Author: IMC

Relating mathematics to questions that are relevant to many students can help address its image problem, argues Eugenia Cheng.

Mathematics has an image problem: far too many people are put off it and conclude that the subject just isn’t for them. There are many issues, including the curriculum, standardised tests and constraints placed on teachers. But one of the biggest problems is how maths is presented, as cold and dry.

Attempts at “real-life” applications are often detached from our daily lives, such as arithmetic problems involving a ludicrous number of watermelons or using a differential equation to calculate how long a hypothetical cup of coffee will take to cool.

I have a different approach, which is to relate abstract maths to questions of politics and social justice. I have taught fairly maths-phobic art students in this way for the past seven years and have seen their attitudes transformed. They now believe maths is relevant to them and can genuinely help them in their everyday lives.

At a basic level, maths is founded on logic, so when I am teaching the principles of logic, I use examples from current events rather than the old-fashioned, detached type of problem. Instead of studying the logic of a statement like “all dogs have four legs”, I might discuss the (also erroneous) statement “all immigrants are illegal”.

But I do this with specific mathematical structures, too. For example, I teach a type of structure called an ordered set, which is a set of objects subject to an order relation such as “is less than”. We then study functions that map members of one ordered set to members of another, and ask which functions are “order-preserving”. A typical example might be the function that takes an ordinary number and maps it to the number obtained from multiplying by 2. We would then say that if x < y then also 2x < 2y, so the function is order-preserving. By contrast the function that squares numbers isn’t order-preserving because, for example, -2 < -1, but (-2)² > (-1)². If we work through those squaring operations, we get 4 and 1.

However, rather than sticking to this type of dry mathematical example, I introduce ones about issues like privilege and wealth. If we think of one ordered set with people ordered by privilege, we can make a function to another set where the people are now ordered by wealth instead. What does it mean for that to be order-preserving, and do we expect it to be so? Which is to say, if someone is more privileged than someone else, are they automatically more wealthy? We can also ask about hours worked and income: if someone works more hours, do they necessarily earn more? The answer there is clearly no, but then we go on to discuss whether we think this function should be order-preserving or not, and why.

My approach is contentious because, traditionally, maths is supposed to be neutral and apolitical. I have been criticised by people who think my approach will be off-putting to those who don’t care about social justice; however, the dry approach is off-putting to those who do care about social justice. In fact, I believe that all academic disciplines should address our most important issues in whatever way they can. Abstract maths is about making rigorous logical arguments, which is relevant to everything. I don’t demand that students agree with me about politics, but I do ask that they construct rigorous arguments to back up their thoughts and develop the crucial ability to analyse the logic of people they disagree with.

Maths isn’t just about numbers and equations, it is about studying different logical systems in which different arguments are valid. We can apply it to balls rolling down different hills, but we can also apply it to pressing social issues. I think we should do both, for the sake of society and to be more inclusive towards different types of student in maths education.

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*Credit for article given to Eugenia Cheng*

Not content with shapes in two or three dimensions, mathematicians like to explore objects in any number of spatial dimensions. Now they have discovered shapes of constant width in any dimension, which roll like a wheel despite not being round.

A 3D shape of constant width as seen from three different angles. The middle view resembles a 2D Reuleaux triangle

Mathematicians have reinvented the wheel with the discovery of shapes that can roll smoothly when sandwiched between two surfaces, even in four, five or any higher number of spatial dimensions. The finding answers a question that researchers have been puzzling over for decades.

Such objects are known as shapes of constant width, and the most familiar in two and three dimensions are the circle and the sphere. These aren’t the only such shapes, however. One example is the Reuleaux triangle, which is a triangle with curved edges, while people in the UK are used to handling equilateral curve heptagons, otherwise known as the shape of the 20 and 50 pence coins. In this case, being of constant width allows them to roll inside coin-operated machines and be recognised regardless of their orientation.

Crucially, all of these shapes have a smaller area or volume than a circle or sphere of the equivalent width – but, until now, it wasn’t known if the same could be true in higher dimensions. The question was first posed in 1988 by mathematician Oded Schramm, who asked whether constant-width objects smaller than a higher-dimensional sphere might exist.

While shapes with more than three dimensions are impossible to visualise, mathematicians can define them by extending 2D and 3D shapes in logical ways. For example, just as a circle or a sphere is the set of points that sits at a constant distance from a central point, the same is true in higher dimensions. “Sometimes the most fascinating phenomena are discovered when you look at higher and higher dimensions,” says Gil Kalai at the Hebrew University of Jerusalem in Israel.

Now, Andrii Arman at the University of Manitoba in Canada and his colleagues have answered Schramm’s question and found a set of constant-width shapes, in any dimension, that are indeed smaller than an equivalent dimensional sphere.

Arman and his colleagues had been working on the problem for several years in weekly meetings, trying to come up with a way to construct these shapes before they struck upon a solution. “You could say we exhausted this problem until it gave up,” he says.

The first part of the proof involves considering a sphere with n dimensions and then dividing it into 2ⁿ equal parts – so four parts for a circle, eight for a 3D sphere, 16 for a 4D sphere and so on. The researchers then mathematically stretch and squeeze these segments to alter their shape without changing their width. “The recipe is very simple, but we understood that only after all of our elaboration,” says team member Andriy Bondarenko at the Norwegian University of Science and Technology.

The team proved that it is always possible to do this distortion in such a way that you end up with a shape that has a volume at most 0.9ⁿ times that of the equivalent dimensional sphere. This means that as you move to higher and higher dimensions, the shape of constant width gets proportionally smaller and smaller compared with the sphere.

Visualising this is difficult, but one trick is to imagine the lower-dimensional silhouette of a higher-dimensional object. When viewed at certain angles, the 3D shape appears as a 2D Reuleaux triangle (see the middle image above). In the same way, the 3D shape can be seen as a “shadow” of the 4D one, and so on.  “The shapes in higher dimensions will be in a certain sense similar, but will grow in complexity as [the] dimension grows,” says Arman.

Having identified these shapes, mathematicians now hope to study them further. “Even with the new result, which takes away some of the mystery about them, they are very mysterious sets in high dimensions,” says Kalai.

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*Credit for article given to Alex Wilkins*

For all the students wondering why they would ever need to use the Pythagorean theorem, Katie Steckles is delighted to report on a real-world encounter.

Recently, I was building a flat-pack wardrobe when I noticed something odd in the instructions. Before you assembled the wardrobe, they said, you needed to measure the height of the ceiling in the room you were going to put it in. If it was less than 244 centimetres high, there was a different set of directions to follow.

These separate instructions asked you to build the wardrobe in a vertical orientation, holding the side panels upright while you attached them to the base. The first set of directions gave you a much easier job, building the wardrobe flat on the floor before lifting it up into place. I was intrigued by the value of 244 cm: this wasn’t the same as the height of the wardrobe, or any other dimension on the package, and I briefly wondered where that number had come from. Then I realised: Pythagoras.

The wardrobe was 236 cm high and 60 cm deep. Looking at it side-on, the length of the diagonal line from corner to corner can be calculated using Pythagoras’s theorem. The vertical and horizontal sides meet at a right angle, meaning if we square the length of each then add them together, we get the well-known “square of the hypotenuse”. Taking the square root of this number gives the length of the diagonal.

In this case, we get a diagonal length a shade under 244 cm. If you wanted to build the wardrobe flat and then stand it up, you would need that full diagonal length to fit between the floor and the ceiling to make sure it wouldn’t crash into the ceiling as it swung past – so 244 cm is the safe ceiling height. It is a victory for maths in the real world, and vindication for maths teachers everywhere being asked, “When am I going to use this?”

This isn’t the only way we can connect Pythagoras to daily tasks. If you have ever needed to construct something that is a right angle – like a corner in joinery, or when laying out cones to delineate the boundaries of a sports pitch – you can use the Pythagorean theorem in reverse. This takes advantage of the fact that a right-angled triangle with sides of length 3 and 4 has a hypotenuse of 5 – a so-called 3-4-5 triangle.

If you measure 3 units along one side from the corner, and 4 along the other, and join them with a diagonal, the diagonal’s length will be precisely 5 units, if the corner is an exact right angle. Ancient cultures used loops of string with knots spaced 3, 4 and 5 units apart – when held out in a triangle shape, with a knot at each vertex, they would have a right angle at one corner. This technique is still used as a spot check by builders today.

Engineers, artists and scientists might use geometrical thinking all the time, but my satisfaction in building a wardrobe, and finding the maths checked out perfectly, is hard to beat.

Katie Steckles is a mathematician, lecturer, YouTuber and author based in Manchester, UK. She is also puzzle adviser for New Scientist’s puzzle column, BrainTwister. Follow her @stecks

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*Credit for article given to Peter Rowlett*

It is said that the ancient Greek philosopher Pythagoras came up with the idea that musical note combinations sound best in certain mathematical ratios, but that doesn’t seem to be true.

Pythagoras has influenced Western music for millennia

An ancient Greek belief about the most pleasing combinations of musical notes – often attributed to the philosopher Pythagoras – doesn’t actually reflect the way people around the world appreciate harmony, researchers have found. Instead, Pythagoras’s mathematical arguments may merely have been taken as fact and used to assert the superiority of Western culture.

According to legend, Pythagoras found that the ringing sounds of a blacksmith’s hammers sounded most pleasant, or “consonant”, when the ratio between the size of two tools involved two integers, or whole numbers, such as 3:2.

This idea has shaped how Western musicians play chords, because the philosopher’s belief that listeners prefer music played in perfect mathematical ratios was so influential. “Consonance is a really important concept in Western music, in particular for telling us how we build harmonies,” says Peter Harrison at the University of Cambridge.

But when Harrison and his colleagues surveyed 4272 people in the UK and South Korea about their perceptions of music, their findings flew in the face of this ancient idea.

In one experiment, participants were played musical chords and asked to rate how pleasant they seemed. Listeners were found to slightly prefer sounds with an imperfect ratio. Another experiment discovered little difference in appeal between the sounds made by instruments from around the world, including the bonang, an Indonesian gong chime, which produces harmonies that cannot be replicated on a Western piano.

While instruments like the bonang have traditionally been called “inharmonic” by Western music culture, study participants appreciated the sounds the instrument and others like it made. “If you use non-Western instruments, you start preferring different harmonies,” says Harrison.

“It’s fascinating that music can be so universal yet so diverse at the same time,” says Patrick Savage at the University of Auckland, New Zealand. He says that the current study also contradicts previous research he did with some of the same authors, which found that integer ratio-based rhythms are surprisingly universal.

Michelle Phillips at the Royal Northern College of Music in Manchester, UK, points out that the dominance of Pythagorean tunings, as they are known, has been in question for some time. “Research has been hinting at this for 30 to 40 years, as music psychology has grown as a discipline,” she says. “Over the last fifteenish years, people have undertaken more work on music in the whole world, and we now know much more about non-Western pitch perception, which shows us even more clearly how complex perception of harmony is.”

Harrison says the findings tell us both that Pythagoras was wrong about music – and that music and music theory have been too focused on the belief that Western views are held worldwide. “The idea that simple integer ratios are superior could be framed as an example of mathematical justification for why we’ve got it right over here,” he says. “What our studies are showing is that, actually, this is not an inviolable law. It’s something that depends very much on the way in which you’re playing music.”

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*Credit for article given to Chris Stokel-Walker*

An unsatisfying caveat in a mathematical breakthrough discovery of a single tile shape that can cover a surface without ever creating a repeating pattern has been eradicated. The newly discovered “spectre” shape can cover a surface without repeating and without mirror images.

The pattern on the left side is made up of the “hat” shape, including reflections. The pattern on the right is made up of round-edged “spectre” shapes that repeat infinitely without reflections

David Smith et al

Mathematicians solved a decades-long mystery earlier this year when they discovered a shape that can cover a surface completely without ever creating a repeating pattern. But the breakthrough had come with a caveat: both the shape and its mirror image were required. Now the same team has discovered that a tweaked version of the original shape can complete the task without its mirror.

Simple shapes such as squares and equilateral triangles can tile a surface without gaps in a repeating pattern. Mathematicians have long been interested in a more complex version of tiling, known as aperiodic tiling, which involves using more complex shapes that never form such a repeating pattern.

The most famous aperiodic tiles were created by mathematician Roger Penrose, who in the 1970s discovered that two different shapes could be combined to create an infinite, never-repeating tiling. In March, Chaim Goodman-Strauss at the University of Arkansas and his colleagues found the “hat”, a shape that could technically do it alone, but using a left-handed and right-handed version. This was a slightly unsatisfying solution and left the question of whether a single shape could achieve the same thing with no reflections remaining.

The researchers have now tweaked the equilateral polygon from their previous research to create a new family of shapes called spectres. These shapes allow non-repeating pattern tiling using no reflections at all.

Until now, it wasn’t clear whether such a single shape, known as an einstein (from the German “ein stein” or “one stone”), could even exist. The researchers say in their paper that the previous discovery of the hat was a reminder of how little understood tiling patterns are, and that they were surprised to make another breakthrough so soon.

“Certainly there is no evidence to suggest that the hat (and the continuum of shapes to which it belongs) is somehow unique, and we might therefore hope that a zoo of interesting new monotiles will emerge in its wake,” the researchers write in their new paper. “Nonetheless, we did not expect to find one so close at hand.”

Sarah Hart at Birkbeck, University of London, says the new result is even more impressive than the original finding. “It’s very intellectually satisfying to have a solution that doesn’t need the mirror image because if you actually had real tiles then a tile and its mirror image are not the same,” she says. “With this new tile there are no such caveats.”

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*Credit for article given to Matthew Sparkes*

There are no rabbits pulled out of hats here – these tricks rely on mathematical principles and will never fail you, says Peter Rowlett.

LOOK, I’ve got nothing up my sleeves. There are magic tricks that work by sleight of hand, relying on the skill of the performer and a little psychology. Then there are so-called self-working magic tricks, which are guaranteed to work by mathematical principles.

For example, say I ask you to write down a four-digit number and show me. I will write a prediction but keep it secret. Write another four-digit number and show me, then I will write one and show you. Now, sum the three visible numbers and you may be surprised to find the answer matches the prediction I made when I had only seen one number!

The trick is that while the number I wrote and showed you appeared random, I was actually choosing digits that make 9 when added to the digits of your second number. So if you wrote 3295, I would write 6704. This means the two numbers written after I made my prediction sum to 9999. So, my prediction was just your original number plus 9999. This is the same as adding 10,000 and subtracting 1, so I simply wrote a 1 to the left of your number and decreased the last digit by 1. If you wrote 2864, I would write 12863 as my prediction.

Another maths trick involves a series of cards with numbers on them (pictured). Someone thinks of a number and tells you which of the cards their number appears on. Quick as a flash, you tell them their number. You haven’t memorised anything; the trick works using binary numbers.

Regular numbers can be thought of as a series of columns containing digits, with each being 10 times the previous. So the right-most digit is the ones, to its left is the tens, then the hundreds, and so on. Binary numbers also use columns, but with each being worth two times the one to its right. So 01101 means zero sixteens, one eight, one four, zero twos and one one: 8+4+1=13.

Each card in this trick represents one of the columns in a binary number, moving from right to left: card 0 is the ones column, card 1 is the twos column, etc. Numbers appear on a card if their binary equivalent has a 1 in that place, and are omitted if it has a 0 there. For instance, the number 25 is 11001 in binary, so it is on cards 0, 3 and 4.

You can work this trick by taking the cards the person’s number appears on and converting them to their binary columns. From there, you can figure out the binary number and convert it to its regular number. But here’s a simple shortcut: the binary column represented by each card is the first number on the card, so you can just add the first number that appears on the cards the person names. So, for cards 0 and 2, you would add 1 and 4 to get 5.

Many self-working tricks embed mathematical principles in card magic, memorisation tricks or mind-reading displays, making the maths harder to spot. The key is they work every time.

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*Credit for article given to Peter Rowlett*

Are we getting closer to solving one of life’s greatest mysteries?

During a lunch in the summer of 1950, physicists Enrico Fermi, Edward Teller and Herbert York were chatting about a recent New Yorker cartoon depicting aliens abducting trash cans in flying saucers. Suddenly, Fermi blurted out, “Where is everybody?”

He reasoned: “Since there are likely many other technological civilisations in the Milky Way galaxy, and since in a few tens of thousands of years at most they could have explored or even colonised many distant planets, why don’t we see any evidence of even a single extraterrestrial civilisation?”

This has come to be known as Fermi’s Paradox.

Clearly the question of whether other civilisations exist is one of the most important questions of modern science. Any discovery of a distant civilisation – say by analysis of microwave data – would rank as among the most far-reaching of all scientific discoveries.

Drake equation

At a 1960 conference regarding extraterrestrial intelligence, Frank Drake (1930 —) sketched out what is now the Drake equation, estimating the number of civilisations in the Milky Way with which we could potentially communicate:

where

N = number of civilisations in our galaxy that can communicate.

R* = average rate of star formation per year in galaxy.

f_p = fraction of those stars that have planets.

n_e = average number of planets that can support life, per star that has planets.

f_l = fraction of the above that eventually develop life.

f_i = fraction of the above that eventually develop intelligent life.

f_c = fraction of civilisations that develop technology that signals existence into space.

L = length of time such civilisations release detectable signals into space.

The result? Drake estimated ten such civilisations were out there somewhere in the Milky Way.

This analysis, led to the Search for Extraterrestrial Intelligence (SETI) project, looking for radio transmissions in a region of the electromagnetic spectrum thought best suited for interstellar communication.

But after 50 years of searching, using increasingly powerful equipment, nothing has been found.

So where is everybody?

Proposed solutions to Fermi’s paradox

Numerous scientists have examined Fermi’s paradox and proposed solutions. The following is a list of some of the proposed solutions, and common rejoinders:

• Such civilisations are here, or are observing us, but are under orders not to disclose their existence.

Common rejoinder: This explanation (known as the “zookeeper’s theory”) is preferred by some scientists including, for instance, the late Carl Sagan. But it falls prey to the fact that it would take just one member of an extraterrestrial society to break the pact of silence – and this would seem inevitable.

• Such civilisations have been here and planted seeds of life, or perhaps left messages in DNA.

Common rejoinder: The notion that life began on Earth from bacterial spores or the like that originated elsewhere, known as the “panspermia theory”, only pushes the origin of life problem to some other star system – scientists see no evidence in DNA sequences of anything artificial.

• Such civilisations exist, but are too far away.

Common rejoinder: A sufficiently advanced civilisation could send probes to distant stars, which could scout out suitable planets, land and construct copies of themselves, using the latest software beamed from home.

So the entire Milky Way galaxy could be explored within, at most, a few million years.

• Such civilisations exist, but have lost interest in interstellar engagement.

Common rejoinder: As with the zookeeper theory, this would require each civilisation to forever lack interest in communication and transportation – and someone would most likely break the pact of silence.

• Such civilisations are calling, but we don’t recognise the signal.

Common rejoinder: This explanation doesn’t apply to signals sent with the direct purpose of communicating to nascent technological societies. Again, it is hard to see how a galactic society could enforce a global ban.

• Civilisations invariably self-destruct.

Common rejoinder: This contingency is already figured into the Drake equation (the L term, above). In any event, we have survived at least 100 years of technological adolescence, and have managed (until now) not to destroy ourselves in a nuclear or biological apocalypse.

Relatively soon we will colonise the moon and Mars, and our long-term survival will no longer rely on Earth.

• Earth is a unique planet in fostering long-lived ecosystems resulting in intelligent life.

Common rejoinder: Perhaps, but the latest studies, in particular the detections of extrasolar planets point in the opposite direction. Environments like ours appear quite common.

• We are alone in the Milky Way galaxy. Some scientists further conclude we are alone in the entire observable universe.

Common rejoinder: This conclusion flies in the face of the “principle of mediocrity,” namely the presumption, popular since the time of Copernicus, that there’s nothing special about human society or environment.

Numerous other proposed solutions and rejoinders are given in by Stephen Webb in his 2002 book, If the Universe Is Teeming with Aliens … Where is Everybody?.

Two of Drake’s key terms – fp (the fraction of stars that have planets) and ne (the average number of planets that can support life, per star that has planets) are subject to measurement.

Scientists once thought stable planetary systems and Earth-like planets were a rarity. But recent evidence suggests otherwise.

Thanks to Kepler and other projects, these two terms have been found to have reasonable values, although not quite as optimistic as Drake and his colleagues first estimated.

With every new research finding in the area of extrasolar planets and possible extraterrestrial living organisms, the mystery of Fermi’s paradox deepens.

“Where is everybody?” is a question that now carries even greater resonance.

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*Credit for article given to Jonathan Borwein (Jon)*

After decades of searching, mathematicians discovered a single shape that can cover a surface without forming repeating patterns, launching a small industry of “aperiodic monotile” merchandise.

The “hat” shape can tile an infinite plane without creating repeating patterns

It is rare for a shape to make a splash, but this year one did just that with the announcement of the first ever single tile that can cover a surface without forming repeating patterns. The discovery of this “aperiodic monotile” in March has since inspired everything from jigsaw puzzles to serious research papers.

“It’s more than I can keep up with in terms of the amount and even, to some extent, the level and depth of the material, because I’m not really a practising mathematician, I’m more of a computer scientist,” says Craig Kaplan at the University of Waterloo, Canada. He is on the team that found the shape, which it called the “hat”. Mathematicians had sought such an object for decades.

After revealing the tile in March, the team unveiled a second shape in May, the “spectre”, which improved on the hat by not requiring its mirror image to tile fully, making it more useful for real surfaces.

The hat has since appeared on T-shirts, badges, bags and as cutters that allow you to make your own ceramic versions.

It has also sparked more than a dozen papers in fields from engineering to chemistry. Researchers have investigated how the structure maps into six-dimensional spaces and the likely physical properties of a material with hat-shaped crystals. Others have found that structures built with repeating hat shapes could be more resistant to fracturing than a honeycomb pattern, which is renowned for its strength.

Kaplan says a scientific instrument company has also expressed an interest in using a mesh with hat-shaped gaps to collect atmospheric samples on Mars, as it believes that the pattern may be less susceptible to problems than squares.

“It’s a bit bittersweet,” says Kaplan. “We’ve set these ideas free into the world and they’ve taken off, which is wonderful, but leaves me a little bit melancholy because it’s not mine any more.”

However, the team has no desire to commercialise the hat, he says. “The four of us agreed early on that we’d much rather let this be free and see what wonderful things people do with it, rather than trying to protect it in any way. Patents are something that, as mathematicians, we find distasteful.”

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*Credit for article given to Matthew Sparkes*

After discovering that a Ralph Rowlett was in charge of the Royal Mint in 1540, Peter Rowlett runs the genealogy calculations to find out if he could be related.

In 1540, Henry VIII’s coins were made in the Tower of London. One of the Masters of the Mint was Ralph Rowlett, a goldsmith from St Albans with six children. I wondered: am I descended from Ralph? My Rowlett ancestors were Sheffield steelworkers, ever since my three-times great grandfather moved north in search of work. The trail goes cold in a line of Bedfordshire farm labourers in the 18th century, offering no evidence of a direct relationship.

My instincts as a mathematician led me to investigate this in a more mathematical way. I have two parents. They each have two parents, so I have four grandparents. So, I have eight great-grandparents, 16 great-great-grandparents and 2ⁿ ancestors n generations ago. This exponential growth doubles each generation and takes 20 generations to reach a million ancestors.

Ralph lived 20 to 25 generations before me in an England of about 2 million people. The exponential growth argument says I have several million ancestors in his generation, so, because we run out of people otherwise, he is one of them.

But this model is based on the assumption that everyone is equally likely to reproduce with anyone else. In reality, especially at certain points in history, people were likely to reproduce with someone from the same geographic area and demographic group as themselves.

But I am not sure this makes a huge difference here because we are dealing with something called a small-world network: most people are in highly clustered groups, tending to pair up with nearby people, but a small number are connected over greater distances. An illegitimate child of a nobleman would have a different social class to their father. A migrant seeking work could reproduce in a different geographic area.

We don’t need many of these more remote connections to allow a great amount of spread around the network. This is the origin of the six degrees of separation concept – that you can link two people through a surprisingly short chain of friend-of-a-friend relationships.

I ran a simulation with 15 towns of a thousand people, where everyone has only a 5 per cent chance of moving to another town to reproduce. It took about 20 generations for everyone to be descended from a specific person in the first generation. I ran the same simulation with 15,000 people living in one town, and the spread took about 18 generations. So the 15-town structure slowed the spread, but only slightly.

What does this mean for Ralph and me? There is a very good chance we are related, whether through Rowletts or another route. And if you have recent ancestors from England, there is a good chance you are too.

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*Credit for article given to Peter Rowlett*

An artificial intelligence that can turn mathematical concepts written in English into a formal proving language for computers could make problems easier for other AIs to solve.

Maths can be difficult for a computer to understand

An artificial intelligence can translate maths problems written in plain English to formal code, making them easier for computers to solve in a crucial step towards building a machine capable of discovering new maths.

Computers have been used to verify mathematical proofs for some time, but they can only do it if the problems have been prepared in a specifically designed proving language, rather than for the mix of mathematical notation and written text used by mathematicians. This process, known as formalisation, can take years of work for just a single proof, so only a small fraction of mathematical knowledge has been formalised and then proved by a machine.

Yuhuai Wu at Google and his colleagues used a neural network called Codex created by AI research company OpenAI. It has been trained on large amounts of text and programming data from the web and can be used by programmers to generate workable code.

Proving languages share similarities with programming languages, so the team decided to see if Codex could formalise a bank of 12,500 secondary school maths competition problems. It was able to translate a quarter of all problems into a format that was compatible with a formal proof solver program called Isabelle. Many of the unsuccessful translations were the result of the system not understanding certain mathematical concepts, says Wu. “If you show the model with an example that explains that concept, the model can then quickly pick it up.”

To test the effectiveness of this auto-formalisation process, the team then applied Codex to a set of problems that had already been formalised by humans. Codex generated its own formal versions of these problems, and the team used another AI called MiniF2F to solve both versions.

The auto-formalised problems improved MiniF2F’s success rate from 29 per cent to 35 per cent, suggesting that Codex was better at formalising these problems than the humans were.

It is a modest improvement, but Wu says the team’s work is only a proof of concept. “If the goal is to train a machine that is capable of doing the same level of mathematics as the best humans, then auto-formalisation seems to be a very crucial path towards it,” says Wu.

Improving the success rate further would allow AIs to compete with human mathematicians, says team member Albert Jiang at the University of Cambridge. “If we get to 100 per cent, we will definitely be creating an artificial intelligence agent that’s able to win an International Maths Olympiad gold medal,” he says, referring to the top prize in a leading maths competition.

While the immediate goal is to improve the auto-formalisation models, and automated proving machines, there could be larger implications. Eventually, says Wu, the models could uncover areas of mathematics currently unknown to humans.

The capacity for reasoning in such a machine could also make it well-suited for verification tasks in a wide range of fields. “You can verify whether a piece of software is doing exactly what you asked it to do, or you can verify hardware chips, so it has applications in financial trading algorithms and hardware design,” says Jiang.

It is an exciting development for using machines to find new mathematics, says Yang-Hui He at the London Institute for Mathematical Sciences, but the real challenge will be in using the model on mathematical research, much of which is written in LaTeX, a typesetting system. “We only use LaTeX because it types nicely, but it’s a natural language in some sense, it has its own rules,” says He.

Users can define their own functions and symbols in LaTeX that might only be used in a single mathematical paper, which could be tricky for a neural network to tackle that has only been trained on the plain text, says He.

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*Credit for article given to Alex Wilkins*