“In mathematics you don’t understand things. You just get used to them.” ― John von Neumann
John von Neumann made important scientific discoveries in physics, computer science, statistics, economics, and mathematics itself. He was, by all accounts, a genius. Yet, here he is saying he “just got used” to mathematics. While this was probably a tongue-in-cheek reply to a friend, there is some truth to it. Mathematics is a language and anyone can eventually learn to speak it.
Indeed, mathematics is the language by which scientists of all fields communicate—from philosophy to physics. And by mathematics, I don’t mean numbers. Scientists communicate ideas through mental pictures which are often represented by symbols invented just for that purpose. Here is an example: think about a ball. Maybe it is a baseball, or a basketball, or—if you are in Europe—a socc…err… football. Maybe it is the Earth or the Sun. Now try to get rid of the details: the stitching, the colors, the size. What is left? A sphere. You just performed the process of abstraction. A sphere is an idea, a mental image that you can’t touch—it doesn’t exist!
Why is this important, anyway? Well, if I can prove things about spheres, then that ought to apply to any ball in the real world. So, formulas for area and volume, for example, equally apply to baseballs, basketballs, soccer balls, and any other ball. Mathematics is a very powerful way of answering infinitely many questions at once!
Now, it is said that to become an expert at anything, you need 10,000 hours of practice. While not a hard-and-fast rule, it seems to work out in terms of acquiring modern language—10,000 hours probably works out to mid-to-late teens for an adept student. Usually, we don’t start practicing real mathematics until well after we have mastered our first language, in late high-school or college. Why not start those 10,000 hours now with your children?
Sounds great, but where do you start? The bad news is that there are no simple rules. The good news is that it doesn’t really matter where you start. With your children, you could practice numeracy, practice puzzles and games, read books, watch science videos, try to code, draw pictures, or just sit in a quiet room and think. As you do these things, encourage generalization and abstraction. Ask questions and let your child ask questions. The correct answers are not important—it is the process that counts.
I was asked recently to share some tips for parents who want their kids to excel at math and do well in the classroom later on. The trouble is, doing well in the classroom—that is, doing well on standardized tests—doesn’t necessarily correlate with understanding the language of mathematics. If you want to do well on standardized tests, then just practice standardized tests. However, if you want your kids to have the powerful tools of abstraction at their disposal and possibly also do well on tests, then teach them the language of mathematics.
The birthday paradox goes… in a room of 23 people there is a 50-50 chance that two of them share a birthday.
OK, so the first step in introducing a paradox is to explain why it is a paradox in the first place. One might think that for each person, there is 1/365 chance of another person having the same birthday as them. Indeed, I can think of only one other person I’ve met that has the same birthday as me—and he is my twin brother! Since I’ve met far more than 23 people, how can this be true?
This reasoning is flawed for several reasons, the first of which is that the question wasn’t asking about if there was another person in the room with a specific birthday—any pair of people (or more!) can share a birthday to increase the chances of the statement being true.
The complete answer gets heavy into the math, but I want to show you how to convince yourself it is true by simulating the experiment. Simulation is programming a computer or model to act as if the real thing was happening. Usually, you set this up so that the cost of simulation is much less than doing the actual thing. For example, putting a model airplane wing in a wind tunnel is a simulation. I’ve simulate the birthday paradox in a computer programming language called Python and this post is available in notebook-style here. Indeed, this is much easier than being in a room with 23 people.
Below I will not present the code (again, that’s over here), but I will describe how the simulation works and present the results.
Call the number of people we need to ask before we get a repeated birthday n. This is what is called a random variable because its value is not known and may change due to conditions we have no control over (like who happens to be in the room).
Now we simulate an experiment realising a value for n as follows.
Pick a random person and ask their birthday.
Check to see if someone else has given you that answer.
Repeat step 1 and 2 until a birthday is said twice.
Count the number of people that were asked and call that n.
Getting to step 4 constitutes a single experiment. The number that comes out may be n = 2 or n = 100. It all depends on who is in the room. So we repeat all the steps many many times and look at how the numbers fall. The more times we repeat, the more data we obtain and the better our understanding of what’s happening.
Here is what it looks like when we run the experiment one million times.
So what do all those numbers mean? Well, let’s look at how many times n = 2 occurred, for example. In these one million trials, the result 2 occurred 2679 times, which is relatively 0.2679%. Note that this is close to 1/365 ≈ 0.274%, which is expected since the probability that the second person has the same as the first is exactly 1/365. So each number of occurrences divided by one million is approximately the probability that we would see that number in a single experiment.
We can then plot the same data considering the vertical axis the probability of needing to see n people before a repeated birthday.
Adding up the value of each of these bars sums to 100%. This is because one of the values must occur when we do an experiment. OK, so now we can just add up these probabilities starting at n = 2 and increasing until we get to 50%. Visually, it is the number which splits the coloured area above into two equal parts. That number will be the number of people we need to meet to have a 50-50 shot at getting a repeated birthday. Can you guess what it will be?
Drum roll… 23! Tada! The birthday paradox simulated and solved by simulation!
But, wait! There’s more.
What about those leap year babies? In fact, isn’t the assumption that birthdays are equally distributed wrong? If we actually tried this experiment out in real life, would we get 23 or some other number?
Happily, we can test this hypothesis with real data! At least for US births, you can find the data over at fivethirteight’s github page. Here is what the actual distribution looks like.
Perhaps by eye it doesn’t look too uniform. You can clearly see 25 Dec and 31 Dec have massive dips. Much has been written about this and many beautiful visualizations are out there. But, our question is whether this has an effect on the birthday paradox. Perhaps the fact that not many people are born on 25 Dec means it is easy to find a shared birthday on the remaining days, for example. Let’s test this hypothesis by simulating the experiment with the real distribution of birthdays.
To do this, we perform the same 4 steps as above, but randomly sampling answers from the actual distribution of birthdays. The result of another one million experiments is plotted below.
And the answer is the same! The birthday paradox persists with the actual distribution of birthdays.
The above discussion is very good evidence that the birthday paradox is robust to the actual distribution of births. However, it does not constitute a mathematical proof. An experiment can only provide evidence. So I will end this with a technical question for those mathematical curiosos out there. (What I am about to do is also called Nerd Sniping.)
Here is the broad problem: quantify the above observation. I think there is more than one question here. For example, it should be possible to bound the 50-50 threshold as a function of the deviation from a uniform distribution.
It is too easy to get trapped into the mentality that—because you know better—you can instantly fix a child’s problem. So why is it that resistance follows? Hint: it’s because you are not listening.
Recently I decided to take the plunge into audiobooks. Since this was a new venture for me, I decided also to listen to something I wouldn’t otherwise read: a parenting book. I’m not implying that I’m a perfect parent by that—I’ve just been blessed by a great partner and four amazing children, such that I haven’t felt the need to seek unnecessary advice.
Whether or not you think you are in the same boat, you should definitely pick up a copy of How to Talk so Your Kids will Listen and Listen so Your Kids will Talk written byAdele Faber and Elaine Mazlish. Now, at this point you might be sarcastically saying, “Thanks for alerting us to the existence of a 30 year old parenting book in its 13th edition that has been referred to as the parenting bible.” But, this post is for those who don’t frequent self-help blogs and the parenting aisle of the bookshop; you might think you have a great relationship with your kids—this book will make it better.
A comment about the audiobook
Before I continue, let me remind you that I listened to the audio version of the book. I have mixed, but still overall positive, feelings about this. First, there are plenty of cartoons in the physical book which obviously don’t translate to audio. I honestly didn’t notice this. Second, there are lots of exercises and places where you are asked to write things down. This is a bit awkward if you are listening to the book while being active. On the other hand, it is narrated bySusan Bennett, otherwise known as Siri. She does an absolutely amazing job capturing the emotion of the authors and, most importantly, the constant dialog in the book between parent and child. In any case, the content is well worth it. The remainder of this discussion will be about the content of the audiobook.
I noticed that much of dialog in the book was dated in terms of language and subject matter. For example, a situation considered in the book was a child who borrowed and scratched a father’s compact disc. I mean, does anyone even own CD’s anymore? Luckily, this is more of a nostalgic amusement than a distraction.
I also didn’t like how the anniversary edition involved appendices of additions rather than a more streamlined approach. It felt a bit lazy and tacked on. In any case, the additional content was a useful addition to an otherwise great book.
Examples, examples, examples
A large chunk of How to Talk consists of real-life example dialogue between parent and child. These are invaluable. Often, the advice seems obvious in hindsight—validate feelings, for example. But, it’s only after hearing the examples when you really see where improvement can be made. The examples usually begin with a fictitious unhelpful response from the parent, followed by a helpful response. Let me give you an example that actually happened to me when I tried to use the skills on my own children after reading the first chapter: “Helping Children Deal with Their Feelings.” My strategy was to default to silent acknowledgment whenever I couldn’t quickly find an appropriate response.
Situation 1: Child (7) is asked to practice math exercises. After doing a few, she becomes bored and says, “I can’t do it.” Clearly, she can. Here is a typical way this would play out.
Child: I can’t do it.
Parent: Yes you can.
Child: I can’t. I don’t know how.
Parent: Well, you are not leaving the table until it is done.
Child (now crying): But I want to watch a movie!
Parent (voice raised): No movies until you are done all your homework!
Child: You’re mean!
And there is no end to the cycle. But, here is how it actually went:
Child: I can’t do it.
Parent: You feel that question is too hard for you?
Child: I think the answer is 17.
I couldn’t believe how well silence works for acknowledging feelings. Here is another example.
Situation 2: Child (3) is struggling to get his footwear on. He is frustrated that the sandal won’t fit on the wrong foot. Here is the typical way this would have gone.
Child (clearly frustrated): I can’t get it on!
Parent: That’s because you are putting it on the wrong foot.
Child: NO! It doesn’t fit!
Parent: Would you like me to do it for you?
Parent: [forces sandal on correct foot]
Here is how it actually went:
Child (clearly frustrated): I can’t get it on!
Parent: I see you are frustrated with that sandal.
Child (calmer): It doesn’t fit.
Parent: That can be so frustrating!
Child (probably knowing all along): It’s the wrong foot.
And he took it off the wrong foot and put it on the correct foot!
How to Talk contains seven chapters, the last being titled, “Putting it All Together.” But, to my mind, the first chapter contains the key insights to a better emotional relationship with my children. In the past, I’ve tried to avoid situations like those exemplified above. Accepting and dealing with my children’s emotions has not been that difficult because I have been fairly well in tune with them. This meant that I was able to easily anticipate meltdowns and avoid them altogether. For example, if I sensed that my 3 year old was irritable, I would not ask him to put his own shoes on, thereby avoiding the situation.
But protecting the children from their own feelings is not doing them any favors in the long run, and is becoming ever more difficult with a growing family. So, rather than doing the hard work of anticipating and preventing the emotional distress of my children, I’m now trying to acknowledge and accept how difficult it is to become an autonomous little person in a grown-up world.
For the first time, physicists have found a new fundamental state of cow, challenging the current standard model. Coined the cubic cow, the ground-breaking new discovery is already re-writing the rules of physics.
A team of physicists at Stanford and Harvard University have nothing to do with this but you are probably already impressed by the name drop. Dr. Chris Ferrie, who is currently between jobs, together with a team of his own children stumbled upon the discovery, which was recently published in Nature Communications*.
The spherical theory of cow had stood unchallenged for over 50 years—and even longer if a Russian physicist is reading this. The spherical cow theory led to many discoveries also based on O(3) symmetries. However, spherical cows have not proven practically useful from a technological perspective. “Spherical cows are prone to natural environmental errors, whereas our discovery digitizes the symmetry of cow,” Ferrie said.
Just as the digital computer has revolutionized computing technology, this new digital cow model could revolutionize innovation disrupting cross-industry ecosystems, or something.
Lead author Maxwell Ferrie already has far-reaching applications for the result. “I like dinosaurs,” he said. Notwithstanding these future aspirations, the team is sure to be milking this new theory for all its worth.
* Not really, but this dumping ground for failed hypesearch has a bar so low you might as well believe it.
I think it was generally well received. Of course it got lots of double takes and laughs, but was it a good scientific poster? One of my senior colleagues was of mixed minds, eventually concluding with some familiar life advice:
Yes, I admit it is funny. But, eventually it will catch up with you. No one is going to take you seriously. You will not be seen as a serious scientist.
Good—because I am not a serious scientist. I am a (hopefully) humorous scientist, but a scientist nonetheless.
I’m going to get straight to the point with my own advice: avoid serious scientists at all costs. They are either psychopaths or sycophants. I can’t find it in me to be either. So I’ll continue doing science, and having a bit of fun while I’m at it. You only science once, right?
Recently, I had a video chat with the kindergarten class of Dragon Bay Kindergarten in Beijing. It was a lot of fun to see how excited the children and teachers were to read my books. They even had an entire Science Fair based around my Physics for Babies books!
During our video call, the students and teachers asked many questions. I’ve transcribed them here.
How can I see atoms in the real world?
We can not see atoms with our eyes. They are too small. We can use other ways to take pictures of atoms. In the labs where I work, physicists shine laser light on the atoms. The electrons take the energy to move up in their energy levels. When they fall back down, they release light that we can see with a camera.
Can you introduce particles and entanglement to us?
Atoms themselves are made of even smaller things called particles. Electrons are one kind of particle. Not all particles make atoms though. Photons, what light is made of, are another kind of particle which is not part of an atom.
Entanglement is tricky to explain in everyday language. It is something we see in the math of quantum physics. Even scientists today argue about how to understand it. But, we can use the math to show us how to build quantum technology where entanglement is used.
What does an atom look like? How are they different?
Electron microscopes take pictures of atoms which look like blurry little balls. Most atoms look the same but some are bigger than others. When electrons move between energy levels, they send out light at very specific colors. Each atom makes a different color, which is how we can tell them apart.
How do you know everything was made by atoms?
We can see them with today’s technology!
How can I touch the atom?
Since everything is made of atoms, you are touching them right now!
Why don’t you wear the clothes of a physicist?
In pictures of scientists, they are often wearing lab coats. In real life, physicists do not wear lab coats. Some work in a lab and others, like me, work in an office with computers and whiteboards.
What made you think that babies need to learn about quantum entanglement?
A lot of science is a language which we learn by listening and talking to other scientists, just like learning your first language. So, the sooner you start to hear the language, the sooner you will speak it.
Will entangled particles always be measured the same or can they just be influenced?
Entanglement has a quality to it which might not make it perfect. Experimental technology is always a bit unreliable. But perfect entanglement, like that described in the book, means that particles will be measured the same every time.
Do your children like and understand your books?
My children like the books and can often repeat some of the sentences. I talk with them about it, but they will not be doing any quantum physics research yet.
How does your work place look like?
There are labs. Some use lasers which means they must be dark. Some have big refrigerators which keep things really really cold. Above the labs is office space. Here it looks like a regular office, but with whiteboards that have lots of math on them.
A few elder-statesmen of quantum theory gathered together while a handful of students listened in eagerly. Paraphrasing, one of them—quite seriously—said, “I don’t think any of the interpretations are logically consistent… but there is this ‘transactional interpretation’, where influences come from the future, that might be the only consistent one.” The students nodded their heads in agreement—I walked away.
Bayesianism is (some would say) a radical alternative philosophy and practice for both understanding probability and performing statistical analysis. So, like all young contrarian students of science, I was intrigued when I first found Bayesian probability. But where is the fun in blaming this on my own faults? I’m going to blame someone else—I’m going to blame it on Chris Fuchs.
On two occasions the Perimeter Institute for Theoretical Physics (PI) hosted lectures on the Foundations of Quantum Mechanics. I was lucky enough to be a graduate student in Waterloo at the time. The first, in 2007, was my first taste of the field. It was exciting to hear the experts at the forefront speaking about deep implications for physics and—indeed—even the philosophy of science itself. I knew then that this was the area I wanted to work in.
However, I quickly became disillusioned. The literature was plagued by lazy physicists posing as armchair philosophers. There was no interest in real problems—only the pandering of borderline pseudoscience. It’s no wonder—why bother doing hard work and difficult mathematics when peddling quantum mysticism is what gets you press?
I stayed, though, because there were several researchers at PI who seemed interested in solving real, technical problems—and, they were doing so using techniques from another field I had already worked in: Quantum Information Theory. I learned an immense amount from Robin Blume-Kohout, Rob Spekkens and Lucien Hardy while there, but the one who left a lasting impression was Chris Fuchs.
Before we get to Fuchs, though, let’s back up for a moment—just what is this Quantum Foundations thing, and what has it got to do with Bayesianism? As you know, quantum theory dictates that the world is uncertain. That is, as a scientific theory, it makes only probabilistic predictions. Many of the philosophical problems and misunderstandings of quantum theory can be traced back to this fact. Thus, if one really wants to understand quantum theory, one ought to understand probability first. Easy, right?
Nope. As it turns out, more people argue about how to interpret the seemingly simple and everyday concept of probability than do our most sophisticated and complex physical theory. Generally speaking, there are two camps in the interpretations of probability: frequentists and Bayesians. As noted, every student begins as one of the former. It was in 2010 when my conversion to the latter was complete.
In 2010, PI hosted its second course on the foundations of quantum theory. This time around I had a few years of experience under my belt and my bullshit detectors were on high alert. My final assignment was to summarize the course, as shown above. The only lectures that didn’t leave me disappointed where Chris Fuchs’. Because I had been reading up on Bayesian probability anyway, his “Quantum Bayesian” interpretation of quantum theory just clicked.
And it wasn’t just about philosophy. Concurrently, I was taking a great course on Stochastic Processes from Matt Scott. Most of this field takes an objective (frequentist) view of probability. Matt was patient with my constant questions on how to phrase the concepts in terms of the subjective Bayesian view. I was starting to feel a bit overwhelmed with the burden of translating everything to the new framework… then it happened.
The assignment question was as follows: Show that , the simplest case of the Chapman-Kolmogorov equation. What you were supposed to do is use the Markov property, , and integrate. It was a straightforward, but tedious, calculation. Here is what I wrote: first . Since is a probability, it integrates to 1. Done.
This was seen as unphysical because is a negative time. So what?—I thought—probabilities are not physical, they are subjective inferences. If I want to consider negative time to help me do my calculation, so be it. After all, I considered negative money to get me into university. But what I couldn’t believe is how difficult it was to convince others the solution was correct. It was at that moment I realized how powerful a slight change of view can be. I was a Bayesian.