By: Franca Driessen,
on 10/19/2015
On 20 October 2015, the global mathematical community is celebrating World Statistics Day. In honour of this, we present here a reading list of OUP books and journal articles that have helped to advance the understanding of these mathematical concepts.
The post World Statistics Day: a reading list appeared first on OUPblog.
By: Amelia Carruthers,
on 10/19/2015
When I started my career as a medical statistician in September 1972, medical research was very different from now. In that month, the Lancet and the British Medical Journal published 61 research reports which used individual participant data, excluding case reports and animal studies. The median sample size was 36 people. In July 2010, I had another look.
The post Better medical research for longer, healthier lives appeared first on OUPblog.
By: Franca Driessen,
on 10/13/2015
Few elementary mathematical ideas arouse the kind of curiosity and astonishment among the uninitiated as does the idea of the “imaginary numbers”, an idea embodied in the somewhat mysterious number i. This symbol is used to denote the idea of , namely, a number that when multiplied by itself yields -1. How come?
The post The real charm of imaginary numbers appeared first on OUPblog.
By: Franca Driessen,
on 10/5/2015
Try googling 'mathematical gem'. I just got 465,000 results. Quite a lot. Indeed, the metaphor of mathematical ideas as precious little gems is an old one, and it is well known to anyone with a zest for mathematics. A diamond is a little, fully transparent structure all of whose parts can be observed with awe from any angle.
The post Diamonds are forever, and so are mathematical truths? appeared first on OUPblog.
By: Sara Pinotti,
on 9/16/2015
A friend of mine picked an argument with me the other day about how people go on about the beauty of mathematics, but this is not only not obvious to non-mathematicians, it cannot be accessed by those outside the field. Unlike, for example, the modern art, which is also not always obvious, mathematical beauty is elusive to all but the mathematicians. Or so he said.
The post Yes, maths can be for the amateur too appeared first on OUPblog.
By: Sara Pinotti,
on 8/18/2015
What is the purpose of mathematics? Or, as many a pupil would ask the teacher on a daily basis: “When are we going to need this?” There is a considerably ruder version of a question posed by Billy Connolly on the internet, but let’s not go there.
The post Will we ever need maths after school? appeared first on OUPblog.
By: Mohamed Sesay,
on 6/20/2015
Elijah Millgram, author of The Great Endarkenment, Svantje Guinebert, of the University of Bremen, to answer his questions and discuss the role of logic in philosophy. On other occasions, you’ve said that logic, at least the logic that most philosophers are taught, is stale science, and that it’s getting in the way of philosophers learning about newer developments. But surely logic is important for philosophers. Would you like to speak to the role of logic in philosophy?
The post The role of logic in philosophy: A Q&A with Elijah Millgram appeared first on OUPblog.
By: Leonard Mlodinow,
on 5/5/2015
When I was in graduate school at Berkeley I was offered a prestigious fellowship to study for a year in Germany, but I decided it would be a disruption, so I wrote a short note declining the offer. As, letter in hand, I stepped to the mailbox, I bumped into a woman from the scholarship [...]
By: Alex Guyver,
on 4/30/2015
Mathematics is used in increasingly sophisticated ways in modern society, explicitly by experts who develop applications and implicitly by the general public who use technological devices. As each of us is taught a broad curriculum in school and then focuses on particular specialisms in our adult life, it is useful to ask the question ‘what does it mean to make sense of mathematics?’.
The post Making sense of mathematics appeared first on OUPblog.
By: Alex Guyver,
on 4/4/2015
The idea of six degrees of separation is now quite well known and posits the appealing idea that any two humans on earth are connected by a chain of at most six common acquaintances. In the movie world this idea has become known as the “Bacon number”; for example Elvis Presley has a Bacon number […]
The post The Erdős number appeared first on OUPblog.
By: SoniaT,
on 3/14/2015
As somebody who loves words and English literature, I have often been assumed to be a natural enemy of the mathematical mind. If we’re being honest, my days of calculus and the hypotenuse are behind me, but with those qualifications under my belt, I did learn that the worlds of words and numbers are not necessarily as separate as they seem. Quite a few expressions use numbers (sixes and sevens, six of one and half a dozen of the other, one of a kind, etc.) but a few are more closely related to mathematics than you’d expect.
The post Putting two and two together appeared first on OUPblog.
By: vschneider46,
on 3/9/2015
Think there’s no need for sepia-toned filters and hashtags in your classroom? Don’t write off the world of #selfies just yet.
Instagram is one of the most popular social media channels among generation Z, or those born after 1995 and don’t know a world without the Internet. It shouldn’t come as much of a surprise that this is a generation of visual learners and communicators, where sharing your life-from the food you’re about to eat to your thoughts about anything and everything-is a part of your everyday routine. So, why allow Instagram in your classroom?
For starters, preparing students to be college and career ready involves helping them build their digital literacy skills on a professional level, and Instagram is a technological tool that offers educators innovative ways to motivate and engage students, opening up a new platform for collaboration, research, and discussion. Secondly, we all know the importance of interest and ownership for getting students excited about learning, and since your students probably already love Instagram you’ve already won half the battle.
Teacher/Classroom Instagram Accounts
Create a private classroom Instagram account that you control and 
can use to connect with your students, their parents and guardians, and other grade team members. Invite them to follow your account and catch a glimpse of your everyday classroom moments and adventures.
Student Instagram Accounts
Asking your students to follow the classroom Instagram account with their personal accounts is one, highly unlikely, and two, probably not the best idea. What you can do is ask your students to create additional Instagram accounts that would only be used for school or classroom purposes. You know how LinkedIn is your professional Facebook? A similar idea applies here.
fictional literary character and share images that he or she believes the specific character would post, highlighting the character’s interests, personality traits, and development throughout the story. The 15-second video option is a great way to really let students get into character through recorded role-playing and even performance reenactments. These activities can also be applied to important figures in history, such as the creator of Honda, Soichiro Honda, or jazz musician, Melba Liston.Note: Instagram, as well as Facebook, Twitter, Pinterest, Tumblr, and Snapchat, has a minimum age limit of 13 to open an account, but according to Instagram’s parents’ guide, there are many younger users on Instagram with their parents’ permission since you don’t have to specify your age. Always check with your school’s administrator and obtain parental permission before sharing photos of students or their work.
Know of any other interesting ways to use Instagram or other social media sites in the classroom? Already using Instagram in the classroom? Let us know in the comments!
Veronica has a degree from Mount Saint Mary College and joined LEE & LOW in the fall of 2014. She has a background in education and holds a New York State childhood education (1-6) and students with disabilities (1-6) certification. When she’s not wandering around New York City, you can find her hiking with her dog Milo in her hometown in the Hudson Valley, NY.
By: Charley,
on 3/2/2015
Modern society requires a reliable and trustworthy Internet infrastructure. To achieve this goal, cybersecurity research has previously drawn from a multitude of disciplines, including engineering, mathematics, and social sciences, as well as the humanities. Cybersecurity is concerned with the study of the protection of information – stored and processed by computer-based systems – that might be vulnerable to unintended exposure and misuse.
The post How do we protect ourselves from cybercrime? appeared first on OUPblog.
By: jilleisenberg14,
on 2/24/2015
I’ll be the first to admit it: I didn’t pay much attention to math. I specialized in literacy and focused on reading, speaking, listening, writing, social studies, and science instruction. Math? My third graders went down the hall each day to the “math classroom.” My co-teacher and I collaborated over best teaching practices, family relationships, and classroom management, but I didn’t spend time delving into the third-grade mathematics standards.
It wasn’t until I entered into our first parent-teachers-student conferences in September that I realized I couldn’t afford to compartmentalize my students’ learning.
In those conferences, we had students who loved math and had excelled in math every year leading up, but were now struggling to advance. They seemed to have hit an invisible wall. What happened?
Two words: Word problems.
Some of our students who were English Language Learners, reluctant readers, or who struggled to read at grade level for other reasons all of a sudden “couldn’t do” math anymore because the vocabulary, text length, and sentence structure were increasing in complexity. Even though they knew what 9 x 5 was, they couldn’t read and decipher the sentence:
Rene enjoys wearing a new outfit every day. His father bought him nine pairs of shorts and five shirts. Rene doesn’t want to wear any outfit twice. How many different outfit combinations does he have?
Now several of my students weren’t only struggling to read in my literacy class, but also struggling to read in math class. This was disheartening and confusing for them because math was a subject they loved, excelled at, and didn’t feel “below their grade level” because of language abilities or background schema. Yet reading challenges were following them down the hall and across instruction periods.
Guess what: Reading teachers are ALSO math teachers.
What?
Let me explain.
So, how can literacy teachers embrace math?
1. Nice to meet you, Math. I’m ELA. The Common Core website also falls victim to sequestering the ELA and math standards. Whether you teach both math and literacy or only one, compare the math standards to the ELA standards of your grade. Open two windows on your computer setting the Reading or Language standards of your grade side by side with the Operations & Algebraic Thinking standards for your grade. What do they have in common?
(Hint, hint: determining central idea of a text, interpreting unknown words or phrases, using context clues, and learning general academic and domain-specific words)
2. Share what read aloud or model text you are reading for the week or unit if you have a separate teacher for math instruction. In word problems, you or the math instructor can write a few of the problems about the characters. Reading In Her Hands: The Story of Sculptor Augusta Savage? Make Augusta the main character in the word problems.
This book has several money references because Augusta earned money from her teaching and from competitions she entered. Use some of the scenes in the book to review the values of currency. For example, Augusta earned a dollar every day from the principal of her school. How many different ways can you make $1.00 using combinations of quarters, dimes, nickels, and pennies?
3. Reward students with a math problem during the reading instruction block. (I’m telling you—students LOVE seeing you break out math during a literacy block). This gives students a break, uses a different part of their brains/thinking, and allows them to display their abilities in another subject (which is especially important if English makes a student feel doubtful or shy). Students can do this if they finish their required assignment early or you are transitioning between periods.
4. Allow students to create a word problem using the setting and characters of a book they are reading as an incentive, extension opportunity, or way to engage reluctant readers. Students can submit problems for you to review at the end of the day and the next day you can post one with the student author’s name. Students will have a chance to model (and observe) high quality writing and thinking, as well as delight in their peers’ recognition.
5. Word problems ARE story problems. Treat a word problem like any other fiction story. Have students identify the main character(s) and the problem. Give the word problem a setting. Encourage students to expand the math problem into a fiction story through writing or drawing.
6. Make a math bin in the classroom library. Whatever gets a student excited to read and pick up a book, right? Just as we will scour web deals and dig through yard sales for books on tiger sharks and poison dart frogs, don’t forget to hunt for math-themed books to add to your classroom library if math is your students’ passion.
7. Pick math-themed books to align with units students are covering in the grade level’s math standards. Great read alouds and leveled readers exist to help teach concepts around counting, money, time, geometry, and mixed operations, such as:
8. Even books without explicit math themes can inspire math conversations.
From Baby Flo: Florence Mills Lights Up the Stage:
From Silent Star: The Story of Deaf Major Leaguer William Hoy:
From Love Twelve Miles Long:
If students can’t read, they will struggle to succeed in math (and science and social studies). These challenges will compound with each year affecting self-confidence and commitment. Bridging math and literacy for students is a powerful way for students to see that learning how to derive meaning from text has real world applications and that you are invested in their entire education.
Jill Eisenberg, our Senior Literacy Expert, began her career teaching English as a Foreign Language to second through sixth graders in Yilan, Taiwan as a Fulbright Fellow. She went on to become a literacy teacher for third grade in San Jose, CA as a Teach for America corps member. She is certified in Project Glad instruction to promote English language acquisition and academic achievement. In her column she offers teaching and literacy tips for educators.
By: Alex Guyver,
on 2/17/2015
A couple of days after seeing Christopher Nolan’s Interstellar, I bumped into Sir Roger Penrose. If you haven’t seen the movie and don’t want spoilers, I’m sorry but you’d better stop reading now.
Still with me? Excellent.
Some of you may know that Sir Roger developed much of modern black hole theory with his collaborator, Stephen Hawking, and at the heart of Interstellar lies a very unusual black hole. Straightaway, I asked Sir Roger if he’d seen the film. What’s unusual about Gargantua, the black hole in Interstellar, is that it’s scientifically accurate, computer-modeled using Einstein’s field equations from General Relativity.
Scientists reckon they spend far too much time applying for funding and far too little thinking about their research as a consequence. And, generally, scientific budgets are dwarfed by those of Hollywood movies. To give you an idea, Alfonso Cuarón actually told me he briefly considered filming Gravity in space, and that was what’s officially classed as an “independent” movie. For big-budget studio blockbuster Interstellar, Kip Thorne, scientific advisor to Nolan and Caltech’s “Feynman Professor of Theoretical Physics”, seized his opportunity, making use of Nolan’s millions to see what a real black hole actually looks like. He wasn’t disappointed and neither was the director who decided to use the real thing in his movie without tweaks.
Black holes are so called because their gravitational fields are so strong that not even light can escape them. Originally, we thought these would be dark areas of the sky, blacker than space itself, meaning future starship captains might fall into them unawares. Nowadays we know the opposite is true – gravitational forces acting on the material spiralling into the black hole heat it to such high temperatures that it shines super-bright, forming a glowing “accretion disk”.
The computer program the visual effects team created revealed a curious rainbowed halo surrounding Gargantua’s accretion disk. At first they and Thorne presumed it was a glitch, but careful analysis revealed it was behavior buried in Einstein’s equations all along – the result of gravitational lensing. The movie had discovered a new scientific phenomenon and at least two academic papers will result: one aimed at the computer graphics community and the other for astrophysicists.
I knew Sir Roger would want to see the movie because there’s a long scene where you, the viewer, fly over the accretion disk–not something made up to look good for the IMAX audience (you have to see this in full IMAX) but our very best prediction of what a real black hole should look like. I was blown away.
Some parts of the movie are a little cringeworthy, not least the oft-repeated line, “that’s relativity”. But there’s a reason for the characters spelling this out. As well as accurately modeling the black hole, the plot requires relativistic “time dilation”. Even though every physicist has known how to travel in time for over a century (go very fast or enter a very strong gravitational field) the general public don’t seem to have cottoned on.
Most people don’t understand relativity, but they’re not alone. As a science editor, I’m privileged to meet many of the world’s most brilliant people. Early in my publishing career I was befriended by Subramanian Chandrasekhar, after whom the Chandra space telescope is now named. Penrose and Hawking built on Chandra’s groundbreaking work for which he received the Nobel Prize; his The Mathematical Theory of Black Holes (1954) is still in print and going strong.
When visiting Oxford from Chicago in the 1990s, Chandra and his wife Lalitha would come to my apartment for tea and we’d talk physics and cosmology. In one of my favorite memories he leant across the table and said, “Keith – Einstein never actually understood relativity”. Quite a bold statement and remarkably, one that Chandra’s own brilliance could end up rebutting.
Space is big – mind-bogglingly so once you start to think about it, but we only know how big because of Chandra. When a giant sun ends its life, it goes supernova – an explosion so bright it outshines all the billions of stars in its home galaxy combined. Chandra deduced that certain supernovae (called “type 1a”) will blaze with near identical brightness. Comparing the actual brightness with however bright it appears through our telescopes tells us how far away it is. Measuring distances is one of the hardest things in astronomy, but Chandra gave us an ingenious yardstick for the Universe.
In 1998, astrophysicists were observing type 1a supernovae that were a very long way away. Everyone’s heard of the Big Bang, the moment of creation of the Universe; even today, more than 13 billion years later, galaxies continue to rush apart from each other. The purpose of this experiment was to determine how much this rate of expansion was slowing down, due to gravity pulling the Universe back together. It turns out that the expansion’s speeding up. The results stunned the scientific world, led to Nobel Prizes, and gave us an anti-gravitational “force” christened “dark energy”. It also proved Einstein right (sort of) and, perhaps for the only time in his life, Chandra wrong.
Why Chandra told me Einstein was wrong was because of something Einstein himself called his “greatest mistake”. When relativity was first conceived, it was before Edwin Hubble (after whom another space telescope is named) had discovered space itself was expanding. Seeing that the stable solution of his equations would inevitably mean the collapse of everything in the Universe into some “big crunch”, Einstein devised the “cosmological constant” to prevent this from happening – an anti-gravitational force to maintain the presumed status quo.
Once Hubble released his findings, Einstein felt he’d made a dreadful error, as did most astrophysicists. However, the discovery of dark energy has changed all that and Einstein’s greatest mistake could yet prove an accidental triumph.
Of course Chandra knew Einstein understood relativity better than almost anyone on the planet, but it frustrates me that many people have such little grasp of this most beautiful and brilliant temple of science. Well done Christopher Nolan for trying to put that right.
Interstellar is an ambitious movie – I’d call it “Nolan’s 2001” – and it educates as well as entertains. While Matthew McConaughey barely ages in the movie, his young daughter lives to a ripe old age, all based on what we know to be true. Some reviewers have criticized the ending – something I thought I wouldn’t spoil for Sir Roger. Can you get useful information back out of a black hole? Hawking has changed his mind, now believing such a thing is possible, whereas Penrose remains convinced it cannot be done.
We don’t have all the answers, but whichever one of these giants of the field is right, Nolan has produced a thought-provoking and visually spectacular film.
Image Credit: “Best-Ever Snapshot of a Black Hole’s Jets.” Photo by NASA Goddard Space Flight Center. CC by 2.0 via Flickr.
The post That’s relativity appeared first on OUPblog.
By: Charley,
on 2/10/2015
Many attempts have been made to explain the historic and current lack of women working in STEM fields. During her two years of service as Director of Policy Planning for the US State Department, from 2009 to 2011, Anne-Marie Slaughter suggested a range of strategies for corporate and political environments to better support women at work. These spanned from social-psychological interventions to the introduction of role models and self-affirmation practices. Slaughter has written and spoken extensively on the topic of equality between men and women. Beyond abstract policy change, and continuing our celebration of women in STEM, there are practical tips and guidance for young women pursuing a career in Science, Technology, Engineering, or Mathematics.
(1) &nsbp; Be open to discussing your research with interested people.
From in-depth discussions at conferences in your field to a quick catch up with a passing colleague, it can be endlessly beneficial to bounce your ideas off a range of people. New insights can help you to better understand your own ideas.
(2) &nsbp; Explore research problems outside of your own.
Looking at problems from multiple viewpoints can add huge value to your original work. Explore peripheral work, look into the work of your colleagues, and read about the achievements of people whose work has influenced your own. New information has never been so discoverable and accessible as it is today. So, go forth and hunt!
(3) &nsbp; Collaborate with people from different backgrounds.
The chance of two people having read exactly the same works in their lifetime is nominal, so teaming up with others is guaranteed to bring you new ideas and perspectives you might never have found alone.
(4) &nsbp; Make sure your research is fun and fulfilling.
As with any line of work, if it stops being enjoyable, your performance can be at risk. Even highly self-motivated people have off days, so look for new ways to motivate yourself and drive your work forward. Sometimes this means taking some time to investigate a new perspective or angle from which to look at what you are doing. Sometimes this means allowing yourself time and distance from your work, so you can return with a fresh eye and a fresh mind!
(5) &nsbp; Surround yourself with friends who understand your passion for scientific research.
The life of a researcher can be lonely, particularly if you are working in a niche or emerging field. Choose your company wisely, ensuring your valuable time is spent with friends and family who support and respect your work.
Image Credit: “Board” by blickpixel. Public domain via Pixabay.
The post Five tips for women and girls pursuing STEM careers appeared first on OUPblog.
By: Charley,
on 1/22/2015
It is becoming widely accepted that women have, historically, been underrepresented and often completely written out of work in the fields of Science, Technology, Engineering, and Mathematics (STEM). Explanations for the gender gap in STEM fields range from genetically-determined interests, structural and territorial segregation, discrimination, and historic stereotypes. As well as encouraging steps toward positive change, we would also like to retrospectively honour those women whose past works have been overlooked.
From astronomer Caroline Herschel to the first female winner of the Fields Medal, Maryam Mirzakhani, you can use our interactive timeline to learn more about the women whose works in STEM fields have changed our world.
With free Oxford University Press content, we tell the stories and share the research of both famous and forgotten women.
Featured image credit: Microscope. Public Domain via Pixabay.
The post Celebrating Women in STEM appeared first on OUPblog.
By: Alex Guyver,
on 1/18/2015
Head hits cause brain damage, but not always. Should we ban sport to protect athletes? Exposure to electromagnetic fields is strongly associated with cancer development. Should we ban mobile phones and encourage old-fashioned wired communication? The sciences are getting more and more specialized and it is difficult to judge whether, say, we should trust homeopathy, fund a mission to Mars, or install solar panels on our roofs. We are confronted with questions about causality on an everyday basis, as well as in science and in policy.
Causality has been a headache for scholars since ancient times. The oldest extensive writings may have been Aristotle, who made causality a central part of his worldview. Then we jump 2,000 years until causality again became a prominent topic with Hume, who was a skeptic, in the sense that he believed we cannot think of causal relationships as logically necessary, nor can we establish them with certainty.
The next major philosophical figure after Hume was probably David Lewis, who proposed quite a controversial account saying roughly that something was a cause of an effect in this world if, in other nearby possible worlds where that cause didn’t happen, the effect didn’t happen either. Currently, we come to work in computer science originated by Judea Pearl and by Spirtes, Glymour and Scheines and collaborators.
All of this is highly theoretical and formal. Can we reconstruct philosophical theorizing about causality in the sciences in simpler terms than this? Sure we can!
One way is to start from scientific practice. Even though scientists often don’t talk explicitly about causality, it is there. Causality is an integral part of the scientific enterprise. Scientists don’t worry too much about what causality is – a chiefly metaphysical question – but are instead concerned with a number of activities that, one way or another, bear on causal notions. These are what we call the five scientific problems of causality:
This does not mean that metaphysical questions cease to be interesting. Quite the contrary! But by engaging with scientific practice, we can work towards a timely and solid philosophy of causality.
The traditional philosophical treatment of causality is to give a single conceptualization, an account of the concept of causality, which may also tell us what causality in the world is, and may then help us understand causal methods and scientific questions.
Our aim, instead, is to focus on the scientific questions, bearing in mind that there are five of them, and build a more pluralist view of causality, enriched by attention to the diversity of scientific practices. We think that many existing approaches to causality, such as mechanism, manipulationism, inferentialism, capacities and processes can be used together, as tiles in a causal mosaic that can be created to help you assess, develop, and criticize a scientific endeavour.
In this spirit we are attempting to develop, in collaboration, complementary ideas of causality as information (Illari) and variation (Russo). The idea is that we can conceptualize in general terms the causal linking or production of effect by the cause as the transmission of information between cause and effect (following Salmon); while variation is the most general conceptualization of the patterns of difference-making we can detect in populations where a cause is acting (following Mill). The thought is that we can use these complementary ideas to address the scientific problems.
For example, we can think about how we use complementary evidence in causal inference, tracking information transmission, and combining that with studies of variation in populations. Alternatively, we can think about how measuring variation may help us formulate policy decisions, as might seeking to block possible avenues of information transmission. Having both concepts available assists in describing this, and reasoning well – and they will also be combined with other concepts that have been made more precise in the philosophical literature, such as capacities and mechanisms.
Ultimately, the hope is that sharpening up the reasoning will assist in the conceptual enterprise that lies at the intersection of philosophy and science. And help decide whether to encourage sport, mobile phones, homeopathy and solar panels aboard the mission to Mars!
The post Why causality now? appeared first on OUPblog.
By: Alex Guyver,
on 1/11/2015
One of the central tasks when reading a mystery novel (or sitting on a jury) is figuring out which of the characters are trustworthy. Someone guilty will of course say they aren’t guilty, just like the innocent – the real question in these situations is whether we believe them.
The guilty party – let’s call her Annette – can try to convince us of her trustworthiness by only saying things that are true, insofar as such truthfulness doesn’t incriminate her (the old adage of making one’s lies as close to the truth as possible applies here). But this is not the only strategy available. In addition, Annette can attempt to deflect suspicion away from herself by questioning the trustworthiness of others – in short, she can say something like:
“I’m not a liar, Betty is!”
However, accusations of untrustworthiness of this sort are peculiar. The point of Annette’s pronouncement is to affirm her innocence, but such protestations rarely increase our overall level of trust. Either we don’t believe Annette, in which case our trust in Annette is likely to drop (without affecting how much we trust Betty), or we do believe Annette, in which case our trust in Betty is likely to decrease (without necessarily increasing our overall trust in Annette).
Thus, accusations of untrustworthiness tend to decrease the overall level of trust we place in those involved. But is this reflective of an actual increase in the number of lies told? In other words, does the logic of such accusations makes it the case that, the higher the number of accusations, the higher the number of characters that must be lying?
Consider a group of people G, and imagine that, simultaneously, each person in the group accuses one, some, or all of the other people in the group of lying right at this minute. For example, if our group consists of three people:
G = {Annette, Betty, Charlotte}
then Betty can make one of three distinct accusations:
“Annette is lying.”
“Charlotte is lying.”
“Both Annette and Charlotte are lying.”
Likewise, Annette and Charlotte each have three choices regarding their accusations. We can then ask which members of the group could be, or which must be, telling the truth, and which could be, or which must be, lying by examining the logical relations between the accusations made by each member of the group. For example, if Annette accuses both Betty and Charlotte of lying, then either (i) Annette is telling the truth, in which case both Betty and Charlotte’s accusations must be false, or (ii) Annette is lying, in which case either Betty is telling the truth or Charlotte is telling the truth (or both).
This set-up allows for cases that are paradoxical. If:
Annette says “Betty is lying.”
Betty says “Charlotte is lying.”
Charlotte says “Annette is lying.”
then there is no coherent way to assign the labels “liar” and “truth-teller” to the three in such a way as to make sense. Since we are here interested in investigating results regarding how many lies are told (rather than scenarios in which the notion of lying versus telling the truth breaks down), we shall restrict our attention to those groups, and their accusations, that are not paradoxical.
The following are two simple results that constraint the number of liars, and the number of truth-tellers, in any such group (I’ll provide proofs of these results in the comments after a few days).
“Accusations of untrustworthiness tend to decrease the overall level of trust we place in those involved”
Result 1: If, for some number m, each person in the group accuses at least m other people in the group of lying (and there is no paradox) then there are at least m liars in the group.
Result 2: If, for any two people in the group p1 and p2, either p1 accuses p2 of lying, or p2 accuses p1 of lying (and there is no paradox), then exactly one person in the group is telling the truth, and everyone else is lying.
These results support an affirmative answer to our question: Given a group of people, the more accusations of untrustworthiness (i.e., of lying) are made, the higher the minimum number of people in the group that must be lying. If there are enough accusations to guarantee that each person accuses at least n people, then there are at least n liars, and if there are enough to guarantee that there is an accusation between each pair of people, then all but one person is lying. (Exercise for the reader: show that there is no situation of this sort where everyone is lying).
Of course, the set-up just examined is extremely simple, and rather artificial. Conversations (or mystery novels, or court cases, etc.) in real life develop over time, involve all sorts of claims other than accusations, and can involve accusations of many different forms not included above, including:
“Everything Annette says is a lie!”
“Betty said something false yesterday!”
“What Charlotte is about to say is a lie!”
Nevertheless, with a bit more work (which I won’t do here) we can show that, the more accusations of untrustworthiness are made in a particular situation, the more of the claims made in that situation must be lies (of course, the details will depend both on the number of accusations and the kind of accusations). Thus, it’s as the title says: accusation breeds guilt!
Note: The inspiration for this blog post, as well as the phrase “Accusation breeds guilt” comes from a brief discussion of this phenomenon – in particular, of ‘Result 2′ above – in ‘Propositional Discourse Logic’, by S. Dyrkolbotn & M. Walicki, Synthese 191: 863 – 899.
The post Accusation breeds guilt appeared first on OUPblog.
By: DanP,
on 1/4/2015
In order to celebrate Trivia Day, we have put together a quiz with questions chosen at random from Very Short Introductions online. This is the perfect quiz for those who know a little about a lot. The topics range from Geopolitics to Happiness, and from French Literature to Mathematics. Do you have what it takes to take on this very short trivia quiz and become a trivia master? Take the quiz to find out…
We hope you enjoyed testing your trivia knowledge in this very short quiz.
Headline image credit: Pondering Away. © GlobalStock via iStock Photo.
The post A very short trivia quiz appeared first on OUPblog.
By: Charley,
on 11/14/2014
Alan Mathison Turing (1912-1954) was a mathematician and computer scientist, remembered for his revolutionary Automatic Computing Engine, on which the first personal computer was based, and his crucial role in breaking the ENIGMA code during the Second World War. He continues to be regarded as one of the greatest scientists of the 20th century.
We live in an age that Turing both predicted and defined. His life and achievements are starting to be celebrated in popular culture, largely with the help of the newly released film The Imitation Game, starring Benedict Cumberbatch as Turing and Keira Knightley as Joan Clarke. We’re proud to publish some of Turing’s own work in mathematics, computing, and artificial intelligence, as well as numerous explorations of his life and work. Use our interactive Enigma Machine below to learn more about Turing’s extraordinary achievements.
Image credits: (1) Bletchley Park Bombe by Antoine Taveneaux. CC-BY-SA-3.0 via Wikimedia Commons. (2) Alan Turing Aged 16, Unknown Artist. Public domain via Wikimedia Commons. (3) Good question by Garrett Coakley. CC-BY-SA 2.0 via Flickr.
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By: Charley,
on 11/2/2014
Are you worried about catching the flu, or perhaps even Ebola? Just how worried should you be? Well, that depends on how fast a disease will spread over social and transportation networks, so it’s obviously important to obtain good estimates of the speed of disease transmission and to figure out good containment strategies to combat disease spread.
Diseases, rumors, memes, and other information all spread over networks. A lot of research has explored the effects of network structure on such spreading. Unfortunately, most of this research has a major issue: it considers networks that are not realistic enough, and this can lead to incorrect predictions of transmission speeds, which people are most important in a network, and so on. So how does one address this problem?
Traditionally, most studies of propagation on networks assume a very simple network structure that is static and only includes one type of connection between people. By contrast, real networks change in time — one contacts different people during weekdays and on weekends, one (hopefully) stays home when one is sick, new University students arrive from all parts of the world every autumn to settle into new cities. They also include multiple types of social ties (Facebook, Twitter, and – gasp – even face-to-face friendships), multiple modes of transportation, and so on. That is, we consume and communicate information through all sorts of channels. To consider a network with only one type of social tie ignores these facts and can potentially lead to incorrect predictions of which memes go viral and how fast information spreads. It also fails to allow differentiation between people who are important in one medium from people who are important in a different medium (or across multiple media). In fact, most real networks include a far richer “multilayer” structure. Collapsing such structures to obtain and then study a simpler network representation can yield incorrect answers for how fast diseases or ideas spread, the robustness level of infrastructures, how long it takes for interaction oscillators to synchronize, and more.
Recently, an increasingly large number of researchers are studying mathematical objects called “multilayer networks”. These generalize ordinary networks and allow one to incorporate time-dependence, multiple modes of connection, and other complexities. Work on multilayer networks dates back many decades in fields like sociology and engineering, and of course it is well-known that networks don’t exist in isolation but rather are coupled to other networks. The last few years have seen a rapid explosion of new theoretical tools to study multilayer networks.
And what types of things do researchers need to figure out? For one thing, it is known that multilayer structures induce correlations that are invisible if one collapses multilayer networks into simpler representations, so it is essential to figure out when and by how much such correlations increase or decrease the propagation of diseases and information, how they change the ability of oscillators to synchronize, and so on. From the standpoint of theory, it is necessary to develop better methods to measure multilayer structures, as a large majority of the tools that have been used thus far to study multilayer networks are mostly just more complicated versions of existing diagnostic and models. We need to do better. It is also necessary to systematically examine the effects of multilayer structures, such as correlations between different layers (e.g., perhaps a person who is important for the social network that is encapsulated in one layer also tends to be important in other layers?), on different types of dynamical processes. In these efforts, it is crucial to consider not only simplistic (“toy”) models — as in most of the work on multilayer networks thus far — but to move the field towards the examination of ever more realistic and diverse models and to estimate the parameters of these models from empirical data. As our review article illustrates, multilayer networks are both exciting and important to study, but the increasingly large community that is studying them still has a long way to go. We hope that our article will help steer these efforts, which promise to be very fruitful.
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By: Rebekah Daniels,
on 10/17/2014
If a “revolution” in our field or area of knowledge was ongoing, would we feel it and recognize it? And if so, how?
I think a methodological “revolution” is probably going on in the science of epidemiology, but I’m not totally sure. Of course, in science not being sure is part of our normal state. And we mostly like it. I had the feeling that a revolution was ongoing in epidemiology many times. While reading scientific articles, for example. And I saw signs of it, which I think are clear, when reading the latest draft of the forthcoming book Causal Inference by M.A. Hernán and J.M. Robins from Harvard (Chapman & Hall / CRC, 2015). I think the “revolution” — or should we just call it a “renewal”? — is deeply changing how epidemiological and clinical research is conceived, how causal inferences are made, and how we assess the validity and relevance of epidemiological findings. I suspect it may be having an immense impact on the production of scientific evidence in the health, life, and social sciences. If this were so, then the impact would also be large on most policies, programs, services, and products in which such evidence is used. And it would be affecting thousands of institutions, organizations and companies, millions of people.
One example: at present, in clinical and epidemiological research, every week “paradoxes” are being deconstructed. Apparent paradoxes that have long been observed, and whose causal interpretation was at best dubious, are now shown to have little or no causal significance. For example, while obesity is a well-established risk factor for type 2 diabetes (T2D), among people who already developed T2D the obese fare better than T2D individuals with normal weight. Obese diabetics appear to survive longer and to have a milder clinical course than non-obese diabetics. But it is now being shown that the observation lacks causal significance. (Yes, indeed, an observation may be real and yet lack causal meaning.) The demonstration comes from physicians, epidemiologists, and mathematicians like Robins, Hernán, and colleagues as diverse as S. Greenland, J. Pearl, A. Wilcox, C. Weinberg, S. Hernández-Díaz, N. Pearce, C. Poole, T. Lash , J. Ioannidis, P. Rosenbaum, D. Lawlor, J. Vandenbroucke, G. Davey Smith, T. VanderWeele, or E. Tchetgen, among others. They are building methodological knowledge upon knowledge and methods generated by graph theory, computer science, or artificial intelligence. Perhaps one way to explain the main reason to argue that observations as the mentioned “obesity paradox” lack causal significance, is that “conditioning on a collider” (in our example, focusing only on individuals who developed T2D) creates a spurious association between obesity and survival.
The “revolution” is partly founded on complex mathematics, and concepts as “counterfactuals,” as well as on attractive “causal diagrams” like Directed Acyclic Graphs (DAGs). Causal diagrams are a simple way to encode our subject-matter knowledge, and our assumptions, about the qualitative causal structure of a problem. Causal diagrams also encode information about potential associations between the variables in the causal network. DAGs must be drawn following rules much more strict than the informal, heuristic graphs that we all use intuitively. Amazingly, but not surprisingly, the new approaches provide insights that are beyond most methods in current use. In particular, the new methods go far deeper and beyond the methods of “modern epidemiology,” a methodological, conceptual, and partly ideological current whose main eclosion took place in the 1980s lead by statisticians and epidemiologists as O. Miettinen, B. MacMahon, K. Rothman, S. Greenland, S. Lemeshow, D. Hosmer, P. Armitage, J. Fleiss, D. Clayton, M. Susser, D. Rubin, G. Guyatt, D. Altman, J. Kalbfleisch, R. Prentice, N. Breslow, N. Day, D. Kleinbaum, and others.
We live exciting days of paradox deconstruction. It is probably part of a wider cultural phenomenon, if you think of the “deconstruction of the Spanish omelette” authored by Ferran Adrià when he was the world-famous chef at the elBulli restaurant. Yes, just kidding.
Right now I cannot find a better or easier way to document the possible “revolution” in epidemiological and clinical research. Worse, I cannot find a firm way to assess whether my impressions are true. No doubt this is partly due to my ignorance in the social sciences. Actually, I don’t know much about social studies of science, epistemic communities, or knowledge construction. Maybe this is why I claimed that a sociology of epidemiology is much needed. A sociology of epidemiology would apply the scientific principles and methods of sociology to the science, discipline, and profession of epidemiology in order to improve understanding of the wider social causes and consequences of epidemiologists’ professional and scientific organization, patterns of practice, ideas, knowledge, and cultures (e.g., institutional arrangements, academic norms, scientific discourses, defense of identity, and epistemic authority). It could also address the patterns of interaction of epidemiologists with other branches of science and professions (e.g. clinical medicine, public health, the other health, life, and social sciences), and with social agents, organizations, and systems (e.g. the economic, political, and legal systems). I believe the tradition of sociology in epidemiology is rich, while the sociology of epidemiology is virtually uncharted (in the sense of not mapped neither surveyed) and unchartered (i.e. not furnished with a charter or constitution).
Another way I can suggest to look at what may be happening with clinical and epidemiological research methods is to read the changes that we are witnessing in the definitions of basic concepts as risk, rate, risk ratio, attributable fraction, bias, selection bias, confounding, residual confounding, interaction, cumulative and density sampling, open population, test hypothesis, null hypothesis, causal null, causal inference, Berkson’s bias, Simpson’s paradox, frequentist statistics, generalizability, representativeness, missing data, standardization, or overadjustment. The possible existence of a “revolution” might also be assessed in recent and new terms as collider, M-bias, causal diagram, backdoor (biasing path), instrumental variable, negative controls, inverse probability weighting, identifiability, transportability, positivity, ignorability, collapsibility, exchangeable, g-estimation, marginal structural models, risk set, immortal time bias, Mendelian randomization, nonmonotonic, counterfactual outcome, potential outcome, sample space, or false discovery rate.
You may say: “And what about textbooks? Are they changing dramatically? Has one changed the rules?” Well, the new generation of textbooks is just emerging, and very few people have yet read them. Two good examples are the already mentioned text by Hernán and Robins, and the soon to be published by T. VanderWeele, Explanation in causal inference: Methods for mediation and interaction (Oxford University Press, 2015). Clues can also be found in widely used textbooks by K. Rothman et al. (Modern Epidemiology, Lippincott-Raven, 2008), M. Szklo and J Nieto (Epidemiology: Beyond the Basics, Jones & Bartlett, 2014), or L. Gordis (Epidemiology, Elsevier, 2009).
Finally, another good way to assess what might be changing is to read what gets published in top journals as Epidemiology, the International Journal of Epidemiology, the American Journal of Epidemiology, or the Journal of Clinical Epidemiology. Pick up any issue of the main epidemiologic journals and you will find several examples of what I suspect is going on. If you feel like it, look for the DAGs. I recently saw a tweet saying “A DAG in The Lancet!”. It was a surprise: major clinical journals are lagging behind. But they will soon follow and adopt the new methods: the clinical relevance of the latter is huge. Or is it not such a big deal? If no “revolution” is going on, how are we to know?
Feature image credit: Test tubes by PublicDomainPictures. Public Domain via Pixabay.
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By: Alice,
on 10/11/2014
Why do we teach students how to prove things we all know already, such as 0.9999••• =1?
Partly, of course, so they develop thinking skills to use on questions whose truth-status they won’t know in advance. Another part, however, concerns the dialogue nature of proof: a proof must be not only correct, but also persuasive: and persuasiveness is not objective and absolute, it’s a two-body problem. Not only to tango does one need two.
The statements — (1) ice floats on water, (2) ice is less dense than water — are widely acknowledged as facts and, usually, as interchangeable facts. But although rooted in everyday experience, they are not that experience. We have firstly represented stuffs of experience by sounds English speakers use to stand for them, then represented these sounds by word-processor symbols that, by common agreement, stand for them. Two steps away from reality already! This is what humans do: we invent symbols for perceived realities and, eventually, evolve procedures for manipulating them in ways that mirror how their real-world origins behave. Virtually no communication between two persons, and possibly not much internal dialogue within one mind, can proceed without this. Man is a symbol-using animal.
Statement (1) counts as fact because folk living in cooler climates have directly observed it throughout history (and conflicting evidence is lacking). Statement (2) is factual in a significantly different sense, arising by further abstraction from (1) and from a million similar experiential observations. Partly to explain (1) and its many cousins, we have conceived ideas like mass, volume, ratio of mass to volume, and explored for generations towards the conclusion that mass-to-volume works out the same for similar materials under similar conditions, and that the comparison of mass-to-volume ratios predicts which materials will float upon others.
Statement (3): 19 is a prime number. In what sense is this a fact? Its roots are deep in direct experience: the hunter-gatherer wishing to share nineteen apples equally with his two brothers or his three sons or his five children must have discovered that he couldn’t without extending his circle of acquaintance so far that each got only one, long before he had a name for what we call ‘nineteen’. But (3) is many steps away from the experience where it is grounded. It involves conceptualisation of numerical measurements of sets one encounters, and millennia of thought to acquire symbols for these and codify procedures for manipulating them in ways that mirror how reality functions. We’ve done this so successfully that it’s easy to forget how far from the tangibles of experience they stand.
Statement (4): √2 is not exactly the ratio of two whole numbers. Most first-year mathematics students know this. But by this stage of abstraction, separating its fact-ness from its demonstration is impossible: the property of being exactly a fraction is not detectable by physical experience. It is a property of how we abstracted and systematised the numbers that proved useful in modelling reality, not of our hands-on experience of reality. The reason we regard √2’s irrationality as factual is precisely because we can give a demonstration within an accepted logical framework.
What then about recurring decimals? For persuasive argument, first ascertain the distance from reality at which the question arises: not, in this case, the rarified atmosphere of undergraduate mathematics but the primary school classroom. Once a child has learned rituals for dividing whole numbers and the convenience of decimal notation, she will try to divide, say, 2 by 3 and will hit a problem. The decimal representation of the answer does not cease to spew out digits of lesser and lesser significance no matter how long she keeps turning the handle. What should we reply when she asks whether zero point infinitely many 6s is or is not two thirds, or even — as a thoughtful child should — whether zero point infinitely many 6s is a legitimate symbol at all?
The answer must be tailored to the questioner’s needs, but the natural way forward — though it took us centuries to make it logically watertight! — is the nineteenth-century definition of sum of an infinite series. For the primary school kid it may suffice to say that, by writing down enough 6s, we’d get as close to 2/3 as we’d need for any practical purpose. For differential calculus we’d need something better, and for model-theoretic discourse involving infinitesimals something better again. Yet the underpinning mathematics for equalities like 0.6666••• = 2/3 where the question arises is the nineteenth-century one. Its fact-ness therefore resembles that of ice being less dense than water, of 19 being prime or of √2 being irrational. It can be demonstrated within a logical framework that systematises our observations of real-world experiences. So it is a fact not about reality but about the models we build to explain reality. Demonstration is the only tool available for establishing its truth.
Mathematics without proof is not like an omelette without salt and pepper; it is like an omelette without egg.
Headline image credit: Floating ice sheets in Antarctica. CC0 via Pixabay.
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By: Sara Pinotti,
on 9/7/2014
Why should you study paradoxes? The easiest way to answer this question is with a story:
In 2002 I was attending a conference on self-reference in Copenhagen, Denmark. During one of the breaks I got a chance to chat with Raymond Smullyan, who is amongst other things an accomplished magician, a distinguished mathematical logician, and perhaps the most well-known popularizer of `Knight and Knave’ (K&K) puzzles.
K&K puzzles involve an imaginary island populated by two tribes: the Knights and the Knaves. Knights always tell the truth, and Knaves always lie (further, members of both tribes are forbidden to engage in activities that might lead to paradoxes or situations that break these rules). Other than their linguistic behavior, there is nothing that distinguishes Knights from Knaves.
Typically, K&K puzzles involve trying to answer questions based on assertions made by, or questions answered by, an inhabitant of the island. For example, a classic K&K puzzle involves meeting an islander at a fork in the road, where one path leads to riches and success and the other leads to pain and ruin. You are allowed to ask the islander one question, after which you must pick a path. Not knowing to which tribe the islander belongs, and hence whether she will lie or tell the truth, what question should you ask?
(Answer: You should ask “Which path would someone from the other tribe say was the one leading to riches and success?”, and then take the path not indicated by the islander).
Back to Copenhagen in 2002: Seizing my chance, I challenged Smullyan with the following K&K puzzle, of my own devising:
There is a nightclub on the island of Knights and Knaves, known as the Prime Club. The Prime Club has one strict rule: the number of occupants in the club must be a prime number at all times.
The Prime Club also has strict bouncers (who stand outside the doors and do not count as occupants) enforcing this rule. In addition, a strange tradition has become customary at the Prime Club: Every so often the occupants form a conga line, and sing a song. The first lyric of the song is:
“At least one of us in the club is a Knave.”
and is sung by the first person in the line. The second lyric of the song is:
“At least two of us in the club are Knaves.”
and is sung by the second person in the line. The third person (if there is one) sings:
“At least three of us in the club are Knaves.”
And so on down the line, until everyone has sung a verse.
One day you walk by the club, and hear the song being sung. How many people are in the club?
Smullyan’s immediate response to this puzzle was something like “That can’t be solved – there isn’t enough information”. But he then stood alone in the corner of the reception area for about five minutes, thinking, before returning to confidently (and correctly, of course) answer “Two!”
I won’t spoil things by giving away the solution – I’ll leave that mystery for interested readers to solve on their own. (Hint: if the song is sung with any other prime number of islanders in the club, a paradox results!) I will note that the song is equivalent to a more formal construction involving a list of sentences of the form:
At least one of sentences S1 – Sn is false.
At least two of sentences S1 – Sn is false.
————————————————
At least n of sentences S1 – Sn is false.
The point of this story isn’t to brag about having stumped a famous logician (even for a mere five minutes), although I admit that this episode (not only stumping Smullyan, but meeting him in the first place) is still one of the highlights of my academic career.
Instead, the story, and the puzzle at the center of it, illustrates the reasons why I find paradoxes so fascinating and worthy of serious intellectual effort. The standard story regarding why paradoxes are so important is that, although they are sometimes silly in-and-of-themselves, paradoxes indicate that there is something deeply flawed in our understanding of some basic philosophical notion (truth, in the case of the semantic paradoxes linked to K&K puzzles).
Another reason for their popularity is that they are a lot of fun. Both of these are really good reasons for thinking deeply about paradoxes. But neither is the real reason why I find them so fascinating. The real reason I find paradoxes so captivating is that they are much more mathematically complicated, and as a result much more mathematically interesting, than standard accounts (which typically equate paradoxes with the presence of some sort of circularity) might have you believe.
The Prime Club puzzle demonstrates that whether a particular collection of sentences is or is not paradoxical can depend on all sorts of surprising mathematical properties, such as whether there is an even or odd number of sentences in the collection, or whether the number of sentences in the collection is prime or composite, or all sorts of even weirder and more surprising conditions.
Other examples demonstrate that whether a construction (or, equivalently, a K&K story) is paradoxical can depend on whether the referential relation involved in the construction (i.e. the relation that holds between two sentences if one refers to the other) is symmetric, or is transitive.
The paradoxicality of still another type of construction, involving infinitely many sentences, depends on whether cofinitely many of the sentences each refer to cofinitely many of the other sentences in the construction (a set is cofinite if its complement is finite). And this only scratches the surface!
The more I think about and work on paradoxes, the more I marvel at how complicated the mathematical conditions for generating paradoxes are: it takes a lot more than the mere presence of circularity to generate a mathematical or semantic paradox, and stating exactly what is minimally required is still too difficult a question to answer precisely. And that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work remains to be done, and a lot of complexity and beauty remaining to be discovered.
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