The post The final piece of the puzzle appeared first on OUPblog.

]]>The Nobel Prize in Physics 2017 was recently awarded to Rainer Weiss, Barry C. Barish, and Kip S. Thorne, “for decisive contributions to the LIGO detector and the observation of gravitational waves”. This has provoked discussion about how Nobel Prizes are awarded. Weiss himself noted that the work that led to the prize involved around a thousand scientists. Martin Rees, the Astronomer Royal, said:

“Of course, LIGO’s success was owed to literally hundreds of dedicated scientists and engineers. The fact that the Nobel committee refuses to make group awards is causing them increasingly frequent problems – and giving a misleading and unfair impression of how a lot of science is actually done.”

There are perhaps two difficult questions here. If a group of people publish a research paper, how do they divide up the credit amongst themselves? And if a breakthrough is a result of a number of papers, by a whole group of people, who gets the acclaim?

The convention in my own subject of pure mathematics is usually that on a paper with multiple authors, the names appear in alphabetical order by last name, regardless of who did the most work, who made the largest contribution, or who is most senior. That, at least, is relatively straightforward – although it does mean for example that when reviewing candidates for a job, it can be hard to identify exactly what someone contributed to their joint publications.

More recently, some mathematicians have started to work in new large-scale online collaborations, and this raises all sorts of questions about how credit is assigned. When the mathematician Tim Gowers first proposed experimenting with such a collaboration, which he called a ‘Polymath’ project, he specified in advance how any resulting research papers would be published. The last of his twelve Polymath rules says

“Suppose the experiment actually results in something publishable. Even if only a very small number of people contribute the lion’s share of the ideas, the paper will still be submitted under a collective pseudonym with a link to the entire online discussion.”

Since, to everyone’s surprise, the first Polymath project did indeed lead to a research paper, this rule was immediately implemented, and it has continued in this way for subsequent Polymath projects. The discussions that led to the paper are all still available online, on various blogs and wikis, so if someone wants to check an individual’s specific contribution, they can do exactly that – which is not the case for more traditional collaborations.

Within pure mathematics, breakthroughs have mostly been attributed to the individual or small group of people who put the final piece in the jigsaw puzzle. That is perhaps unsurprising. If a person gives a solution to a problem or proves a conjecture, then they should get the credit, shouldn’t they? But mathematical arguments don’t usually exist in isolation: most often one piece of work builds on many previous ideas. Isaac Newton famously said “If I have seen further it is by standing on the shoulders of Giants”. If other mathematicians contributed ideas that were crucial ingredients but that don’t have the glamour of a complete solution to a famous problem, how can they receive appropriate credit for their work?

The public nature of the Polymath projects makes it possible to track progress on some problems in a way that has not previously been possible. In 2010, there was a Polymath collaborative project on the ‘Erdős discrepancy problem’, but the project did not reach a solution. The problem was subsequently solved by Terry Tao in 2015. Tao had been one of the participants in the Polymath5 project on the problem, and in his paper he acknowledged the role that Polymath5 had played in his work. He also built on work by Kaisa Matomäki and Maksym Radziwiłł, and a suggestion by Uwe Stroinski that a recent paper of Matomäki, Radziwiłł, and Tao might be linked to the Erdős discrepancy problem. I am not commenting on this example because I think that anyone has behaved badly. On the contrary, Tao was scrupulous about acknowledging all of this in his paper. Rather, the public collaborative aspect of the story has made it easier than usual to trace the journey that eventually led to a solution. Without a doubt it was Tao who put it all together, added his own crucial ideas and insights, and came up with a solution, but it does seem that others’ contributions were key to the breakthrough coming at that particular moment. I suspect that history will record that “the Erdős discrepancy problem was solved by Tao”, without the nuances.

Two of the mathematicians I have mentioned, Tim Gowers and Terry Tao, are winners of the Fields Medal, one of the highest honours to be awarded in mathematics. The Fields Medal is awarded “to recognize outstanding mathematical achievement for existing work and for the promise of future achievement”. I am curious about whether in the future the Fields Medal, the Abel Prize, or any of the other accolades in mathematics might be awarded to a Polymath collaboration that has achieved something extraordinary.

*Featured image credit: Pay by geralt. Public domain via **Pixabay**. *

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]]>The post On serendipity, metals, and networks appeared first on OUPblog.

]]>The first run of metals data through complex networks algorithms happened the night before we were due to deliver our ideas on how networks research can benefit early Balkan metallurgy research at a physics conference. Needless to say, we (over)committed ourselves to delivering a fresh view on this topic without giving it a thorough thought, rather, we hoped that our enthusiasm would do the job. That night, the best and the worst thing happened: our results presented the separation of modules, or most densely connected structures in our networks, as statistically, archaeologically, and spatiotemporally significant. The bad news was that we stumbled upon it in a classic serendipitous manner – we did not know what was it that we pursued, but it looked too good to let go. It subsequently took us three years to get to the bottom of networks analysis we made that night.

In simple terms, what we did here is present ancient societies as a network. A large number of systems can be represented as a network. For example, human society is a network where the nodes are people and the links are social or genetic ties between them. A lot of real-world networks exhibit nontrivial properties that we do not observe in a regular lattice or in the network where we connect the nodes randomly. For example social networks have the property called ‘six degrees of separation’, which means that the distance between any of us to anybody else on the planet is less than six steps of friendships. So any of us knows somebody, who knows somebody etc (six times) who knows Barack Obama or fishermen on a small island in Indonesia. Another property that is common in complex networks is so-called modularity. This means that some parts of the network are more densely connected with each other than with other parts of the network. Successful investigation of modularity or community structure property of networks includes detecting modules in citation networks, or pollination systems – in our case we used this property to shed light on the connections between prehistoric societies that traded copper. It turned out that they did not do it randomly, but within their own network of dense social ties, which are remarkably consistent with the distribution of known archaeological phenomena at the time (c. 6200- 3200 BC), or cultures.

What we managed to capture were properties of highly interconnected systems based on copper supply networks that also reflected organisation of social and economic ties between c. 6200 and c. 3200 BC.

Our example is the first 3,000 years of development of metallurgy in the Balkans. The case study includes more than 400 prehistoric copper artefacts: copper ores, beads made of green copper ores, production debris like slags, and a variety of copper metal artefacts, from trinkets to massive hammer axes weighing around 1kg each. Although our database was filled with detailed archaeological, spatial, and temporal information about each of 400+ artefacts used to design and conduct networks analyses, we only employed chemical analysis, which is the information acquired independently, and can be replicated. Importantly, we operated under the premise that networks of copper supply can reveal information relevant for the specific histories of people behind these movements, and hence reflect human behaviour.

Our initial aim was to see how supply networks of copper artefacts were organised in the past, and as the last step of analysis we planned to utilize geographical location only to facilitate visual representation of our results. Basically, if two artefacts from the same chemical cluster were found in two different sites, we placed a link between them. In the final step, the so-called Louvain algorithm was applied in order to identify structures in our networks, and we used it as a good modularity optimization method. Another advantage is this approach is that we can test its statistical significance and put a probability figure to the obtained modules.

What we managed to capture were properties of highly interconnected systems based on copper supply networks that also reflected organisation of social and economic ties between c. 6200 and c. 3200 BC. The intensity of algorithmically calculated social interaction revealed three main groups of communities (or modules) that are archaeologically, spatiotemporally, and statistically significant across the studied period (and represented in different colours in Figure 1). These communities display substantial correlation with at least three dominant archaeological cultures that represented main economic and social cores of copper industries in the Balkans during these 3,000 years (Figure 2). Basically, such correlation shows that known archaeological phenomena can be mathematically evaluated using modularity approach.

Although serendipity marked the beginnings of our research, our plan is to take it from here with a detailed research strategy plan, which now includes looking at other aspects of material culture (not only metals), testing the model on datasets across prehistoric Europe, or indeed different chronological periods. We can say that the Balkan example worked out well because metal supply and circulation played a great role in the lives of societies within an observed period, but it may not apply in cases where this economy was not as developed. The most exciting part for us though was changing our perspective on what archaeological culture might represent. Traditional systematics is commonly looking at cultures as depositions of similar associations of materials, dwelling and subsistence forms across distinct space-time, and debates come down to either grouping or splitting distinctive archaeological cultures based on expressions of similarity and reproduction across the defined time and space. But now we have the opportunity to change this perspective and look at the strength of links between similar material culture, rather than their accumulation patterns. This is a game changer for us. And we hope that this research inspires colleagues to pursue this idea of measuring connectedness amongst past societies in order to shed more light on how people in the past cooperated, and why.

*Featured image credit: Mountains in Bulgaria by Alex Dimitrov. CC BY-SA 4.0 via Wikimedia Commons. *

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]]>The post Mathematical reasoning and the human mind [excerpt] appeared first on OUPblog.

]]>Mathematics is more than the memorization and application of various rules. Although the language of mathematics can be intimidating, the concepts themselves are built into everyday life. In the following excerpt from *A Brief History of Mathematical Thought*, Luke Heaton examines the concepts behind mathematics and the language we use to describe them.

There is strong empirical evidence that before they learn to speak, and long before they learn mathematics, children start to structure their perceptual world. For example, a child might play with some eggs by putting them in a bowl, and they have some sense that this collection of eggs is in a different spatial region to the things that are outside the bowl. This kind of spatial understanding is a basic cognitive ability, and we do not need symbols to begin to appreciate the sense that we can make of moving something into or out of a container. Furthermore, we can see in an instant the difference between collections containing one, two, three or four eggs. These cognitive capacities enable us to see that when we add an egg to our bowl (moving it from outside to inside), the collection somehow changes, and likewise, taking an egg out of the bowl changes the collection. Even when we have a bowl of sugar, where we cannot see how many grains there might be, small children have some kind of understanding of the process of adding sugar to a bowl, or taking some sugar away. That is to say, we can recognize particular acts of adding sugar to a bowl as being examples of someone ‘adding something to a bowl’, so the word ‘adding’ has some grounding in physical experience.

Of course, adding sugar to my cup of tea is not an example of mathematical addition. My point is that our innate cognitive capabilities provide a foundation for our notions of containers, of collections of things, and of adding or taking away from those collections. Furthermore, when we teach the more sophisticated, abstract concepts of addition and subtraction (which are certainly not innate), we do so by referring to those more basic, physically grounded forms of understanding. When we use pen and paper to do some sums we do not literally add objects to a collection, but it is no coincidence that we use the same words for both mathematical addition and the physical case where we literally move some objects. After all, even the greatest of mathematicians first understood mathematical addition by hearing things like ‘If you have two apples in a basket and you add three more, how many do you have?’

As the cognitive scientists George Lakoff and Rafael Núñez argue in their thought-provoking and controversial book *Where Mathematics Comes From*, our understanding of mathematical symbols is rooted in our cognitive capabilities. In particular, we have some innate understanding of spatial relations, and we have the ability to construct ‘conceptual metaphors’, where we understand an idea or conceptual domain by employing the language and patterns of thought that were first developed in some other domain. The use of conceptual metaphor is something that is common to all forms of understanding, and as such it is not characteristic of mathematics in particular. That is simply to say, I take it for granted that new ideas do not descend from on high: they must relate to what we already know, as physically embodied human beings, and we explain new concepts by talking about how they are akin to some other, familiar concept.

Conceptual mappings from one thing to another are fundamental to human understanding, not least because they allow us to reason about unfamiliar or abstract things by using the inferential structure of things that are deeply familiar. For example, when we are asked to think about adding the numbers two and three, we know that this operation is like adding three apples to a basket that already contains two apples, and it is also like taking two steps followed by three steps. Of course, whether we are imagining moving apples into a basket or thinking about an abstract form of addition, we don’t actually need to move any objects. Furthermore, we understand that the touch and smell of apples are not part of the facts of addition, as the concepts involved are very general, and can be applied to all manner of situations. Nevertheless, we understand that when we are adding two numbers, the meaning of the symbols entitles us to think in terms of concrete, physical cases, though we are not obliged to do so. Indeed, it may well be true to say that our minds and brains are capable of forming abstract number concepts because we are capable of thinking about particular, concrete cases.

Mathematical reasoning involves rules and definitions, and the fact that computers can add correctly demonstrates that you don’t even need to have a brain to correctly employ a specific, notational system. In other words, in a very limited way we can ‘do mathematics’ without needing to reflect on the significance or meaning of our symbols. However, mathematics isn’t only about the proper, rule-governed use of symbols: it is about *ideas *that can be expressed by the rule-governed use of symbols, and it seems that many mathematical ideas are deeply rooted in the structure of the world that we perceive.

*Featured image credit: “mental-human-experience-mindset” by johnhain. CC0 via **Pixabay.
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]]>The post Who needs quantum key distribution? appeared first on OUPblog.

]]>Should we be impressed? Yes – scientific breakthroughs are great things.

Does this revolutionise the future of cyber security? No – sadly, almost certainly not.

At the heart of modern cyber security is cryptography, which provides a kit of mathematically-based tools for providing core security services such as confidentiality (restricting who can access data), data integrity (making sure that any unauthorised changes to data are detected), and authentication (identifying the correct source of data). We rely on cryptography every day for securing everything we do in cyberspace, such as banking, mobile phone calls, online shopping, messaging, social media, etc. Since everything is in cyberspace these days, cryptography also underpins the security of the likes of governments, power stations, homes, and cars.

Cryptography relies on secrets, known as keys, which act in a similar role to keys in the physical world. *Encryption*, for example, is the digital equivalent of locking information inside a box. Only those who have access to the key can open the box to retrieve the contents. Anyone else can shake the box all they like – the contents remain inaccessible without access to the key.

A challenge in cryptography is *key distribution*, which means getting the right cryptographic key to those (and only those) who need it. There are many different techniques for key distribution. For many of our everyday applications key distribution is effortless, since keys come preinstalled on devices that we acquire (for example, mobile SIM cards, bank cards, car key fobs, etc.) In other cases it is straightforward because devices that need to share keys are physically close to one another (for example, you read the key on the label of your Wi-Fi router and type it into devices you permit to connect).

Key distribution is more challenging when the communicating parties are far from one another and do not have any business relationship during which keys could have been distributed. This is typically the case when you buy something from an online store or engage in a WhatsApp message exchange. Key distribution in these situations is tricky, but very solvable, using techniques based on a special set of cryptographic tools known as *public-key cryptography*. Your devices use such techniques every day to distribute keys, without you even being aware it is happening.

There is yet another way of distributing keys, known as *quantum key distribution*. This uses a quantum channel such as line of sight or fibre-optic cable to exchange light particles, from which a cryptographic key can eventually be extracted. Distance limitations, poor data rates, and the reliance on specialist equipment have previously made quantum key distribution more of a scientific curiosity than a practical technology. What the Chinese scientists have done is blow the current distance record for quantum key distribution from around 100kms to 1000kms, through the use of a satellite. That’s impressive.

However, the Chinese scientists have not significantly improved the case for using quantum key distribution in the first place. We can happily distribute cryptographic keys today without lasers and satellites, so why would we ever need to? Just because we can?

Well, there’s a glimmer of a case. For the likes of banking and mobile phones, it seems unlikely we will ever need quantum key distribution. However, for applications which currently rely on public-key cryptography, there is a problem brewing. If anyone gets around to building a practical quantum computer (and we’re not talking tomorrow), then current public-key cryptographic techniques will become insecure. This is because a quantum computer will efficiently solve the hard mathematical problems on which today’s public-key cryptography relies. Cryptographers today are thus developing new types of public-key cryptography that will resist quantum computers. I am confident they will succeed. When they do, we will be able to continue distributing keys in similar ways to today.—in other words, without quantum key distribution.

Who needs quantum key distribution then? Frankly, it’s hard to make a case, but let’s try. One possible advantage of quantum key distribution is that it enables the use of a highly secure form of encryption known as the *one-time pad*. One reason almost nobody uses the one-time pad is that it’s a complete hassle to distribute its keys. Quantum key distribution would solve this. More importantly, however, nobody uses the one-time pad today because modern encryption techniques are so strong. If you don’t believe me, look how frustrated some government agencies are that we are using them. We don’t use the one-time pad because we don’t need to. The same argument applies to quantum key distribution itself.

Finally, let’s just suppose that there is an application which somehow merits the use of the one-time pad. Do the one-time pad and quantum key distribution provide the ultimate security that physicists often claim? Here’s the really bad news. We have just been discussing all the wrong things. Cyber security rarely fails due to problems with encryption algorithms or the ways that cryptographic keys are distributed. Much more common are failures in the systems and processes surrounding cryptography. These include poor implementations and misuse. For example, one-time pads and quantum key distribution don’t protect data after it is decrypted, or if a key is accidentally used twice, or if someone forgets to turn encryption on, etc. We already have good encryption and key distribution techniques. We need to get much better at building secure systems.

So, I’m very impressed that a cryptographic key can be distributed via satellite. That’s great – but I don’t think this will revolutionise cryptography. And I certainly don’t feel any more secure as a result.

*Featured image credit: Virus by geralt. CC0 public domain via **Pixabay**.*

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]]>The post The life and work of Alan Turing appeared first on OUPblog.

]]>Pioneering the field of ‘machine intelligence’, today we celebrate all of Turing’s achievements and the legacy his research left. Find out more about some of the key events that shaped his investigations with this interactive timeline.

*Featured image credit: Enigma by Rama. CC BY-SA 3.0 via **Wikimedia Commons**.*

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]]>The post Suspected ‘fake results’ in science appeared first on OUPblog.

]]>Because it is based on random sampling model, a ‘P value’ implies that the probability of a treatment being truly better in a large idealized study is very near to ‘1 – P’ *provided* that it is calculated by using a symmetrical (e.g. Gaussian) distribution, that the study is described accurately so that someone else can repeat in exactly the same way, the study is performed with no hidden biases, and there are no other study results that contradict it. It should also be borne in mind that ‘truly better’ in this context includes differences of just greater than ‘no difference’, so that ‘truly better’ may not necessarily mean a big difference. However, if the above conditions of accuracy etc. are not met then the probability of the treatment being truly better than placebo in an idealized study will be lower (i.e. it will range from an upper limit of ‘1 – P’ [e.g. 1 – 0.025 = 0.975] down to zero). This is so because the possible outcomes of a very large number of random samples are always equally probable, this being a special property of the random sampling process. I will explain.

Figure 1 represents a large population two mutually exclusive subgroups. One contains people with ‘appendicitis’ numbering 80M + 20M = 100M; the other group has ‘no appendicitis’ numbering 120M + 180M = 300M. Now, say that a single computer file contains all the records of *only one* of these groups and we have to guess which group it holds. In order to help us, we are told that 80M/(80M+20M) = 80% of those with appendicitis have RLQ pain and that 120M/(120M+180M) = 40% of those without appendicitis have RLQ pain as shown in figure 1. In order to find out which one of the group’s records is in the computer file, we could perform an ‘idealised’ study. This would involve selecting an individual patient’s record at random from the unknown group and looking to see if that person had RLQ pain or not. If the person had RLQ pain we could write ‘RLQ pain’ on a card and put it into a box. We could repeat this process an ideally large number (N) times (e.g. thousands).

If we had been selecting from the group of people with appendicitis then we would get the result in Box A where 80N/100N = 80% of the cards had ‘RLQ pain’ written on them. However, if we had been selecting from people without appendicitis, we would get the result in Box B, with 120N/300N = 40% of the cards bearing ‘RLQ pain’. We would then be able to tell immediately from which group of people we had been selecting. Note that random sampling only ‘sees’ the *proportion* with RLQ pain in each group (i.e. either 80% or 40%). It is immaterial that the size of the group of people in figure 1 with appendicitis (100M) is different to the group without appendicitis (300M).

The current confusion about ‘P values’ is because this ‘fact’ is overlooked and that it is assumed wrongly that a difference in size of the source populations affects the sampling process. A scientist would be interested in the possible long term outcome of an idealised study (in this case the possible contents of the two boxes A and B) not in the various proportions in the unknown source population.

Making a large number of ‘N’ random selections would represent an idealized study. In practice we cannot do such idealized studies but have to make do with a smaller number of observations. For example, we would have to try to predict from which of these possible boxes with N cards representing ideal study outcomes we would have selected a smaller sample. If we selected 24 cards at random from the box of cards drawn from the computer file containing details of the unknown population and found that 15 by chance had ‘RLQ pain’, we can work out the probability (from the binomial distribution e.g. when n=24, r=15 and p=0.8) of getting 15/24 exactly from each possible box A and B. From Box A it would be 0.023554 and from Box B it would be 0.0141483. The proportions in box A and B are not affected by the numbers with and without appendicitis in the source population and are therefore equally probable before the random selections were made. This allows us to work out the probability that the computer file contained records of patients with appendicitis by dividing 0.023554 by (0.023554 + 0.0141483) = 0.6247. The probability of the computer file containing the ‘no appendicitis’ group would thus be 1- 0.6247 = 0.3753.

It does not matter how many possible idealized study results we have to consider; they will always be equally probable. This is because each possible idealized random selection study result is not affected by differences in sizes of the source populations. So, if a ‘P value’ is 0.025 based on a symmetrical (e.g. Gaussian) distribution, the probability of a treatment being better than placebo will be 1 – P = 0.975 or less if there are inaccuracies, biases, or other very similar studies that give contrary results, etc. These factors will have to be taken into account in most cases.

*Featured image credit: Edited STATS1_P-VALUE originally by fickleandfreckled. CC BY 2.0 via Flickr.*

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]]>The post What is game theory? appeared first on OUPblog.

]]>Despite the theory’s origins dating back to Neumann and Morgenstern’s work, the economists John Nash, John Harsanyi, and Reinhard Selten received the Nobel Prize for Economics in 1994 for further developing game theory in relation to economics. Here are some interesting facts on the field; from its key influencers and terms, to how it applies in everyday life and examples.

- Game theory can be thought of as an extension of decision theory. In standard decision theory, each agent has utilities associated with outcomes. However, in game theory each agent also has to consider the utilities of other agents and how they will affect the other agent’s decisions and the overall outcome.
- The term ‘Tit for Tat’ is a concept used in the mathematical side of game theory. It is used to describe when a player responds with the same action or move used by an opponent in the previous action or move.

- One of the most celebrated theorems of game theory is referred to as the minimax theorem. This theorem explains that there is always a solution to a conflict between two people with opposing interests.
- “Common Knowledge” is widely used in game theory. This refers to the assumption in games that everyone knows a piece of information but does not essentially know if everyone else knows it too.
- Focal point or Schelling point is one of the many key terms used in game theory. It was developed by the American economist Thomas Schelling in his book
*The Strategy of Conflict*which was published in 1960. Thomas Schelling and Robert J. Aumann both were awarded a noble Prize in economics for developing game theory analysis in 2005. - Prisoner’s Dilemma is one of the best known examples of games analysed in game theory. The name’s origin comes from a situation that involves two prisoners who would have to choose either ‘confess’ or ‘don’t confess’ without knowing what the other person will choose. This game aims to illustrate how people behave in tactical situations.
- Another widely used example is known as the Battle of the Sexes Game. For instance, two partners would like to share an evening together. However, they have two different ideas of what they would like to do but still would prefer to be together than attend two separate events. This game is used to demonstrate the pros and cons of coordination.
- John Forbes Nash Jr was an American mathematician renowned for his contribution to game theory. The phrase Nash equilibrium used in game theory is named after him.

*Featured image credit: checkmate chess by Stevepb. Public domain via Pixabay.*

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]]>The post Mathematics Masterclasses for young people appeared first on OUPblog.

]]>In fact the idea really goes back to Michael Faraday, who gave Christmas lectures about science for young people at The Royal Institution of Great Britain in London in 1826. Sir Christopher Zeeman, following upon Porter’s initiative, gave the first series of six one-hour lectures (Mathematics Masterclasses) to young people at The Royal Institution in 1981, about “The Nature of Mathematics and The Mathematics of Nature”.

A consequence has been initiatives, widespread now throughout the United Kingdom, of Mathematics Masterclasses, in particular for age groups from 8 to 18 years of age, and with enthusiastic local organisers. I served for several years on the Committee at The Royal Institution whose role was to encourage those Masterclasses nationally.

A reasonable definition of a Masterclass might be that it is devised for “students” (of whatever age level) who have a ready curiosity about what goes on around them, and an interest in identifying an explanation of what they observe, even if that explanation is not immediately obvious but requires, perhaps, a two- or three-stage process to arrive at a solution. The “speaker” will have an intrinsic interest in drawing out an answer from such students, and also of devising problems from any circumstances that lie within the area just stated. In mathematics, the solution process will normally require the identification of appropriate “variables” to describe the problem, the formulation of suitable relations (equations) between those variables, and then the “solution” of those equations in a way which expresses an unknown quantity entirely in terms of known quantities. That is how mathematics “works”.

Every year in the 1990s in Berkshire, England, sixty 12-year-old pupils were gathered at Mathematics Masterclasses at the University of Reading. Attendees were nominated by their schools and showed an aptitude for maths. Two parallel sessions were held, each containing 30 pupils, a lecturer, and qualified helpers.

A typical Masterclass might last for up to three hours (with refreshment breaks, and tutorial sessions, interspersing the lecture material) and broken up into three sessions. Ideally there will be several volunteer teachers circulating to give advice during the tutorial sessions. Teachers from the participating schools were readily found to be enthusiastic to volunteer for this role.

Examples of topics treated in Masterclasses have been “Weather” (the atmosphere and forces therein) by Sir Brian Hoskins, “Water Waves” (in deep and shallow water, and in groups) by Winifred Wood, and the “Dynamics of Dinosaurs” (e.g. their weight and speed) by Michael Sewell. I also gave a Masterclass about “Balloons and Bubbles”, which used mathematics allied to classroom demonstrations to illustrate an associated sequence of topics: pressure, equilibrium of a spherical bubble, tension in a soap film, tension in rubber, pressure peaks and pits, and cylindrical balloons.

The long-term benefit of a Masterclass, and one of its objectives, is to encourage a lasting enthusiasm and curiosity about how to devise a “model” of a natural phenomenon by using mathematics, and thereby to develop the capacity for original thinking about an observed situation in nature, and which is still within the scope of schoolchildren.

An example of an everyday problem suitable for a Masterclass is the following “Coffee Shop Problem”, actually posed to me by my wife in that situation. Given eight points equally distributed around a circle, how many differently shaped triangles can be drawn using only three of those points as vertices? Now generalise the problem by introducing more equally spaced points, and looking for different polygons (not just triangles). This teaches one how to realise that any given problem may be the start of a much larger problem, which is an important part of any mathematical investigation, and which may not be at first apparent.

A further example of a Masterclass problem is the following. Draw a right-angled triangle with unequal shorter sides. Draw three circles, each using one of those sides as the diameter. The two external regions between the larger circle and (in turn) the two smaller circles are called lunes (because they each have the shape of a crescent Moon). Now, prove Hippocrates Theorem (c. 410 B.C.), that the sum of the areas of external lunes is equal to the area of the right-angled triangle.

*Featured image: Calculator by 422737. Public domain via **Pixabay**. *

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]]>The post Coincidences are underrated appeared first on OUPblog.

]]>The unreasonable popularity of pseudosciences such as ESP or astrology often stems from personal experience. We’ve all had that “Ralph” phone call or some other happening that seems well beyond the range of normal probability, at least according to what we consider to be common sense. But how accurately does common sense forecast probabilities and how much of it is fuzzy math? As we will see, fuzzy math holds its own.

Let’s try to de-fuzz the math a bit, starting with a classic example: the birthday problem. Perhaps you’ve encountered this problem in a math class that dealt with probabilities. In a group of people, what is the probability that two people would have the same birthday? Certainly it must depend on the size of the group. If we start with only two people, the chance would be one out of 365 – well, OK, one in 366 for leap years. If the group included more people, common sense might suggest that the probability would just increase linearly. So, to get a 50% chance, you might think it would take 183 people in the group. Wrong. That’s where common sense goes off the rails. It turns out that, in a group of only 23 people, the probability of two having the same birthday is 50%.

Details of the logic required to arrive at this result are unnecessary here, but a clue is given by a group of three. The third person might match the birthday of either of the first two, so you might think to just double the first probability. But think about this from the inverse point of view. The probability of the second person’s birthday NOT matching the first is 364/365. But the third person could match either of the first two, so the probability of NOT matching is only 363/365. Since NOT matching is thus less probable, matching becomes more probable. Working this out involves a bit of number crunching, but math classes have calculators galore, and since many classes have 23 or more members, real data are available to support the probability calculation. As you can see, what we take to be common sense often yields inaccurate solutions.

Meanwhile, back at the “Ralph” problem, a math textbook might tackle this problem in terms of drawing different colored pebbles from a large urn. Let’s forego that approach, and set the “Ralph” problem in more realistic terms. Suppose you know N people. During the course of a single day, a number of those people, k, cross your mind on a purely random basis. For this illustration, let’s agree to ignore close relatives and friends that you think about almost every day. Next, a certain number of people, L, contact you in a given day by any means, including phone calls, e-mails and electronic messages, social media, snail mail, and random meetings.

Working though this problem (actually kind of fun if you like mathematical puzzles) yields an equation for the number of days that will elapse before the probability of getting a contact from someone you thought about reaches a given level. Of course, it depends on the variables N, k, and L, not the easiest quantities to obtain.

An estimate of N, the number of people that an average person knows, is available from various sources, and ranges from 200 to 1500, but k, the number of people one would think about is highly subjective, as is L, the number of contacts one receives in an average day. Yet, all these numbers are necessary to find an estimate of the time required for someone you thought about to contact you shortly after you thought about them. Unscientific surveys of students, neighbors, and friends produced numbers of thoughts from 10 to 100 and contacts from 5 to 30. Substituting these numbers into the appropriate derived equation and requiring that it be 90% probable yields a remarkable result. Such coincidences would happen anywhere from once a week to once every other month. Most people’s fuzzy math would probably have estimated a much longer time period.

If you are curious about how often you might expect such coincidences to occur, e-mail your numbers for N, k, and L to me and I’ll calculate the estimate for your case and send it to you.

Next time you get that “Ralph” call, rather than attributing it to ESP, you might tell Ralph: “Hey, I was just thinking about you, so you can consider yourself my coincidence of the month.”

*Featured image: Ancient Planet by PIRO4D. Public domain via **Pixabay**. *

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]]>The post Prime numbers and how to find them appeared first on OUPblog.

]]>A prime number is always bigger than 1 and can only be divided by itself and 1 – no other number will divide in to it. So the number 2 is the first prime number, then 3, 5, 7, and so on. Non-prime numbers are defined as composite numbers (they are composed of other smaller numbers).

Prime numbers are so tantalizing because they seem to be in never ending supply, and are distributed somewhat randomly throughout all the other numbers. Also, no-one has (yet) found a simple and quick way to find a specific (new) prime number.

Because of this, very large prime numbers are used every day when encrypting data to make the online world a safer place to communicate, move money, and control our households. But could we ever run out of prime numbers? How can we find new, incredibly large prime numbers? Below is a brief explanation about how we can do this:

This got us interested in learning more about primes, so we’ve collected together some facts about these elusive numbers:

- A simple way to find prime numbers is to write out a list of all numbers and then cross off the composite numbers as you find them – this is called the
*Sieve of Eratosthenes*. However, this can take a long time! - In 2002 a quicker way to test whether a number is prime was discovered – an algorithm called the ‘AKS primality test’, published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena.
- Even though prime numbers seem to be randomly distributed, there are fewer large primes than smaller ones. This is logical, as there are more ways for large numbers to not be prime, but mathematicians ask: how much rarer are larger primes?
- In 2001 a group of computer scientists from IBM and Stanford University showed that a quantum computer could be programmed to find the prime factors of numbers.
- The RSA enciphering process, published in 1978 by Ron Rivest, Adi Shamir, and Leonard Adleman, is used to hide plaintext messages using prime numbers. In this process every person has a private key which is made up of three numbers, two of which are very large prime numbers.
- At any moment in time, the largest known prime number is also usually the largest known Mersenne prime.

*Featured image credit: numbers by morebyless. CC-BY-2.0 via **Flickr**.*

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]]>The post Opening the door for the next Ramanujan appeared first on OUPblog.

]]>It is still possible to learn mathematics to a high standard at a British university but there is no doubt that the fun and satisfaction the subject affords to those who naturally enjoy it has taken a hit. Students are constantly reminded about the lifelong debts they are incurring and how they need to be thoroughly aware of the demands of their future employers. The fretting over this within universities is relentless. To be fair, students generally report high to very high levels of satisfaction with their courses right throughout the university system. Certainly university staff are kept on their toes by being annually called to account by the National Student Survey, which is a data set that offers no hiding place. We should bear in mind, however, that this key performance indicator does not measure the extent to which students have mastered their degree subject. What is important here is getting everyone to say they are happy, which is quite another matter.

This all contrasts with the life of the main character, Sri Ramanujan in the recent film *The Man Who Knew Infinity*. The Indian genius of the early twentieth century had a reasonable high school education after which he was almost self-taught. It seems he got hold of a handful of British mathematics books, amongst them *Synopsis of Pure Mathematics* by G. S. Carr, written in 1886. I understand that this was not even a very good book in the ordinary sense for it merely listed around five thousand mathematical facts in a rather disjointed fashion with little in the way of example or proof. This compendium, however, suited the young Ramanujan perfectly for he devoured it, filling in the gaps and making new additions of his own. Through this process of learning he emerged as a master of deep and difficult aspects of mathematics, although inevitably he remained quite ignorant of some other important fields within the subject.

It would therefore be a very good thing if everyone had unfettered online access to the contents of a British general mathematics degree. Mathematics is the subject among the sciences that most lends itself to learning through books and online sources alone. There is nothing fake or phoney when it comes to maths. The content of the subject, being completely and undeniably true, does not date. Mathematics texts and lectures from many decades ago remain as valuable as ever. Indeed, older texts are often refreshing to read because they are so free from clutter. There are new developments of course but learning from high quality older material will never lead you astray.

I had thought this had already been taken care of as for ten years or more, many universities, for example MIT in the United States, have granted open online access to all their teaching materials, completely free of charge. There is no need to even register your interest — just go to their website and help yourself. Modern day Ramanujans would seem not to have a problem coming to grips with the subject.

The reality, however, is somewhat different and softer barriers remain. The attitude of these admirable institutions is relaxed but not necessarily that helpful to the private student who is left very much to their own devices. There is little guidance as to what you need to know, and what is available online depends on the decisions of individual lecturers so there is no consistency of presentation. Acquiring an overall picture of mainstream mathematics is not as straightforward as one might expect. It would be a relatively easy thing to remedy this and the rather rigid framework of British degrees could be useful. In Britain, a degree normally consists of 24 modules (eight per year), each demanding a minimum of 50 hours of study (coffee breaks not included). If we were to set up a suite of 24 modules for a general mathematics degree that met the so-called QAA Benchmark and placed the collection online for anyone on the planet to access, it would be welcomed by poor would-be mathematicians from everywhere around the globe. The simplicity and clarity of that setting would be understood and appreciated.

This modern day Ramanujan Project would require some work by the mathematical community but it would largely be a one-off task. As I have explained, the basic content of a mathematical undergraduate degree has no need to change rapidly over time for here we are talking about fundamental advanced mathematics and not cutting-edge research. Everyone, even a Ramanujan, needs to learn to walk before they can run and the helping hand we will be offering will long be remembered with gratitude and be a force for good in the world.

*Featured image credit: Black-and-white by Pexels. CC0 public domain via Pixabay.*

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]]>The post The historian and the longitude appeared first on OUPblog.

]]>When Harrison arrived in London in the 1730s with ambitions to build a successful longitude timepiece, he was supported and encouraged by Fellows of the Royal Society, who occasioned the very first meeting of the Board of Longitude (formed some 20 years previously), at which a clock presented by Harrison was the only item of business. He requested, and was granted, the very considerable sum of £500 to work on a second timepiece, to be finished in two years (the annual salary of the Astronomer Royal was £100). This was the first of a series of grants that had amounted to £4,000 by the time Harrison announced in 1760 that his third timepiece was ready for testing. It had taken 19 years to complete and the Board were, not unreasonably, becoming doubtful whether this was the route to a practical solution to the problem. To say that such a sum was inadequate is to ignore completely the simple fact that this was the 18th century, long before the accepted notion of government grants for research and development, but this is just one example of where a historian becomes frustrated with the popular narrative.

In the event, Harrison asked for a fourth timepiece – quite unlike the first three – to be given the statutory test of keeping time on a voyage to the West Indies. Many difficulties and arguments had to be overcome before a satisfactory test was completed in 1764, when everyone agreed that ‘the watch’ had kept time within the limits required for the maximum award of £20,000. It was now that the Board’s difficulties began in earnest. Faced with the real prospect of parting with their major award, they needed to know that the longitude problem really had been solved – anything less would have been a very public failure to fulfilling their central responsibility. The original Act of Parliament of 1714 offered the reward for a method that was ‘Practicable and Useful at Sea’, while stipulating that the test was to be a single voyage. The Board were troubled over whether these two criteria were compatible, and such doubts were being aired in the popular press. Was the legislation itself inadequate?

So far the Board had not been given a detailed account of the watch’s manufacture and operation, and they wanted to know what principles or manufacturing procedures had resulted in its outstanding performance. Could these be explained and communicated to other makers? Could such watches be manufactured in numbers, in a reasonable time, at a reasonable cost, by moderately competent makers? Might the success of Harrison’s watch have been a matter of chance in a single instance? Had it depended on the achievement of a wholly exceptional, individual talent? All of these considerations were relevant to the question of a ‘practicable and useful’ method, notwithstanding the recent performance of the watch.

The Board decided to separate the components of the legislation by granting Harrison half the full reward, once he had explained the watch and its operation, while retaining the other half until it could be proved that watches of this type could go into routine production. Harrison did ‘discover’ his watch, as it was said (that is, literally, he removed the cover and explained its working), and so was granted £10,000, but gave up on the Board and appealed to Parliament and the King for the remainder.

In many ways the Board were left, as they had feared, without a practical solution. Harrison’s watch did not go into regular production. He had shown that a timepiece could keep time as required, but the design of the successful marine chronometer, as it emerged towards the end of the century, was quite different from his work. Other makers, in France for example, had been making independent advances, and two English makers, John Arnold and Thomas Earnshaw, brought the chronometer to a manageable and successful format. It is difficult to claim without important qualification that Harrison solved the longitude problem in a practical sense. In the broad sweep of the history of navigation, Harrison was not a major contributor.

The Harrison story seems to attract challenge and controversy. The longitude exhibition at the National Maritime Museum in 2014 was an attempt to offer a more balanced account than has been in vogue recently. The Astronomer Royal Nevil Maskelyne, for example, has been maligned without justification. A recent article in *The Horological Journal* takes a contrary view and offers ‘An Antidote to John Harrison’, and we seem set for another round of disputation. From a historian’s point of view, one of the casualties of the enthusiasm of recent years has been an appreciation of the context of the whole affair, while a degree of partisanship has obscured the legitimate positions of many of the characters involved. There is a much richer and more interesting story to be written than the one-dimensional tale of virtue and villainy.

*Featured image credit: Pocket watch time of Sand by annca. Public domain via Pixabay. *

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]]>The post In defense of mathematics [excerpt] appeared first on OUPblog.

]]>Once reframed in its historical context, mathematics quickly loses its intimidating status. As a subject innately tied to culture, art, and philosophy, the study of mathematics leads to a clearer understanding of human culture and the world in which we live. In this shortened excerpt from *A Brief History of Mathematical Thought*, Luke Heaton discusses the reputation of mathematics and its significance to human life.

Mathematics is often praised (or ignored) on the grounds that it is far removed from the lives of ordinary people, but that assessment of the subject is utterly mistaken. As G. H. Hardy observed in *A Mathematician’s Apology*:

Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.

The considerable popularity of sudoku is a case in point. These puzzles require nothing but the application of mathematical logic, and yet to avoid scaring people off, they often carry the disclaimer “no mathematical knowledge required!” The mathematics that we know shapes the way we see the world, not least because mathematics serves as “the handmaiden of the sciences.” For example, an economist, an engineer, or a biologist might measure something several times, and then summarize their measurements by finding the mean or average value. Because we have developed the symbolic techniques for calculating mean values, we can formulate the useful but highly abstract concept of “the mean value.” We can only do this because we have a mathematical system of symbols. Without those symbols we could not record our data, let alone define the mean.

Mathematicians are interested in concepts and patterns, not just computation. Nevertheless, it should be clear to everyone that computational techniques have been of vital importance for many millennia. For example, most forms of trade are literally inconceivable without the concept of number, and without mathematics you could not organize an empire, or develop modern science. More generally, mathematical ideas are not just practically important: the conceptual tools that we have at our disposal shape the way we approach the world. As the psychologist Abraham Maslow famously remarked, “If the only tool you have is a hammer, you tend to treat everything as if it were a nail.” Although our ability to count, calculate, and measure things in the world is practically and psychologically critical, it is important to emphasize that mathematicians do not spend their time making calculations.

The great edifice of mathematical theorems has a crystalline perfection, and it can seem far removed from the messy and contingent realities of the everyday world. Nevertheless, mathematics is a product of human culture, which has co-evolved with our attempts to comprehend the world. Rather than picturing mathematics as the study of “abstract” objects, we can describe it as a poetry of patterns, in which our language brings about the truth that it proclaims. The idea that mathematicians bring about the truths that they proclaim may sound rather mysterious, but as a simple example, just think about the game of chess. By describing the rules we can call the game of chess into being, complete with truths that we did not think of when we first invented it. For example, whether or not anyone has ever actually played the game, we can prove that you cannot force a competent player into checkmate if the only pieces at your disposal are a king and a pair of knights. Chess is clearly a human invention, but this fact about chess must be true in any world where the rules of chess are the same, and we cannot imagine a world where we could not decide to keep our familiar rules in place.

Mathematical language and methodology present and represent structures that we can study, and those structures or patterns are as much a human invention as the game of chess. However, mathematics as a whole is much more than an arbitrary game, as the linguistic technologies that we have developed are genuinely fit for human purpose. For example, people (and other animals) mentally gather objects into groups, and we have found that the process of counting really does elucidate the plurality of those groups. Furthermore, the many different branches of mathematics are profoundly interconnected, to art, science, and the rest of mathematics.

In short, mathematics is a language and while we may be astounded that the universe is at all comprehensible, we should not be surprised that science is mathematical. Scientists need to be able to communicate their theories and when we have a rule-governed understanding, the instructions that a student can follow draw out patterns or structures that the mathematician can then study. When you understand it properly, the purely mathematical is not a distant abstraction – it is as close as the sense that we make of the world: what is seen right there in front of us. In my view, math is not abstract because it has to be, right from the word go. It actually begins with linguistic practice of the simplest and most sensible kind. We only pursue greater levels of abstraction because doing so is a necessary step in achieving the noble goals of modern mathematicians.

In particular, making our mathematical language more abstract means that our conclusions hold more generally, as when children realize that it makes no difference whether they are counting apples, pears, or people. From generation to generation, people have found that numbers and other formal systems are deeply compelling: they can shape our imagination, and what is more, they can enable comprehension.

*Featured image credit: Image by Lum3n.com – Snufkin. CC0 public domain via Pexels.*

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]]>The post Alan Turing’s lost notebook appeared first on OUPblog.

]]>The yellowing notebook — from Metcalfe and Son, just along the street from Turing’s rooms at King’s College in Cambridge — contains 39 pages in his handwriting. The auction catalogue (which inconsequentially inflated the page count) gave this description:

“Hitherto unknown wartime manuscript of the utmost rarity, consisting of 56 pages of mathematical notes by Alan Turing, likely the only extensive holograph manuscript by him in existence.”

A question uppermost in the minds of Turing fans will be whether the notebook gives new information about his famous code-cracking breakthroughs at Bletchley Park, or about the speech-enciphering device named “Delilah” that he invented later in the war at nearby Hanslope Park. The answer may disappoint. Although most probably written during the war, the notebook has no significant connection with Turing’s work for military intelligence. Nevertheless it makes fascinating reading: Turing titled it “Notes on Notations” and it consists of his commentaries on the symbolisms advocated by leading figures of twentieth century mathematics.

My interest in the notebook was first piqued more than 20 years ago. This was during a visit to Turing’s friend Robin Gandy, an amiable and irreverent mathematical logician. In 1944-5 Gandy and Turing had worked in the same Nissen hut at Hanslope Park. Gandy remembered thinking Turing austere at first, but soon found him enchanting — he discovered that Turing liked parties and was a little vain about his clothes and appearance. As we sat chatting in his house in Oxford, Gandy mentioned that upstairs he had one of Turing’s notebooks. For a moment I thought he was going to show it to me, but he added mysteriously that it contained some private notes of his own.

In his will Turing left all his mathematical papers to Gandy, who eventually passed them on to King’s College library — but not the notebook, which he kept with him up till his death in 1995. Subsequently the notebook passed into unknown hands, until its reappearance in 2015. Gandy’s private notes turned out to be a dream diary. During the summer and autumn of 1956, two years after Turing’s death, he had filled 33 blank pages in the center of the notebook with his own handwriting. What he said there was indeed personal.

Only a few years before Gandy wrote down these dreams and his autobiographical notes relating to them, Turing had been put on trial for being gay. Gandy began his concealed dream diary: “It seems a suitable disguise to write in between these notes of Alan’s on notation; but possibly a little sinister; a dead father figure and some of his thoughts which I most completely inherited.”

**Mathematical reformer **

Turing’s own writings in the notebook are entirely mathematical, forming a critical commentary on the notational practices of a number of famous mathematicians, including Courant, Eisenhart, Hilbert, Peano, Titchmarsh, Weyl, and others. Notation is an important matter to mathematicians. As Alfred North Whitehead — one of the founders of modern mathematical logic — said in his 1911 essay “The Symbolism of Mathematics”, a good notation “represents an analysis of the ideas of the subject and an almost pictorial representation of their relations to each other”. “By relieving the brain of all unnecessary work”, Whitehead remarked, “a good notation sets it free to concentrate on more advanced problems”. In a wartime typescript titled “The Reform of Mathematical Notation and Phraseology” Turing said that an ill-considered notation was a “handicap” that could create “trouble”; it could even lead to “a most unfortunate psychological effect”, namely a tendency “to suspect the soundness of our [mathematical] arguments all the time”.

This typescript, which according to Gandy was written at Hanslope Park in 1944 or 1945, provides a context for Turing’s notebook. In the typescript Turing proposed what he called a “programme” for “the reform of mathematical notation”. Based on mathematical logic, his programme would, he said, “help the mathematicians to improve their notations and phraseology, which are at present exceedingly unsystematic”. Turing’s programme called for “An extensive examination of current mathematical … books and papers with a view to listing all commonly used forms of notation”, together with an “[e]xamination of these notations to discover what they really mean”. His “Notes on Notations” formed part of this extensive investigation.

Key to Turing’s proposed reforms was what mathematical logicians call the “theory of types”. This reflects the commonsensical idea that numbers and bananas, for example, are entities of different types: there are things which makes sense to say about a number — e.g. that it has a unique prime factorization — that cannot meaningfully be said of a banana. In emphasizing the importance of type theory for day-to-day mathematics, Turing was as usual ahead of his time. Today, virtually every computer programming language incorporates type-based distinctions.

**Link to the real Turing**

Turing never displayed much respect for status and — despite the eminence of the mathematicians whose notations he was discussing — his tone in “Notes on Notations” is far from deferential. “I don’t like this” he wrote at one point, and at another “this is too subtle and makes an inconvenient definition”. His criticisms bristle with phrases like “there is obscurity”, “rather abortive”, “ugly”, “confusing”, and “somewhat to be deplored”. There is nothing quite like this blunt candor to be found elsewhere in Turing’s writings; and with these phrases we perhaps get a sense of what it would have been like to sit in his Cambridge study listening to him. This scruffy notebook gives us the plain unvarnished Turing.

*Featured image credit: Enigma by Tomasz_Mikolajczyk. CC0 Public domain via Pixabay. *

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]]>The post Really big numbers appeared first on OUPblog.

]]>Of course, in real life you’ll die before you get to any really *big* numbers that way. So here’s a more interesting way of asking the question: what is the biggest whole number that you can uniquely describe on a standard sheet of paper (single spaced, 12 point type, etc.) or, more fitting, perhaps, in a single blog post?

In 2007 two philosophy professors – Adam Elga (Princeton) and Agustin Rayo (MIT) – asked essentially this question when they competed against each other in the *Big Number Duel*. The contest consisted of Elga and Rayo taking turns describing a whole number, where each number had to be larger than the number described previously. There were three additional rules:

- Any unusual notation had to be explained.
- No primitive semantic vocabulary was allowed (i.e. “the smallest number not mentioned up to now.”)
- Each new answer had to involve some new notion – it couldn’t be reachable in principle using methods that appeared in previous answers (hence after the second turn you can’t just add 1 to the previous answer)

Elga began with “1”, Rayo countered with a string of “1”s, Elga then erased bits of some of those “1”s to turn them into factorials, and they raced off into land of large whole numbers. Rayo eventually won with this description:

The least number that cannot be uniquely described by an expression of first-order set theory that contains no more than a googol (10^{100}) symbols.

A more detailed description of the *Duel*, along with some technical details about Rayo’s description, can be found here.

Fans of paradox will recognize that Rayo’s winning move was inspired by the Berry paradox:

The least number that cannot be described in less than twenty syllables.

This expression leads to paradox since it seems to name the least number that cannot be described in less than twenty syllables, and to do so using less than twenty syllables! Rayo’s description, however, is not paradoxical, since although it uses far fewer than a googol symbols to describe the number in English, this doesn’t contradict the fact that, in the expressively much less efficient language of set theory, the number cannot be described in fewer than a googol symbols.

The number picked out by Rayo’s description has come to be called, appropriately enough, Rayo’s number. And it is big – *really* big. But can we come up with short descriptions of even bigger numbers?

Notice that Rayo’s construction implicitly provides us with a description of a function:

*F*(*n*) = The least number that cannot be uniquely described by an expression of first-order set theory that contains no more than *n* symbols.

Rayo’s number is then just *F*(10^{100}). So one way to answer the question would be to construct a function *G*(*n*) such that *G*(*n*) grows more quickly than *F*(*n*). Here’s one way to do it.

First, we’ll define a two place function *H*(*m*, *n*) as follows. We’ll just let *H*(0, 0) be 0. Now:

*H(0, n)* = The least number that cannot be uniquely described by an expression of first-order set theory that contains no more than *n* symbols.

So *H*(0, *n*) is just the Rayo function, and *H*(0, 10^{100}) is Rayo’s number. But now we let:

*H(m, n)* = The least number that cannot be uniquely described by an expression of first-order set theory supplemented with constant symbols for:

*H*(*m*-1, *n*), *H*(*m*-2, *n*),… *H*(1, *n*), *H*(0, *n*)

that contains no more than *n* symbols.

In other words, *H*(1, 10^{100}) is the least number that cannot be described in first-order set theory supplemented with a constant symbol that picks out Rayo’s number. Note that, in this new theory, Rayo’s number can now be described very briefly, in terms of this new constant! So *H*(1, 10^{100}) will be *much* larger than Rayo’s number.

But then we can consider *H*(2, 10^{100}), which is the least the least number that cannot be described in first-order set theory supplemented with a constant symbol that picks out Rayo’s number and a second constant symbol that picks out *H*(1, 10^{100}). This number is *much*, *much* bigger than *H*(1, 10^{100})!

And then we have *H*(3, 10^{100}), which is the least the least number that cannot be described in first-order set theory supplemented with a constant symbol that picks out *H*(0, 10^{100}), a second constant symbol that picks out *H*(1, 10^{100}) and a third constant symbol that picks out *H*(2, 10^{100}). This number is *much*, *much*, *much* bigger than *H*(2, 10^{100})!

And so on…

We can now get our quickly growing unary function *G*(*n*) by just identifying *m* and *n*:

*G*(*n*) = *H*(*n*, *n*).

And finally, our big, huge, enormous, number is:

*G*(10^{100})

*G*(10^{100}) is the least number that cannot be described in first-order set theory supplemented with googol-many constant symbols – one for each of *H*(0, 10^{100}), *H*(1, 10^{100}), … *H*(10^{100}-1, 10^{100}).

This number really is big. Can you come up with a bigger one?

*Featured image: “Infant Stars in Orion” Public domain via Wikimedia Commons. *

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]]>Imagine that Alice is walking towards a point – call it *A* – and will continue walking past *A* unless something prevents her from progressing further.

There is also an infinite series of gods, which we shall call *G*_{1}, *G*_{2}, *G*_{3}, and so on. Each god in the series intends to erect a magical barrier preventing Alice from progressing further if Alice reaches a certain point (and each god will do nothing otherwise):

(1) *G*_{1} will erect a barrier at exactly ½ meter past *A* if Alice reaches that point.

(2) *G*_{2} will erect a barrier at exactly ¼ meter past *A* if Alice reaches that point.

(3) *G*_{3} will erect a barrier at exactly ^{1}/_{8} meter past *A* if Alice reaches that point.

And so on.

Note that the possible barriers get arbitrarily close to *A*. Now, what happens when Alice approaches *A*?

Alice’s forward progress will be mysteriously halted at *A*, but no barriers will have been erected by any of the gods, and so there is no explanation for Alice’s inability to move forward. Proof: Imagine that Alice did travel past *A*. Then she would have had to go some finite distance past *A*. But, for any such distance, there is a god far enough along in the list who would have thrown up a barrier before Alice reached that point. So Alice can’t reach that point after all. Thus, Alice has to halt at *A*. But, since Alice doesn’t travel past *A*, none of the gods actually do anything.

Some responses to this paradox argue that the Gods have individually consistent, but jointly inconsistent intentions, and hence cannot actually promise to do what they promise to do. Other responses have suggested that the fusion of the individual intentions of the gods, or some similarly complex construction, is what blocks Alice’s path, even though no individual God actually erects a barrier. But it turns out that we can construct a version of the paradox that seems immune to both strategies.

Image that *A*, *B*, and *C* are points lying exactly one meter from the next, in a straight line (in that order). A particle *p* leaves point *A*, and begins travelling towards point *B* at exactly one second before midnight. The particle *p* is travelling at exactly one meter per second. The particle *p* will pass through *B* (at exactly midnight) and continue on towards *C* unless something prevents it from progressing further.

There is also an infinite series of force-field generators, which we shall call *G*_{1}, *G*_{2}, *G*_{3}, and so on. Each force-field generator in the series will erect an impenetrable force field at a certain point between *A* and *B*, and at a certain time. In particular:

(1) *G*_{1} will generate a force-field at exactly ½ meter past *B* at ¼ second past midnight, and take the force-field down at exactly 1 second past midnight.

(2) *G*_{2} will generate a force-field at exactly ¼ meter past *B* at exactly ^{1}/_{8} second past midnight, and take the force-field down at exactly ^{1}/_{2} second past midnight.

(3) *G*_{3} will generate a force-field at exactly ^{1}/_{8} meter past *B* at exactly ^{1}/_{16} second past midnight, and take the force-field down at exactly ^{1}/_{4} second past midnight.

And so on. In short, for each natural number *n*:

(n) *G*_{n} will generate a force-field at exactly ^{1}/_{2}^{n} meter past *B* at exactly ^{1}/_{2}^{n+1 }second past midnight, and take the force-field down at exactly ^{1}/_{2}^{n-1 }second past midnight.

Now, what happens when *p* approaches *B*?

Particle *p*’s forward progress will be mysteriously halted at *B*, but *p* will not have impacted any of the barriers, and so there is no explanation for *p*’s inability to move forward. Proof: Imagine that particle *p* did travel to some point *x* past *B*. Let *n* be the largest whole number such that ^{1}/_{2}^{n} is less than *x*. Then *p* would have travelled at a constant speed between the point ½^{n+2} meter past *B* and ^{1}/_{2}^{n} meter past *B* during the period from ½^{n+2} second past midnight and ^{1}/_{2}^{n} second past midnight. But there is a force-field at ^{1}/_{2}^{n+1 }meter past *B* for this entire duration, so *p* cannot move uniformly from ½^{n+2} meter past *B* and ^{1}/_{2}^{n} meter past *B* during this period. Thus, *p* is halted at *B*. But *p* does not make contact with any of the force-fields, since the distance between the *m*^{th} force-field and *p* (when it stops at *B*) is ^{1}/_{2}^{m} meters, and the *m*^{th} force-field does not appear until ^{1}/_{2}^{m+1} second after the particle halts at *B*.

Notice that since there are no gods (or anyone else) in this version of the puzzle, no solution relying on facts about intentions will apply here. More generally, unlike the original puzzle, in this set-up the force-fields are generated at the appropriate places and times regardless of how the particle behaves – there are no instructions or outcomes that are dependent upon the particle’s behavior. In addition, arguing that, even though no individual force-field stops the particle, the fusion or union of the force-fields does stop the particle will be tricky, since although at any point during the first ½ second after midnight two different force-fields will exist, there is no time at which all of the force-fields exist.

Thanks go to the students in my Fall 2016 Paradoxes and Infinity course for the inspiration for this puzzle!

*Featured image credit: Photo by Nicolas Raymond, CC BY 2.0 via Flickr.*

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]]>The post Is elementary school mathematics “real” mathematics? appeared first on OUPblog.

]]>There is little doubt that elementary students should know the multiplication tables, be able to do simple calculations mentally, develop fluency in using algorithms to carry out more complex calculations, and so on. Indeed, these topics are fundamental to students’ future learning of mathematics and important for everyday life. Yet, is elementary students’ engagement with these topics in itself engagement with “real” mathematics?

I suggest that classroom discourse in an elementary school classroom where students engage with “real” mathematics should satisfy two major considerations. First, it should be meaningful and important to the students. Elementary students’ engagement with the topics I mentioned earlier can offer a productive context in which to satisfy this first consideration, especially if students’ work is characterized by an emphasis not only on procedural fluency but also on conceptual understanding.

Second, the classroom discourse in an elementary school classroom where students engage with “real” mathematics should be a rudimentary but genuine reflection of the broader mathematical practice. One might interpret the second consideration as asking us to treat elementary students as little mathematicians. That would be a misinterpretation. The point is that some aspects of mathematicians’ work that are fundamental to what it means to do mathematics in the discipline should also be represented, in pedagogically and developmentally appropriate forms, in elementary students’ engagement with the subject matter.

In its typical form, classroom discourse in elementary school classrooms fails to satisfy the second consideration. A main reason for this is the limited attention it pays to issues concerning the epistemic basis of mathematics, including what counts as evidence in mathematics and how new mathematical knowledge is being validated and accepted. The notion of *proof* lies at the heart of these epistemic issues and is a defining feature of authentic mathematical work. Yet the notion of proof has a marginal place (if any at all) in many elementary school classrooms internationally, thus jeopardizing students’ opportunities to engage with “real” mathematics.

Consider, for example, a class of eight–nine-year-olds who have been writing number sentences for the number ten and have begun to develop the intuitive understanding that there are infinitely many number sentences for ten when subtracting two whole numbers (e.g., 15-5=10). In most elementary school classrooms the activity would finish here, possibly with the teacher ratifying students’ intuitive understanding thus giving it the status of public knowledge in the classroom. However, in a classroom that aspires to engage students with “real” mathematics, new mathematical knowledge isn’t established by appeal to the authority of the teacher, but rather on the basis of the logical structure of mathematics. Thus the teacher of this classroom may help the students think how they can prove their intuitive understanding.

Given appropriate instructional support, students of this age can prove that there are infinitely many number sentences for ten when subtracting two whole numbers. For example, a student called Andy in a class of eight–nine-year-olds I studied for my research generated an argument along the following lines:

To generate infinitely many subtraction number sentences for ten, you can start with 11-1=10. For each new number sentence you can add one to both terms of the previous subtraction sentence. This looks like this: 12-2=10, 13-3=10, 14-4=10, 15-5=10, and so on. This can go on forever and will maintain a constant difference of ten.

Andy’s argument used mathematically accepted ways of reasoning, which were also accessible to his peers, to establish convincingly the truth of an intuitive understanding. This argument illustrates what a proof can look like in the context of elementary school mathematics. The process of developing this argument contributed also a powerful element of mathematical sense making to Andy’s work with number sentences for ten: As he carried out calculations to write the various number sentences, he thought deeply about key arithmetical properties (e.g., how to maintain a constant difference) and he put everything together in a coherent line of reasoning. Thus an elevated status of proof in elementary students’ work can play a pivotal role in students’ meaningful engagement with mathematics. This presents a connection with the first consideration I discussed earlier.

To conclude, elementary school mathematics as reflected in typical classroom work internationally falls short of being “real.” Yet it has the potential to become “real” if the learning experiences currently offered to elementary students are transformed. A major part of this transformation needs to concern the epistemic basis of mathematics, with more opportunities offered for students to engage with proof in the context of mathematics as a sense-making activity. The teacher has an important role to play as the representative of the discipline of mathematics in the classroom and as the person with the responsibility to induct students into mathematically acceptable ways of reasoning and standards of evidence. This is a complex role that cannot be fully understood without a strong research basis about the kind of teaching practices and curricular materials that can facilitate elementary students’ access to “real” mathematics.

*Featured image credit: Math by Pixapopz. Public domain via Pixabay.*

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]]>The post Measuring up appeared first on OUPblog.

]]>My interest was further aroused by complications arising from the interactions between statistics and the results of different kinds of measurement. Many textbooks say it’s meaningless to calculate the arithmetic mean of ordinal measurements — those where the numbers reflect only the order of the objects being measured — and yet a glance at scientific and medical practice shows that this is commonplace. Clearly, although measurement was ubiquitous throughout the entire world (or, as I have put it elsewhere, we view the world through the lens of measurement), there was more to it than met the eye. Things were not always as simple as they might seem. Indeed, it would not be stretching things to say that occasionally, consideration of measurement issues revealed apparent rips in the fabric of reality.

A simple example arises from the *Daily Telegraph* report of 8 February 1989, which said that “Temperatures in London were still three times the February average at 55 °F (13 °C) yesterday”, prompting the natural question: what is the average February temperature? The answer is obvious — we just divide the temperature by three. So the February average is a third of 55 °F, equal to 18⅓ °F. Alternatively, it is a third of 13 °C, equal to 4⅓ °C. But this is very odd, because these two results are different. Indeed, the first is below freezing, while the second is above. In fact, in this example a little thought shows where things have done wrong, and which average temperature is right. But things are not always so straightforward, and occasionally deep thought about the nature of measurement is needed to work out what is going on. This reveals that there are different kinds of measurement. At one extreme we have so-called representational measurement, and at the other pragmatic measurement, with most being a mixture of the two extremes.

The aim of representational measurement is to construct a simplified model of some aspect of the world. In particular, we assign numbers to objects so that the relations between the numbers correspond to the relations between the objects. This rock extends the spring further than that, so we say it is heavier, and assign it a larger weight number. These two rocks together stretch the spring the same distance as a third one alone, so we give them numbers which add up to the number we give the third rock. And so on.

Representational measurement is essentially based on certain symmetries in the mapping from the world to the numbers, and understanding of these symmetries can be very revealing about properties of the world — about the way the world works. A familiar example is through the use of dimensional analysis in physics, engineering, and elsewhere. In contrast, a provocative way of describing pragmatic measurement is that “we don’t know what we are talking about.” What this really means is that we must define the characteristic we aim to measure before we can measure it. Or, more precisely, we define it at the same time as we measure it. The definition is implicit in the measurement procedure, and it is only through the measurement procedure that we know precisely what it is we are talking about. At first this strikes some people as strange. But take the economic example of inflation rate. Inflation can be defined in various different ways. None is “right.” Rather, it depends what properties you want the measurement to have, and what questions you want to answer. It depends on what you want to use the concept and the measured numbers for.

The bottom line to all this is that decisions and understanding are (or at least should be!) based on evidence. Evidence comes from data. And data come from measurements. Given how central measurement is to our understanding of the universe about us, to education, to government, to medicine, to technology, and so on, it is entirely fitting that it should be the topic of the 500^{th} volume in the *Very Short Introduction *series.

*Featured image credit: Scale kitchen measure by Unsplash. CC0 Public Domain via Pixabay.*

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]]>The post Very short facts about the Very Short Introductions appeared first on OUPblog.

]]>- VSIs have been translated into 50 languages, including Gujarati (an Indo-Aryan language) and Belarusian. Arabic is the most popular translated language.
- When their VSIs published, the oldest VSI author was Stanely Wells at age 85, author of
*William Shakespeare: A Very Short Introduction*. - The first VSI,
*Classics*, was published 21 years ago, in 1995 and remains in its first edition. - The highest selling VSI is
*Globalization*, which will soon be on its fourth edition! When it was first proposed people were worried it might not be a success. - Someone once wrote in suggesting we needed a VSI to Olivia Newton John. Other suggestions have included a very short introduction to coconuts and a very short introduction to Harry Potter.
- One VSI author had a tie made to match his jacket cover. Unfortunately his cover then needed to be flipped around, so his tie is now upside down.
- There are 84 VSI titles starting with “The”.
- Discounting the word “The”, the most common initial letter of a VSI is ‘A’ (55 titles), followed closely by ‘C’ and ‘M’ (52 titles each). Between them the VSI titles cover every letter of the alphabet, with the letter ‘e’ appearing over 600 times.

So, where’s the gap in your knowledge?

*Featured image credit: Very Short Introductions © Jack Campbell-Smith, for Oxford University Press.*

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]]>The post Just because all philosophers are on Twitter… appeared first on OUPblog.

]]>It is not all bad news of course. The expansion and ready availability of communication technologies has meant that it is far easier for serious ideas to be tested and refined, far easier to develop diverse communities of scholarship, and far easier for new discoveries, theories and data to permeate beyond ivory towers. A modern counterpoint is that it is also far easier to spread misleading and self-serving theories, far easier to spread messages of hate and violence, and far easier for discourse to polarise as, with so many options available, people gravitate towards those sources which reinforce and intensify existing prejudices.

This explosion of available information and opinions also presents a challenge to traditional notions of education and citizenry. There may have been a time when the purpose of education was primarily to create an informed citizenry – to give them the relevant information – but that time is certainly not now. Now, information is more freely available and a far more important skill is the ability to independently discern reliable from unreliable sources, fact from fiction, genuine authority from charlatanism, feline ecology from lolcats. Where once scholarship meant perseverance and a dedication to tracking down otherwise inaccessible information, an increasingly vital skill in modern scholarship is a well-tuned bullshit detector.

A useful distinction to bear in mind here is between message and medium. (Philosophers love distinctions!) Each has its own hype cycle, and they are not always in sync. At the top of each cycle is the peak of inflated expectations. It is at this point in the cycle of new media technologies that we hear grand transformational claims, such as the view that virtual reality will end inequality, that the internet will kill traditional publishers and bookshops, that social networking will scupper academic peer-review, or that Massive Open Online Courses (MOOCs) will turn University campuses into ghost towns. After a tough period of disillusionment when initial enthusiasts and investors come to terms with the failure of their over-inflated expectations, the cycle reaches a plateau of productivity where the new medium is embraced by an increasingly significant portion of the population who see genuine usefulness beyond the hype.

Though the cycle repeats with each new medium – from telephones to Twitter – it is not futile. For what is gained through each iteration is a deeper understanding of the phenomenon which the technology was due to replace. So, for example, we learn something about the true (and changing) value of publishers, bookshops, peer-review and universities by understanding that they cannot be wholly replaced by new technologies. A strong theme running through these particular values is the notion of a discerning eye. With so many pieces of information, opinions and lolcats out there, we would simply be lost if we did not have some way of filtering reliable from unreliable research, scientific from wishful thinking, well-reasoned interpretations from self-serving propaganda.

This is not to say, of course, that the best way to navigate modern media seas is by blind deference to authority. All authorities are fallible (with the notable exception of OUP, of course). Far more important is the ability to critically evaluate pedigree for oneself. This is where universities can come in. My own engagement with MOOCs (through the Open University’s FutureLearn platform) has taught me that while large online courses are fantastic at bringing together a diverse range of students, they work best when those students are encouraged to engage critically with the ideas and experience they and others bring to the community. Inculcating and refining these skills is something that smaller scale teaching and face-to-face education are, im my experience, uniquely placed to do. So while everyone being on Twitter might not mean that everyone has interesting things to say, the resulting flood of information and opinion does mean that educators still have interesting things to do.

*Featured image: Mobile Phone by geralt. Public domain via Pixabay.*

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]]>The post Teaching teamwork appeared first on OUPblog.

]]>I think we can improve undergraduate and graduate students’ educational experiences by giving them the benefit of working in teams. This can be implemented in short-term (two-hour to two-week) or longer term (2-12 week) projects. I believe that working on a larger project with 2-4 other students, for at least 15-35% of their coursework in several courses, would build essential professional and personal skills. I agree that it is easier to plan and execute team projects in smaller graduate courses than larger undergraduate courses.

Unfortunately, many faculty members were trained through lecture, individual homework, and strictly solitary testing. They have weak teamwork skills and are little inclined to teach teamwork. In fact, they have many fears that increase their resistance. Some believe that teamwork takes extra effort for faculty or that teams naturally lead to one person doing most of the work.

Teamwork projects may require fresh thinking by faculty members, but it may be easier to supervise and grade ten teams of four students, than to mentor and grade 40 individuals. Moreover, well-designed teamwork projects could lead to published papers or start-up companies in which faculty are included as co-authors or advisers. In my best semester, five of the seven teams in my graduate course on information visualization produced a final report that led to a publication in a refereed journal or conference.

Another possible payoff is that teamwork courses may create more engaged students with higher student retention rates. Of course teams can run into difficulties and conflict among students. These are teachable moments when students can learn lessons that will help them in their professional and personal lives. These difficulties and conflicts may be more visible than individual students failing or dropping out, but I think they are a preferable alternative.

So if faculty members are ready to move towards teaching with team projects, there are some key decisions to be made. Sometimes two-person teams are natural, but larger teams of 3-5 allow more ambitious projects, while increasing the complexity. I’ve also run projects where the entire class acts as a team to produce a project such as the Encyclopedia of Virtual Environments (EVE), in which the 20 students wrote about 100 web-based articles defining the topic. Colleagues have told me about their teamwork projects that had their French students create an online newspaper for French alumni describing campus sports events or a timeline of the European philosophical movements leading up to the framing of the US Constitution.

**Team formation:** I have moved to assigning team membership (rather than allow self-formation) using a randomization strategy, which is recommended in the literature. This helps ensure diversity among the team members, speeds the process of getting teams started, and eliminates the problem of some students having a hard time finding a team to join.

**Project design (student-driven): **Well-designed team projects take on more ambitious efforts, giving students the chance to learn how to deal with a larger goal. I prefer student designed projects with an outside mentor, where the goal is to produce an inspirational pilot project that benefits someone outside the classroom and survives beyond the semester. I’ve had student teams work on software to schedule the campus bus routes or support a local organization that brings hundreds of foreign students for summer visits in people’s homes. Other teams helped a marketing company to assess consumer behavior in a nearby shopping mall or an internet provider to develop a network security monitor. Two teams proposed novel visualizations for the monthly jobs report of the US Bureau of Labor Statistics, which they presented to the Commissioner and her staff. I give a single grade to the team, but do require that their report includes a credits section in which the role of each person is described.

**Project design (faculty-driven):** Another approach is for the teacher to design the team projects, which might be the same ones for every team. With a four-person team, distinct roles can be assigned to each person, so it becomes easier to grade students individually. Just getting students to talk together, resolve differences, agree to schedules, etc. gives them valuable skills.

**Milestones:** Especially in longer projects, there should be deliverables every week, e.g. initial proposal, first designs, test cases, mid-course report, preliminary report, and final report.

**Deliverables:** With teams there can be multiple deliverables, e.g. in my graduate information visualization course, students produce a full conference paper, 3-5 minute YouTube video, working demo, and slide deck & presentation.

**Teamwork strategies:** For short-term teams (a few weeks to a semester), simple strategies are probably best. I use: (1) “Something small soon,” which asks students to make small efforts that validate concepts before committing greater energy and (2) “Who does what by when,” which clarifies responsibilities on an hourly basis, such as “If Sam and I do the draft by 6pm Tuesday, will Jose and Marie give us feedback by noon on Wednesday?” Teamwork does not require any meetings at all; it is a management strategy to coordinate work among team members.

**Critiques and revisions:** I ask students to post their preliminary reports on the class’s shared website two weeks before the end of the semester. Then students sign up to read and critique one of the reports, which they send to me and the report authors. They write one paragraph about what they learned and liked, then as much as constructive suggestions for improvements to the report’s overall structure, to proposed references and improved figures, to grammar and spelling fixes. When students realize that their work will be read by other students they are likely to be more careful. When students read another team’s project report, they reflect on their own project report, possibly seeing ways to improve it. I grade the critiques which can be 3-6% of their final grade. My goal is to help every team to improve the quality of their work. Sometimes the process of preparing their preliminary reports early and then revising does much to improve quality.

**Concerns:** I know that some faculty members worry that one person in a team will do the majority of the work, but if projects are ambitious enough then that possibility is reduced. Grading remains an issue that each faculty member has to decide on. I find that having students include a credits box in their final report helps, but other instructors require peer rating/reporting for team members.

In summary, anything novel takes some thinking, but embracing team projects could substantially improve education programs, engage more marginal students, and improve student retention rates. Learning to use teamwork tools such as email, videoconferencing, and shared documents provides students with valuable skills. Working in teams can be fun for students and satisfying for teachers.

*Featured image credit: Harvard Business School classroom by HBS 1908. CC BY-SA 3.0 via Wikimedia Commons.*

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]]>The post What is combinatorics? appeared first on OUPblog.

]]>- How many possible sudoku puzzles are there?
- Do 37 Londoners exist with the same number of hairs on their head?
- In a lottery where 6 balls are selected from 49, how often do two winning balls have consecutive numbers?
- In how many ways can we give change for £1 using only 10p, 20p, and 50p pieces?
- Is there a systematic way of escaping from a maze?
- How many ways are there of rearranging the letters in the word “ABRACADABRA”?
- Can we construct a floor tiling from squares and regular hexagons?
- In a random group of 23 people, what is the chance that two have the same birthday?
- In chess, can a knight visit all the 64 squares of an 8 × 8 chessboard by knight’s moves and return to its starting point?
- If a number of letters are put at random into envelopes, what is the chance that no letter ends up in the right envelope?

What do you notice about these problems?

First of all, unlike many mathematical problems that involve much abstract and technical language, they’re all easy to understand – even though some of them turn out to be frustratingly difficult to solve. This is one of the main delights of the subject.

Secondly, although these problems may appear diverse and unrelated, they mainly involve selecting, arranging, and counting objects of various types. In particular, many of them have the forms. Does such-and-such exist? If so, how can we construct it, and how many of them are there? And which one is the ‘best’?

The subject of combinatorial analysis or combinatorics (pronounced *com-bin-a-tor-ics*) is concerned with such questions. We may loosely describe it as the branch of mathematics concerned with selecting, arranging, constructing, classifying, and counting or listing things.

To clarify our ideas, let’s see how various sources define combinatorics.

Oxford Dictionaries describe it briefly as:

“The branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints, such as those of graph theory.”

While the Collins dictionary present it as:

“the branch of mathematics concerned with the theory of enumeration, or combinations and permutations, in order to solve problems about the possibility of constructing arrangements of objects which satisfy specified conditions.”

Wikipedia introduces a new idea, that combinatorics is:

“a branch of mathematics concerning the study of finite or countable discrete structures.”

So the subject involves finite sets or discrete elements that proceed in separate steps (such as the numbers 1, 2, 3 …), rather than continuous systems such as the totality of numbers (including π, √2, etc.) or ideas of gradual change such as are found in the calculus. The Encyclopaedia Britannica extends this distinction by defining combinatorics as:

“the field of mathematics concerned with problems of selection, arrangement, and operation within a finite or discrete system … One of the basic problems of combinatorics is to determine the number of possible configurations (e.g., graphs, designs, arrays) of a given type.”

Finally, Wolfram Research’s *MathWorld* presents it slightly differently as:

“the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties,”

adding that:

“Mathematicians sometimes use the term ‘combinatorics’ to refer to a larger subset of discrete mathematics that includes graph theory. In that case, what is commonly called combinatorics is then referred to as ‘enumeration’.”

The subject of combinatorics can be dated back some 3000 years to ancient China and India. For many years, especially in the Middle Ages and the Renaissance, it consisted mainly of problems involving the permutations and combinations of certain objects. Indeed, one of the earliest works to introduce the word ‘combinatorial’ was a *Dissertation on the combinatorial art* by the 20-year-old Gottfried Wilhelm Leibniz in 1666. This work discussed permutations and combinations, even claiming on the front cover to ‘prove the existence of God with complete mathematical certainty’.

Over the succeeding centuries the range of combinatorial activity broadened greatly. Many new types of problem came under its umbrella, while combinatorial techniques were gradually developed for solving them. In particular, combinatorics now includes a wide range of topics, such as the geometry of tilings and polyhedra, the theory of graphs, magic squares and Latin squares, block designs and finite projective planes, and partitions of numbers.

Much of combinatorics originated in recreational pastimes, as illustrated by such well-known puzzles such as the Königsberg bridges problem, the four-colour map problem, the Tower of Hanoi, the birthday paradox, and Fibonacci’s ‘rabbits’ problem. But in recent years the subject has developed in depth and variety and has increasingly become a part of mainstream mathematics. Prestigious mathematical awards such as the Fields Medal and the Abel Prize have been given for ground-breaking contributions to the subject, while a number of spectacular combinatorial advances have been reported in the national and international media.

Undoubtedly part of the reason for the subject’s recent importance has arisen from the growth of computer science and the increasing use of algorithmic methods for solving real-world practical problems. These have led to combinatorial applications in a wide range of subject areas, both within and outside mathematics, including network analysis, coding theory, probability, virology, experimental design, scheduling, and operations research.

*Featured image credit: ‘Sudoku’ by Gellinger. CC0 public domain via Pixabay.*

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]]>The post A brief history of crystallography appeared first on OUPblog.

]]>So, what is crystallography? Put simply, it is the study of crystals. Now, let’s be careful here. I am not talking about all those silly websites advertising ways in which crystals act as magical healing agents, with their chakras, auras, and energy levels. No, this is a serious scientific subject, with around 26 or so Nobel prizes to its credit. And yet, despite this, it remains a largely hidden subject, at least in the public mind.

Crystallography as a science has a long and venerable history going back to the 17^{th} century when the sheer beauty of the symmetry of crystals suggested an underlying order of some kind. For the next three centuries, our knowledge of what crystals actually were was based on conjecture and argument, with a few simple experiments thrown in. From their symmetry and shapes it was argued that crystals must consist of ordered arrangements of minute particles: today we know them as atoms and molecules.

But it was the discovery of X-rays in 1895 that changed all that, for a few years later in 1912 in Germany, Max Laue, Walter Friedrich, and Paul Knipping showed that an X-ray beam incident on a crystal was scattered to form a regular pattern of spots on a film (we call this diffraction). Thus it was proved that X-rays consisted of waves and furthermore this gave direct evidence of the underlying order of atoms in the crystal. Hence Nobel Prize number 1 went to Laue in 1914. However, it was William Lawrence Bragg (WLB) who in 1912 at the age of 22 showed how the observed diffraction pattern could be used to determine the positions of atoms in the crystal, thus launching a completely new scientific discipline, X-ray crystallography. Working with his father, William Henry Bragg (WHB), they quickly determined the crystal structures of several materials starting with that of common salt and diamond. Both father and son shared Nobel prize number 2 in 1915. William Henry Bragg and William Lawrence Bragg went on to create world-class research groups working on a huge range of solid materials and incidentally they were active in encouraging women into science.

Since then X-ray crystallography, which today is used throughout the world, has been the method of choice for determining the crystal structures of organic and inorganic solids, pharmaceuticals, biological substances such as proteins and viruses, and indeed all kinds of solid substances. Crick and Watson’s determination of the double helix of DNA is probably the most well-known example of the use of crystallography, incidentally a discovery made in William Lawrence Bragg’s laboratory in Cambridge. Had it not been for X-ray (and later neutron and electron) crystallography we probably would not have today much of an electronics industry, computer technology, new pharmaceuticals, new materials of all sorts, nor the modern field of genetics. The Braggs left a huge legacy which today continues to make astonishing progress.

*Featured image credit: Protein Crystals Use in XRay Crystallography by CSIRO. CC BY 3.0 via Wikimedia Commons *

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]]>The post Today’s Forecast: Cloudy with a chance of seizures appeared first on OUPblog.

]]>For many years, experts in neurology, computer science, and engineering have worked toward developing algorithms to predict a seizure before it occurs. If an algorithm could detect subtle changes in the electrical activity of a person’s brain (measured by electroencephalography (EEG)) before a seizure occurs, people with epilepsy could take medications only when needed, and possibly reclaim some of those daily activities many of us take for granted. But algorithm development and testing requires substantial quantities of suitable data, and progress has been slow. Many early research reports developed and tested algorithms on relatively short intracranial EEG data segments from patients with epilepsy undergoing intracranial EEG before surgery. There are a number of problems with this. First, patients undergoing pre-surgical monitoring for epilepsy typically have their medications reduced to encourage seizures to occur, which causes a progressive decrease in the blood levels of medications which have been shown to affect the normal baseline pattern in a patient’s EEG. Second, hospital stays for pre-surgical monitoring by necessity rarely last more than two weeks, providing a very limited amount of data for any single patient. These short data segments with changing baseline EEG characteristics are particularly problematic when algorithm scientists attempt to measure an algorithm’s false positive rate, or the number of false alarms that a seizure forecasting algorithm might raise. Development of robust, reliable seizure prediction algorithms requires data on many seizures and many periods of baseline, non-seizure EEG with enough time between the seizures to allow the brain to recover. In addition, researchers are often reluctant to share algorithm data and programs; privacy concerns and the high cost of sharing large data sets makes testing and comparison very difficult.

In 2013 a group of physicians and scientists from Melbourne Australia reported a successful trial of an implanted device capable of measuring EEG from intracranial electrode strips, and telemetering the EEG data to a small external device about the size of a smart phone that could run seizure forecasting algorithms and provide warnings of impending seizures. The device used a proprietary seizure forecasting algorithm that performed well enough to be helpful for some patients in the trial, raising hopes that seizure forecasting might soon become clinically possible.

We recently made an effort to use Kaggle.com — a website that runs data science competitions to develop algorithms to predict everything from insurance rates to the Higgs Boson — to develop new algorithms for seizure forecasting. Our competition used intracranial EEG data from the same device in the Australian trial (implanted in eight dogs with naturally occurring epilepsy) as well as data from two human patients undergoing intracranial monitoring. In hope of winning $15,000 in prize money, plus bragging rights among elite data science circles, hundreds of algorithm developers, most with little or no experience with epilepsy or EEG, worked countless hours to build, test, and rebuild algorithms for seizure forecasting, and tested their algorithms on nearly 350 seizures recorded over more than 1,500 days. After four months, over half of these “crowdsourced” algorithms performed better than random predictions, and the winning algorithms accurately predicted over 70% of seizures with a 25% false positive rate. The data are available for researchers to continue developing new algorithms for predicting seizures, and can serve as a benchmark for new algorithms to be compared directly to one another and to the algorithms developed in this competition. The best performing algorithms in the competition used a mixture of conventional and complex approaches drawn from physics, engineering, and computer science, sometimes in unorthodox ways that proved to be surprisingly effective. The winning teams also made the source code for their algorithms publicly available, providing a benchmark and starting point for future algorithm developers.

While we applaud the talented algorithm scientists who took home the prize money, we hope the real winners of the contest will be our patients.

*Featured image: Sky cloudy. Uploaded by Carla Nunziata. CC-BY-SA-3.0 via Wikimedia Commons*

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]]>The most recent time this happened, it reminded me of a startling academic paper, first published in 1978, in the *New England Journal of Medicine.* Dr Ward Casscells and colleagues reported something very disturbing: that most doctors can’t calculate risks correctly.

The question they posed was this. Imagine a disease (let’s call it Gobble’s disease*), which has a prevalence of 1 in 1000 in your population. There is a test for Gobble’s disease, and you know it has a false positive rate of 5%. You meet a patient in your clinic, who has tested positive. What is the probability that the patient has Gobble’s disease?

A member of the public could be forgiven for thinking the answer is 100%. After all, medical tests are always reliable, right? Someone a bit savvier, say a doctor, might look at that 5% false positive rate, and decide the answer is 95%. That’s what most of the respondents in Casscells’ study said, and he offered his question to senior doctors, junior doctors, and medical students. (And, if you had offered it to me as a medical student or a junior doctor, that’s almost certainly what I would have said – even though, in all fairness, my medical school tried hard to teach us the truth).

But they would all be hopelessly wrong. A statistician would say this: suppose you test the population for Gobble’s disease; a 5% false positive rate means that 5% of your population will test positive for Gobble’s disease, *even when they don’t have it.* 5% of your population is 50 per 1000. But we know that only 1 in 1000 of the people in your population has Gobble’s disease; therefore your test will be wrong for those 50 people, and right only for that last 1 person. So the probability of your patient – who tested positive for Gobble’s disease – actually having Gobble’s disease is only 1 in 50, or 2%.

This result is so unexpected, so counter-intuitive, that it’s worth looking at more closely.

All medical tests have two basic properties. These are known as *sensitivity* and *specificity*. The sensitivity is the probability that the patient will test positive for the disease, if they actually have it. Our fictitious Gobble’s test is, we assume, 100% sensitive it will always detect someone with Gobble’s disease. In practice, few medical tests approach 100% sensitivity.

The specificity is the probability that the patient will test negative for the disease if they haven’t got the disease. Our Gobble’s test is 95% specific: if the patient doesn’t have Gobble’s disease, there is a 95% likelihood that they will test negative for the disease. That sounds great, until we remember that there’s a 5% likelihood they will test positive, which is the cause of all our problems. Sadly, in reality, few medical tests approach 95% specificity.

In reality, sensitivity and specificity are two sides of the same coin. One cannot improve the sensitivity of any test without including more false positives (which might, as we can see, drown out the true positives we are actually interested in). An extreme example is to make every test a positive result: you would never miss anyone with the disease, but there would be so many false positives that your test would be useless.

The reason our test for Gobble’s disease is so unhelpful is that Gobble’s disease is rare. The test becomes much more valuable if Gobble’s disease is more common. Therefore to make it more useful, we shouldn’t apply the test indiscriminately, but we should try to narrow down our focus to people with risk factors. If Gobble’s disease is rare in the young but gets more common in the elderly (as many cancers do), then we can improve the usefulness of the test by applying it only to the elderly.

The other way in which we can improve the usefulness of our test is to combine it with other tests. Say our test is quick and safe. We can apply it easily to a large number of people. But to those who test positive, we can then go on and apply a different test, perhaps one which is more invasive or more expensive. Patients who test positive for both are much more likely to actually *have* Gobble’s disease.

That security guard, having a quick look through my bag, is applying a diagnostic test: do I have a dangerous item in there, or not? Unfortunately his test isn’t very sensitive, since he might easily miss something down at the bottom. And, since most people going to the concert are there to enjoy the music, the prevalence of miscreants is low. Therefore the simple mathematics of the test tells us it is likely to be worthless. The effectiveness of the test is multiplied by applying a different test: an X-ray scan of my bag, or even of my body. These are much more expensive than a quick visual check, but airports, understandably, are prepared to foot the bill.

There are powerful lessons to be learned here. The first is that applying a single test to a whole population is likely to be very unhelpful, especially if what you are looking for is rare. The second is that medical tests seldom give a clear-cut answer; instead they lengthen or shorten the likelihood of a particular diagnosis being true. Finally, quite a lot of other tests (such as concert security) are subject to exactly the same mathematical rules as medical tests. A thorough understanding of the mathematics of probability will help no end in this endeavour. In the words of William Osler (often described as the father of modern medicine), “Medicine is a science of uncertainty and an art of probability”!

*‘Gobble’s Disease’ is an invented illness from the *Oxford Handbook of Clinical Medicine*.

*Featured Image Credit: ‘Dice, Die, Probability’ by Jody Lehigh. CCO Public Domain via Pixabay.*

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