Friends of the Aperiodical, nerd-comedy troupe Festival of the Spoken Nerd, are currently on tour around the UK. As part of their show, questionably titled You Can’t Polish a Nerd, Matt Parker attempts to calculate the value of $\pi$ using only a length of string and some meat encased in pastry. He’s previously done this on YouTube, and the idea was inspired by the Aperiodical’s 2015 Pi Approximation Challenge, and in particular my own attempt to approximate $\pi$ with a (more conventional) pendulum.
For our $\pi$ approximate-off, we wanted to derive values for $\pi$ using a suite of methods that mostly didn’t involve measuring the circumference and diameter of a circle. This included evaluating as many terms of some infinite series as the length of our room booking permitted, and a recreation of the famous Buffon’s Needle experiment. But surely the most satisfying method is to just swing a heavy ball on a bit of string.
As many people learn in high school, the formula governing how long a pendulum takes to do one complete swing is \[ T = 2\pi\sqrt{\frac l g} \] where $l$ is the length of the pendulum and $g$ is the acceleration due to gravity (about $9.8ms^{-2}$). Slightly counterintuitively, the swing time doesn’t depend on the weight of the bob (or if you prefer, the mass of the weight), nor on how far back you swing it from. So just from timing a swing and measuring the string, you can solve the equation to get a numerical value for $\pi$.
But all that calculation sounds pretty boring. We can choose the length of the string, so why not choose it so that it cancels the $l$ and the 2, to give $T=\pi$? Conveniently that requires a length of $g/4 \approx 2.45$ metres (if you’ll excuse my abuse of dimensionality), which is just about doable in a tallish room or a stairwell or a theatre stage. When we tried this timing ten swings to reduce the impact of imperfect human reaction time, we ended up with a value of 3.133. (I speculate that the overwhelming source of inaccuracy is the difficulty in measuring the length of the string from the precise point of pivot at the top to the precise centre of mass of the bob/weight/pie.) On his video, Matt, using a fancy slow-mo camera, got a value of 3.128 seconds per swing. The Aperiodical has yet to take its annual pilgrimage to the Festival, so it remains to be seen how successfully the feat is accomplished live.
You might think that using a pendulum in an attempt to find $\pi$ without using circles is cheating a bit: surely a pendulum’s swing time depends on $\pi$ because it moves along the arc of a circle? Well, no. In fact the opposite is true. I lied a bit when I said the time doesn’t depend on how far back you pull the pendulum. The formula above is only an approximation for small initial angles, when a pendulum approximates simple harmonic motion. The full formula is this:
So in fact the more of a circle you make your pendulum trace out, the worse your approximation will be. Circles kill pi.
If you’d like to see Matt attempt this live on stage, the Spoken Nerd show is still on tour until the end of November, and as well as swinging food, it includes songs, live experiments, comedy and many nerdy references.
Calculating π with a pendulum, by Matt Parker on YouTube
Aperiodical’s π approximation challenge, on YouTube
Festival of the Spoken Nerd tour
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At the start of his HLF lecture on Asymptotic Group Theory on Thursday morning, Fields medalist Efim Zelmanov described the ‘group’ as: “the great unifying concept in mathematics,” remarking “if you go for a trip, and you are allowed to take only two or three mathematical concepts with you, give serious consideration to this one.” Very loosely defined, a group is a set of things (its ‘elements’) that you can ‘multiply’ together, with this multiplication behaving in certain helpful ways. Think of numbers being added, functions composed together or rotations and reflections of a shape being carried out one after the other. I doubt any mathematician would accuse Zelmanov of overstating their importance in mathematics.
In his talk he discussed residually finite groups. These are groups which are infinite in size but still just a little bit finite-y. In technical terms, the group has a set of homomorphisms with finite kernels having trivial intersection. Although the group is too large to see all at once, as Zelmanov put it, we have “photos from all sides of the group”. He contrasted this to “hopelessly infinite groups”, for which no such photo album is possible.
A common way to look at a group is to find a set of ‘generators’: these are elements of the group which you can multiply together to create any element of a group (the elements ‘generate’ the entire group). Some infinite groups can’t be generated from a finite set — consider trying to find a set of rational numbers that you can multiply together to create any rational number. Those that can be generated from a finite set are unexcitingly called ‘finitely generated’. Of course, finite groups are also finitely generated.
Zelmanov considered under what circumstances finitely generated groups can be proved to be finite. One immediate way this won’t happen is if one of the generators is not periodic: if you keep multiplying it by itself you keep getting new elements forever, never ‘looping back’ to the original generator. (Imagine starting with 1 and continually adding 1…) The Burnside problem asks whether there are any other ways to make a finitely-generated, yet infinite, group. In 1991, Zelmanov proved that for residually finite groups, there aren’t. However, this isn’t the case for the ‘hopelessly infinite’ groups.
In his lecture Zelmanov, accompanied by his excellent hand-drawn slides, discussed this before moving on to related topics such as the growth of groups (if you start with a generating set, and create new elements by multiplying them together, how quickly does the set grow?) and ‘approximate groups’ (which, as the name suggests, are things that are like, but not quite, groups).
]]>The text reads:
An orchestra of 120 players takes 40 minutes to play Beethoven’s 9th Symphony. How long would it take for 60 players to play the Symphony? Let P be the number of players and T the time playing.
Well, once you’re done laughing, we’ve done some investigative journalism and found the origin of this question. And it turns out it’s quite nice!
I wrote this!! How did you get this??? I am a maths teacher in Nottingham UK. Wrote this 10 years ago. Here is the original whole worksheet pic.twitter.com/jYX55GSBKz
— Claire Longmoor (@LongmoorClaire) October 11, 2017
The question is from a worksheet developed by maths teacher Claire Longmoor (who is, based on current evidence, brilliant) ten years ago. Claire put together a selection of example questions with relationships in direct and inverse proportion, and deliberately included the orchestra question as an example of something where it doesn’t work that way. It’s a nice activity to help reinforce the difference, and in context the question works nicely.
Other examples on the sheet include a bricklaying example with creditably diverse gender representation, a car with terrifyingly low fuel efficiency, good cow names and a delightful insight into the bygone world of fruit picking.
]]>Paul and I have spent this week blogging from the Heidelberg Laureate Forum, an international event for PhD/postdoc students and top-level maths and computer science researchers.
It was a long week of extravagant dinners, incredible talks and press conferences, (maths) celeb spotting, branded conference freebies, hilarious quotes and exceptional hospitality. Oh, and blogging. Here’s a round-up of what we wrote, in case you’ve missed it this week, as well as some of the other posts the rest of the HLF blog team wrote.
Blog posts by Katie
Blog posts by Paul
A few more posts may appear on the blog over the coming week or so, and we’ll post them across here as well.
Katie and Paul were only part of the blog team at the HLF – they were joined by Math With Bad Drawings’ Ben Orlin, a maths teacher, prolific blogger and author; Constanza (Coni) Rojas-Molina, who draws amazing sketch summaries of talks, blogs at The RAGE of the Blackboard and researches mathematical physics at the University of Bonn; Nana Liu, a quantum computing researcher there to cover the event’s ‘hot topic’ session on Quantum Computing; and Alaina Levine, a freelance writer, speaker and consultant based in Arizona. They were joined by (not pictured) Tobias Maier (biologist) and Markus Possel (physicist), who both blog in English and German.
Here are some of our favourite posts from the rest of the blog team:
You can see the rest of the HLF blog where it’s hosted at Spektrum SciLogs.
]]>The Open University and UK Mathematics Trust have teamed up to launch Perplex, a mobile app containing mathematical puzzles and games. It’s available for iPhone and Android, and can also be played directly on their website.
The description promises 8 main puzzles and over 40 daily challenges to keep you occupied, and with the mobile versions it’s also possible to share your scores and compete against your friends. It looks to be a fairly pretty game, with fun cartoony graphics and (thankfully muteable) music. It seems like exactly the kind of thing my friends and I got hooked on to distract ourselves from work while we were at uni.
The puzzle games cover various different areas of maths – including colouring problems, arithmetic challenges, and variants of classic river-crossing puzzles. Some will be pretty familiar already to fans of Henry Dudeney puzzles or Simon Tatham’s games, but for anyone who hasn’t seen this kind of thing before it should be a nice introduction – and even if you have, they’re still fun to play.
Sadly, the game has limited content, at least for now – the ‘daily challenges’ will presumably be added daily, and further puzzles are promised, but as it stands there’s only one version of each type of puzzle, and your challenge is to complete each in the most efficient way to earn all three stars. Given that it’s a free game, I’d say that’s still good value, but it would surely be easy enough to generate more versions of each?
Now if you’ll excuse me, I still haven’t got three stars on the bubbles one…
Perplex, on the iTunes app store
Perplex, on the Android app store
Perplex, on the Open University website
A wonderful potted history of the theory of communication was capably presented by 2002 Nevanlinna Prize winner Madhu Sudan, who talked us through from the earliest mathematical thinking on the subject through to the present day, and his team’s work. It was also almost a love letter to one of his mathematical heroes, the father of information theory, Claude Shannon.
Most human communication has historically used speech or writing, but the digital age has changed this into the communication of bits and bytes of digital data. Of course, speech and writing can both also be encoded in this way, but communication theory as a subject is more concerned with the pure bandwidth of data for any type of communication.
There are many problems when you attempt to communicate any kind of information from one place to another. Communication is expensive, depending on how much data you are transferring; it can be noisy, as communication lines aren’t always perfect in transmitting the data; it sometimes needs to be interactive, so systems need to be able to communicate both ways; and it’s often contextual, and data transfer needs to be robust to people speaking different languages, or using different hardware/software.
Sudan gave a simple example of an early method (still used today in simple cases!) to encode a message before sending through a noisy system, so as to improve the likelihood of the message arriving intact – this is to simply repeat each character of the message three times. In his example, to prevent the message WE ARE NOT READY being catastrophically mis-received as WE ARE NOW READY (the opposite meaning!), you could send it in this form:
WWW EEE AAA RRR EEE NNN OOO TTT RRR EEE AAA DDD YYY
Now, even if some of your data is corrupted, as below:
WXW EEA ARA SSR EEE BNN OOO PTT RSR EEE AAS DFR IYY
Errors can still make it through – if, for example, as in the second-to-last group, two different errors happen, you can’t tell what the original letter was meant to be. In the fourth group there are two errors both going to the same incorrect letter, which would lead to an incorrect conclusion about that letter if you simply looked at the most common letter in the group.
So how do we improve this system? Of course, we could use groups of 5 or 10 characters – but as Sudan points out, increasing the number of repeats decreases the probability that all the symbols in a set might be corrupted, but it decreases the amount of information you’re actually able to communicate. For 100 characters of data sent, instead of sending a 100-letter message, you can now only send 10 or 20 letters. As the number of repeats increases to infinity, usefulness of your system drops to zero – and equally, as your message gets longer, the number of repeats you can afford drops so your likelihood of errors increases.
This fine balance between methods which are more likely to preserve your message with a higher certainty, and methods which will allow your system to send the largest amount of information, was long believed to be an unwinnable fight. That is, until the hero of Madhu Sudan’s talk came along – Claude Shannon. In the late 1940s Shannon worked out a theory of communication that changed the game completely.
He envisioned ways of encoding messages which would allow a good compromise, and worked out formulae to compare rates of communication for different encoders and decoders. Put simply, if your encoder and decoder are functions E and D which you can apply to a message M, you need:
M = D(E(M) + error) : with a high probability
If you know the probability that any given bit of data will be damaged by the transmission or storage method you’re using, you can find the kind of function you’ll need.
If the probability of a bit being flipped is 0, you can send the messages as they already are with no errors. If it’s as large as ½, you may as well send anything because the message received at the other end will be essentially random, so the rate of data transfer is 0.
Even with a probability of 0.4999, there would still be a non-zero rate of data transfer, and it’s possible to design systems that work under these conditions. Shannon’s work was revolutionary, and considered a major leap of faith, since at the time no computers existed, and even the functions he was imagining weren’t known yet.
He was the first to use a probabilistic approach to this kind of mathematics, and invented many deep concepts including that of entropy, of information and even coined the word ‘bit’ – a binary digit. His work was non-constructive, as the maths it has since been applied to was yet to be invented, but all subsequent technology has kept his ideas in mind.
Sudan also explained some later contributions from Chomsky (on language structure and human communication), Yao (on designing protocols for two-way communication, which won him a Gödel prize), and Schulman (on interactive coding schemes, useful for collaborative shared document editing).
He finished by mentioning some of his group’s research into communicating in situations where there is a large shared body of information, or ‘context’. For example, in giving his talk he’d assumed everyone in the room spoke English, and had given the talk in English on this basis.
However, if the shared context isn’t quite perfect, his system is still robust to this – if there’s a word he uses nobody else in the room knows, he can take a little extra time to define it. This will increase the length of the talk (and decrease the amount he can communicate in a given time), which must be taken into account. The systems he’s working on have this same kind of shared context, but must be robust to imperfectly shared context, and this has been explored in his recent work.
]]>As part of the HLF, the Laureates are participating in press conferences throughout the week, and being bombarded with questions by well-meaning journalists and bloggers. Unlike most press conferences, where participants often have a specific topical thing they’re there to speak to the press about, the Laureates can be asked about any of their past projects, on any area of maths they’ve worked on, and many of them have a very long and illustrious career to speak of.
It can be difficult then, to be put on the spot by a taxing question, especially if you’re not expecting it. I’ve been surprising the topologists whose press conferences I’ve attended with a deceptively deep but simple question: What’s your favourite manifold?
This question has had mixed results – known Poincaré-botherer Stephen Smalewas unwilling to provide an answer, essentially because he loves all manifolds equally (he wouldn’t even narrow it down to one infinite family, although we know he probably really has a soft spot for the n-sphere, in the case n>5).
The best answer I’ve had so far came from algebraic topologist, Abel Prize winner and Fields Medalist Sir Michael Atiyah. After a moment’s thought, he stated that he definitely did have a favourite manifold, and that it’s a particular K3 surface he’s studied recently.
You may well be wondering what a manifold is – and to a topologist, it’s as fundamental as what a number or an equation is. A manifold is a surface which locally looks like ordinary Euclidean space – if you look close-up at a small part of the surface, it behaves normally. For example, if your manifold is 2-dimensional, this means that each small part of it looks like a flat piece of 2D surface, with coordinates in the usual way, and the distance between two points defined the way you’d expect.
A great example of a manifold is a sphere – it’s locally, in very small patches, pretty much a sensible surface you could put a coordinate system on. In fact, we very much do this, as the earth itself is a sphere, and yet we tend to think about it in small 2D patches which we represent as flat on maps, and measure distance in a straight line, even though we know that they join up in a more complicated way to cover the whole surface of the sphere.
Manifolds come in all different kinds – without holes in, like the sphere; with holes in, like a torus or donut shape. There even exist some 2D manifolds which can’t be realised in 3D space without intersecting themselves, like a Klein bottle or a real projective plane.
It’s thought that manifolds are especially useful because this property of being easy to understand in small local patches means we can apply knowledge of how, for example, functions behave on those simpler parts to understand the whole shape, even when the complete structure is much more complex.
They often crop up as the solution spaces to problems in physics or as graphs of functions. A nice example is the space of configurations a two-part robot arm could achieve – if it’s hinged at the shoulder and elbow, and each joint can be positioned at any angle from 0 to 360 degrees, the position of the arm is determined by a point on each of two circles; if you imagine extruding one of these circles along the length of the other, with the two circles running perpendicular to each other, you get a torus shape, and one single point on the surface of this torus corresponds to one possible configuration of the robotic arm.
K3 surfaces were first discovered by Ramanujan, back in the 1910s – while studying the diophantine equation a^{3} +b^{3} = c^{3} + d^{3}, his writings reveal he anticipated the structures of K3 surfaces – although this work was never published. The name was coined in 1958 by André Weil, who named them after three revered algebraic geometers: Kummer, Kähler and Kodaria (and the mountain K2 in Kashmir).
A K3 surface is a type of Calabai-Yau manifold, which is a kind of manifold named after Eugenio Calabi (who first conjectured that they might exist) and Shing-Tung Yau. Calabai-Yau manifolds have many applications in theoretical physics, and it’s conjectured that the extra dimensions of space-time take the form of a Calabai-Yau manifold in 6 dimensions.
K3 surfaces are a particular subset of Calabai-Yau manifolds. They can often be found as the intersection or the product of other objects, and are closely related to elliptic curves, and to higher-dimensional analogues of the torus. The image below shows a Kummer surface, which is a type of K3 surface.
The particular K3 surface Michael Atiyah specified as his favourite is one made as a product of two elliptic curves. He came across it as part of some work he did recently studying atoms and their isotopes, and using complex geometrical surfaces to model the shape of the atoms.
Helium-4 is a stable isotope of helium, meaning it’s a version of the helium atom with two neutrons, so (along with its two protons) it has a total mass of 4. Helium-4 is the most commonly occurring isotope, although Helium-3 also exists (having only one neutron).
The elliptic curve corresponding to the shape of Helium-4 is given by y^{2} + x^{4} − 1 = 0, and it’s easy to see why this manifold has captured Atiyah’s imagination. Studying the topological properties of atoms, molecules and even larger biological structures such as DNA and proteins has been incredibly fruitful and in recent years has led to some interesting results.
What’s your favourite manifold?
The 1729 K3 Surface (ArXiV paper on Ramanujan’s work on K3 surfaces)
Geometric Models of Helium (PDF, ArXiV paper)
Having extensively covered the talks and press conferences of the Laureates so far, we thought it was time to talk to some of the Young Researchers at this year’s HLF about the work they’re doing.
We took the opportunity of the Wednesday afternoon boat trip – a captive audience – to ask the researchers to tell us a bit about what they’re working on. Since we’re mathematicians, we thought our best plan of attack was to present the researchers with a simple challenge:
We found a selection of researchers prepared to explain their work, by sharing with us the numbers behind what they’re doing.
Alexandru Mihai is studying flows in fluid dynamics – patterns of movement within liquids and gases, which can be modelled using differential equations relating properties of the fluid, such as temperature, pressure and viscosity.
The Reynolds Number is used to help predict flow patterns in different fluid flow situations. Initially discovered in 1851, the concept was popularised by fluid dynamics innovator Osborne Reynolds, and named after him in 1908.
The Reynolds number of a situation can be used to predict when the flow is about to change from a smooth to a turbulent type of motion. This is useful in designing aircraft, and modelling weather patterns. Alexandru mentioned it’s particularly useful because it can be applied to similar situations at different scale levels – so you can study a scale model of a plane in a wind tunnel, and apply your conclusions to a full-size plane.
The Catalan Numbers crop up in lots of different ‘counting’ problems, and are the only sequence we were given that has an entry in the Online Encyclopedia of Integer Sequences.
One way to think about them is that the nth Catalan number gives you the number of ways to bracket a statement with n+1 terms. So for example, 1-2-3-4 could be bracketed to give you (1-2)-(3-4), 1-(2-(3-4)), 1-((2-3)-4), (1-(2-3))-4, or ((1-2)-3)-4): so the third Catalan number is 5. The numbers grow reasonably quickly: the sequence begins 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012…
Pavel uses these numbers in his graph-theoretic research in bioinformatics to study the structure of DNA, and ancestral genome reconstruction.
Mabel’s research is into the controllability of stochastic differential equations, ie, equations that incorporate some random ‘noise’. Controllability is concerned with how a function’s outputs can be controlled by manipulating its state.
The numbers Mabel chose to represent her research are the random ones that the Matlab function ‘randn’ provides. This function generates random numbers drawn from a standard normal distribution. For those of you who want something concrete, here is a sample acquired from the equivalent function in a different software package (spot the massive outlier): -1.0261, -0.60365, 1.0681, -0.68539, -2.4482, -0.62760.
Matthew studies approximation algorithms, and he told us about the counting problems he’s studying. Counting problems are questions like, if we take a given set of integers, how many subsets can we find which add up to zero? This type of problem is computationally difficult, and falls into a class called #P-complete (sharp-P complete). The answers to such problems are the numbers Matthew is trying to find.
These are more difficult than the better known class of NP-complete problems, and the particular problems Matthew is working on include computing the permanent of a matrix. This is a property of a matrix very similar to the determinant but much more difficult to compute in general. One of this year’s Laureates, Leslie Valiant, proved in 1979 that this problem is #P-complete.
Saul Freedman is a master’s student doing research in group theory. He’s constructing p-groups – groups whose size is a power of a prime number – related to certain exceptional groups of Lie type.
He wanted to tell us about one (well, one infinite family) in particular, of size p^{14} for p an odd prime, related to the group G_{2}(p). So that’s 4,782,969 elements in the smallest case. His work involves constructing these groups and studying the types of transformations you can apply to them preserving the group structure.
In mathematical physics, ħ (‘h-bar’) is known as the reduced Planck constant, or Dirac constant and is given by the Planck constant h, divided by 2π. It can be used to calculate the energy and the linear momentum of a photon, and is part of the Schrödinger Equation in quantum mechanics.
Eric is studying the mathematical foundations of electronic structure theory, and the solvability of time-dependent Schrödinger equations.
]]>The HLF, like all good conference events, has involved a large number of extravagant dinners, serving a variety of delicious food and drink to sustain the high levels of serious mathematical and research conversation. At last night’s Bavarian evening, I noticed a particularly mathematically interesting foodstuff was on the menu, and it’s inspired me enough to write about it.
Alongside the traditional Bavarian wurst and noodles, there was a pile of cooked vegetables, most of which were pretty standard – broccoli, and cauliflower. But in amongst was also a very special type of vegetable – known in German as Pyramidenblumenkohl, or “pyramid cauliflower”, and in English called Romanesco broccoli. In fact, even botanists can’t agree whether it’s really a broccoli or a cauliflower, but everyone who’s ever seen one agrees that it’s beautiful, and mathematically interesting.
One of the first things you notice when you look at this plant is that it’s made up of a repeated pattern of shapes – each part is a small version of the whole plant, and they get smaller and smaller as you go up to the top.
This property of a shape is called self-similarity, and it simply means that small parts of the object have the same structure as the whole shape. This is a property which fractals have, and in fact is one of the defining characteristics of a fractal – if you look at any example of a fractal you should be able to see smaller copies of the shape inside itself, repeated at different sizes all the way down.
In the case of a fractal, this property needs to continue to infinity, and sadly in the case of food it doesn’t carry on forever – but it does mimic the structure of a fractal.
As well as having the excellent fractal property of self-similarity, the Romanesco broccoli also contains a beautiful spiral structure. The spirals go from the tip of each part of the plant and continue around to the base of each floret.
The shape of the spiral is a logarithmic spiral – a spiral whose rate of curving changes along the length of the curve in such a way that the shape of the curve stays the same. So, if you zoom in on a small part of this spiral, it’ll look like the whole curve – it’s also self-similar.
There are many examples of logarithmic spirals in nature, including in nautilus shells, and in the way spiral galaxies form. But this is obviously the most delicious.
Just when you thought you’d discovered all the interesting mathematical properties contained within your dinner – there’s another. If you count the number of spirals running around the florets in each direction, with a small amount of variation due to natural mutations, you’ll find there’s a Fibonacci number of spirals running each way.
The Fibonacci series is a list of numbers dating back to 1202, when Leonardo of Pisa (aka Fibonacci) discovered them as a pattern in various systems. Each Fibonacci number is the sum of the previous two, so if you start with 0 and 1 as the first two numbers, the sequence continues 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on.
The same property of having a Fibonacci number of spirals running in each direction crops up in many other types of plant also – pine cones, pineapples, flower petals and the seeds in sunflower heads have all also exhibited this property – the latter being noted by Alan Turing, and thoroughly researched in a large-scale sunflower growing and observation project that took place in 2012.
Mathematician Vi Hart has produced an excellent series of YouTube videos on the subject, starting with this one.
]]>Bad news: The Turing award winner and father of LaTeX thinks the proofs you (and everyone else) are writing are sloppy, non-rigorous and quite likely flat-out wrong. But there’s good news too: Sir Michael Atiyah is not quite so sure.
In one of the more combative talks at this year’s HLF, on Tuesday morning Lamport outlined his issues with the current state of proofs in mathematics, and gave us a glimpse of his preferred system. Essentially, while much about maths notation has moved on since the 17th century, he thinks the prevalent style of writing proofs has not, and that it’s over-reliant on prose, wilfully terse, and fatally prone to obscuring errors. For Lamport, the purpose of a proof is to make sure that the result you’re trying to prove is actually true, and current proofs are simply not fit for this purpose.
I’m sure anyone who’s waded through a typical published proof, trying in desperation to tie together the unexplained inferences, can sympathise. If you do, then Lamport’s half-hour talk How to Write a 21st Century Proof is well worth a watch. (He also has a paper explaining the same ideas.) He reworks a proof from Spivak’s Calculus of a corollary to the Mean Value Theorem, and his hierarchical structure and thorough referencing of the justifications for each step certainly seemed like they’d be appreciated by a reader new to the result. He imagines a possible future of proofs in hypertext, with collapsible layers of explanation, optional sketches and other extras to help guide you through a dense proof.
But Lamport’s ideas haven’t been met with universal agreement at the HLF. When he asked rhetorically whether a student should be expected to hunt through a textbook to find a statement 36 pages earlier justifying a step of the proof, I thought I heard a muttered “Yes!” from the audience.
Sir Michael Atiyah was not wholly convinced either. During his press conference later in the day, the Fields medalist said in passing that he had agreed with 90% of Lamport’s talk, and disagreed with 90% of it.
Later in the conference, I asked Atiyah about his recent work on the Feit-Thompson Theorem (a result from my area of maths, finite group theory). He has a proposed new proof, an order of magnitude shorter than the current 255-page behemoth. He took the opportunity of my question to return to his thoughts on proofs, jokingly saying in reference to the 12-page one he was carrying a copy of, “If you want to know how to write a proof for the twenty-first century […] this is it!”
Atiyah likened his view of a theorem to a building with foundations and inner structure, rather than a chain in which each link is crucial to prevent the whole thing falling apart. “A good theorem has structural stability”, and if you really understand a proof, a hole in one window will not cause the whole edifice to collapse.
Perhaps both ways of looking at these things are necessary. Understanding the high-level structure behind a theorem is bound to be necessary to place the result in its context and learn about the wider field. But one wonders how many erroneous results are published by authors who thought that they understood the grand structure of their ‘proof’: according to Lamport’s talk, anecdotal evidence suggests one-third of published papers are contain incorrect results. Checking the nitty-gritty details will always be necessary, and as long as published proofs exist that help with neither of these very successfully, advances in either area are to be welcomed.
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