Not having access to such modern tools, the Stoics had a much harder time establishing their results. But establish them they did.
By singling out propositions as the building blocks for reasoning and identifying some of the abstract patterns involved in reasoning with propositions, including modus ponens, the Stoics’ contribution to logic was a major intellectual achievement. Together with Aristotelean logic, it paved the way for all subsequent work in logic, right up to the present day, and led to much of twentieth century logic and computer science.
Clearly, it’s high time the other Zeno was given proper credit for his role in the development of modern logic.
And in case you are wondering, yes, these are the Stoics from whom we get our present-day word “stoical”. The Stoics were such enthusiastic believers in the power of formal logic that if formal reasoning led them to adopt a particular course of action, they would pursue that course even if it involved hardship, pain, or suffering.
The above article is adapted from the book “Goodbye Descartes: The End of Logic and the Search for a New Cosmology of Mind”, by Keith Devlin, to be published by John Wiley and Sons in January 1997.
Devlin’s Angle is updated at the start of each month.
JULY 1996
Tversky’s Legacy Revisited
Amos Tversky died earlier this year. To mathematicians, the Stanford-based psychologist is best known for the research he did with his colleague Daniel Kahneman in the early 1970s, into the way people judge probabilities and estimate likely outcomes of events.
The following problem is typical of the scenarios considered by Tversky and Kahneman.
Imagine you are a member of a jury judging a hit-and-run driving case. A taxi hit a pedestrian one night and fled the scene. The entire case against the taxi company rests on the evidence of one witness, an elderly man who saw the accident from his window some distance away. He says that he saw the pedestrian struck by a blue taxi. In trying to establish her case, the lawyer for the injured pedestrian establishes the following facts:
1.There are only two taxi companies in town, ‘Blue Cabs’ and ‘Black Cabs’. On the night in question, 85% of all taxis on the road were black and 15% were blue.
2.The witness has undergone an extensive vision test under conditions similar to those on the night in question, and has demonstrated that he can successfully distinguish a blue taxi from a black taxi 80% of the time.
If you were on the jury, how would you decide? If you are at all typical, faced with eye-witness evidence from a witness who has demonstrated that he is right 4 times out of 5, you might be inclined to declare that the pedestrian was indeed hit by a blue taxi, and assign damages against the Blue Taxi Company. Indeed, if challenged, you might say that the odds in favor of the Blue Company being at fault were exactly 4 out of 5, those being the odds in favor of the witness being correct on any one occasion.
The facts are quite different. Based on the data supplied, the mathematical probability that the pedestrian was hit by a blue taxi is only 0.41, or 41%. Less than half. In other words, the pedestrian was MORE likely to have been hit by a black taxi than a blue one. The error in basing your decision on the accuracy figures for the witness is that this ignores the overwhelming probability, based on the figures, that any taxi in the town is likely to be black.
If the witness had been unable to identify the color of the taxi, but had only been able to state—with 100% accuracy, let us suppose—that the accident was caused by a taxi, then the probability that it had been a black taxi would have been 85%, the proportion of taxis in the town that are black. So before the witness testifies to the color, the chances are low—namely 15%—that the taxi in question was blue. This figure is generally referred to as the ‘prior probability’, the probability based purely on the way things are, not the particular evidence pertaining to the case in question.
When the witness then testifies as to the color, that evidence increases the odds from the 15% prior probability figure, but not all the way to the 80% figure of the witness’s tested accuracy. Rather the reliability figure for the witness’s evidence must be combined with the prior probability to give the real probability. The exact mathematical manner in which this combination is done is known as ‘Bayes’ law’, which gives the answer to be 41%.
(Okay experts, I’m being a bit loose in how I expressed things in the above paragraph. But MAA Online is open to all-comers, not just those with a course in probability theory under their belts. This is a column, not a lesson. Indeed, it’s a column about people’s intuitions, not about mathematical precision.)
The moral many writers try to draw from problems like the taxi-cab scenario is that human beings are innumerate, and as a result are not always able to make rational decisions. “Improve our math classes in the schools,” say these experts, “and we will all be better equipped to make sound judgments.”
Now, improving school mathematics instruction may or may not have a beneficial effect on society. But to my mind, the fact that the vast majority of people get the taxicab problem ‘wrong’ is not an argument in favor of teaching mathematics, and it certainly does not show that “innumerate” people make “poor” judgments. What that particular example shows, I would suggest, is that there can be a big difference between rational behavior and numerically or logically based behavior. The jurist who assigns blame to the Blue Cab Company might display a form of innumeracy, but he or she could still be acting rationally. Indeed, the jurist probably is acting rationally. Here’s why.
First, let’s be clear about what the use of Bayes’ law tells us in this case. Providing that the stated proportions of blue and black taxis, namely 15% blue and 85% black, is uniform throughout the town, or at least that these figures are reliable in the region where the accident occurred, then the 41% figure for the chance of the pedestrian being hit by a blue taxi is accurate. Given the right circumstances, Bayes’ law is totally correct. So, assuming that the various figures quoted are
reliable, the probability that the Blue Comany is guilty is indeed a mere 0.41, and the ‘chances are’ that they are not to blame. Clearly, there is more than enough ‘reasonable doubt’ here, and a rational jurist to whom this application of Bayes’ law is explained should act accordingly.
On what basis then do I claim that the jurist who, ignorant of the use of Bayes’ law, decides the Blue Company is at fault, can be said to be acting rationally? Well, over thousands of years of evolution, human beings have learned to make decisions that are beneficial—beneficial firstly to themselves and their nearest and dearest, then to others in the society. Only very rarely are they in full possession of enough information to make what an analyst would declare to be the ‘best’ decision. Typically, humans have to make decisions based on very limited information, quite often only the information provided by their own eyes, ears, and sense of smell.
During the course of our evolutionary history, we have become very good at making optimum decisions based on such evidence. Those that were not adept at identifying potential danger often did not survive long enough to pass on their genes. Moreover, seeing, hearing, and smelling have a conscious immediacy that gives us overwhelming faith in information we acquire through those senses, far more than information we read about or are told of.
Seeing is particularly strong in this regard. Thus, both on evolutionary grounds and our own conscious experience, we tend to put great significance on information acquired first hand through sight, and what is more it is entirely rational to do so. It is entirely consistent with our rational tendency to rely on information acquired by actually seeing something to likewise place great reliance on information that is directly reported to us by others who have acquired it by seeing. “I saw it with my own eyes,” amounts to a personal guarantee of truth when someone reports something he has seen.
By contrast, neither evolution nor our own experience has equipped us to have a ‘feel’ for highly abstract information based on numerical data about a large population we cannot possibly see. The information that 15% of the taxicabs in the town are blue and 85% are black and that their distribution through the town is uniform is mathematically precise but entirely abstract—we do not SEE it.
In our daily lives, though we are constantly faced with evaluating information and making decisions, in many cases decisions on which our lives quite literally depend (crossing the street, driving a car, etc.), we rarely do so on the basis of statistical data of the taxicab variety. It is then hardly surprising that most of us, when faced with the kind of problem facing the jurist in the taxicab case, tend to downplay evidence based on statistical data, and put great significance on eye-witness accounts.
In the taxicab case, it is undoubtedly wrong to reason that if a series of tests show that the eye-witness is right 4 out of 5 times, then the probability of what he says being true on the occasion in question is also 4 in 5. But it is not at all irrational to reason in this way, and those of us whom society considers ‘numerate’ should not, I suggest, sneer at those who “get the problem wrong.” In many ways, and certainly in human terms, the popular answer is the “right” one. As a jurist, you could only be accused of irrationality if, faced with a clear explanation of the application of Bayes’ law to the case, you refuse to change your original evaluation of the eye-witness’s evidence.
The above is adapted from the book Goodbye Descartes: The End of Logic and the Search for a New Cosmology of Mind, to be published by John Wiley and Sons in January 1997.
Devlin’s Angle is updated at the start of each month.
Keith Devlin (devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA. He is the Dean of Science at Saint Mary’s College of California, and the author of Mathematics: The Science of Patterns, published by W. H. Freeman in 1994.
AUGUST 1996
Of Men, Mathematics, and Myths
With the Unabomber now firmly implanted in the public psyche as “a mathematician”, the caricature of the mathematician as the misunderstood, antisocial loner has received a considerable boost. That much of mathematics these days is done by two or more mathematicians working in collaboration is overlooked when such caricatures are being bandied about. Also forgotten is that fact that practically all present-day mathematicians earn their daily crust by working alongside dozens of other academics
at a college or university. Even for those mathematicians who spend most of their time engaged in highly esoteric, pure research, interaction with colleagues is an essential ingredient. The myth of the mathematical loner—the misunderstood genius who cannot relate to his (this mythical creature is always male) colleagues—is surely belied by the eagerness with which the contemporary mathematician heads for the airport before the red ink on the last exam script is barely dry, jetting off to meet old acquaintances and catch up on the latest gossip. But so what?
The idea of the mathematical genius dropping out of society to contemplate abstract, Platonic worlds has a nice Hollywood ring to it, but it’s hardly the truth. For all the press coverage given to Andrew Wiles’ New Jersey attic, the British-born solver of Fermat’s Last Theorem has a wife, a teaching position at Princeton, and a drawer full of airline-ticket stubbs. In the case of the alleged Unabomber, the fact is that this societal misfit GAVE UP mathematics to pursue his life of isolation. But who cares?
Please don’t spoil a good story.
Still and all, it’s hard to condemn journalists for milking a popular conception when mathematicians themselves create and perpetuate romantic myths. Perhaps the most famous mathematical fairy story is the tale of Evariste Galois, the man who gave the world group theory.
Killed in a duel at the tender age of twenty, the final hours of this undoubted mathematical genius were described in the following breathless prose by the mathematician Eric Temple Bell in his book Men of Mathematics, first published in 1937:
“All night long he had spent the fleeting hours feverishly dashing off his scientific last will and testament, writing against time to glean a few of the great things in his teeming mind before the death he saw could overtake him. Time after time he broke off to scribble in the margin “I have not time; I have not time,” and passed on to the next frantically scrawled outline. What he wrote in those last desperate hours before the dawn will keep generations of mathematicians busy for hundreds of years.
He had found, once and for all, the true solution of a riddle which had tormented mathematicians for centuries: under what circumstances can an equation be solved?”
It’s a great story. The stuff of legends. Which one of us, when we were starting out on our mathematical careers, on hearing the story for the first time—or perhaps reading Bell’s words ourselves—did not find our emotions stirred? The more so if we took into account the tragic details of the short life that preceded the young Frenchman’s violent end. At least, the details as described to us by Bell, who tells us how this young genius suffered the frustrations of his work being shunned by a
mathematical establishment that was too stupid to see true genius when it was presented to them.
The facts, sadly to say, do not support this romantic picture—not that anything as mundane as hard fact will stop a good story once it gets going. The facts in this case have been around for a long time. They were thrust firmly in front of the mathematical community in 1982, when Tony Rothman published an article in the February issue of the American Mathematical Monthly entitled “Genius and Biographers: The Fictionalization of Evariste Galois”. An expanded version of the article appears
as Chapter 6 in Rothman’s 1989 book Science a la Mode, published by Princeton University Press, on which I am basing most of this column.
What then, is the true story of young Galois?
He was born on October 25, 1811, in Bourg-la-Reine, near Paris. His parents were well-educated. He started his formal schooling in October 1823, at first doing well in all subjects. In February 1827 he took his first real mathematics course, studying the work of Legendre on geometry and of Lagrange on algebra. The immediate passion with which he embraced mathematics soon led to problems with his other subjects, and the beginning of a paranoid antagonism toward educational authority that was to be characteristic of the rest of his academic life.
Galois’ paranoia was given a further boost when, in 1828, he took the examination for entry to the prestigious Ecole Polytechnique a year early and failed. That his failure was due to lack of adequate preparation, and not an absence of talent, is clear from the fact that only the next year, in April 1829, he published his first paper, on continued fractions. But that paper was little more than a diversion from his major work—the analysis of the solvability of equations that would lead to what we now call Galois Theory. On May 25 and June 1 of 1829, Galois sent his results to the Academy Francais for publication. Cauchy was appointed referree.
According to the Bell account, and the popular legend that Bell in large part inspired, Cauchy lost the papers. But as Rothman indicates, there is evidence in Cauchy’s own hand that he not only read Galois’ work, he was sufficiently impressed by it to offer to present it at a meeting of the Academy in January 1830. (Illness prevented Cauchy from attending the meeting on the day in question. He did not take the opportunity to present the work at a subsequent meeting, but it seems likely
that this was because he encouraged Galois to turn his notes into a submission for the Grand Prize in Mathematics instead, for which the submission deadline was March 1.)
In 1829, Galois once again tried to gain entrance to the Ecole Polytechnique, but again failed. This time, the problem seemed to be more one of a rapidly growing academic arrogance rather than lack of preparedness. He reportedly told the examiner that he would not answer his questions because they were trivial. He might well have been right, but the result was predictable. Galois ended up enrolling at the Ecole Normale.
In 1830, Galois sent in his submission for the Grand Prize. As secretary for mathematics and physics at the Academy, Fourier handled the submission. The referees were to be Lacroix, Poisson, Legendre, and Poinsot. That such a high caliber review team was involved puts paid to another aspect of the Galois legend, one started by the paranoid Galois himself, that Fourier “buried” his paper. It was however the case that when Fourier died in May 1830, no one could find Galois’ submission, and he did miss out on the prize.
Prize or no prize, however, during the same year, 1830, Galois carried out the research that would give immortality to his name. That year, he published three papers. Their titles were “An analysis of a memoir on the algebraic resolution of equations”, “Notes on the resolution of numerical equations”, and “On the theory of numbers”. In these three papers he developed what we now call Galois Theory.
In all three papers he used the word “group” to mean a “group of permutations”. So much for that famous last night of feverish writing, when the mathematical world came within a few scribbled pages of never having group theory!
Whether or not his feeling of rejection by the mathematical community spurred him on, at the same time he was developing his mathematical theory, Galois’ long standing interest in radical politics intensified. On May 10, 1831, he was arrested for making a seditious remark at a banquet the day before, but when his trial came on June 15 he was acquitted. A month later he was arrested again, this time for publically wearing the banned uniform of the Artillery Guard, an act of political defiance that can only have been done in order to provoke arrest and the six months of prison that followed.
And so to that tragic end. It seems clear that the deadly duel was over a woman. However, the popular accounts give us two alternatives. One version has the dispute a straightforward quarrel between two young men vying for the same female. The other has the whole affair politically motivated, with the woman a prostitute hired to maneuver Galois to his untimely death. Purveyors of both versions usually add mystery to the drama by claiming that the identity of the woman in question is not known. She was Stephanie-Felice Poterin du Motel, a woman from a good and respected family.
What does seem in doubt—oddly enough given the amount of evidence for most other aspects of Galois’ life—is who was the other duelist. Rothman thinks it was Galois’ revolutionary friend Duchatelet. However, in his memoirs, Galois’ contemporary Alexander Dumas gives the name of the duelist as Pescheux d’Herbinville. At last, we do seem to have a bit of mystery! But whoever the other man was, it seems likely that Galois entered into the final duel with some kind of (romantic?) death wish.
The night before the duel Galois did indeed do a lot of writing. In particular, in a long letter to his friend Chevalier, he outlined his mathematical work. However, though it was still not accepted by the Academy, it had pretty well all been published, and Cauchy for one had clearly recognized its importance. At one point, Galois remarked that a published proof was missing a key detail, adding (just that once) that “I have not the time [to give the details]”.
Ans so the world lost a great mathematical talent. Of that there is no doubt. At the same time, the seeds were set for the creation of a great legend. But of that legend, there is not just considerable doubt, in most respects there is concrete evidence to show that the story is just plain false.
But the truth, as I remarked once already, is hardly likely to stop a good story. No more than we can hope to stop journalists from describing the alleged Unabomber as “a mathematician”.
Devlin’s Angle is updated on the first of each month.
Keith Devlin (devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA. He is the Dean of Science at Saint Mary’s College of California, and the author of Mathematics: The Science of Patterns, published by W. H. Freeman in 1994.
SEPTEMBER 1996
Dear New Student
This is the month when, all across the nation, and indeed around the world, a new generation of students enters college or university for the first time. Among them, I hope, are large numbers of mathematics majors, or at least students with sufficient interest in mathematics to be regular readers of MAA Online. This month’s column is directed at those students. At least, that is my overt audience. One of the aims of a regular column such as this is to be provocative and controversial from time to time—to take an alternative look at something familiar and relevant (in this case relevant to MAA members). You may not agree with what I say below. If so, please e-mail me with your views, and I’ll try to pass on your views in a future column.
Dear New Student:
What awaits you in the coming years—your college years?
My college freshman year was 1965, in London, England. I came from Hull, a city of a third of a million people some two-hundred miles to the north. The year before I left for college, the local newspaper carried a story that Hull University had just received delivery of its first computer, the first in the entire city. (It was a large, room-filling beast with a gigantic 8 kilobytes of memory.)
With computers just on the horizon, there was no way, in my freshman year, that I or anyone else could have envisaged the kind of world I would be working in five years after graduation.
Today’s college freshman takes it for granted that he or she will make regular use of a personal computer, both as a student and then later in employment. PCs are so ubiquitous, it seems they have always been with us. But they are little more than fifteen years old.
As an entering college student, the likelihood is the job you will be doing ten years from now does not yet exist. You will be doing something that at present no one is doing, or hardly anyone.
After you graduate, the chances are you will have as many as five or six different careers in your lifetime. That’s not jobs. It’s different careers. Your individual jobs might last only three or four years. Then that job will disappear.
How can you possibly prepare for such a future? How can we, as college educators, help you to prepare for that future?
The first thing is not to think of a college education as training for a particular career. Think of it as helping to prepare you for the remainder of your life. For part of that life you will be employed, though no one can be quite sure in what. For other parts, you may not be “employed,” at least in the traditional sense of the word.
For many universities, this represents a major change. For the liberal arts colleges, this is how they have always viewed their educational mission. In any event, what I am saying does not apply to just a few college or university students; these days it applies to most of them; probably to you.
What about the major? How important is that choice? Well, despite the uncertainty of people’s career paths, some things are as true today as they always were. A science major or a business major will have a broader choice of careers than, say, an English major, and a statistically greater chance of getting a good job early in their career. But I would never advise a student to choose a major on that basis. That would have been poor advice for my generation, when steady careers were expected. So it certainly would be poor advice for today’s student.
My strongest advice would be to value the breadth of the education available to you. Choose your major according to what interests you most, and realize that it is only a part of your education. Don’t fall into the trap of regarding all your other courses as just “graduation requirements” to be gotten out of the way as quickly as possible. It is the entire spread of your courses that will be of value to you in your later life and careers.
If you downgrade your non majors courses, you will be effectively throwing away a significant part of your education. [As an aside, I should remark that, to the best of my knowledge, education is the only business where many of the customers complain if they are given more of what they have paid for.]
Let me get back to careers after graduation. There is one ability that you will need above all others in your various careers. If you think about it, it’s obvious what it is. The ability to adapt to new conditions and to learn new skills rapidly and effectively. If you ever find yourself using public transport (trains, subways, airplanes) at the same time as professional people, take a look round at what they are doing. Many of them will be reading instructional books or training manuals. If you were to visit them at them homes in the evening, you would find them doing the same thing. Many of the major cities in the United States now have huge convention centers that cater to the multitude of professional conferences that people attend to learn new skills and keep up to date with their rapidly changing work environment. A significant part of their job is learning. All the time.
The main key to being successful in the professional world of today or tomorrow is an ability to learn. Not to be taught, notice. To learn. At high school you had a teacher. When you are out at work and you need to acquire new information or learn a new skill, you will probably have to go it alone–at least if you want to get ahead. College is a half-way house. The professors are there to help and guide you. But as teachers, the main thing they are trying to “teach” you is how to learn. Your mathematics professor is not there to teach you mathematics. He or she is there to show you how to learn mathematics, and to help you in the process. That’s a big difference from high school. And the sooner you realize the distinction, the sooner you will start to get the most out of your college or university education.
Second piece of advice. Which course should you put the most effort into? Well, if you follow my logic about future life and careers, what will count for most in the long run is your ability to learn a new skill in an area that is quite unfamiliar to you, which you might not like (at least at first), and which you find hard and are sure you can never master. So my advice is to work hardest at those subjects you don’t like or think you can never do.
Though it pains me to say it as a mathematician, I realize that for many students, that subject will turn out to be mathematics. However, since this letter is being published on MAA Online, the chances are you are a future math major. But what I am saying applies just as well to you, when you find yourself faced with a language requirement or a performing arts class or a creative writing class. Colleges and universities don’t have all those graduation requirements in order to make you suffer. They are there to help you broaden your mind and your horizons, and to prepare you to live your life to the fullest.
One final piece of advice. (Sorry about all this advice. But one of my two children has already graduated and the other is a college sophomore, and I still have that overwhelming parental urge to give advice, regardless of whether it is asked for or not.) I often hear people say that college is not the real world; that the purpose of your college years is to prepare you for your subsequent life “in the real world.” That’s just nonsense. You won’t stop living for the next four years. Your time at college or university is not pre-life. It’s four years of your life. For four years, it will be your “real world.” So my final piece of advice is, enjoy your time as a student and live your new “real life” to the full.
-Keith Devlin
Devlin’s Angle is updated on the first of each month.
Keith Devlin (devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA. He is the Dean of Science at Saint Mary’s College of California, and the author of Mathematics: The Science of Patterns, published by W. H. Freeman in 1994.
OCTOBER 1996
Wanted: A New Mix
One of the features of a column—as opposed to a regular slot to publish an article—is that the columnist has the freedom to be self-indulgent from time to time, to push a particular pet line of thought. Here goes.
One of my own favorite themes for over a decade-and-a-half now is that, as a whole we (university) mathematicians should be more open to embracing, as a useful and bona fide part of the enterprise, collaborative investigations with experts in other disciplines, where we have to be willing to descend from our Platonic mountaintop and flounder around in the mud of our ignorance and confusion.
That is to say, we should be more open to collaborations that go beyond us peddling our wares on our own terms, or entering a collaboration with the intention of simply “doing the math parts”, or providing expertise in the application of established mathematical techniques. There is a place for such kind of collaboration, and I think that by and large the mathematical community has always been very good at that kind of thing. But such collaborations are essentially risk-free. The mathematician might have to become familiar with part of another discipline, but in the final analysis what the mathematician does is what he or she is familiar with: mathematics.
Indeed, not only are such collaborations risk-free, they often pay dividends in the form of funding. What is missing, I would claim—or at least, what is relatively rare—is for the mathematician to make a full-blown attempt to meet his or her collaborator(s) on the no-mans-land between their disciplines.
In such a collaboration, the mathematician brings his or her experience in mathematics into the project, but the work done is probably not going to constitute mathematics in any currently recognizable sense, and is unlikely to lead to publications in mathematical journals.
It seems to me that the world of information management is an area where significant benefit can come from such collaborations.
A modern company has many assets that have to be properly managed. The most obvious—at least from a traditional perspective—are physical plant, personnel, and the company’s financial assets. Management of each of those assets requires different kinds of expertise. The appropriate expertise may be provided by the company’s regular employees, or it may be outsourced, either in the form of subcontracting or by the use of expert consultants.
In today’s commercial environment, information is another asset that requires proper management. Of course, information has always been important to any organization. But it is only within the last fifty years or so that it has become a clearly identifiable asset that requires proper management. The reason for the growing importance of information within the organization is the growth of computer and communications technologies, and the increasing size and complexity of organizations that has in large part been facilitated by those technologies. It is both a cliche and a fact that information is the glue that holds together most of today’s organizations.
Actually, that last metaphor is often apt in a negative way. In many cases, information acts as the glue that causes things to stick fast when it should be the oil that keeps the wheels turning. A familiar scenario in the industrial world of the late twentieth century is for a company to introduce a new computer system to improve its information management, only to discover that, far from making things better and more efficient, the new system causes an array of problems that had never arisen with the old way of doing things. The shining new system provides vastly more information than was previously available, but it is somehow of the wrong kind, or presented in the wrong form, at the wrong time, or delivered to the wrong person, or there is simply too much of it for anyone to be able to use. What used to be a simple request for information to one person over the phone becomes a tortuous battle with a seemingly uncooperative computer system that can take hours or even days, eventually drawing in a whole team of people.
Why does this happen? The answer is that, for all that the newspapers tell us we are living in the Information Age, what we have is an information technology, or rather a collection of information technologies. We do not yet have the understanding or the skill to properly design or manage the information flow that our technologies make possible. In fact, it is often worse. In many cases, companies are not even aware that they need such skill. Faced with the persuasive marketing of ever-more powerful and glitzy computer systems, there is a great temptation to go for the ‘technological fix’. If the present information system is causing problems, get a bigger, better, faster system. This approach is like saying that the key to Los Angeles’ traffic problem is to build even more, and still bigger, roads.
The solution? Just as the company has experts to manage its other assets, so too it needs experts to manage its information assets. Alongside the lawyers who handle and advise on the company’s contracts and the accountants who handle and advise on the company’s financial assets, should be the ‘information scientists’ who handle and advise on the company’s information assets.
But there is one problem. There are, at present, no such ‘information scientists’. The world of information flow does not yet have the equivalent of a lawyer or an accountant. There is not even an established body of knowledge that can be used to train such people. To become a lawyer, you go to law school and follow a well-established educational path. To become an accountant, you learn about the various principles and theories of accounting and finance. But there is no established ‘information science’ (in the sense I am using this term). It is, I would suggest, only a matter of time before such a science develops. The real question, I think, is what will that science look like? My guess is it will not resemble any of today’s sciences.
It may be presumptious of me, as a mathematician, to say that I think mathematics will play a role in the development of this new ‘science of information’. On the other hand, I suspect it will not look very much like anything we see in today’s mathematics books. I can’t see it coming from the highly constrained cross-disciplinary research collaborations I alluded to at the start of this column. What is needed, it seems to me, are radical, new interdisciplinary efforts, where mathematicians leave behind practically all the mathematics they know, taking with them only their mathematical skill and a pair of mathematician’s eyes.
Given the constraints on our careers, I doubt that many of us in the profession already could take the necessary steps. But we can help to prepare some members of the next generation of mathematicians to do so. They’ll do it anyway, of course. New generations always march boldly onto the ground their predecessors thought inaccessible. But if we help them, they’ll do it faster. And we’ll all benefit from that. So too will mathematics—even if we change the borders of mathematics in the process.
FOOTNOTE
Some of the thoughts in this column are taken from the book Language at Work: Analyzing Communication Breakdown in the Workplace to Inform Systems Design, by Keith Devlin and Duska Rosenberg, just published by CSLI Publications (Stanford University) and distributed by Cambridge University Press.
Devlin’s Angle is updated at the start of each month.
NOVEMBER 1996
Spreading the word
In a splendid article in September’s Math Horizons, William Dunham celebrated the three-hundred year anniversary of the appearance of the world’s first calculus textbook. That’s right, the first calculus text hit the shelves in 1696. They have been growing steadily in size (if not mathematical content) ever since.
That first genre setter was Guillaume Francois Antoine de l’Hospital’s Analysis of the Infinitely Small. Written for the mathematical community, l’Hospital’s book contained no problem sets, no color-highlighted definitions, and no full-color photographs, diagrams, and illustrations. But as Dunham points out, it was a calculus textbook, designed to “spread the word” about the then new techniques of the differential calculus.
Invented (or, if you prefer, discovered) just a few years earlier by Isaac Newton and later Gottfried Leibniz, a great deal of the early development work in calculus had been done by the Bernoulli brothers, Jakob and Johann. (Among the early uses to which the calculus was put was Johann Bernoulli’s discovery of the catenary curve, the shape assumed by a chain suspended between two supports.) In fact, until the appearance of l’Hospital’s book, Newton, Leibniz, and the two Bernoullis were pretty well the only people on the face of the earth who knew much about calculus.
Born in 1661, l’Hospital was a French nobleman of fairly minor rank who developed a keen interest in mathematics at an early age. He met Johann Bernoulli in 1691, shortly after the latter had made his discovery of the catenary. Eager to learn about this marvellous new technique calculus, l’Hospital hired Bernoulli to teach him.
L’Hospital learned rapidly, and before long was able to assemble all he had discovered into a comprehensive expository text, much as Euclid had done with geometry some two thousand years earlier. As was the case with Euclid and Elements, it is not clear that l’Hospital’s text contained any significant discoveries he had made himself. Most if not all the “new” results and techniques were due to Leibniz and the Bernoullis, and the treatment was largely that of Johann Bernoulli. L’Hospital’s contribution was to sift, to organize, to explain, and to assemble into a cohesive whole.
In short, l’Hospital was an expositor. Until relatively recently, that was a derogatory word in mathematical circles. Never mind the empirically-observable fact that first-rate expositors of mathematics are as rare as first-rate (i.e., Fields Medal standard) mathematicians, for centuries the mathematical community regarded its primary task to be proving theorems and solving problems.
In his classic book A Short Account of the History of Mathematics, Cambridge (England) mathematics historian W. W. Rouse Ball wrote,
“Leaving for a moment the English mathematicians of the first half of the eighteenth century, we come next to a number of continental writers who barely escape mediocrity, and to whom it will be necessary to devote but few words. Their writing mark the steps by which analytical geometry and the differential calculus were perfected and made familiar to mathematicians.” [p.369]
When you stop and think about it, that’s an extraordinary passage. The contrast between “English mathematicians” and “continental writers” is particularly stark, and speaks volumes for the value systems of the mathematics community at the time (value systems still dominant when I learned my mathematics in the England of the 1960s, and not totally dead to this day). Rouse Ball’s statement is the more remarkable when you read on. The first of those “continental writers who barely escape mediocrity” that Rouse Ball mentions is l’Hospital. Of l’Hospital’s book, Rouse Ball says,
“… the credit of putting together the first treatise which explained the principles and use of the method is due to l’Hospital. … This work had a wide circulation; it brought the differential notation into general use in France, and helped make it known in Europe.” [pp.371-372]
For that service to mankind, I would hope that contemporary historians of mathematics give rather more credit to l’Hospital than did Rouse Ball. It is, of course, an interesting question whether, in these post-Boyer [Scholarship Reconsidered] times, l’Hospital’s contribution would be enough to earn tenure at a present day American university.
Personally, I would think that, as the author of the first calculus textbook, a work that was largely responsible for the initial dissemination of the techniques of calculus, l’Hospital was a fairly significant player on the mathematical scene. On the other hand, as far as Rouse Ball is concerned, l’Hospital comes of little worse than Descartes, of whom the eminent Cambridge historian wrote:
“As to his philosophical theories, it will be sufficient to say that he discussed the same problems which have been debated for the last two thousand years, and probably will be debated with equal zeal two thousand years hence.” [pp.271-272]
Five sentences later, Rouse Ball ends the brief paragraph with:
“Whether this is a correct historical generalization of the views which have been successively prevalent I do not care to discuss here, but the statement as to the scope of modern philosophy marks the limitations of Descartes’s writings.” [p.272]
But now I have strayed from expository writing onto the mathematical community’s traditional view of philosophy (mind you, those philosophers do write a lot, you know). So I’d better stop.
References
W. W. Rouse Ball, A Short Account of the History of Mathematics, fourth edition (1908), Dover, New York, 1960.
Ernset L. Boyer, Scholarship Reconsidered; Priorities of the Professoriate, Carnegie Foundation, Princeton, 1990.
Devlin’s Angle is updated on the first of each month.
DECEMBER 1996
Moment of truth
This month’s column first appeared as an editorial in my regular “Computers and Mathematics” column in the AMS Notices, Volume 39, Number 9, November 1992.
The scene is a familiar one. The brightest student in the mathematics class (who, naturally enough, intends to double-major in philosophy and business administration) has cornered the professor at the end of the third week.
Student: I’m not sure exactly what a proof is?
Professor: It’s exactly what I said in class. A proof is a sound, logical argument that establishes the truth of the statement in question.
S: But how do you know an argument is sound and logical? What do those words mean?
P: Good heavens, surely you can recognize a logical argument when you see one, can’t you? Weren’t you convinced by any of the examples I did in class?
S: Well, I was convinced that your examples of false proofs were indeed false. In each case, once you pointed out the logical error, I could see why that particular argument was not a proof. But I’m not so sure about the examples you gave that you said were valid proofs. I admit I could’t see any logical errors, and the arguments did seem pretty convincing. But how can we know for sure that the argument was sound, and that there was not some hidden error that we all missed?
P: Well, you know, those proofs have all been around for hundreds of years, and lots of very clever mathematicians have examined them, and no one has found any errors. Surely, we can’t all be wrong, can we?
S: Probably not. But doesn’t that mean that the notion of a valid proof is a socially defined one; that what makes a proof valid is that the majority of mathematicians agree it is valid?
P: Good Lord, no. To be valid, a proof has to follow the laws of logic. You make a series of statements, each of which follows from the previous ones by the laws of logic.
S: What rules of logic? You never told us what they were. It seemed to me that for each of your examples, you presented a series of statements in which each one sure seemed reasonable, given the previous ones. But where were the rules of logic? How is what you did different from a clever political argument?
P: Well, of course, to make it easy for you to follow the proof, I did not write down every step. But it could be done. Logicians sorted all this out early this century. They developed a formal language for expressing proofs, ‘predicate calculus’ they call it, figured out the basic axioms of logic, specified the rules of deduction—I think one of them is called ‘modus ponens’ or something like that—and then you get a total rigorous notion of a proof. In fact, it is so rigorous that in some areas of mathematics—not mine, I hasten to add—you can program a computer to do it for you, to generate proofs or to check them.
S: So what you are saying is that any one of the ordinary, everyday proofsmathematicians deal with in their work can be re-written in a formal way that fits into this framework the logicians have axiomatized.
P: Right.
S: So why didn’t you do that in class? Why not tell us what the axioms and rules of logic are and do it that way?
P: Good grief, that would be hopeless. If I tried to make even the simplest proof that precise, it would be incredibly long, and quite unreadable. You would never be able to follow it. It would be far less convincing than the version I gave.
S: You mean that no one ever does write a proof down in formal logic?
P: Yes, now you’re getting it. In principle it could be done. Any valid proof could be written out in full in formal logic, and then it would satisfy the logician’s definition of a proof.
S: Only, it would be so long that no one could actually read it and check that it was right.
P: Absolutely.
S: So let me see if I have got this right. What you are saying is that,strictly speaking, a proof is valid if it is written out in predicate calculus and has the right structure according to the rules of logic. But such proofs are far too long and complicated for anyone to read, so no one ever does it. Whathappens in practice is that a proof is declared to be valid if it could, in principle, be written out in this formal way.
P: Yes, that’s about it.
S: But how do you know that your proof, the one you write on the board, could be written out as a formal proof if you never actually do it?
P: Well, you just look at it. The irrationality of root-two for example. The argument I gave is quite obviously valid, so you know it could be translated into a valid predicate calculus proof.
S: But how do you know, if no one has ever done it?
P: (Impatiently). Well, you know, those logicians were clever folk, and we know from their work that any valid proof could be written out in that way, at least in principle.
S: But how do you know your proof is valid in the first place?
P: Look, I really don’t have time for this. You’re just going round in circles.You clearly haven’t really understood what is involved in proving something. Maybe mathematics is not your subject. Have you thought about taking a philosophy course instead?
End of scenario.
But not, of course, the end of the debate. And it is a debate that has achieved increasing significance over the past decade or so, with the arrival on the scene of computer proofs, the development and use of automatic theorem provers, and the attempts to provide verification of computer systems. Just what do we mean by a `proof’?
It seems that mathematicians—and to avoid generating too many letters of complaint, let me hasten to rephrase that as “we mathematicians”—are somewhat schizophrenic when it comes to answering this question. When pressured by the persistent student, we fall back on the logician’s definition and the “translatable in principle” defense. But in practice, we work happily with what is quite clearly a socially determined notion of proof.
Though we might harbor doubts about particularly long and novel proofs, we generally feel confident in our ability to tell a sound argument from an invalid one. Moreover, we tend to feel that it really is not an issue of judgement, and that for all their surface brevity, the proofs we construct and publish are, in an absolute sense, genuine proofs. I certainly feel that way. But I also know from experience that the certainty (we) mathematicians feel about this issue is not easily explained to others outside the field, to whom, I suspect, it really does seem as though we are simply playing a game by rules we determine, decided by some sort of secret poll.
Devlin’s Angle is updated on the first of each month.
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