A Company Induction and Celebration of Found Spaces in Toronto
by Vanessa Smythe
the way I see it we have two sources of light: the stories we tell and the sun what if all the people in our city disappeared. you woke up went outside but heard nothing not only people but the animals too, the bugs, even flowers bent down and stopped breathing. all the clocks struck and they echoed so far, because no one was there to muffle the sound If all of us vanished, think of the things we would lose. like the time I walked home talking to you on the phone, so it was like you were walking there with me, and it took me several minutes to even realize it had started to snow. or the time it was New Years Eve! and in the middle of the street I met students underdressed and holding instruments and all of us sang made up songs that would never again be repeated as we walked to get breakfast in a diner that pulled five tables together just so we could eat side by side. if it was gone I’d miss the guy in the donut store around the corner I don’t even like donuts but I go in because what happens to all those donuts at the end of the day, my mom once asked me. I feel so sad for the man, she said, tearing. I’d miss the lit up corner of the department store too big to go into, the one that’ll be torn down and that even though I never much went in I’ll be sad to see gone I’d miss the way each pocket of the city has its own note and when you go there you see young couples pushing babies in strollers or old men with missing teeth cracking jokes, where your cab driver tells you the secrets of living a happy life and what you need to do to keep a man and what happened when somebody got into the back of his car with a gun and forced him to drive outside the city, and how he, the two of you leaning close together now, how he’d crashed the car as a way to survive. I’d miss the guy making shawarmas at 3am on Saturday night and how you have an entire conversation with just your eyes. and the old woman in your building who always complains: what’s wrong with the weather, how long it’s taking to get the elevator fixed, how ridiculous it is they’d be doing construction on a Tuesday, and even though she croaks like an animal you wish would just shoo away you hear her voice and start to wonder who she’s lost, why her hair’s always hidden under that scarf, what she’s seen or the man and his daughter at the Laundromat and he asks if you have any advice on writing a book and you can’t tell if he’s serious, but then he tells you how long he’s lived here, what it was like to leave his family in his country but how this is his country now and his teeth look so white you think as you put your darks and lights together in the same load because what difference does it make anyway One time I found a street I’d never walked down before, it parted the city like a forest. I saw the museum and the university and the wall where in graffiti someone’s painted: Embrace What You Are. I walked, made a left, saw the shops, the ones that’ve been here since my grandparents were teenagers kissing, I went to the CN tower like it was an x on a map that I could dig out of the ground. I got to the dome where the blue jays live, where the steam whistles brew, where tourists take pictures in front of train tracks as new couples pose in cold weather in their wedding dresses and suits and I wandered around, paused on the bridge, remembered coming here as a kid and touching the base of the CN tower with my hand as if, I don’t know what I expected, but as if it could give me some charge like lightning. but nothing happened, so like then once again I closed my eyes and leaned back to the sun When the party ends and it’s time to go home forever, what do we want to feel? what do we want to gaze behind us to see? what colour should our fireworks be exploding in the distance like spurs? in those moments when we wonder what the hell it’s all for is there any meaning, any beauty or is everything just people trying to sell us something like the man on the street who handed me a pamphlet about a harp and I nodded and hurried away I don’t want to buy a harp, but then when I looked at it closer I saw that in faded letters in the corner was a poem he’d written that stopped me in my tracks so that when I looked back I didn’t see the shiny shops or spinning cars but I only saw this man with a very long beard, holding his harp in his arms like a child. What if a city is not just a place, but a suggestion of what we place in our hearts? A series of traces we might see ourselves through how else might we wonder what else we could be without first reinventing our own walls and roofs? listen, lean in to the space that we’re in so we know where we are and recall what we have to lose this is what I try to remember when I walk along your paths and underneath your bridges and sit on your streetcar right by the window so the breeze funnels through and makes me feel like I’m on a ship heading home for just this summerI remember walking around with a friend at the end of an afternoon, we wandered an east corridor, one we’d never been before. We walked until we got to the top of a field with a bench and we sat there We watched the people playing baseball badly, laughing with friends that’d made t-shirts they could wear so they could all be part of the same team before the sun gleaned into the shadows. We sat there eating fast-food takeout sandwiches as we looked at the city skyline from this other side of where we came from, We sat there just us and the warm summer air and I swear: we had everything we needed.
Canadian actor and performance poet Vanessa Smythe composed a 100-line love letter to Toronto. 100 wonderful OtM supporters commissioned each line. And 100 OtM artists performed the poem in a special location of their choosing. 100 Outside the March Artists Celebrate Toronto
\( \begin{bmatrix} 3 &2 \\ 5 &3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 14 \\ 22 \end{bmatrix} \)
\( AX=B \)
We have a linear map from \( \mathbf{\mathbb {R^2}} \) to \( \mathbf{\mathbb{R^2}} \) and we want to know which vectors x, if any, map to the vector b.
for 2×2 matrices \( A = \begin{bmatrix}a&b \\ c&d\end{bmatrix} \),
The determinant of the matrix \(\det A = ad - bc. \)
If the determinant is zero, the matrix is not invertible.
A matrix with determinant 0 is not invertible. The converse of this turns out to be true as well: a matrix with nonzero determinant is invertible. Thus the determinant gives us a way of finding out whether a matrix can be inverted. (The Princeton Companion to Mathematics - Part III Mathematical Concepts - III.15 Determinants)
The starting point of Linear algebra is Gaussian elimination, a procedure that can solve n linear equations in n unknowns using on the order of \( n^3 \) arithmetic operations. Equivalently, it solves equations of the form \( Ax = b \), where A is an \( n×n \) matrix and x and b are column vectors of size n. Gaussian elimination is invoked on computers around the world almost every time a system of linear equations is solved. (The Princeton Companion to Mathematics - IV.21 Numerical Analysis, Lloyd N. Trefethen - 4 Numerical Linear Algebra)
Let’s take a look at this clay tablet. It might not look like much, but it’s actually some of the oldest mathematics we have. It’s a 4,000-year-old message in a bottle from ancient Babylon – a precursor to the quadratic equation. And for four millennia, people have been doing math basically the same way. Someone will have a brilliant idea, they’ll write it down, and their peers will discuss and check it. It’s a process built on creativity, communication and, most importantly, trust between people. And what might seem like a humble or simple process is anything but. It’s not just been successful. It’s been, as the physicist Eugene Wigner famously put it, unreasonably effective.
Wigner was pondering and trying to unravel a deep mystery. Why should the abstract, creative, and often bizarre ideas that spring from a mathematician’s imagination so often be the perfect language with which we understand the universe? Why should the strange laws of non-Euclidean geometry, which were originally conceived of as a thought experiment in the 19th century, turn out to be the exact mathematics that Einstein needed for general relativity? Why should the esoteric math of group theory, which was originally designed to study the abstract nature of symmetry, be fundamental to understanding everything from particle physics to the patterns in crystals? Well, there’s no logical reason it has to be this way. This strange connection between pure mathematical thought and the real world has actually been the invisible engine driving human progress.
Every piece of technology that defines our lives was ignited with a mathematical spark. If you take the device in your phone, its brain is based on the quantum mechanics of semiconductors. And that’s a theory built on linear algebra and complex numbers. The wireless signals that get data to it, they’re just a concrete manifestation of Maxwell’s equations. And finally, the security that protects your data online is based on number theory, which for a long time was truly considered the most pure and least applicable possible branch of mathematics. And now it safeguards trillions of dollars in the global economy.
And now we come to AI. Modern AI is not just built with math, it’s forged from it. A neural network is just a monumental structure of applied mathematics. And when AIs learn, they’re using the tools of calculus to navigate vast landscapes of possibilities with billions of dimensions. So AI is, in its soul, a mathematical idea that’s given life through computation.
So we agree that math is the foundation that modern civilization is based on. But that foundation is starting to show some signs of strain. The very process of human-led discovery that’s gotten us to this point is nearing a breaking point, buckling under the weight of its own success. And now AI, which is one of mathematics’ greatest creations, is accelerating us towards that breaking point faster than the world’s ready for. So let’s just look at some evidence.
Consider the Poincaré conjecture. This is a legendary problem. It’s a fundamental question about the nature of three-dimensional shapes originally posed in 1904. And for nearly a century, it stood as an unconquered Everest of mathematics. Until in 2002, a Russian mathematician working in isolation named Grigori Perelman posted a series of three short, cryptic papers online. He didn’t bother submitting them to a journal – he just put them on the internet and walked away. His fellow mathematicians had to stop what they were doing and try to decipher it. And several teams working independently of the best of colleges in the world, took the next four years to try to unpack the arguments, fill in the logical gaps and eventually, at the end, after they really reviewed it, declare that yes, he did it. He proved the Poincaré conjecture. But that’s interesting because it took one person to write a proof and a global, multi-year intellectual mobilization to check it. And that’s in the best case, when the proof is correct.
Consider Andrew Wiles’s proof of Fermat’s Last Theorem. With the electrifying announcement in 1993 in Cambridge, the world celebrated. But during the peer-review process, deep in it, a single thread was found out of place in that magnificent tapestry of a proof, and when we started to pull on it, the proof started to unravel. And this wasn’t a small mistake. Andrew Wiles and his collaborator Richard Taylor took two years of heroic, secret effort to try to fix it. And that effort included some insights that Andrew Wiles said were among the most important in his life. And that’s before we throw AI into the mix.
Two short years ago, AI could barely solve entry-level high school math-contest problems. They were very clever, but brittle. Now, in 2025, they can compete with the best of us at the International Math Olympiad, which is the premier precollege math competition. But the interesting bit is the following. The AI might work for four hours and produce a purported solution, which takes an expert human mathematician maybe up to an hour to check. And we all know the exponential trend that AI is on. So we can expect it’s not going to be one proof in an afternoon – it’s going to be a thousand pretty soon. And they’re not going to be attempts to solve math-contest problems. They’re going to be attacks on the most fundamental and important questions of the day, whether it’s the Riemann hypothesis, Navier-Stokes or P versus NP, just to pick a few. We simply don’t have the human bandwidth to review all these proofs. There’s only a couple thousand mathematicians that are qualified to do it, and they already have day jobs. And it’s not just a verification bottleneck. The very process by which we train these AIs is taking the data off the internet, which is from humans, post-training them with human feedback, and so we’re essentially baking in the cognitive biases and the flawed reasoning of humans into these future engines of discovery.
So the conclusion is in some sense obvious. Humans are becoming the bottleneck of verification for AI. And now the question is, where does that leave us? Is this the end of the road for reliable mathematical discovery? Are we resigned to drowning in a sea of unverified claims where we can’t really tell truth from fiction? And are we about to squander the opportunity for AI to revolutionize math? Well, the good news is no. But it does mean it’s time to upgrade the 4,000-year-old operating system of math, and move away from the imprecise and ambiguous nature of human language, and towards a language that computers can understand.
The solution is formal mathematics. But before I tell you how this futuristic idea works, we should first recognize that it has a deep and fascinating history dating back to the 17th century, where a mathematician actually laid out the road map with stunning foresight. Four hundred years ago, in a Europe torn by religious and political conflict, a polymath named Gottfried Wilhelm Leibniz had a vision of breathtaking ambition. He was a contemporary of Newton and a cocreator of calculus, but his dreams went far beyond that. He dreamed of something called a universal characteristic, which was a system for perfectly encoding all scientific and philosophical thought. And the system had three parts. First, you need a perfect logical language. Second, you need a grand encyclopedia written in language that contains all verified human thought. And third, and this is the masterstroke, you need a so-called engine of reason, a system of mechanical rules by which you can automatically derive new facts from that library as surely as a calculator performs arithmetic.
Now, Leibniz thought this would revolutionize humanity. With a system like this, if two people had an intellectual conflict, they would resort to logic and not rhetoric to resolve it. They would simply sit down, say “calculemus” – “let us calculate,” and get to the bottom of it. In some sense, it was meant to be a universal calculator for truth. Now, Leibniz was a bit of an optimist. He thought this would take a small group of people five years to build, and he was off by several centuries. But what I think is really remarkable is that in 2025, truly for the first time in history, it’s actually possible to realize this philosopher’s dream.
So what do we need? Well, we need a perfect, logical language. Turns out we’ve got it. It’s called Lean. Lean is a programming language, but it’s also what’s known as a proof assistant. You can think of it as a programming environment for mathematical proofs, where it doesn’t just give you feedback if you have a syntax error here or there – it’s actually looking at the core of the mathematical argument and telling you if you have any problems anywhere in it. Great.
What’s the second thing we need? We need the grand encyclopedia. Well, the good news is we’ve got that too. It’s called Mathlib. Mathlib is an open-source project. It’s about two million lines of code in Lean, and it covers a lot of the undergraduate and graduate math curriculum. You can think of it like a Wikipedia for proven truth, where every edit is computationally certified for correctness.
OK, we’ve got the language, we’ve got the encyclopedia, what about the engine of reason? Well, we could try to have humans do it, but you’ve got to write a lot of Lean code and the level of robotic precision you need to write a formal proof is not something that human creativity is so well suited for. And that’s how we’ve come full-circle. It turns out that AI is the key to making this whole thing work.
In the future, AI is not just going to be writing math papers in English for humans to read. They’re going to be writing math proofs in Lean for computers to check. And that is the fundamental key that makes it possible to use Leibniz’s vision to unlock the full potential of AI in mathematics. Because when a math AI spits out a proof in Lean of, let’s say, the Riemann hypothesis, we’re not going to need humans to go through every single line of the proof in painstaking detail, check every single case, and understand the possibly strange and alien logic of the proof just to see if it’s correct. Instead, all we’re going to do is we’re going to take those files, we’re going to give them to a Lean compiler, and if it builds, we can know with absolute certainty it’s correct. And this is what fundamentally alters our relationship with AI.
AI can now become a true collaborator, one whose word we don’t have to take on blind faith. We get to trade in the tedium of checking for the creative joy of discovery. Humans get to use our intuition and judgment, we ask the questions, we chart the course, we propose the brilliant conjectures and then we delegate to AI to explore the vast oceans of logic, to find the correct answer, and then a computer confirms that we’ve gotten to the destination. And the amazing thing is that this isn’t just some far-off science-fiction dream. It turns out that at this year’s International Math Olympiad, automated systems were able to find solutions to five of the six problems in a way that computers could check and require no human review whatsoever. And that’s enough to get a gold-medal-level performance. So the transition is already happening.
So are humans going to be the bottleneck for math research? Well, the answer is yes, but only if we refuse to change. Only if we insist on being the only thinkers and the only checkers. But if we’re able to realize this 400-year-old vision, we’re not going to replace ourselves, we’re going to elevate ourselves. We’re going to put ourselves in the driver’s seat as the explorers, the architects and the question askers. And that means that formal mathematics is the key to this new era of discovery, based on the powerful and essential partnership between human imagination and mathematical superintelligence.
Some mathematicians even believe that computer-assisted and, more importantly, computer-generated proofs are the future of the entire discipline. Under this (currently minority) view, our present views about what counts as an acceptable mathematical proof will soon become obsolete. (The Princeton Companion to Mathematics - II.6 The Development of the Idea of Proof, Leo Corry - Epilogue: Proof in the Twentieth Century)
In this paper, we show that every \( (2^{n-1}+1) \)-vertex induced subgraph of the n-dimensional cube graph has maximum degree at least \( \sqrt{n} \). This result is best possible, and improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.
A non-technical exposition of the Kakeya conjecture, why it matters, and the road to the solution of this conjecture in three dimensions by Hong Wang and Joshua Zahl.
Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, Vincent Tassion
We prove that for supercritical percolation on every infinite transitive graph, the probability that the origin belongs to a finite cluster of size at least n decays exponentially in \( \Phi(n) \), where \( \Phi \) is the isoperimetric function of the graph.
The object of a typical paper is to establish mathematical statements. mathematical statements are the main currency of mathematics.
A conjecture is a mathematical statement that the author firmly believes but cannot prove.
A mathematical statement is established by means of a proof.
Theorems
The most important of these statements are usually called theorems.
The statements that are the main goals of a paper are usually called theorems.
A theorem is a statement that you regard as intrinsically interesting, a statement that you might think of isolating from the paper and telling other mathematicians about in a seminar, for instance. The statements that are the main goals of a paper are usually called theorems.
The main aim of an article in mathematics is usually to prove theorems.
A proposition is a bit like a theorem, but it tends to be slightly “boring.”
Lemmas
Lemma 1.1.15 Let A be a square matrix that has a right inverse, a matrix R such that AR = I and also a left inverse, a matrix L such that LA = I. Then R = L. So A is invertible and R is its inverse.
Often, if you are trying to prove a theorem, the proof becomes long and complicated, in which case if you want anybody to read it you need to make the structure of the argument as clear as possible. One of the best ways of doing this is to identify subgoals, which take the form of statements intermediate between your initial assumptions and the conclusion you wish to draw from them. These statements are usually called lemmas.
One can draw a parallel with computer programming: if you are writing a complicated program, it is good practice to divide your main task into subtasks and write separate mini-programs for them, which you can then treat as “black boxes,” to be called upon by other parts of the program whenever they are useful.
Some lemmas are difficult to prove and are useful in many different contexts, so the most important lemmas can be more important than the least important theorems. However, a general rule is that a result will be called a lemma if the main reason for proving it is in order to use it as a stepping stone toward the proofs of other results.
A corollary of a mathematical statement is another statement that follows easily from it. Sometimes the main theorem of a paper is followed by several corollaries, which advertise the strength of the theorem.
Fermat’s little theorem, which states that if p is a prime and \( a \not \equiv 0 \hspace{2mm} (mod \hspace{1mm} p) \), then \( a^{p-1} \equiv 1 \hspace{2mm} (mod \hspace{1mm} p) \).
Euler’s theorem states that if m is a positive integer and a is another positive integer that is coprime to m(this means that a and m have no common factor), then \( a^{\phi(m)} \equiv 1 \hspace{2mm} (mod \hspace{1mm} m) \). Here \( \phi \) is Euler’s totient function: \( \phi(m) \) is the number of integers less than m that are coprime to m.
Fermat’s little theorem is a corollary of Euler’s theorem. Euler’s theorem is a generalization of Fermat’s little theorem to composite moduli.
Lagrange’s theorem, which states that the size of a group is always divisible by the size of any of its subgroups.
Fermat’s little theorem is also a corollary of Lagrange’s theorem.
Here is the argument in outline. The first step is to show that the numbers 1, 2, . . . , p−1 form a group under multiplication mod p. (This means multiplication followed by taking the remainder on division by p. For example, if p = 7 then the “product” of 3 and 6 is 4, since 4 is the remainder when you divide 18 by 7.) The next step is to note that if \( 1 \leq a \leq p−1 \) then the powers of a (mod p) form a subgroup of this group. Moreover, the size of the subgroup is the smallest positive integer m such that \( a^m \) is congruent to 1 mod p. One then applies Lagrange’s theorem, which states that the size of a group is always divisible by the size of any of its subgroups. In this case, the size of the group is p − 1, from which it follows that p − 1 is divisible by m. But then, since \( a^m = 1 \), it follows that \( a^{p−1} = 1 \). This argument shows that Fermat’s little theorem is, when viewed appropriately, just one special case of Lagrange’s theorem. (The word “just” is, however, a little misleading, because it is not wholly obvious that the integers mod p form a group in the way stated. This fact is proved using Euclid’s algorithm.) - (The Princeton Companion to Mathematics - Part I Introduction - I.4 The General Goals of Mathematical Research - 3 Generalizing - 3.2 Proving a More Abstract Result)
Definitions
Another very important component of mathematical papers is definitions.
Some mathematicians will tell you that the main aim of their research is to find the right definition, after which their whole area will be illuminated. Yes, they will have to write proofs, but if the definition is the one they are looking for, then these proofs will be fairly straightforward.
The Princeton Companion to Mathematics, Part I Introduction, I.4 The General Goals of Mathematical Research, 10 What Do You Find in a Mathematical Paper?
The apparent breathtaking strength of the approximation li(x) to π(x) is uncanny. As you can plainly see, Gauss’s guess is right on the money! - Carl Pomerance
His prediction that it would rain was right on the money.
live paycheck to paycheck
About a quarter of Americans today live paycheck to paycheck, spending more than 95% of their income on necessities. — The Week Us, TheWeek, 9 Mar. 2026
you can say that again
“She’s in a bad mood.” “You can say that again.”
you can’t teach an old dog new tricks
I tried to get my mother to start using a computer, but she says you can’t teach an old dog new tricks.
the good old days
In the 1960s, everything seemed possible. Those were the good old days.
don’t mention it
“Thanks for your help.” “Oh, don’t mention it. I was happy to do it.”
speak/talk of the devil
“Well, speak of the devil! We were just talking about you!”
be/become one’s own man
He left home and moved to the city to become his own man.
kick the bucket
He had things he wanted to accomplish before he kicked the bucket.
kicks upstairs
After years of hard work she was kicked upstairs to an executive role.
do (one’s) grocery shopping
But people are still venturing out to do grocery shopping and to visit parks. — BostonGlobe.com, 28 Apr. 2021
no worries
“What if we miss the bus?” “No worries, there’s another one in seven minutes.”
no wonder
(It’s) No wonder you’re hungry; you didn’t have any breakfast.
Raymond is an autistic savant in the film Rain Man.
autism spectrum disorder
noun
Autism spectrum disorder includes conditions previously called autism, pervasive developmental disorder and Asperger’s syndrome. … Because of the range of symptoms, this condition is now called autism spectrum disorder (ASD). — Jody Holton
Thanks very much. I am Hannah Fry, the badass. And today I’m asking the question: Is life really that complex? Now, I’ve only got nine minutes to try and provide you with an answer, so what I’ve done is split this neatly into two parts: part one: yes; and later on, part two: no. Or, to be more accurate: no?
So first of all, let me try and define what I mean by “complex.” Now, I could give you a host of formal definitions, but in the simplest terms, any problem in complexity is something that Einstein and his peers can’t do. So, let’s imagine – if the clicker works … there we go. Einstein is playing a game of snooker. He’s a clever chap, so he knows that when he hits the cue ball, he could write you an equation and tell you exactly where the red ball is going to hit the sides, how fast it’s going and where it’s going to end up. Now, if you scale these snooker balls up to the size of the solar system, Einstein can still help you. Sure, the physics changes, but if you wanted to know about the path of the Earth around the Sun, Einstein could write you an equation telling you where both objects are at any point in time. Now, with a surprising increase in difficulty, Einstein could include the Moon in his calculations. But as you add more and more planets, Mars and Jupiter, say, the problem gets too tough for Einstein to solve with a pen and paper. Now, strangely, if instead of having a handful of planets, you had millions of objects or even billions, the problem actually becomes much simpler, and Einstein is back in the game. Let me explain what I mean by this, by scaling these objects back down to a molecular level.
If you wanted to trace the erratic path of an individual air molecule, you’d have absolutely no hope. But when you have millions of air molecules all together, they start to act in a way which is quantifiable, predictable and well-behaved. And thank goodness air is well-behaved, because if it wasn’t, planes would fall out of the sky. Now, on an even bigger scale, across the whole of the world, the idea is exactly the same with all of these air molecules. It’s true that you can’t take an individual rain droplet and say where it’s come from or where it’s going to end up. But you can say with pretty good certainty whether it will be cloudy tomorrow. So that’s it. In Einstein’s time, this is how far science had got. We could do really small problems with a few objects with simple interactions, or we could do huge problems with millions of objects and simple interactions. But what about everything in the middle?
Well, just seven years before Einstein’s death, an American scientist called Warren Weaver made exactly this point. He said that scientific methodology has gone from one extreme to another, leaving out an untouched great middle region. Now, this middle region is where complexity science lies, and this is what I mean by complex. Now, unfortunately, almost every single problem you can think of to do with human behavior lies in this middle region. Einstein’s got absolutely no idea how to model the movement of a crowd. There are too many people to look at them all individually and too few to treat them as a gas. Similarly, people are prone to annoying things like decisions and not wanting to walk into each other, which makes the problem all the more complicated. Einstein also couldn’t tell you when the next stock market crash is going to be. Einstein couldn’t tell you how to improve unemployment. Einstein can’t even tell you whether the next iPhone is going to be a hit or a flop. So to conclude part one: we’re completely screwed. We’ve got no tools to deal with this, and life is way too complex.
But maybe there’s hope, because in the last few years, we’ve begun to see the beginnings of a new area of science using mathematics to model our social systems. And I’m not just talking here about statistics and computer simulations. I’m talking about writing down equations about our society that will help us understand what’s going on in the same way as with the snooker balls or the weather prediction. And this has come about because people have begun to realize that we can use and exploit analogies between our human systems and those of the physical world around us.
Now, to give you an example: the incredibly complex problem of migration across Europe. Actually, as it turns out, when you view all of the people together, collectively, they behave as though they’re following the laws of gravity. But instead of planets being attracted to one another, it’s people who are attracted to areas with better job opportunities, higher pay, better quality of life and lower unemployment. And in the same way as people are more likely to go for opportunities close to where they live already – London to Kent, for example, as opposed to London to Melbourne – the gravitational effect of planets far away is felt much less.
So, to give you another example: in 2008, a group in UCLA were looking into the patterns of burglary hot spots in the city. Now, one thing about burglaries is this idea of repeat victimization. So if you have a group of burglars who manage to successfully rob an area, they’ll tend to return to that area and carry on burgling it. So they learn the layout of the houses, the escape routes and the local security measures that are in place. And this will continue to happen until local residents and police ramp up the security, at which point, the burglars will move off elsewhere. And it’s that balance between burglars and security which creates these dynamic hot spots of the city. As it turns out, this is exactly the same process as how a leopard gets its spots, except in the leopard example, it’s not burglars and security, it’s the chemical process that creates these patterns and something called “morphogenesis.” We actually know an awful lot about the morphogenesis of leopard spots. Maybe we can use this to try and spot some of the warning signs with burglaries and perhaps, also to create better crime strategies to prevent crime. There’s a group here at UCL who are working with the West Midlands police right now on this very question. I could give you plenty of examples like this, but I wanted to leave you with one from my own research on the London riots.
Now, you probably don’t need me to tell you about the events of last summer, where London and the UK saw the worst sustained period of violent looting and arson for over twenty years. It’s understandable that, as a society, we want to try and understand exactly what caused these riots, but also, perhaps, to equip our police with better strategies to lead to a swifter resolution in the future. Now, I don’t want to upset the sociologists here, so I absolutely cannot talk about the individual motivations for a rioter, but when you look at the rioters all together, mathematically, you can separate it into a three-stage process and draw analogies accordingly.
So, step one: let’s say you’ve got a group of friends. None of them are involved in the riots, but one of them walks past a Foot Locker which is being raided, and goes in and bags himself a new pair of trainers. He texts one of his friends and says, “Come on down to the riots.” So his friend joins him, and then the two of them text more of their friends, who join them, and text more of their friends and more and more, and so it continues. This process is identical to the way that a virus spreads through a population. If you think about the bird flu epidemic of a couple of years ago, the more people that were infected, the more people that got infected, and the faster the virus spread before the authorities managed to get a handle on events. And it’s exactly the same process here.
So let’s say you’ve got a rioter, he’s decided he’s going to riot. The next thing he has to do is pick a riot site. Now, what you should know about rioters is that, um … Oops, clicker’s gone. There we go. What you should know about rioters is, they’re not prepared to travel that far from where they live, unless it’s a really juicy riot site.
So you can see that here from this graph, with an awful lot of rioters having traveled less than a kilometer to the site that they went to. Now, this pattern is seen in consumer models of retail spending, i.e., where we choose to go shopping. So, of course, people like to go to local shops, but you’d be prepared to go a little bit further if it was a really good retail site. And this analogy, actually, was already picked up by some of the papers, with some tabloid press calling the events “Shopping with violence,” which probably sums it up in terms of our research. Oh! – we’re going backwards.
OK, step three. Finally, the rioter is at his site, and he wants to avoid getting caught by the police. The rioters will avoid the police at all times, but there is some safety in numbers. And on the flip side, the police, with their limited resources, are trying to protect as much of the city as possible, arrest rioters wherever possible and to create a deterrent effect. And actually, as it turns out, this mechanism between the two species, so to speak, of rioters and police, is identical to predators and prey in the wild. So if you can imagine rabbits and foxes, rabbits are trying to avoid foxes at all costs, while foxes are patrolling the space, trying to look for rabbits. We actually know an awful lot about the dynamics of predators and prey. We also know a lot about consumer spending flows. And we know a lot about how viruses spread through a population.
So if you take these three analogies together and exploit them, you can come up with a mathematical model of what actually happened, that’s capable of replicating the general patterns of the riots themselves. Now, once we’ve got this, we can almost use this as a petri dish and start having conversations about which areas of the city were more susceptible than others and what police tactics could be used if this were ever to happen again in the future. Even twenty years ago, modeling of this sort was completely unheard of. But I think that these analogies are an incredibly important tool in tackling problems with our society, and perhaps, ultimately improving our society overall.
So, to conclude: life is complex, but perhaps understanding it need not necessarily be that complicated.
Today I want to talk to you about the mathematics of love. Now, I think that we can all agree that mathematicians are famously excellent at finding love. But it’s not just because of our dashing personalities, superior conversational skills and excellent pencil cases. It’s also because we’ve actually done an awful lot of work into the maths of how to find the perfect partner.
Now, in my favorite paper on the subject, which is entitled, “Why I Don’t Have a Girlfriend” – Peter Backus tries to rate his chances of finding love. Now, Peter’s not a very greedy man. Of all of the available women in the UK, all Peter’s looking for is somebody who lives near him, somebody in the right age range, somebody with a university degree, somebody he’s likely to get on well with, somebody who’s likely to be attractive, somebody who’s likely to find him attractive. And comes up with an estimate of 26 women in the whole of the UK.
It’s not looking very good, is it Peter? Now, just to put that into perspective, that’s about 400 times fewer than the best estimates of how many intelligent extraterrestrial life forms there are. And it also gives Peter a 1 in 285,000 chance of bumping into any one of these special ladies on a given night out. I’d like to think that’s why mathematicians don’t really bother going on nights out anymore.
The thing is that I personally don’t subscribe to such a pessimistic view. Because I know, just as well as all of you do, that love doesn’t really work like that. Human emotion isn’t neatly ordered and rational and easily predictable. But I also know that that doesn’t mean that mathematics hasn’t got something that it can offer us, because, love, as with most of life, is full of patterns and mathematics is, ultimately, all about the study of patterns. Patterns from predicting the weather to the fluctuations in the stock market, to the movement of the planets or the growth of cities. And if we’re being honest, none of those things are exactly neatly ordered and easily predictable, either. Because I believe that mathematics is so powerful that it has the potential to offer us a new way of looking at almost anything. Even something as mysterious as love. And so, to try to persuade you of how totally amazing, excellent and relevant mathematics is, I want to give you my top three mathematically verifiable tips for love.
OK, so Top Tip #1: How to win at online dating. So my favorite online dating website is OkCupid, not least because it was started by a group of mathematicians. Now, because they’re mathematicians, they have been collecting data on everybody who uses their site for almost a decade. And they’ve been trying to search for patterns in the way that we talk about ourselves and the way that we interact with each other on an online dating website. And they’ve come up with some seriously interesting findings. But my particular favorite is that it turns out that on an online dating website, how attractive you are does not dictate how popular you are, and actually, having people think that you’re ugly can work to your advantage.
Let me show you how this works. In a thankfully voluntary section of OkCupid, you are allowed to rate how attractive you think people are on a scale between one and five. Now, if we compare this score, the average score, to how many messages a selection of people receive, you can begin to get a sense of how attractiveness links to popularity on an online dating website.
This is the graph the OkCupid guys have come up with. And the important thing to notice is that it’s not totally true that the more attractive you are, the more messages you get. But the question arises then of what is it about people up here who are so much more popular than people down here, even though they have the same score of attractiveness? And the reason why is that it’s not just straightforward looks that are important. So let me try to illustrate their findings with an example. So if you take someone like Portia de Rossi, for example, everybody agrees that Portia de Rossi is a very beautiful woman. Nobody thinks that she’s ugly, but she’s not a supermodel, either. If you compare Portia de Rossi to someone like Sarah Jessica Parker, now, a lot of people, myself included, I should say, think that Sarah Jessica Parker is seriously fabulous and possibly one of the most beautiful creatures to have ever have walked on the face of the Earth. But some other people, i.e., most of the Internet … seem to think that she looks a bit like a horse.
Now, I think that if you ask people how attractive they thought Jessica Parker or Portia de Rossi were, and you ask them to give them a score between one and five I reckon that they’d average out to have roughly the same score. But the way that people would vote would be very different. So Portia’s scores would all be clustered around the four because everybody agrees that she’s very beautiful, whereas Sarah Jessica Parker completely divides opinion. There’d be a huge spread in her scores. And actually it’s this spread that counts. It’s this spread that makes you more popular on an online Internet dating website. So what that means then is that if some people think that you’re attractive, you’re actually better off having some other people think that you’re a massive minger. That’s much better than everybody just thinking that you’re the cute girl next door.
Now, I think this begins to make a bit more sense when you think in terms of the people who are sending these messages. So let’s say that you think somebody’s attractive, but you suspect that other people won’t necessarily be that interested. That means there’s less competition for you and it’s an extra incentive for you to get in touch. Whereas compare that to if you think somebody is attractive but you suspect that everybody is going to think they’re attractive. Well, why would you bother humiliating yourself, let’s be honest? But here’s where the really interesting part comes. Because when people choose the pictures that they use on an online dating website, they often try to minimize the things that they think some people will find unattractive. The classic example is people who are, perhaps, a little bit overweight deliberately choosing a very cropped photo, or bald men, for example, deliberately choosing pictures where they’re wearing hats. But actually this is the opposite of what you should do if you want to be successful. You should really, instead, play up to whatever it is that makes you different, even if you think that some people will find it unattractive. Because the people who fancy you are just going to fancy you anyway, and the unimportant losers who don’t, well, they only play up to your advantage.
OK, Top Tip #2: How to pick the perfect partner. So let’s imagine then that you’re a roaring success on the dating scene. But the question arises of how do you then convert that success into longer-term happiness, and in particular, how do you decide when is the right time to settle down? Now generally, it’s not advisable to just cash in and marry the first person who comes along and shows you any interest at all. But, equally, you don’t really want to leave it too long if you want to maximize your chance of long-term happiness. As my favorite author, Jane Austen, puts it, “An unmarried woman of seven and twenty can never hope to feel or inspire affection again.”
Thanks a lot, Jane. What do you know about love? So the question is then, how do you know when is the right time to settle down, given all the people that you can date in your lifetime? Thankfully, there’s a rather delicious bit of mathematics that we can use to help us out here, called optimal stopping theory. So let’s imagine, then, that you start dating when you’re 15 and ideally, you’d like to be married by the time that you’re 35. And there’s a number of people that you could potentially date across your lifetime, and they’ll be at varying levels of goodness. Now the rules are that once you cash in and get married, you can’t look ahead to see what you could have had, and equally, you can’t go back and change your mind. In my experience at least, I find that typically people don’t much like being recalled years after being passed up for somebody else, or that’s just me. So the math says then that what you should do in the first 37 percent of your dating window, you should just reject everybody as serious marriage potential.
And then, you should pick the next person that comes along that is better than everybody that you’ve seen before. So here’s the example. Now if you do this, it can be mathematically proven, in fact, that this is the best possible way of maximizing your chances of finding the perfect partner. Now unfortunately, I have to tell you that this method does come with some risks. For instance, imagine if your perfect partner appeared during your first 37 percent. Now, unfortunately, you’d have to reject them.
Now, if you’re following the maths, I’m afraid no one else comes along that’s better than anyone you’ve seen before, so you have to go on rejecting everyone and die alone. Probably surrounded by cats … nibbling at your remains.
OK, another risk is, let’s imagine, instead, that the first people that you dated in your first 37 percent are just incredibly dull, boring, terrible people. That’s OK, because you’re in your rejection phase, so that’s fine, you can reject them. But then imagine the next person to come along is just marginally less boring, dull and terrible … than everybody that you’ve seen before. Now, if you are following the maths, I’m afraid you have to marry them … and end up in a relationship which is, frankly, suboptimal. Sorry about that. But I do think that there’s an opportunity here for Hallmark to cash in on and really cater for this market. A Valentine’s Day card like this. “My darling husband, you are marginally less terrible than the first 37 percent of people I dated.” It’s actually more romantic than I normally manage.
OK, so this method doesn’t give you a 100 percent success rate, but there’s no other possible strategy that can do any better. And actually, in the wild, there are certain types of fish which follow and employ this exact strategy. So they reject every possible suitor that turns up in the first 37 percent of the mating season, and then they pick the next fish that comes along after that window that’s, I don’t know, bigger and burlier than all of the fish that they’ve seen before. I also think that subconsciously, humans, we do sort of do this anyway. We give ourselves a little bit of time to play the field, get a feel for the marketplace or whatever when we’re young. And then we only start looking seriously at potential marriage candidates once we hit our mid-to-late 20s. I think this is conclusive proof, if ever it were needed, that everybody’s brains are prewired to be just a little bit mathematical.
OK, so that was Top Tip #2. Now, Top Tip #3: How to avoid divorce. OK, so let’s imagine then that you picked your perfect partner and you’re settling into a lifelong relationship with them. Now, I like to think that everybody would ideally like to avoid divorce, apart from, I don’t know, Piers Morgan’s wife, maybe?
But it’s a sad fact of modern life that one in two marriages in the States ends in divorce, with the rest of the world not being far behind. Now, you can be forgiven, perhaps for thinking that the arguments that precede a marital breakup are not an ideal candidate for mathematical investigation. For one thing, it’s very hard to know what you should be measuring or what you should be quantifying. But this didn’t stop a psychologist, John Gottman, who did exactly that. Gottman observed hundreds of couples having a conversation and recorded, well, everything you can think of. So he recorded what was said in the conversation, he recorded their skin conductivity, he recorded their facial expressions, their heart rates, their blood pressure, basically everything apart from whether or not the wife was actually always right, which incidentally she totally is. But what Gottman and his team found was that one of the most important predictors for whether or not a couple is going to get divorced was how positive or negative each partner was being in the conversation.
Now, couples that were very low-risk scored a lot more positive points on Gottman’s scale than negative. Whereas bad relationships, by which I mean, probably going to get divorced, they found themselves getting into a spiral of negativity. Now just by using these very simple ideas, Gottman and his group were able to predict whether a given couple was going to get divorced with a 90 percent accuracy. But it wasn’t until he teamed up with a mathematician, James Murray, that they really started to understand what causes these negativity spirals and how they occur. And the results that they found, I think, are just incredibly impressively simple and interesting. So these equations predict how the wife or husband is going to respond in their next turn of the conversation, how positive or negative they’re going to be. And these equations depend on the mood of the person when they’re on their own, the mood of the person when they’re with their partner, but most importantly, they depend on how much the husband and wife influence one another.
Now, I think it’s important to point out at this stage, that these exact equations have also been shown to be perfectly able at describing what happens between two countries in an arms race. So that an arguing couple spiraling into negativity and teetering on the brink of divorce is actually mathematically equivalent to the beginning of a nuclear war.
But the really important term in this equation is the influence that people have on one another, and in particular, something called “the negativity threshold.” Now, the negativity threshold, you can think of as how annoying the husband can be before the wife starts to get really pissed off, and vice versa. Now, I always thought that good marriages were about compromise and understanding and allowing the person to have the space to be themselves. So I would have thought that perhaps the most successful relationships were ones where there was a really high negativity threshold. Where couples let things go and only brought things up if they really were a big deal. But actually, the mathematics and subsequent findings by the team have shown the exact opposite is true. The best couples, or the most successful couples, are the ones with a really low negativity threshold. These are the couples that don’t let anything go unnoticed and allow each other some room to complain. These are the couples that are continually trying to repair their own relationship, that have a much more positive outlook on their marriage. Couples that don’t let things go and couples that don’t let trivial things end up being a really big deal.
Now of course, it takes a bit more than just a low negativity threshold and not compromising to have a successful relationship. But I think that it’s quite interesting to know that there is really mathematical evidence to say that you should never let the sun go down on your anger.
So those are my top three tips of how maths can help you with love and relationships. But I hope, that aside from their use as tips, they also give you a little bit of insight into the power of mathematics. Because for me, equations and symbols aren’t just a thing. They’re a voice that speaks out about the incredible richness of nature and the startling simplicity in the patterns that twist and turn and warp and evolve all around us, from how the world works to how we behave. So I hope that perhaps, for just a couple of you, a little bit of insight into the mathematics of love can persuade you to have a little bit more love for mathematics.
Thank you.
Hannah Fry is awarded the Leelavati Prize for her creative approach to communicating that mathematics is very powerful and also a lot of fun, and for her uncanny skill to attract particularly girls and young women to mathematics. Her work has already influenced many millions and promises to inspire many more.
