Two contrasting experiences stick in mind from my first year at university.

First, I spent a lot of time in lectures that I did not understand. I don’t mean lectures in which I got the general gist but didn’t quite follow the technical details. I mean lectures in which I understood not one thing from the beginning to the end. I still went to all the lectures and wrote everything down – I was a dutiful sort of student – but this was hardly the ideal learning experience.

Second, at the end of the year, I was awarded first class marks. The best thing about this was that later that evening, a friend came up to me in the bar and said, “Hey Lara, I hear you got a first!” and I was rapidly surrounded by other friends offering enthusiastic congratulations. This was a revelation. I had attended the kind of school at which students who did well were derided rather than congratulated. I was delighted to find myself in a place where success was celebrated.

Looking back, I think that the interesting thing about these two experiences is the relationship between the two. How could I have done so well when I understood so little of so many lectures?

I don’t think that there was a problem with me. I didn’t come out at the very top, but obviously I had the ability and dedication to get to grips with the mathematics. Nor do I think that there was a problem with the lecturers. Like the vast majority of the mathematicians I have met since, my lecturers cared about their courses and put considerable effort into giving a logically coherent presentation. Not all were natural entertainers, but there was nothing fundamentally wrong with their teaching.

I now think that the problems were more subtle, and related to two issues in particular.

First, there was a communication gap: the lecturers and I did not understand mathematics in the same way. Mathematicians understand mathematics as a network of axioms, definitions, examples, algorithms, theorems, proofs, and applications.  They present and explain these, hoping that students will appreciate the logic of the ideas and will think about the ways in which they can be combined. I didn’t really know how to learn effectively from lectures on abstract material, and research indicates that I was pretty typical in this respect.

Students arrive at university with a set of expectations about what it means to ‘do mathematics’ – about what kind of information teachers will provide and about what students are supposed to do with it. Some of these expectations work well at school but not at university. Many students need to learn, for instance, to treat definitions as stipulative rather than descriptive, to generate and check their own examples, to interpret logical language in a strict, mathematical way rather than a more flexible, context-influenced way, and to infer logical relationships within and across mathematical proofs. These things are expected, but often they are not explicitly taught.

My second problem was that I didn’t have very good study skills. I wasn’t terrible – I wasn’t lazy, or arrogant, or easily distracted, or unwilling to put in the hours. But I wasn’t very effective in deciding how to spend my study time. In fact, I don’t remember making many conscious decisions about it at all. I would try a question, find it difficult, stare out of the window, become worried, attempt to study some section of my lecture notes instead, fail at that too, and end up discouraged. Again, many students are like this. I have met a few who probably should have postponed university until they were ready to exercise some self-discipline, but most do want to learn.

What they lack is a set of strategies for managing their learning – for deciding how to distribute their time when no-one is checking what they’ve done from one class to the next, and for maintaining momentum when things get difficult. Many could improve their effectiveness by doing simple things like systematically prioritizing study tasks, and developing a routine in which they study particular subjects in particular gaps between lectures.  Again, the responsibility for learning these skills lies primarily with the student.

Personally, I never got to a point where I understood every lecture. But I learned how to make sense of abstract material, I developed strategies for studying effectively, and I maintained my first class marks. What I would now say to current students is this: take charge. Find out what lecturers and tutors are expecting, and take opportunities to learn about good study habits. Students who do that should find, like I did, that undergraduate mathematics is challenging, but a pleasure to learn.

Lara Alcock is a Senior Lecturer in the Mathematics Education Centre at Loughborough University. She has taught both mathematics and mathematics education to undergraduates and postgraduates in the UK and the US. She conducts research on the ways in which undergraduates and mathematicians learn and think about mathematics, and she was recently awarded the Selden Prize for Research in Undergraduate Mathematics Education. She is the author of How to Study for a Mathematics Degree (2012, UK) and How to Study as a Mathematics Major (2013, US).

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In the heyday of the British Empire, Britain’s second most-widely-read book, after the Bible, was: (a) Richard III (b) Robinson Crusoe (c) The Elements (d) Beowulf ? Why do I ask?

“Since late medieval or early modern time,” Michael Walzer writes in Exodus and Revolution, “there has existed in the West a characteristic way of thinking about political change, a pattern that we commonly impose upon events, a story that we repeat to one another. The story has roughly this form: oppression, liberation, social contract, political struggle, new society…. Because of the centrality of the Bible in Western thought and the endless repetition of the story, the pattern has been etched deeply into our political culture. It isn’t only the case that events fall, almost naturally, into an Exodus shape; we work actively to give them that shape.”

The second-most-widely-read book plays that role in Western thought too: (c) The Elements by Euclid. Since late medieval or early modern time, there has existed in the West a characteristic way of organizing knowledge, a pattern that we commonly impose upon observations, concepts, and ideas, a pattern we teach our children. Because of the centrality of Euclid in Western education and the endless repetition of his axioms, definitions, theorems and proofs, the pattern has been etched deeply into our intellectual culture. It isn’t only the case that knowledge falls, almost naturally, into a Euclidean shape; we work actively to give it that shape.

Euclid was the geometry of the medieval university and the bedrock of European education for centuries. It wasn’t just about the triangles; Euclid sharpened your mind, trained your logic. His clever proofs were the very model of argument. To master Euclid was to master the world, the world around you and beyond. “Nature and Nature’s laws lay hid in night; God said, Let Newton be! and all was light.” And what did Newton’s lamp look like? See for yourself in the Principia Mathematica. “All human knowledge begins with intuitions,” said Kant, “proceeds from there to concepts, and ends with ideas.” Where do you think he got that? Euclideana even permeates our politics, but for this blog I’ll stick to science.

Non-Euclidean geometries put an end to that? No, they didn’t. Non-Euclidean geometries substituted one axiom for another, but they kept Euclid’s vision of organized knowledge, his faith in deductive reasoning. Non-Euclidean geometry is as Euclidean as Euclid’s! So is the new, improved axiom set David Hilbert proposed for geometry in the 19th century. (It turned out that Euclid’s wasn’t perfect.) So is the quixotic Russell-Whitehead program, in the early 20th century, to reduce mathematics to logic. Modern mathematics is consciously Euclidean to the core. In 1900, in a still-influential address, David Hilbert proposed rewriting Newton for modern physics along this vision of organized knowledge.

Born in 1894, Dorothy Wrinch grew up in a London suburb. She aced the mathematics program at Cambridge University and then studied logic with Bertrand Russell. The naturalist D’Arcy Thompson was another mentor and friend; his Growth and Form was her bible. Tugged by philosophy, mathematics, and biology for a decade, she cast her lot with biology, determined to unravel it through the powerful lens of logic. The model of protein architecture she came up with catalyzed protein chemists despite or because of its weaknesses. Why?

Figure 1. Dorothy Wrinch’s model of scientific development: First stage: brute facts. Second: sorting facts into classes. Third: theories linking the classes and explaining the facts. Fourth, finding the relationships among the theories and deducing their logical consequences. Fifth, Euclid-style axioms.

With this map to guide her, she found what she was looking for. “A number of new sciences have passed from the embryonic stage,” she wrote in 1934. “Discarding description as their ultimate purpose, they are now ready to take their places in the world state of science. The thesis which I wish now to develop is but a logical consequence of the thorough-going application of this principle.” Her protein model was one such consequence.

Biology ripe for logic? Some natives were not amused. (Or they were.) “Her idea of science is completely different from theirs,” Linus Pauling put it. You betcha!

Euclid fell from his curricular throne and the British Empire collapsed at about the same time. Quantum mechanics scotched Hilbert’s program and Gödel scotched Russell’s. Biology has resisted Euclid too. Though the structures of thousands of proteins are now known in exact detail, their inner logic remains where Dorothy left it, the brass ring on the Nobel carousel.

Marjorie Senechal is the Louise Wolff Kahn Professor Emerita in Mathematics and History of Science and Technology, Smith College, author of I Died for Beauty: Dorothy Wrinch and the Cultures of Science, and Editor-in-Chief of The Mathematical Intelligencer. At the Join Mathematics Meeting, AMS-MAA Special Session on the History of Mathematics, II, Room 9, Upper Level, San Diego Convention Center, she is speaking on 5:00 p.m. on Saturday, 12 January, on Biogeometry, 1941.

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In the last few years algorithmic thinking has become somewhat of a buzzword among computer science educators, and with some justice: ubiquity of computers in today’s world does make algorithmic thinking a very important skill for almost any student. Although at the present time there are few colleges and universities that require non-computer science majors to take a course exposing them to important issues and methods of algorithmic problem solving, one should expect the number of such schools to grow significantly in the near future.

Algorithmic puzzles, i.e., puzzles that involve clearly defined procedures for solving problems, provide an ideal vehicle to introduce students to major ideas and methods of algorithmic problem solving:

• Algorithmic puzzles force students to think about algorithms on a more abstract level, divorced from programming and computer language minutiae. In fact, puzzles can be used to illustrate major strategies of the design and analysis of algorithms without any computer programming — an important point, especially for courses targeting non-CS majors.
• Solving puzzles helps in developing creativity and problem-solving skills — the qualities any student should strive to acquire.
• Puzzles are fun, and students are usually willing to put more effort into solving them than in doing routine exercises.
• Puzzles provide attractive topics for student research because many of them don’t require an extensive mathematical or computing background.

It’s important to stress that algorithmic puzzles is a serious topic. A few algorithmic puzzles such as Fibonacci’s Rabbits and Königsberg’s Bridges played an important role in history of mathematics. Such well-known and intriguing problems as the Traveling Salesman and the Knapsack Problem, which clearly have a puzzle flavor, lie at the heart of the so-called P ≠ NP conjecture, the most important open question in modern computer science and mathematics.

So reader, I would like to challenge you to an algorithmic puzzle, #136, “Catching a Spy”:

In a computer game, a spy is located on a one-dimensional line. At time 0, the spy is at location a. With each time interval, the spy moves b units to the right if b≥0 and |b| units to the left if b<0. a and b are fixed integers, but they are unknown to you. Your goal is to identify the spy’s location by asking at each time interval (starting at time 0) whether the spy is currently at some location of your choosing. For example, you can ask whether the spy is currently at location 19, to which you will receive a truthful yes/no answer.  If the answer is “yes,” you reach your goal; if the answer is “no,” you can ask the next time whether the spy is at the same or another location of your choice. Devise an algorithm that will find the spy after a finite number questions.

Leave the answer in the comments below.

Anany Levitin is a professor of Computing Sciences at Villanova University. He is the co-author of Algorithmic Puzzles with Maria Levitin. He is the author of Introduction to the Design and Analysis of Algorithms, Third edition, a popular textbook on design and analysis of algorithms, which has been translated into Chinese, Greek, Korean, and Russian. He has also published papers on mathematical optimization theory, software engineering, data management, algorithm design, and computer science education.

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Image credit: Leonardo da Pisa, Liber abbaci, Ms. Biblioteca Nazionale di Firenze, Codice magliabechiano cs cI, 2626, fol. 124r Source: Heinz Lüneburg, Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, 2. überarb. und erw. Ausg., Mannheim et al.: BI Wissenschaftsverlag, 1993. Public domain via Wikimedia Commons.

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Writing in 1866, the British mathematician John Venn wrote, in reference to the branch of mathematics known as probability theory, “To many persons the mention of Probability suggests little else than the notion of a set of rules, very ingenious and profound rules no doubt, with which mathematicians amuse themselves by setting and solving puzzles.” I suspect many of my students would extend Venn’s quip to the entirety of mathematics. Often they seem to believe, upon entering my classroom for the first time, that a tacit agreement exists between us. They will dutifully memorize whatever rules I give them and apply them with machine-like accuracy at test-time, but to expect anything beyond that is considered a serious breach of etiquette.

I held such views myself, once upon a time. That is why my first visit to the annual Joint Mathematics Meetings, as an undergraduate student in the early nineties, was such an eye-opening experience. This is the largest mathematics conference of the year, held every January in a different city. Almost two decades later, I am still consistently amazed by the sheer variety of things that mathematicians study. Browsing through the program for this year’s edition, which is being held in San Diego, I notice that there are sessions on complex dynamics and celestial mechanics. Continued fractions get their own session, as do coverings of the integers, and frontiers in geomathematics. Financial mathematics gets a session. So does graph theory, and also the history of mathematics. If you prefer, you can go in for the real jawbreakers. They have titles like, “Advances in General Optimization and Global Optimality Conditions for Multiobjective Fractional Programming Based on Generalized Invexity.” For me, reading the program is like listening to opera. I may not understand all the words, but it sure sounds good!

This conference is called the Joint Mathematics Meetings, because it is held jointly between the two major mathematics organizations in the United States: The American Mathematical Society (AMS) and the Mathematical Association of America (MAA). The AMS generally concerns itself with the profession of mathematics and publishes several highly prestigious research journals. The MAA, by contrast, generally focuses on the educational aspects of mathematics. The sessions I listed above are directed towards researchers and are organized by the AMS. MAA sessions tend to have gentler titles. This year they are hosting a session on the beauty and power of number theory; another one on writing, talking, and sharing mathematics; still another on mathematics in industry; and, my personal favorite, a session called, “Where Have All the Zeros Gone?”

The sessions, however, are only the tip of the iceberg. There are also keynote talks featuring the alphas of our profession. In my experience, the main purpose of these talks is to remind you that, your PhD notwithstanding, there are mathematicians out there who are way smarter than you are. There is also the employment center, populated by eager job-seekers who stand out clearly from the other conference attendees, because they are well-dressed. There is also the exhibition center, in which every mathematical publisher on the planet shows off its latest books. For an impulse buyer like me, this is a dangerous place.

Which brings me back to the John Venn quote with which I started and the question at the top of this essay. Yes, I suppose we do spend a lot of time setting and solving puzzles. We dutifully apply the rules of proper inference to the abstract objects that have caught our fancy, thereby producing publishable theorems. That, however, is really a very small part of what mathematicians do.

You see, more than anything else, to be a mathematician is to be part of a community. Whatever else it is, mathematics is a social activity undertaken by human beings to further human goals and purposes. The main point of the conference is not to transact mathematical business, though that is certainly important. Rather, the point is to socialize, to renew old friendships, and to engage in casual conversations. The point is to remind you that mathematics is not about ivory tower theorizing, but about being part of a community that is united by its love for, and its belief in the importance of, mathematics. This applies whether your focus is on pure mathematics or applied mathematics. It does not matter whether you prefer teaching, research or community outreach. It includes elementary school teachers showing grade-schoolers the mechanics of basic arithmetic, high school teachers giving students their first taste of higher-level math, and graduate school professors at the frontiers of modern research. It also includes the students who will form the next generation not just of professional mathematicians, but of mathematically informed lay people as well.

All are part of the same community, and all are essential to the continued health of our discipline.

Jason Rosenhouse is Associate Professor of Mathematics at James Madison University. His most recent book is Among The Creationists: Dispatches from the Anti-Evolutionist Front Lines. He is also the author of Taking Sudoku Seriously: The Math Behind the World’s Most Popular Pencil Puzzle with Laura Taalman and The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser. Read Jason Rosenhouse’s previous blog articles.

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This year, 2012, marks the 325th anniversary of the first publication of the legendary Principia (Mathematical Principles of Natural Philosophy), the 500-page book in which Sir Isaac Newton presented the world with his theory of gravity. It was the first comprehensive scientific theory in history, and it’s withstood the test of time over the past three centuries.

Unfortunately, this superb legacy is often overshadowed, not just by Einstein’s achievement but also by Newton’s own secret obsession with Biblical prophecies and alchemy. Given these preoccupations, it’s reasonable to wonder if he was quite the modern scientific guru his legend suggests, but personally I’m all for celebrating him as one of the greatest geniuses ever. Although his private obsessions were excessive even for the seventeenth century, he was well aware that in eschewing metaphysical, alchemical, and mystical speculation in his Principia, he was creating a new way of thinking about the fundamental principles underlying the natural world. To paraphrase Newton himself, he changed the emphasis from metaphysics and mechanism to experiment and mathematical analogy. His method has proved astonishingly fruitful, but initially it was quite controversial.

He had developed his theory of gravity to explain the cause of the mysterious motion of the planets through the sky: in a nutshell, he derived a formula for the force needed to keep a planet moving in its observed elliptical orbit, and he connected this force with everyday gravity through the experimentally derived mathematics of falling motion. Ironically (in hindsight), some of his greatest peers, like Leibniz and Huygens, dismissed the theory of gravity as “mystical” because it was “too mathematical.” As far as they were concerned, the law of gravity may have been brilliant, but it didn’t explain how an invisible gravitational force could reach all the way from the sun to the earth without any apparent material mechanism. Consequently, they favoured the mainstream Cartesian “theory”, which held that the universe was filled with an invisible substance called ether, whose material nature was completely unknown, but which somehow formed into great swirling whirlpools that physically dragged the planets in their orbits.

The only evidence for this vortex “theory” was the physical fact of planetary motion, but this fact alone could lead to any number of causal hypotheses. By contrast, Newton explained the mystery of planetary motion in terms of a known physical phenomenon, gravity; he didn’t need to postulate the existence of fanciful ethereal whirlpools. As for the question of how gravity itself worked, Newton recognized this was beyond his scope — a challenge for posterity — but he knew that for the task at hand (explaining why the planets move) “it is enough that gravity really exists and acts according to the laws that we have set forth and is sufficient to explain all the motions of the heavenly bodies…”

What’s more, he found a way of testing his theory by using his formula for gravitational force to make quantitative predictions. For instance, he realized that comets were not random, unpredictable phenomena (which the superstitious had feared as fiery warnings from God), but small celestial bodies following well-defined orbits like the planets. His friend Halley famously used the theory of gravity to predict the date of return of the comet now named after him. As it turned out, Halley’s prediction was fairly good, although Clairaut — working half a century later but just before the predicted return of Halley’s comet — used more sophisticated mathematics to apply Newton’s laws to make an even more accurate prediction.

Clairaut’s calculations illustrate the fact that despite the phenomenal depth and breadth of Principia, it took a further century of effort by scores of mathematicians and physicists to build on Newton’s work and to create modern “Newtonian” physics in the form we know it today. But Newton had created the blueprint for this science, and its novelty can be seen from the fact that some of his most capable peers missed the point. After all, he had begun the radical process of transforming “natural philosophy” into theoretical physics — a transformation from traditional qualitative philosophical speculation about possible causes of physical phenomena, to a quantitative study of experimentally observed physical effects. (From this experimental study, mathematical propositions are deduced and then made general by induction, as he explained in Principia.)

Even the secular nature of Newton’s work was controversial (and under apparent pressure from critics, he did add a brief mention of God in an appendix to later editions of Principia). Although Leibniz was a brilliant philosopher (and he was also the co-inventor, with Newton, of calculus), one of his stated reasons for believing in the ether rather than the Newtonian vacuum was that God would show his omnipotence by creating something, like the ether, rather than leaving vast amounts of nothing. (At the quantum level, perhaps his conclusion, if not his reasoning, was right.) He also invoked God to reject Newton’s inspired (and correct) argument that gravitational interactions between the various planets themselves would eventually cause noticeable distortions in their orbits around the sun; Leibniz claimed God would have had the foresight to give the planets perfect, unchanging perpetual motion. But he was on much firmer ground when he questioned Newton’s (reluctant) assumption of absolute rather than relative motion, although it would take Einstein to come up with a relativistic theory of gravity.

Einstein’s theory is even more accurate than Newton’s, especially on a cosmic scale, but within its own terms — that is, describing the workings of our solar system (including, nowadays, the motion of our own satellites) — Newton’s law of gravity is accurate to within one part in ten million. As for his method of making scientific theories, it was so profound that it underlies all the theoretical physics that has followed over the past three centuries. It’s amazing: one of the most religious, most mystical men of his age put his personal beliefs aside and created the quintessential blueprint for our modern way of doing science in the most objective, detached way possible. Einstein agreed; he wrote a moving tribute in the London Times in 1919, shortly after astronomers had provided the first experimental confirmation of his theory of general relativity:

“Let no-one suppose, however, that the mighty work of Newton can really be superseded by [relativity] or any other theory. His great and lucid ideas will retain their unique significance for all time as the foundation of our modern conceptual structure in the sphere of [theoretical physics].”

Robyn Arianrhod is an Honorary Research Associate in the School of Mathematical Sciences at Monash University. She is the author of Seduced by Logic: Émilie Du Châtelet, Mary Somerville and the Newtonian Revolution and Einstein’s Heroes. Read her previous blog posts.

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29 November 2012 is the 140th anniversary of the death of mathematician Mary Somerville, the nineteenth century’s “Queen of Science”. Several years after her death, Oxford University’s Somerville College was named in her honor — a poignant tribute because Mary Somerville had been completely self-taught. In 1868, when she was 87, she had signed J. S. Mill’s (unsuccessful) petition for female suffrage, but I think she’d be astonished that we’re still debating “the woman question” in science. Physics, in particular — a subject she loved, especially mathematical physics — is still a very male-dominated discipline, and men as well as women are concerned about it.

Of course, science today is far more complex than it was in Somerville’s time, and for the past forty years feminist critics have been wondering if it’s the kind of science that women actually want; physics, in particular, has improved the lives of millions of people over the past 300 years, but it’s also created technologies and weapons that have caused massive human, social and environmental destruction. So I’d like to revisit an old debate: are science’s obstacles for women simply a matter of managing its applications in a more “female-friendly” way, or is there something about its exclusively male origins that has made science itself sexist?

To manage science in a more female-friendly way, it would be interesting to know if there’s any substance behind gender stereotypes such as that women prefer to solve immediate human problems, and are less interested than men in detached, increasingly expensive fundamental research, and in military and technological applications. Either way, though, it’s self-evident that women should have more say in how science is applied and funded, which means it’s important to have more women in decision-making positions — something we’re still far from achieving.

But could the scientific paradigm itself be alienating to women? Mary Somerville didn’t think so, but it’s often argued (most recently by some eco-feminist and post-colonial critics) that the seventeenth-century Scientific Revolution, which formed the template for modern science, was constructed by European men, and that consequently, the scientific method reflects a white, male way of thinking that inherently preferences white men’s interests and abilities over those of women and non-Westerners. It’s a problematic argument, but justification for it has included an important critique of reductionism — namely, that Western male experimental scientists have traditionally studied physical systems, plants, and even human bodies by dissecting them, studying their components separately and losing sight of the whole system or organism.

The limits of the reductionist philosophy were famously highlighted in biologist Rachel Carson’s book, Silent Spring, which showed that the post-War boom in chemical pest control didn’t take account of the whole food chain, of which insects are merely a part. Other dramatic illustrations are climate change, and medical disasters like the thalidomide tragedy: clearly, it’s no longer enough to focus selectively on specific problems such as the action of a drug on a particular symptom, or the local effectiveness of specific technologies; instead, scientists must consider the effect of a drug or medical procedure on the whole person, whilst new technological inventions shouldn’t be separated from their wider social and environmental ramifications.

In its proper place, however, reductionism in basic scientific research is important. (The recent infamous comment by American Republican Senate nominee Todd Akin — that women can “shut down” their bodies during a “legitimate rape”, in order not to become pregnant — illustrates the need for a basic understanding of how the various parts of the human body work.) I’m not sure if this kind of reductionism is a particularly male or particularly Western way of thinking, but either way there’s much more to the scientific method than this; it’s about developing testable hypotheses from observations (reductionist or holistic), and then testing those hypotheses in as objective a way as possible. The key thing in observing the world is curiosity, and this is a human trait, discernible in all children, regardless of race or gender. Of course, girls have traditionally faced more cultural restraints than boys, so perhaps we still need to encourage girls to be actively curious about the world around them. (For instance, it’s often suggested that women prefer biology to physics because they want to help people — and yet, many of the recent successes in medical and biological science would have been impossible without the technology provided by fundamental, curiosity-driven physics.)

Like Mary Somerville, I think the scientific method has universal appeal, but I also think feminist and other critics are right to question its patriarchal and capitalist origins. Although science at its best is value-free, it’s part of the broader community, whose values are absorbed by individual scientists. So much so that Yale researchers Moss-Racusin et al recently uncovered evidence that many scientists themselves, male and female, have an unconscious sexist bias. In their widely reported study, participants judged the same job application (for a lab manager position) to be less competent if it had a (randomly assigned) female name than if it had a male name.

In Mary Somerville’s day, such bias was overt, and it had the authority of science itself: women’s smaller brain size was considered sufficient to “prove” female intellectual inferiority. It was bad science, and it shows how patriarchal perceptions can skew the interpretation not just of women’s competence, but also of scientific data itself. (Without proper vigilance, this kind of subjectivity can slip through the safeguards of the scientific method because of other prejudices, too, such as racism, or even the agendas of funding bodies.) Of course, acknowledging the existence of patriarchal values in society isn’t about hating men or assuming men hate women. Mary Somerville met with “the utmost kindness” from individual scientific men, but that didn’t stop many of them from seeing her as the exception that proved the male-created rule of female inferiority. After all, it takes analysis and courage to step outside a long-accepted norm. And so, the “woman question” is still with us — but in trying to resolve it, we might not only find ways to remove existing gender biases, but also broaden the conversation about what sort of science we all want in the twenty-first century.

Robyn Arianrhod is an Honorary Research Associate in the School of Mathematical Sciences at Monash University. She is the author of Seduced by Logic: Émilie Du Châtelet, Mary Somerville and the Newtonian Revolution and Einstein’s Heroes.

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Three words to sum up Alan Turing? Humour. He had an impish, irreverent and infectious sense of humour. Courage. Isolation. He loved to work alone. Reading his scientific papers, it is almost as though the rest of the world — the busy community of human minds working away on the same or related problems — simply did not exist. Turing was determined to do it his way. Three more words? A patriot. Unconventional — he was uncompromisingly unconventional, and he didn’t much care what other people thought about his unusual methods. A genius. Turing’s brilliant mind was sparsely furnished, though. He was a Spartan in all things, inner and outer, and had no time for pleasing décor, soft furnishings, superfluous embellishment, or unnecessary words. To him what mattered was the truth. Everything else was mere froth. He succeeded where a better furnished, wordier, more ornate mind might have failed. Alan Turing changed the world.

What would it have been like to meet him? Turing was tallish (5 feet 10 inches) and broadly built. He looked strong and fit. You might have mistaken his age, as he always seemed younger than he was. He was good looking, but strange. If you came across him at a party you would notice him all right. In fact you might turn round and say “Who on earth is that?” It wasn’t just his shabby clothes or dirty fingernails. It was the whole package. Part of it was the unusual noise he made. This has often been described as a stammer, but it wasn’t. It was his way of preventing people from interrupting him, while he thought out what he was trying to say. Ah – Ah – Ah – Ah – Ah. He did it loudly.

If you crossed the room to talk to him, you’d probably find him gauche and rather reserved. He was decidedly lah-di-dah, but the reserve wasn’t standoffishness. He was a man of few words, shy. Polite small talk did not come easily to him. He might if you were lucky smile engagingly, his blue eyes twinkling, and come out with something quirky that would make you laugh. If conversation developed you’d probably find him vivid and funny. He might ask you, in his rather high-pitched voice, whether you think a computer could ever enjoy strawberries and cream, or could make you fall in love with it. Or he might ask if you can say why a face is reversed left to right in a mirror but not top to bottom.

Once you got to know him Turing was fun — cheerful, lively, stimulating, comic, brimming with boyish enthusiasm. His raucous crow-like laugh pealed out boisterously. But he was also a loner. “Turing was always by himself,” said codebreaker Jerry Roberts: “He didn’t seem to talk to people a lot, although with his own circle he was sociable enough.” Like everyone else Turing craved affection and company, but he never seemed to quite fit in anywhere. He was bothered by his own social strangeness — although, like his hair, it was a force of nature he could do little about. Occasionally he could be very rude. If he thought that someone wasn’t listening to him with sufficient attention he would simply walk away. Turing was the sort of man who, usually unintentionally, ruffled people’s feathers — especially pompous people, people in authority, and scientific poseurs. He was moody too. His assistant at the National Physical Laboratory, Jim Wilkinson, recalled with amusement that there were days when it was best just to keep out of Turing’s way. Beneath the cranky, craggy, irreverent exterior there was an unworldly innocence though, as well as sensitivity and modesty.

Turing died at the age of only 41. His ideas lived on, however, and at the turn of the millennium Time magazine listed him among the twentieth century’s 100 greatest minds, alongside the Wright brothers, Albert Einstein, DNA busters Crick and Watson, and the discoverer of penicillin, Alexander Fleming. Turing’s achievements during his short life were legion. Best known as the man who broke some of Germany’s most secret codes during the war of 1939-45, Turing was also the father of the modern computer. Today, all who click, tap or touch to open are familiar with the impact of his ideas. To Turing we owe the brilliant innovation of storing applications, and all the other programs necessary for computers to do our bidding, inside the computer’s memory, ready to be opened when we wish. We take for granted that we use the same slab of hardware to shop, manage our finances, type our memoirs, play our favourite music and videos, and send instant messages across the street or around the world. Like many great ideas this one now seems as obvious as the wheel and the arch, but with this single invention — the stored-program universal computer — Turing changed the way we live. His universal machine caught on like wildfire; today personal computer sales hover around the million a day mark. In less than four decades, Turing’s ideas transported us from an era where ‘computer’ was the term for a human clerk who did the sums in the back office of an insurance company or science lab, into a world where many young people have never known life without the Internet.

B. Jack Copeland is the Director of the Turing Archive for the History of Computing, and author of Turing: Pioneer of the Information Age, Alan Turing’s Electronic Brain, and Colossus. He is the editor of The Essential Turing. Read the new revelations about Turing’s death after Copeland’s investigation into the inquest.

Visit the Turing hub on the Oxford University Press UK website for the latest news in theCentenary year. Read our previous posts on Alan Turing including: “Maurice Wilkes on Alan Turing” by Peter J. Bentley, “Turing : the irruption of Materialism into thought” by Paul Cockshott, “Alan Turing’s Cryptographic Legacy” by Keith M. Martin, and “Turing’s Grand Unification” by Cristopher Moore and Stephan Mertens, “Computers as authors and the Turing Test” by Kees van Deemter, and “Alan Turing, Code-Breaker” by Jack Copeland.

For more information about Turing’s codebreaking work, and to view digital facsimiles of declassified wartime ‘Ultra’ documents, visit The Turing Archive for the History of Computing. There is also an extensive photo gallery of Turing and his war at www.the-turing-web-book.com.

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As well as Halloween, Guy Fawkes, and All Saints’s day, this time of the year used to see another day of fun and frenzy. ‘Almanac Day’, towards the end of November, saw the next year’s almanacs go on sale. It generally came round on or about 22 November: St Cecilia’s Day. In London, Stationers’ Hall would be crammed to the rafters:

The clock strikes, wide asunder start the gates, and in they come, a whole army of porters, darting hither and thither, and seizing the said bags, in many instances as big as themselves. Before we can well understand what is the matter, men and bags have alike vanished – the hall is clear … they will be dispersed through every city and town, and parish, and hamlet of England; the curate will be glancing over the pages of his little book to see what promotions have taken place in the church, and sigh as he thinks of rectories, and deaneries, and bishoprics; the sailor will be deep in the mysteries of tides and new moons that are learnedly expatiated upon in the pages of his; the believer in the stars will be finding new draughts made upon that Bank of Faith impossible to be broken or made bankrupt — his superstition, as he turns over the pages of his Moore — but we have let out our secret. Yes, they are all almanacks — those bags contained nothing but almanacks.

Two hundred or three hundred years ago you could choose from twenty or more almanacs every year. Unlike most of the modern ones they were slim things, with a couple of dozen pages. There were almanacs for Whigs, almanacs for Tories, almanacs for people who believed in astrology and almanacs for those who didn’t, almanacs for farmers, sailors, merchants.

My own journey into the wonderful world of early modern almanacs began with Poor Robin’s Almanac. Robin was a fictional character, invented in the 1660s as a way to lampoon astrologers and their almanacs. He went on to write a long-running spoof almanac, clocking up 164 annual issues. He did prognostication –

If on the second of February, thou go either to Fair or Market with store of money in thy pocket, and there have thy purse picked of it all, then that is an unfortunate day.

and history –

1367 BC: Women first invented kissing

and the year’s calendar –

23 June: Friar Tuck’s Day.

Poor Robin’s intellectual descendants included Punch (it copied part of his title page) and Poor Richard, pseudonym of Benjamin Franklin and author of The Way to Wealth. In his day he was loved and very widely read, but he was killed off in the 1820s by a combination of mismanagement, waning popularity, and attacks from the Society for the Diffusion of Useful Knowledge.

Others were less uproarious, but just as much fun. The Ladies’ Diary, or Woman’s Almanack specialized in genteel mathematical puzzles. ‘If I’m a year younger than one-twentieth the square of my age, how old am I?’ ‘If the sun takes four minutes to cross the horizon on New Year’s Day’, where am I? It attracted questions and answers sent in from all over Britain, and gave prizes for the best ones. It ran for over 130 years.

Old Moore provided predictions political, social, and meteorological based on the movements of the heavens.

Let my Muse raise, and tell what News she hears
Amongst the Stars, and Motions of the Spheres.

But it combined them with some remarkable popular science writing, on subjects ranging from astronomy to ancient history, compiled by authors who had one eye on the Philosophical Transactions and the other on the public’s taste for sensationalism.

Another scientifically-minded production was the Nautical Almanac, started in the 1760s by Longitude’s villain Nevile Maskelyne (he was actually rather a pleasant chap). It gave the moon’s position at three-hour intervals for the whole year, and instructions for working out your longitude from an observation. At two shillings and sixpence, plus the price of a sextant, it came in a good bit cheaper than a Harrison chronometer.

At times nearly one Briton in six was buying an almanac: ‘the greatest triumph of journalism until modern times’ according to historian Bernard Capp. Almanac day may be no more, but almanacs have been circulating for nearly as long as calendars, and if the genre has waxed and waned over the years it seems in no danger of extinction. Partly eclipsed in the early nineteenth century by other forms of popular instruction, the almanac blazed forth again from the 1830s, with sales rising to a million a year for the most popular. Today, Old Moore is still with us, though somewhat transformed; so is the Nautical Almanac. Whitaker and Schott have given almanacs a new lease of life as annual reference books. Their survival seems a safe prediction.

Benjamin Wardhaugh is a historian and fellow of Wolfson College, Oxford. His book, Poor Robin’s Prophecies: A curious Almanac, and the everyday mathematics of Georgian Britain, publishes this month.

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An interesting, if somewhat uncommon, lens through which to view politics is that of mathematics. One of the strongest arguments ever made in favor of democracy, for example, was in 1785 by the political philosopher-mathematician, Nicolas de Condorcet. Because different people possess different pieces of information about an issue, he reasoned, they predict different outcomes from the same policy proposals, and will thus favor different policies, even when they actually share a common goal. Ultimately, however, if the future were perfectly known, some of these predictions would prove more accurate than others. From a present vantage point, then, each voter has some probability of actually favoring an inferior policy. Individually, this probability may be rather high, but collective decisions draw information from large numbers of sources, mistaking mistakes less likely.

To clarify Condorcet’s argument, note that an individual who knows nothing can identify the more effective of two policies with 50% probability; if she knows a lot about an issue, her odds are higher. For the sake of argument, suppose that a citizen correctly identifies the better alternative 51% of the time. On any given issue, then, many will erroneously support the inferior policy, but (assuming that voters form opinions independently, in a statistical sense) a 51% majority will favor whichever policy is actually superior. More formally, the probability of a collective mistake approaches zero as the number of voters grows large.

Condorcet’s mathematical analysis assumes that voters’ opinions are equally reliable, but in reality, expertise varies widely on any issue, which raises the question of who should be voting? One conventional view is that everyone should participate; in fact, this has a mathematical justification, since in Condorcet’s model, collective errors become less likely as the number of voters increases. On the other hand, another common view is that citizens with only limited information should abstain, leaving a decision to those who know the most about the issue. Ultimately, the question must be settled mathematically: assuming that different citizens have different probabilities of correctly identifying good policies, what configuration of voter participation maximizes the probability of making the right collective decision?

It turns out that, when voters differ in expertise, it is not optimal for all to vote, even when each citizen’s private accuracy exceeds 50%. In other words, a citizen with only limited expertise on an issue can best serve the electorate by ignoring her own opinion and abstaining, in deference to those who know more. Mathematically, it might seem that more information is always better, if only slightly. This would indeed be the case, except that each vote takes weight away from other votes, which may be better informed.

If voters recognize the potential harm of an uninformed vote, this could explain why many citizens vote in some races, but skip others on the same ballot, or vote in general elections, but not in primaries, where information is more limited. This raises a new question, however, which is who should continue voting: if the least informed citizens all abstain, then a moderately informed citizen now becomes the least informed voter; should she abstain, as well?

Mathematically, it turns out that for any distribution of expertise, there is a threshold above which citizens should continue voting, no matter how large the electorate grows. A citizen right at this threshold is less knowledgeable than other voters, but nevertheless improves the collective electoral decision by bolstering the number of votes. The formula that derives this threshold is of limited practical use, since voter accuracies cannot readily be measured, but simple example distributions demonstrate that voting may well be optimal for a sizeable majority of the electorate.

The dual message that poorly informed votes reduce the quality of electoral decisions, but that moderately informed votes can improve even the decisions made even by more expert peers, may leave an individual feeling conflicted as to whether she should express her tentative opinions, or abstain in deference to those with better expertise. Assuming that her peers vote and abstain optimally, it may be useful to first predict voter turnout, and then participate (or not) accordingly: when half the electorate votes, it should be the better-informed half; when voter turnout is 75%, all but the least-informed quartile should participate.

An important caveat of Condorcet’s probability analysis is that disagreements are actually illusory: if voters envisioned the same policy outcomes, they would largely support the same policies. Whether this is accurate or not is an open philosophical question, but voters seem implicitly to embrace this assumption when they attempt to persuade and convert one another via debate, endorsements, or policy research: such efforts are only worthwhile if an individual expects others, once convinced, to abandon their former policy positions, in favor of her own. Some policies also do receive overwhelming public support.

If Condorcet’s basic premise is right, an uninformed citizen’s highest contribution may actually be to abstain from voting, trusting her peers to make decisions on her behalf. At the same time, voters with only limited expertise can rest assured that a single, moderately-informed vote can improve upon the decision made by a large number of experts. One might say that this is the true essence of democracy.

Joseph C. McMurray is Assistant Professor in the Department of Economics at Brigham Young University. His recent paper, Aggregating Information by Voting: The Wisdom of the Experts versus the Wisdom of the Masses, has been made freely available for a limited time by the Review of Economic Studies journal.

The Review of Economic Studies is widely recognised as one of the core top-five economics journals. The Review is essential reading for economists and has a reputation for publishing path-breaking papers in theoretical and applied economics.

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Image credit: Voting card. Photo by rrmf13, iStockphoto.

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In recent days, the pro-mathematics portion of the Internet has been buzzing over the following paragraph, taken from the website of Christian publishing company A Beka Book:

Unlike the “modern math” theorists, who believe that mathematics is a creation of man and thus arbitrary and relative, A Beka Book teaches that the laws of mathematics are a creation of God and thus absolute….A Beka Book provides attractive, legible, and workable traditional mathematics texts that are not burdened with modern theories such as set theory.

As a result of recent legislative activity in the state of Louisiana, these curricular materials will now be supported with taxpayer money.

In more than a decade of socializing with creationists and other religious fundamentalists, I frequently encountered blinkered arguments about mathematics. This attack on set theory, however, was new to me. I cannot even imagine why anyone would think set theory is relevant to discussions of whether it is man or God who creates math. Perhaps the problem is that set theorists often speak a bit casually about infinity, which some people think is tantamount to discussing God. Alas, this line of criticism is too blinkered to take seriously.

Whatever their objection, they are really missing out on something great. Set theory is fascinating.

By a “set” we mean simply any collection of objects. You walk into a grocery store and see a pile of grapefruits over here and a pile of apples over there. A mathematician might then refer to the set of grapefruits on the one hand and the set of apples on the other. This provides a useful way of talking about all the apples (or grapefruits) combined as one unit, as opposed to discussing any specific apple (or grapefruit).

Of course, we can identify many other sets. We might wish to distinguish the set of Gala apples from the set of Granny Smiths. Or we might want to make a larger set by combining the set of apples and the set of grapefruits together to form part of the set of all fruit. For any description you would care to give, it is reasonable to talk about the set of all things that fit that description.

This seemed obvious, for example, to Gottlob Frege, a German mathematician/philosopher who did pioneering work in logic and set theory in the late nineteenth and early twentieth centuries. Bertrand Russell pointed out that this notion is fundamentally flawed. He first observed that some sets answer to their own descriptions while others don’t. The set of all grapefruits isn’t itself a grapefruit. Therefore, this set doesn’t contain itself among its members. On the other hand, the set of all abstract ideas is, indeed, an abstract idea. So it contains itself.

Russell now considered the set whose members are precisely the sets that are not contained within themselves. The set of all grapefruits is contained in Russell’s set, for example, while the set of all abstract ideas isn’t. He now wondered whether his set did or didn’t answer to its own description. If we suppose that it does so answer, then it must be contained within itself. But Russell’s set only contains sets that aren’t contained within themselves. This is a contradiction. You see, if we assume that Russell’s set answers to its own description then it both contains itself and doesn’t contain itself. Impossible.

Alas, the alternative assumption fares no better. If we suppose that Russell’s set doesn’t answer to its own description, then it must be among the sets that aren’t contained within themselves. But this is precisely the criterion you must satisfy to get into Russell’s set in the first place. Either way you have a contradiction, meaning this isn’t a properly defined set.

Nor is this the only way to get into trouble with sets. Consider the set of counting numbers {1, 2, 3, 4, …} that cannot be uniquely identified with fewer than two hundred characters. For example, a number such as 1000 can be identified by writing “ten multiplied by one hundred,” but I can do it more efficiently by writing “one thousand,” and more efficiently still by writing “ten cubed.”

Now, since there are only finitely many phrases having fewer than 200 characters, and infinitely many counting numbers, it is clear that my set must contain something. And since it must contain something, it must also contain a smallest number. (In the math biz, this curious fact of counting numbers is known as the “well-ordering principle.”) That smallest number in the set is therefore uniquely identified by the phrase, “The smallest counting number that cannot be described with fewer than two hundred characters.” But did I not just describe it with fewer than 200 characters? Prolonged consideration of such things can be harmful to your mental health.

Actually, my favorite application of the well-ordering principle is this: Consider the set of all the boring counting numbers. This set must have a smallest member, let us call it X. But then X is the smallest boring counting number, which makes it very interesting indeed! Surely this contradiction shows that all counting numbers are interesting?

Indeed they are. And sets are as well. Just don’t be too ingenious about how you define them.

Jason Rosenhouse is Associate Professor of Mathematics at James Madison University. His most recent book is Among The Creationists: Dispatches from the Anti-Evolutionist Front Lines. He is also the author of Taking Sudoku Seriously: The Math Behind the World’s Most Popular Pencil Puzzle with Laura Taalman and The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser. Read Jason Rosenhouse’s previous blog articles.

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The world famous Edinburgh International Festival has kicked off, beginning three weeks of the best the arts world has to offer. The Fringe Festival has countless alternative, weird, and wacky events happening all over the city, and the Edinburgh International Book Festival is underway. Throughout the Book Festival we’ll be bringing you sneak peeks of our authors’ talks and backstage debriefs so that, even if you can’t make it to Edinburgh this year, you won’t miss out on all the action.

By David Acheson

I’ve just had a great time at the 2012 Edinburgh International Book Festival, even though it was a rather strange experience for a mathematician.

In the Author’s Yurt (sic), for example, I was surrounded by fiction writers, with lots of pointy beards and wild hair.

As it happens, I used to write detective stories when I was a young boy, so once had vague dreams of becoming a fiction writer myself. But it was not to be. And now, after an academic career at Oxford, I find that I am an author of a rather different kind.

Who says maths can't be taught through the medium of music?

I’ve lectured on this book before but Edinburgh was a bit different, because I was part of the RBS Schools Programme, so I knew in advance that the audience would be pupils and their teachers. But my lecture had been advertised for a wide age range, so I wasn’t entirely sure what to expect.

Twenty minutes before the lecture, in Charlotte Square Gardens, the audience started to arrive, and I peered out of the Author’s Yurt.

The children looked very small. The average age was about 10.

I began to get nervous. My lecture had number tricks, practical demonstrations, a bit of audience participation, and, at the end, the electric guitar. But it also involved some quite deep mathematical ideas.

So, five minutes later, I peered out again.

They’d got even smaller.

Thankfully, it all seemed to go well enough in the end. But what really surprised me were the questions afterwards. There was no stopping them.

Some, like “How long have you been playing the guitar?” (53 years), were predictable. But there were several on maths (“What’s your favourite equation?”) and many more on the actual process of writing the book. In fact I began to wonder afterwards how many budding young authors there had been in the audience. And, in retrospect, I wish I’d asked them.

In any event, it seemed to come to a happy end, and I was whisked off to a book-signing and then to a radio interview with BBC Scotland, which turned out to be conducted by three pupils from a local primary school.

So, for me, the whole experience was a memorable one, to say nothing of the spectacular view of Edinburgh Castle from my hotel window, complete with night-time illumination and festival fireworks.

And while I could ramble on further, it seems to me that it was all summed up, really, a long time ago, by the satirical novelist Peter de Vries, who wrote: “I love being an author; what I can’t stand is the paperwork.”

David Acheson is the author of 1089 and All That: A Journey into Mathematics. He is an Emeritus Fellow of Jesus College, Oxford, and was recently President of the Mathematical Association. He is also the author of From Calculus to Chaos: An Introduction to Dynamics.

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Alan Turing’s work was so important and wide-ranging that it is difficult to think of a more broadly influential scientist in the last century. Our understanding of the power and limitations of computing, for example, owes a tremendous amount to his work on the mathematical concept of a Turing Machine. His practical achievements are no less impressive. Some historians believe that the Second World War would have ended differently without his contributions to code-breaking. Yet another part of his work is the Turing test — Turing’s answer to a momentous question: What’s essential about human intelligence?

The inspiration for the Turing test came from a conversation game in which one player (the deceiver) tries to fool another player (the detective) about the deceiver’s gender. To win, a male deceiver would need to answer the detective’s questions in a way that suggests that he, the deceiver, is female. It doesn’t suffice for the deceiver to answer direct questions about gender. He should also show good knowledge about feminine topics and get properly upset over male chauvinism. What’s more, he should use turns of phrase that are typical of women. All this without overdoing it, of course.

Turing realised that this conversation game could be turned on its head if the role of the deceiver is played by a computer, not a person. The task for a computer deceiver is to fool the detective into believing that the deceiver is a person of flesh and blood. Analogous to the original game, the computer can win by thinking like a human. Now suppose that, playing this modified game, a computer was able to fool human detectives into believing it to be human. (A deceiver wins if the detectives are unable to get the computer/human decision correct more often than would be expected by chance.) Surely, so Turing argued, this would mean that the computer has managed to think like a human. Hence, if this happened, one would have to conclude that the computer displays real human thinking; the makers of the deceiver program would have captured human intelligence. The link between the Turing Test and intelligence has often been questioned, but the idea of the Test itself is very much alive.

Natural Language Generation (NLG) systems are computer programs that convert numerical or symbolic information into ordinary language. Weather forecasting, medical decision support, and other applications are starting to use systems of this kind. How should NLG programs be tested? No single method has all the answers, but human behaviour is still a gold standard to which many of these systems aspire. As in the Turing Test, researchers try to make their NLG systems produce text that resembles human-written text, partly because they believe that this may be the shortest route to making them effective. More and more often, NLG systems are tested in international evaluation contests that focus on one or more particular aspects of language use.

One of the most important challenges for NLG is to let computers talk in a human way about numbers. Numbers play an important role in many areas, including the medical domain. When nurses write about a patient — producing a shift report for instance — they have many numbers at their disposal (body temperature, oxygen saturation rates, etc.). However, they frequently suppress these numbers, replacing them by terms that are qualitative and vague. Instead of citing concrete oxygen saturation figures, they simply write “The SATS have remained OK”, for example. When talking about episodes of decreased heart rate, they throw in words like “temporary,” “prolonged,” and “significant.. Interestingly, doctors suppress numbers even more than nurses. Computational NLG systems in this area, by contrast, tend to stick with the numbers, producing stilted bits of text like the following: “By 10:40 SaO2 had decreased to 87. As a result, Fraction of Inspired Oxygen (FIO2) was set to 36%. SaO2 increased to 93.” The challenge is to do better, emulating human writers.

Unfortunately, the writings of doctors are rather difficult to mimic. The challenge is not just to decide when numerical information is useful. (Texts written by doctors contain numbers too, though fewer.) The hardest challenge for the NLG system is to “interpret” the numbers and this can involve difficult judgment calls, deciding whether a certain pattern of numbers should be summarized as “OK,” for instance, and deciding whether an episode of slow heart rhythm is merely “temporary” or “prolonged.” The medics’ texts are not dumbed-down versions of the computer-generated ones. They are highly sophisticated, despite their apparent simplicity. It will take research in NLG years before its computer programs stand a chance at winning a Turing Test in this area.

Kees van Deemter is a Reader in Computing Science at the University of Aberdeen. He is interested in getting computers to speak or write, and in the logical, linguistic, and philosophical issues that this raises. His book, Not Exactly: In Praise of Vagueness, puts the spotlight on vague and qualitative concepts, viewing them from a variety of angles and making a highly technical literature easily accessible to a wide audience. It explores how vague and qualitative concepts play a role in all areas of life, including even the exact sciences, where they are mostly unwelcome; how vague concepts fit into our current understanding of language and logic; the practical applications; and when and why vague language can be effective. Find out more about Not Exactly: In Praise of Vagueness. Kees van Deemter is also the author of approximately 120 peer-reviewed research publications.

OUPblog is celebrating Alan Turing’s 100th birthday with blog posts from our authors all this week. Read our previous posts on Alan Turing including: “Maurice Wilkes on Alan Turing” by Peter J. Bentley, “Turing : the irruption of Materialism into thought” by Paul Cockshott, “Alan Turing’s Cryptographic Legacy” by Keith M. Martin, and “Turing’s Grand Unification” by Cristopher Moore and Stephan Mertens.

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Many of the central moments in science have been unifications: realizations that seemingly disparate phenomena are all aspects of one underlying structure. Newton showed that the same laws of motion and gravity govern apples and planets, creating the first explanatory framework that joins the terrestrial to the celestial. Maxwell showed that a single field can explain electricity, magnetism, and light. Darwin realized that natural selection shapes all forms of life. And Einstein demonstrated that space and time are shadows of a single, four-dimensional spacetime.

This quest for unity drives us to this day, as we hunt for a Grand Unified Theory that combines gravity with quantum mechanics. But while it is less well-known, computer science had its own grand unification in 1936, thanks to Alan Turing.

At the dawn of the 20th century, mathematicians and logicians were focused on the axiomatic underpinnings of mathematics. Shaken by paradoxes — like Russell’s set of all sets that don’t contain themselves (does it or doesn’t it?) — they wanted to re-build mathematics from the ground up, creating a foundation free of paradox. This stimulated a great deal of interest in axiomatic systems, their power to establish truth, and the difficulty of finding proofs or disproofs of open questions in mathematics.

In 1928, David Hilbert posed the Entscheidungsproblem, asking whether there is a “mechanical procedure” that can decide whether or not any given mathematical statement is true — say the Twin Prime Conjecture, that there are an infinite number of pairs of primes that differ by two. Such a procedure would complete the mathematical adventure, providing a general method to determine the truth or falsehood of any statement. To us, this sounds horribly final, but to Hilbert it was a glorious dream.

But what exactly is a “mechanical procedure”? Or, in modern terms, an algorithm? Intuitively, it is a procedure that can be carried out according to a fixed computer program, like a recipe followed by a dutiful cook. But what kinds of steps can this computer perform? What kind of information does it have access to, and how is it allowed to transform this information?

Several models of computation had been proposed, with different attitudes towards what it means to compute. The recursive functions build functions from simpler ones using rules like composition and induction, starting with “atomic” functions like x+1 that we can take for granted. Another model, Church’s λ-calculus, repeatedly transforms a string of symbols by substitutions and rearrangements until only the answer remains.

Each of these models has its charms. Indeed, each one lives on in today’s programming languages. Recursive functions are much like subroutines and loops in C and Java, and the λ-calculus is at the heart of functional programming languages like Lisp and Scheme. Every computer science student knows that we can translate from each of these languages to the others, and that while their styles are radically different, they can ultimately perform the same tasks. But in 1936, it was far from obvious that these models are equivalent, or that either one is capable of everything we might reasonably call a computation. Why should a handful of ways to define functions in terms of simpler ones, or a particular kind of symbol substitution, be enough to carry out any conceivable computation?

Turing settled this issue with a model that is both mathematically precise and intuitively complete. He began by imagining a human computer, carrying out a procedure with pencil and paper. If we had to, he argued, we could boil any such procedure down into a series of steps, each of which reads and writes a single symbol. We don’t need to remember much ourselves, since we can use notes on the paper to keep track of what to do next. And although it might be inconvenient, a one-dimensional roll of paper is enough to write down anything we might need to read later.

At that point, we have a Turing machine: a controller with a finite number of internal states, and a tape on which it can read and write symbols in a finite alphabet. Nothing has been left out. Any reasonable attempt to augment the Turing machine with two-dimensional tapes, multiple controllers, etc. can be simulated by the original model.

Famously, Turing then showed that there are problems that no Turing machine can solve. The Halting Problem asks whether a given Turing machine will ever halt; in modern terms, whether a program will return an answer, or “hang” and run forever. If there were a machine that could answer this question, we could ask it about itself, demanding that it predict its own behavior. We then add a twist, making it halt if it will hang, and hang if it will halt. The only escape is to accept that no such machine exists in the first place.

This also gives a nice proof of Gödel’s Theorem that there are unprovable truths, or to be more precise, that no axiomatic system can prove all mathematical truths. (Unless it also “proves” some falsehoods, in which case we can’t trust it!) For if every truth of the form “this Turing machine will never halt” had a finite proof, we could solve the Halting Problem by doing two things in parallel: running the machine to see if it halts, while simultaneously looking for proofs that it won’t. Thus, no axiomatic system can prove every truth of this form.

Turing showed that his machines are exactly as powerful as the recursive functions and the λ-calculus, unifying all three models under a single definition of computation. What we can compute doesn’t depend on the details of our computers. They can be serial or parallel, classical or quantum. One kind of computer might be much more efficient than another, but given enough time, each one can simulate all the others. The belief that these models capture everything that could reasonably be called an algorithm, or a mechanical procedure, is called the Church-Turing Thesis.

Turing drew the line between what is computable and what is not, if we have an unbounded amount of time and memory. Since his death, computer science has focused on what we can compute if these resources are limited. For instance, the P vs. NP question asks whether there are problems where we can check a solution in polynomial time (as a function of problem size) but where actually finding a solution is much harder. For instance, mathematical proofs can be checked in time roughly proportional to their length — that’s the whole point of formal proofs — but they seem difficult to find. Despite our strong intuition that finding things is harder than checking them, we have been unable to prove that P ≠ NP, and it remains the outstanding open question of the field.

The fact that Turing didn’t live to see how computer science grew and flowered — that he wasn’t there to play a role in its development, as he did at its foundations – is one of the tragedies of the 20th century. He received, too late, an apology from the British government for the persecution that led to his death. Let’s wish him a happy birthday, and raise a glass to his short but brilliant life.

Cristopher Moore and Stephan Mertens are the authors of The Nature of Computation. Cristopher Moore is a professor at Santa Fe Institute and previously, a professor in the Computer Science Department and Department of Physics and Astronomy, University of New Mexico. Stephan Mertens is a a theoretical physicist at the Institute of Theoretical Physics, Otto-von-Guericke University, Magdeburg, and external professor at the Santa Fe Institute.

OUPblog is celebrating Alan Turing’s 100th birthday with blog posts from our authors all this week. Read our previous posts on Alan Turing including: “Maurice Wilkes on Alan Turing” by Peter J. Bentley, “Turing : the irruption of Materialism into thought” by Paul Cockshott, and “Alan Turing’s Cryptographic Legacy” by Keith M. Martin. Look for “Computers as authors and the Turing Test” by Kees van Deemter tomorrow.

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This year is being widely celebrated as the Turing centenary. He is being hailed as the inventor of the computer, which perhaps overstates things, and as the founder of computing science, which is more to the point. It can be argued that his role in the actual production of the first generation computers, whilst real, was not vital. In 1946 he designed the Automatic Computing Engine (ACE), a very advanced design of computer for its day, but because of its challenging scale, initially only a cut down version (the Pilot ACE) was built (and can now be seen in the Science Museum). From 1952 to 1955, the Pilot ACE was the fastest computer in the world and it went on to be sucessfully commercialised as the Deuce. In engineering terms though, none of the distinctive features of Turing’s ACE survive in today’s computer designs. The independent work of Zuse in Germany or Atanasoff in the US indicates that electronic computers were a technology waiting to be discovered across the industrial world.

What distinguished Turing from the other pioneer computer designers was his much greater philosophical contribution. Turing thought deeply about what computation is, what its limits are, and what it tells us about the nature of intelligence and thought itself.

Turing’s 1936 paper on the computable real numbers marks the epistemological break between idealism and a materialism in mathematics. Prior to Turing it was hard to get away from the idea that through mathematical reason, the human mind gained access to a higher domain of Platonic truths. Turing’s first proposal for a universal computing machine is based on an implicit rejection of this view. His machine is intended to model what a human mathematician does when calculating or reasoning, and by showing what limits this machine encounters, he identifies constraints which bind mathematical reasoning in general (whether done by humans or machines).

From the beginning, he emphasises the limited scope of our mental abilities and our dependence on artificial aids — pencil and paper for example — to handle large problems. We have, he asserted, only a finite number of ‘states of mind’ that we can be in when doing calculation. We have in our memories a certain stock of what he calls ‘rules of thumb’ that can be applied to a problem. Our vision only allows us to see a limited number of mathematical symbols at a time and we can only write down one symbol of a growing formula or growing number at a time. The emphasis here, even when he looks at the human mathematician, is on the mundane, the material, the constraining.

In his later essays on artificial intelligence Turing doesn’t countenance any special pleading for human reason. He argues with his famous Turing Test that the same criteria that we use to impute intelligence and consciousness to other human beings could in principle be used to impute them to machines (provided that these machines communicate in a way that we can not distinguish from human behaviour). In his essay ‘Computing Machinery and Intelligence,’ he confronts the objection that machines can never do anything new, only what they are programmed to do. “A better variant of the objection says that a machine can never ‘take us by surprise’…. Machines take me by surprise with great frequency. This is largely because I do not do sufficient calculation to decide what to expect them to do, or rather because although I do a calculation, I do it in a hurried, slipshod fashion, taking risks.”

Turing starts a philosophical tradition of grounding mathematics on the material and hence ultimately on what can be allowed by the laws of physics. The truth of mathematics become truths like those of any other science — statements about sets of possible configurations of matter. So the truths of arithmetic are predictions about the behaviour of actual physical calculating systems, whether these be children with chalks and slates or microprocessors. In this view it makes no more sense to view mathematical abstractions as Platonic ideals than it does to posit the existence of ideal doors and cups of which actual doors and cups are partial manifestations. Mathematics then becomes a technology of modeling one part of the material world with another. In Deutch’s formulation of the Turing Principle, any finite physical system can be simulated to an arbitrary degree of accuracy by a universal Turing machine.

Paul Cockshott is a computer scientist and political economist working at the University of Glasgow. His most recent books are Computation and its Limits (with Mackenzie and Michaelson) and Arguments for Socialism (with Zachariah). His research includes programming languages and parallelism, hypercomputing and computability, image processing, and experimental computers.

OUPblog is celebrating Alan Turing’s 100th birthday with blog posts from our authors all this week. Read the previous post in our Turing series: “Maurice Wilkes on Alan Turing” by Peter J. Bentley. Look for “Alan Turing’s Cryptographic Legacy” by Keith M. Martin, “Turing’s Grand Unification” by Cristopher Moore and Stephan Mertens, and “Computers as authors and the Turing Test” by Kees van Deemter later this week.

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Among mathematicians, it is always a happy moment when a long-standing problem is suddenly solved. The year 2012 started with such a moment, when an Irish mathematician named Gary McGuire announced a solution to the minimal-clue problem for Sudoku puzzles.

You have seen Sudoku puzzles, no doubt, since they are nowadays ubiquitous in newspapers and magazines. They look like this:

Your task is to fill in the vacant cells with the digits from 1-9 in such a way that each row, column and three by three block contains each digit exactly once. In a proper puzzle, the starting clues are such as to guarantee there is only one way of completing the square.

This particular puzzle has just seventeen starting clues. It had long been believed that seventeen was the minimum number for any proper puzzle. Mathematician Gordon Royle maintains an online database which currently contains close to fifty thousand puzzles with seventeen starting clues (in fact, the puzzle above is adapted from one of the puzzles in that list). However, despite extensive computer searching, no example of a puzzle with sixteen or fewer clues had ever been found.

The problem was that an exhaustive computer search seemed impossible. There were simply too many possibilities to consider. Even using the best modern hardware, and employing the most efficient search techniques known, hundreds of thousands of years would have been required.

Pure mathematics likewise provided little assistance. It is easy to see that seven clues must be insufficient. With seven starting clues there would be at least two digits that were not represented at the start of the puzzle. To be concrete, let us say that there were no 1s or 2s in the starting grid. Then, in any completion of the starting grid it would be possible simply to change all the 1s to 2s, and all the 2s to 1s, to produce a second valid solution to the puzzle. After making this observation, however, it is already unclear how to continue. Even a simple argument proving the insufficiency of eight clues has proven elusive.

McGuire’s solution requires a combination of mathematics and computer science. To reduce the time required for an exhaustive search he employed the idea of an “unavoidable set.” Consider the shaded cells in this Sudoku square:

Now imagine a starting puzzle having this square for a solution. Can you see why we would need to have at least one starting clue in one of those shaded cells? The reason is that if we did not, then we would be able to toggle the digits in those cells to produce a second solution to the same puzzle. In fact, this particular Sudoku square has a lot of similar unavoidable sets; in general some squares will have more than others, and of different types. Part of McGuire’s solution involved finding a large collection of certain types of unavoidable sets in every Sudoku square under consideration.

Finding these unavoidable sets permits a dramatic reduction in the size of the space that must be searched. Rather than searching through every sixteen-clue subset of a given Sudoku square, desperately looking for one that is actually a proper puzzle, we need only consider sets of sixteen starting clues containing at least one cell from each known unavoidable set. Finding those particular sets of starting clues is a specific instance of a more general problem, known to mathematicians as the “hitting set problem.” The really clever part of McGuire’s work is the development of algorithms for solving the hitting set problem in a reasonable amount of time. Solving the minimum-clue problem for Sudoku was just an application of this new algorithm.

Of course, caution is required until researchers have had time to check carefully the details of McGuire’s proof. It is one of the cruelties of mathematics that subtle errors can elude even the most careful of practitioners. We can certainly say, though, that the techniques being employed here are very plausible and interesting. They might also be useful for polishing off other “minimal clue” problems in Sudoku. For example, if we require that the starting clues be placed symmetrically, then eighteen clues is the smallest that is known (as shown below left; puzzle taken from Taking Sudoku Seriously). On the other hand, in Sudoku X, in which the two main diagonals join the rows, columns and blocks as valid Sudoku regions, the current minimum is twelve (see below right; puzzle adapted from one in Ruud’s online collection of 12-clue Sudoku X puzzles).

Sudoku X

But the second is the pace of mathematical discovery. We learned of McGuire’s work as we were traveling to a mathematics conference at which we were organizing a session on Sudoku puzzles. One of us had a talk prepared in which this problem was described as unsolved. It was very exciting to be able to use the conference to report on this development. Our book on the mathematics of Sudoku has only been out for a few weeks, but it might already be outdated, at least with regard to this one problem.

Jason Rosenhouse is Associate Professor of Mathematics at James Madison University.  He is the author of Taking Sudoku Seriously: The Math Behind the World’s Most Popular Pencil Puzzle and the forthcoming Among the Creationists: Dispatches from the Anti-Evolutionist Front.

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October 1, 1847

Mitchell discovers a comet

Each evening that weather permitted, Maria (pronounced Mah-RYE-uh) Mitchell mounted the stairs to the roof of her family’s Nantucket home to sweep the sky with a telescope looking for a comet. Mitchell—who had been taught mathematics and astronomy by her father—began the practice in 1836. Eleven years later, on October 1, 1847, her long labors finally paid off. When she saw the comet, she quickly summoned her father, who agreed with her conclusion.

Years previously, in 1831, Denmark’s king Frederic VI had offered to give a prize to anyone who discovered a comet using a telescope. While Mitchell’s discovery qualified, she had not followed the rules of notifying the king of her finding. Harvard’s president Edward Everett—a family friend—took up the Mitchell cause and used his position to argue that she deserved the medal. The king consented, and Mitchell had her medal.

The discovery and recognition made Mitchell’s career. In the spring of 1848, she was named an honorary member of the American Academy of Arts and Sciences. For the next dozen or so years, she remained active in astronomy and prominent in international scientific circles. In the 1860s, when brewer Henry Vassar was preparing to create a new college for women, he tapped Mitchell as professor of astronomy. She was one of only two women in Vassar’s first faculty. Mitchell taught at Vassar from 1865 until 1888, when poor health forced her to retire. She also served several years as president of the Association for Advancement of Women.

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Falling cats can turn over in mid-air. Well, most cats can. Our first cat, Seamus, didn’t have a clue. My wife, worried he might fall off a fence and hurt himself, tried to train him by holding him over a cushion and letting go. He enjoyed the game, but he never learned how to flip himself over.

Maybe Seamus was puzzled for the same reason scientists were, a hundred years ago. On the face of it, the feat seems impossible. Flapping your paws and hoping that air resistance will do the trick is over-optimistic. Since the cat has nothing to push against, how can it make itself rotate?

In fact, there seems to be a good mathematical reason why it can’t be possible. It’s called the law of conservation of angular momentum, which is a fancy way to say that you can’t create spin from nothing. How can an upside-down cat turn over without spinning?

In 1894 a French doctor, Étienne Jules Marey, took a series of photos of a falling cat and discovered that it doesn’t. Spin, that is. But it does turn over. How come?  

Now they had pictures to look at, the scientists figured it out. What really matters isn’t little bits of spin here and there. What can’t change is the overall spin of the entire cat. It starts with a total spin of zero: motionless. The same is true at the end. So the cat doesn’t have to create or lose any spin. It just has to wiggle various appendages so that the spin remains zero, but the cat flips over.

That wouldn’t be possible if the cat were a rigid body, but it’s actually very flexible. It can change its shape. And shape-changing is what makes the trick work. Some bits of the animal can turn one way, while other bits simultaneously turn the other way, keeping the overall spin at zero. Fit it all together, and the cat can flip over.    

Here’s how. First, the cat sticks out its back legs and pulls in its front legs. Then it twists its rear end slightly one way, and twists its front end in the opposite direction. The total spin remains zero, but the front end twists a lot more than the rear because of the positions of the legs. Then the cat pulls in its back legs, sticks out its front legs, and twists everything back the way it came. However, the back end now twists more than the front, because of the changes in which legs stick out and which don’t. So neither end of the cat goes back to its original position.

The net effect is that the cat flips over, but the total spin at all stages is zero. The cat has to do all this in mid-air, while falling. I doubt it does the sums: the technique is instinctive. The ability must have evolved over millions of years. Somewhere along the line, Seamus missed out.

Ian Stewart is Emeritus Professor and Digital Media Fellow in the Mathematics Department at Warwick University, with special responsibility for public awareness of mathematics and science. He has written many books, including How to Cut a Cake and Cows in the Maze for OUP. This post first appeared on the BBC Focus/Oxford University Press microsite.

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By Michael Humphreys

Baseball fans love to compare the players of today to the players who came before, but one must wonder how great the margin of error in these comparisons is. Is there any way of knowing who the real baseball greats are, and whose legend should stand the test of time?

Let’s take Omar Vizquel as an example. So says Wikipedia, “Vizquel is considered one of baseball’s all-time best fielding shortstops.” It’s true, Vizquel “is considered” a great fielder. Of shortstops, he

-holds the highest career fielding percentage of those with a long career.
-has participated in more double plays (and his primary double play partner just entered the Hall of Fame)
-is third in career assists
-has played more games at shortstop than anyone in major league history.

On top of all that, Vizquel has received more Gold Gloves than any other shortstop except for Ozzie “Wizard of Oz” Smith. Indeed, writers have described Omar and Ozzie as the “graceful Fred Astaire” and “acrobatic Gene Kelly,” respectively, of shortstops.

Vizquel has something of a signature play—fielding ordinary grounders (not just bunts) with his bare hand and throwing in one motion. He was the starting shortstop for the most successful American League team of the 1990s, second only to the Yankees. He hasn’t been much of a hitter, even for a shortstop, so it’s not unreasonable to infer he must have been a great fielder to hang on as long as he has.

But, after all that, how do we really Vizquel actually is one of baseball’s all-time best fielding shortstops? With metrics.

Let’s start with the question: What is the job of a fielder? To help his team prevent runs. At shortstop, this mainly involves converting ground balls into outs and getting the second out on double plays—in other words, recording assists. (It is very rare that shortstops catch fly balls or pop ups that couldn’t be fielded by at least two and as many as five other fielders. Most of the differences in putout rates for shortstops reflect how much they ‘hog’ these easy chances, not how many marginal hits they help their teams prevent. And line drive putouts at short are mostly dumb-luck plays.)

It is not the job of a shortstop (or any fielder) to look “graceful” or make trick plays. It’s not even a fielder’s job to avoid errors. In fact, a fielder who makes ten more successful plays but also ten more errors has just the same value as the fielder who makes an average number of plays and errors, because an error is no worse than a play not made.

Any fielding metric for shortstop needs to estimate how many assists a shortstop generated above or below what an average shortstop would have, playing for the same team. My system uses some arithmetic and the statistical technique of “regression analysis,” resulting in what I call Defensive Regression Analysis, or DRA.

DRA estimates the number of assists the league average shortstop would have recorded in place of the shortstop you’re rating by starting with the average number of shortstop assists per team that year and adjusting that number up or down based on statistically significant relationships between shortstop assists and other defensive statistics of the player’s team that are

1. not influenced by the shortstop himself,
2. as little influenced by the fielding quality of his teammates as possible, and
3. independent (approximately) of each other.

These defensive statistics are designed to take into account

1. the total number of batted balls allowed by the shortstop’s pitcher,
2. the relative number that were hit by right- and left-handed batters,
3. the tendency of the right-handed batters to hit ground balls,
4. the tendency of the left-handed batters to hit ground balls, and
5. the relative number of runners on first base, which would impact double-play opportunities.

DRA also uses regression analysis to estimate the number of runs statistically associated with each play above or below the DRA estimate of expected plays. Net plays multiplied by runs per play yields runs saved (if negative, allowed) relative to the league average rate, or “defensive runs.”

Taking all of these factors into account, DRA finds that Vizquel has been . . . average. Actually, ‒19 defensive runs, which is just noise over the course of a two-decade career. Now he was a very fine shortstop his first three seasons (1989-91), and he seems to have had a miraculous age-40 season (2007), but he’s never been anything special in-between.

DRA is not alone in estimating that Vizquel has been just average overall. Noted baseball analyst Tom Tango, who maintains one of the best baseball blogs around, has his own system for evaluating career fielding value. Tango calls his system With Or Without You (“WOWY”). As applied at shortstop, it compares the rate at which pitchers get outs at shortstop per batted ball in play with the shortstop you’re trying to rate and without the shortstop you’re trying to rate (that is, when any other shortstop is fielding behind the same pitcher). Tango’s WOWY has Vizquel just slightly above average throughout his career, probably about +17 defensive runs. Again, just noise.

Michael A. Humphreys advises on tax aspects of international capital markets transactions at Ernst & Young LLP and is author of Wizardry: Baseball’s All-Time Greatest Fielders Revealed.

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By Jason Rosenhouse

A recent satirical essay in the Huffington Post reports that congressional Republicans are trying to legislate the value of pi. Fearing that the complexity of modern geometry is hurting America’s performance on international measures of mathematical knowledge, they have decreed that from now on pi shall be equal to three. It is a sad commentary on American culture that you must read slowly and carefully to be certain the essay is just satire.

It has been wisely observed that reality is that which, when you stop believing in it, doesn’t go away. Scientists are especially aware of this, since it is sometimes their sad duty to inform people of truths they would prefer not to accept. Evolution cannot be made to go away by folding you arms and shaking your head, and the planet is warming precipitously regardless of what certain business interests claim to believe. Likewise, the value of pi is what it is, no matter what a legislative body might think.

That value, of course, is found by dividing the circumference of a circle by its diameter. Except that if you take an actual circular object and apply your measuring devices to it you will obtain only a crude approximation to pi. The actual value is an irrational number, meaning that it is a decimal that goes on forever without repeating itself. One of my middle school math teachers once told me that it is just crazy for a number to behave in such a fashion, and that is why it is said to be irrational. Since I rather liked that explanation, you can imagine my disappointment at learning it was not correct.

In this context, the word “irrational” really just means “not a ratio.” More specifically, it is not a ratio of two integers. You see, if you divide one integer by another there are only two things that can happen. Either the process ends or it goes on forever by repeating a pattern. For example, if you divide one by four you get .25, while if you divide one by three you get .3333… . That these are the only possibilities can be proved with some elementary number theory, but I shall spare you the details of how that is done. That aside, our conclusion is that since pi never ends and never repeats, it cannot be written as one integer divided by another.

Which might make you wonder how anyone evaluated pi in the first place. If the number is defined geometrically, but we cannot hope to measure real circles with sufficient accuracy, then why do we constantly hear about computers evaluating its first umpteen million digits? The answer is that we are not forced to define pi in terms of circles. The number arises in other contexts, notably trigonometry. By coupling certain facts about right triangles with techniques drawn from calculus, you can express pi as the sum of a certain infinite series. That is, you can find a never-ending list of numbers that gets smaller and smaller and smaller, with the property that the more of the numbers you sum the better your approximation to pi. Very cool stuff.

Of course, I’m sure we all know that pi is a little bit larger than three. This means that any circle is just over three times larger around than it is across. The failure of most people to be able to visualize this leads to a classic bar bet. Take any tall, thin, drinking glass, the kind with a long stem, and ask the person sitting nearest you if its height is greater than its circumference. When he answers that it is, bet him that he is wrong. Optically, most such glasses appear to be much taller than they are fat, but unless your specimen is very tall and very thin you will win the bet every time. The circumference is more than three times larger than the diameter at the top of the glass. A vessel so proportioned that this length is nonetheless smaller than its height would be physically unpleasant to drink from.

Somehow, though, when people told you there was money to be made from mathematics I don’t think this is what they had in mind!

Jason Rosenhouse is Associate Professor of Mathematics at James Madison University and the voice behind EvolutionBlog. He is author of The Monty Hall Problem: The Remarkable Story Behind Math’s Most Contentious Brainteaser, as well as two forthcoming books Among the Creationists: Dispatches From the Anti-Evolution Frontline and Taking Sudoku Seriously: The Math Behind the World’s Most Popular Pencil Puzzle. These are his cats

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By Lauren Appelwick, Blog Editor

Raise your hand if you celebrate Valentine’s Day! OK…no one… Raise your hand if you like fun facts and history! Yay! Everyone! No matter how you feel about this holiday (or non-holiday, as it might be), I think the following might be of interest. Let’s start off with an excerpt from The Oxford Companion to the Year:

Valentine (3rd c.), martyr. In the fourth century, two martyrs of the name were honoured on the Via Flamninia, one at the second milestone, later identified as a Roman priest martyred on 14 February 269, the other at the sixty-third, near Terni, where he was later said to have been bishop. The two martyrs may be the same man, the bishop’s cult having spread from Terni to the capital. They have no obvious connection with lovers, though in 1998 the Irish tourist board, Bord Fáilte, wishing to promote Dublin as the international capital of romantic love, asserted that ‘St Valentine’ had conducted weddings for Roman soldiers against an order of Claudius II (268-70) forbidding them to marry (such a prohibition had been imposed by Augustus but repealed by Septimius Severus) and when condemned to death cured the judge’s daughter of blindness and sent her a letter signed ‘your Valentine’. The tradition that birds began to sing about this time gave rise in the late fourteenth century to a belief, attested by Chaucer and contemporaries both English and French, that they chose their mates on 14 February; the association of this time of year with the spring renewal of fertility goes back to the Roman festival of Lupercalia (15 Feb). This amorous behaviour passed from birds to human beings; in modern times it has been exported to other countries, even Japan, where it has mutated into a requirement for women to give chocolates to men, in particular their superiors at work. However, in 1994 the Holy Synod of the Greek Orthodox Church, observing that Valentine was a Western saint not recognized in the Eastern calendar, denounced his recently imported day and declared that if such celebrations of love were needed they should take place on St Hyacinthus’ day (3 July); and in Germany, where St Andrew is the patron of lovers, Sankt Velten is a euphemism for the Devil.

Now, how to choose a valentine? Historically, one might choose a valentine for the coming year in one of three ways: according to true desire, by the drawing of names on 13 Feb, or as the first person of the opposite sex encountered on the day. Thereby, the exchange of love-tokens became obligatory in some places, and the traditions of name-drawing evolved into complex rituals. In the passage below, one elaborate ritual is described by an anonymous girl dubbed “Arabella Whimsey” in the London paper The Connoisseur (20 Feb. 1755).

Last Friday…was Valentine’s Day; and I’ll tell you what I did the night before. I got five bay-leaves, and pinned four of them to the four corners of my pillow, and the fifth to the middle; and then if I dreamt of my sweetheart, Betty said we should be married before the year was out. But to make it more sure, I boiled an egg hard, and took out the yolk, and filled it up with salt; and when I went to bed, eat it shell and all, without speaking or drinking after it…We also wrote our lovers names upon bits of paper, and rolled them up in clay, and put them into water; and the frist that rose up, was to be our Valentine. Would you think it?–Mr. Blossom was my man: and I lay a-bed and shut my eyes all the morning, till he came to our house; for I would not have seen another man before him for all the world.

But this is modern day–you say–I would never go through so much trouble! Nor would I, and I rarely find myself giving even platonic, silly paper valentines after the I’m-bananas-about-you debacle of 1992. (That is a story for another time.) But perhaps you’re feeling festive anyway, and will appreciate the following math trick:

Happy…erm…Monday, everyone!

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By David Acheson

Why do so many people think they hate mathematics?
All too often, I suspect, the truth is that they were never let anywhere near it, but were fobbed off instead with something that was called mathematics, but which had none of the attractions of the real thing. In particular, they may have had no ‘big picture’ of the subject to help them along.

For what it’s worth, my own big picture of mathematics can be summed up in just six words: (i) surprising theorems, (ii) beautiful proofs and (iii) great applications.

While the subject started with applications to the physical world, mathematicians soon found beauty, too, in mathematics for its own sake, not least because of some of the elegant logical reasoning involved. My own personal view of mathematics at its best is in fact (i), (ii) and (iii) all at once, in one piece of work. That, in my view, is when you really open the champagne.

But above all, perhaps, it is the element of surprise that characterises much of mathematics at its best, and I got my first big mathematical surprise at the age of ten, in 1956. I was keen on conjuring at the time, and one day I came across the following ‘mind-reading’ trick in a magic book.

- Write down a 3-figure number. Any such number will do, provided the first and last figures differ by two or more.

- Now reverse your number, and subtract the smaller 3-figure number from the larger.

- Finally, reverse the result of that calculation, and add.

- Then the final answer will always be 1089, no matter which number you start with!

Okay, it’s not very ‘serious’ mathematics, but I have to tell you this: if you first see it as a 10-year old boy in 1956, it blows your socks off.

David Acheson is a Fellow at Jesus College, Oxford. He is the author of 1089 and All That: A Journey into Mathematics, which aims to make mathematics accessible to everyone. On the way, via Kepler and Newton, he explains what calculus really means, gives a brief history of pi, and even takes us to chaos theory and imaginary numbers, but ensures that no one gets lost along the way.

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