One is the loneliest prime number

You can’t prove 1 is, or is not, prime. You have the freedom to choose whether to include 1 as a prime or not and this choice is either guided by convenience or credulity.

I occasionally get some cruel and bitter criticism from an odd source. I’m putting my response here for two reasons: (1) so I that I can simply refer them to it and not have to repeat myself or engage in the equally impersonal displeasure of internet arguments, and (2) I think there is something interesting to be learned about mathematics, logic, and knowledge more generally.

It all started when I wrote a very controversial book about an extremely taboo topic: mathematics. In my book ABCs of Mathematics, “P is for Prime”. The short, child-friendly description I gave for this was:

A prime number is only divisible by 1 and itself.

I thought I did a pretty good job of reducing the concept and syllables down to a level palatable by a young reader. Oh, boy, was I wrong. Enter: the angriest group of people I have met on the internet.

You see, by the given definition, I had to include 1 as a prime number since, as we should all agree, it is divisible only by 1 and itself.

Big mistake. Because, apparently, it has been drilled into people’s heads that this is a grave error, a misconception that can eventually lead young impressionable minds to a life of crime and possibly even death! It might even end up on a list of banned books!

By a vast majority, people love the book. I am generally happy with the reponse. The baby books I write are not for everyone—I get that. And I do try to take advice from all the feedback I receive on my books. There is always room for improvement. But the intense emotions some people have with the idea of 1 being a prime number is truly perplexing. Here are some examples:

I actually love the book, but there is a big mistake. The number 1 is not a prime number! The book should not be sold like this and needs to be reprinted.

and

1 IS NOT PRIME! How could a supposed math book have an error like this in it? I am disgusted!

Yikes. So what gives? Is 1 prime, or not? The answer is: that’s not a valid question.

Let me explain.

First, let’s look at a typical definition. Compare to, for example, Wikipedia’s entry on prime numbers:

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

Much more precise—no denying that. It’s grammatically correct, but probably hard to parse. I wanted to avoid negative definitions as much as I could in my books. But that’s beside the point. The reason 1 is not a prime is that the definition of prime itself is contorted to exclude it!

OK, so why is that? Well, the answer is probably not as satisfying as you might like: convenience. By excluding 1 as prime, one can state other theorems more concisely. Take the Fundamental Theorem of Arithmetic, for example:

Every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

Now, this statement would not be true if 1 were a prime since, for example, 6 = 2 × 3 but also 6 = 2 × 3 × 1 and also 6 = 2 × 3 × 1 × 1, etc. That is, if 1 were prime, the representation would not be unique and the theorem would be false.

However, if we do chose to include 1 as a prime number, all is not lost. Then the Fundamental Theorem of Arithmetic would still be true if it were stated as:

Every integer is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors and the number of 1’s.

Which version do you prefer? In either case, both the definition and theorem treat 1 as a special number. I’d argue that in this context, the number 1 is more of an annoyance that gets in the way of the deeper concept behind the theorem. But in mathematics you must be precise with your language. And so 1 must be dealt with as an awkward special case no matter which way you slice it.

So, is 1 prime, or not? Well, it depends on how you define it. But in the end it doesn’t really matter, so long as you are consistent. And understanding that is a much bigger lesson than memorizing some fact you were told in grade school.

The definition given in ABCs of Mathematics is not wrong” any more than all of the other simplifications and analogies I have made are “wrong”. But, in case you were wondering, the second printing will be modified with the hope that everyone can enjoy the book. Even the angry people on the internet deserve to be happy.

Why are there so many symbols in math?

“Mathematics is the language of the universe.” — every science popularizer ever

I am a mathematician. In fact, I have a PhD in mathematics. But, I am terrible at arithmetic. Confused? I certainly would have been if a self-proclaimed mathematician told me that 15 years ago.

The answer to this riddle is simple: math is not numbers. Whenever a glimpse of my research is seen by nearly anyone but another mathematician, they ask where the numbers are. It’s just a bunch of gibberish symbols, they say.

IMG_20171212_090428
The whiteboard in my office on 12 December 2017.

Well, they are right. Without speaking the language, it is just gibberish. But why—why all these symbols?

The symbols are necessary because communicating the ideas requires it. A simple analogy is common human language.

Mandarin Chinese, for example, has many more like-sounding syllables than English. This has led to a great number of visual puns, which have become a large part of Chinese culture. For example, the phrase 福到了(“fortune has arrived”) sounds the same as 福倒了(“fortune is upside down”). Often you will see the character 福 (“fortune”, fú, which you pronounce as ‘foo’ with an ascending pitch) upside down. While, like most puns, this has no literal meaning, it denotes fortune has arrived.

Fudao
Fudao, CC BY-SA 3.0, https://en.wikipedia.org/w/index.php?curid=30725479

Not laughing? OK, well, not even English jokes are funny when they have to be explained, but you get the idea. This pun just doesn’t translate to English. (Amusingly, there is also no simple common word for pun in Chinese.)

The point here is that upside down 福, with its intended emotional response, is not something you can even convey in English. The same is true in mathematics. Ideas can be explained in long-winded and confusing English sentences, but it is much easier if symbols are used.

And, there really is a sense in which the symbols are necessary. Much like the example of 福, most mathematicians use symbols in a way that is just impossible to translate to English, or any other language, without losing most of the meaning.

Here is a small example. In the picture above you will see p(x|θ). First, why θ? (theta, the eighth letter of the Greek alphabet, by the way). That’s just convention—mathematicians love Greek letters. So, you could replace all the θ’s by another symbol and the meaning wouldn’t change. It’s like the difference between writing Chinese using characters or pinyin: 拼音 = pīnyīn.

You might think that it is weird to mix symbols, such as Roman and Greek, but it now very common in many languages, particularly in online conversations. For example, Chinese write 三Q to mean “thank you”, because 三 is 3 and, in English, 3Q sounds like ‘thank you”. In English, and probably all languages now, emojis are mixed with the usual characters to great effect. You could easily write, “Have a nice day. By the way, my mood is happy and I am trying to convey warmth while saying this.” But, “Have a nice day :)” is much easier, and actually better at conveying the message.

OK, so we are cool with Greek letters now, how about  p(x|θ)? That turns out to be easy to translate—it means “the probability of x given θ.” Unfortunately, much like any statement, context is everything. In this case, not even a mathematician could tell you exactly what p(x|θ) means since they have not been told what x or θ means. It like saying “Bob went to place to get thing that she asked for.” An English speaker recognises this as a grammatically correct sentence, but who is “she”, what is the “thing”, and what is the “place”? No one can know without context.

What the English speaker knows is that (probably) a man, named Bob, went to store to purchase something for a woman, whose name we don’t know. The amazing thing is that many more sentences could follow this and an English speaker could easily understand without the context. Have you ever read or listened to a story in which the characters are never named or described? You probably filled in your own context to make the story understandable for you. Maybe that invented context is fluid and changes as you hear more of the story.

The important point is that such actions are not taught. They come from experience—from being immersed in the language and a culture built from it. The same is true in mathematics. A mathematician with experience in probability theory could follow most of what is written on that whiteboard, or at least get the gist of it, without knowing the context. This isn’t something innate or magical—it’s just experience.

Quantum Physics for Babies

This talk was given at the University of Sydney School of Physics Colloquium 19 June 2017.

It’s​ great to be back here. That feels a bit awkward to say since it’s only been 6 months since I left and I’m only 10 minutes away. But King and Broadway might as well be the Pacific Ocean for academics. I’m Chris Ferrie. I’m just down the road at the Centre for Quantum Software and Information. It’s an awesome new Centre. We’re on Twitter. You should check us out.

Now, though the title of the talk doesn’t make it obvious, I am a serious, well… maybe not serious, but I am an academic. But I also have a hobby… tennis. No, I write children’s books. Yes, it is a real book. And, yes, I wrote it and self published it several years ago when I was a postdoc. Why, and how, and for what purpose, well… that is the purpose of this talk.

measure-1509707_1920Measure twice, cut once. So the old proverb goes. It certainly it makes sense if you only have enough material to build a thing. However, and I see this all too often in otherwise very smart people, too much measuring leads to over optimisation and inaction, not enough cutting. Whereas, I like cut several times, toss things out, try new cutting instruments, and so on. I almost never measure. Ultimately, this is the story of Quantum Physics for Babies. I just did it. It wasn’t carefully planned, nor was there a spark or ah-ha moment which spawned the idea. I started, I failed, I started again.

And, for better or worse, the book became popular. Journalists starting asking me, “why did you write this book?” and, more seriously, “why teach quantum physics to babies, why is that important?”

brain-2062057_1920So, I started to rationalize. Why did I write this book? And, is it important? In particular, is it important for all children, not just my own? (because it is always important to find a way to discuss your passion with your own kids.) I think the answer to “is it important?” is yes. In this talk I’ll walk you through the various levels of rationalisation I’ve went through. Each has an element of truth to it, both for myself personally and what the experts on the topic of early childhood education espouse.

But let me start at the same place I start most things, with a joke. Someone that has known me for only a short time probably wouldn’t be too surprised that I was voted “class clown” in high school. Humor plays a crucial part of almost every aspect of my life. I laugh with my partner, I laugh with my children, I laugh with my friends, and I laugh with other scientists. (Einstein didn’t think it was very funny—but, then again, he never liked quantum physics.) Happiness is the difference between your reality and your expectations. Humor often defies expectation and happiness ensues. So, hopefully you didn’t come to this talk with too many expectations and you’ll leave a little happier then when you came in. At least there’s cake.

There is no denying that I saw the irony as good for a laugh the first time the title popped into my head. Of course, the level of humour I’m talking about is not at all for the advertised audience. I’ve never seen a child laugh at the title of the book. Adults, on the other hand, love the juxtaposition of quantum physics with “for babies”. So I knew that at least a few people would buy it as a gag gift for a nerdy friend having a baby. What I didn’t expect was this nerdy friend getting a copy.

I’ve joked with various people about making other goofy “for babies” books. Why not “contract law for babies” or “geopolitical policy for babies”? Though, the only person in the world that needs to read such a book is too busy tweeting insults at women. But quantum physics—yeah—people seem to agree that is worth being more than a joke, and hopefully I knew something about it.

In the end, I put real thought and effort into the content. The goal became clear enough: how to fill a baby book out with short sentences, no jargon and a coherent description of quantum physics. It was a challenge and there is still probably room for improvement. But I’ve already had people say, “we all had a good laugh, then I started to read it and there was real quantum physics inside.” Many adults even claimed they learned something. But were the children learning?

The unanimous advice for new parents is to read to your newborn. Most say it doesn’t even matter what it is, just read. But, let’s play a little game here. Suppose a parent does read to their child and has no time to add a new book to the rotation. Then, Quantum Physics for Babies needs to replace a book. What book should it be? First, I don’t think it should replace fiction. Fiction and fairy tales serve many purposes and, besides, variety is the spice of life. So we are left with nonfiction, which for baby books is limited solely to only a few types of reference material.

Quantum Physics for Babies - Sydney Uni 19 June 2017 (1)A huge fraction of any newborn’s library will begin with the word “first”: “First Words”, “First book of numbers”, “First alphabet book”, and so on. One quickly gets the impression that these are essential reference books for the early learner. But beyond the obvious things—letters, numbers, shapes, three letter words—are a myriad of books about animals, and mostly farm animals.

Now, learning is tricky concept to define even for adults. There are numerous models of early childhood cognitive development, and so it is hard to say conclusively what is being “learned” and at what level, but something is clearly happening since every 3 year in the world knows what sound a cow makes. Do you? I think I do. But I have never heard one myself. Maybe there was a time when that was important, or at least relevant, but I don’t think that time is today.

Quantum Physics for Babies - Sydney Uni 19 June 2017 (2)Here is another example. Do you know what these birds are? My children seem to know and can identify the difference between a penguin and a puffin. Why? Why are there more books about puffins than there are puffins and no books about transistors when you are probably sitting on a billion of them right now. In your phone lives a few billion transistors making up, by the standards of only decade ago anyway, a supercomputer. A child today will probably spend their entire life closer to computer than they will an animal of comparable size. I’m not suggesting than all books on animals be replaced by physics for babies books, but we could maybe replace a few.

I won’t claim my children understand quantum physics, but they certainly understand it at the same level they understand anything else gotten from a book. They will tell you that everything in the world is made of atoms and atoms are made of neutrons, protons and electrons and electrons have energy. I think that is about the same level of understanding as being able to identify a puffin, or should I say Fratercula corniculata for the baby ornithologists in the crowd.

So it seems then that Quantum Physics for Babies is here to stay. But we’re all scientists here and we love nothing more than free cake and to categorize things. So where does Quantum Physics fit? In what aisle of the bookshop does it sit on the shelf? Well, it turns out that it has been shoehorned into the new educational buzzword de jour: STEM.

Quantum Physics for Babies - Sydney Uni 19 June 2017 (3).png

STEM (Science Technology Engineer and Mathematics) started out as an initiative to focus on its namesake topics with the goal of training a workforce ready for the careers that were assumed new technologies would create. Interestingly, the first press mention of the acronym seems to go back to 2008 when The Bill and Melinda Gates Foundation donated $12 million dollars to the Ohio STEM Learning Network, which is still going strong today. Most never looked back. [By the way, much backlash ensued over leaving out the Arts, for example. So you might see STEAM or even STREAM (Reading) out there.]

Now governments all over the world currently have numerous initiatives at all levels of the curriculum to enhance what they called STEM-based learning. This is vaguely and variably defined and can mean anything from simply having access to more technology in the classroom to the design and building of simple machines to solve practical problems. But the motivation and directives that follow are often based on decade-old studies suggesting rises in STEM-related jobs. One recent state-level education​ department cited a study with data collected prior to the release of the first iPhone (that was only 10 years ago, by the way). The often cited report of the Chief Scientist of Australia contained recommendations citing data accumulated from 1964-2005. Policy is good, but it cannot keep up with the pace of technology.

Disruption! The fear today—fueled by startups, makers, and ever younger entrepreneurs—is that we just have no idea what jobs will look like in the future. And so STEM, at least for the trailblazers, is now a movement with the audacious goal of graduating creators and innovators. We no longer want graduates who simply have more and integrated technical skills.

What does this look like? Let me give you an example. Here is Taj Parabi, now 17, CEO of his own business which ships DIY tablets. His company, fiftysix, also visits schools and puts on extracurricular workshops for students on technology and… entrepreneuring! That’s right. Your children are competing with 8-year-olds trained to be CEOs of their own companies!

On a topic near and dear to my own heart, a now veteran effort from the Institute for Quantum Computing is the Quantum Cryptography School for Young Students (QCSYS)Quantum Physics for Babies - Sydney Uni 19 June 2017 (4), which invites international high-school students for a week of intensive training on quantum technology. Indeed, many of these students eventually become PhD students in Quantum Information Theory. Other efforts include school incursions and the new QUANTUM: The Exhibition which is an all-ages, hands-on exploratory exhibit.

Quantum Physics for Babies - Sydney Uni 19 June 2017 (5)On the other side of the border (remember: the wall is on the souther border), IBM has recently released the “Quantum Experience”, an app that lets you program a quantum computer, a real quantum computer. You create an algorithm and then jump in the queue for it to run on real device housed in IBM’s labs. Here they are video conferencing with a school in South Africa and hosting local students.

So that is the tiniest snapshot of STEM education today. Is Quantum Physics for Babies on par with these efforts? Are the children learning the skills necessary to be quantum engineering start-up entrepreneurs? Of course not. Quantum Physics for Babies, at least as far as reading to actual babies is concerned, is about the parents.

20 years from now, your child might be sitting in an interview for the job of Quantum Communication Analyst or Quantum Software Engineer. How long will it be before such topics feature in the report of the Chief Scientist on the curriculum? How long before it is mainstream in public schools? I’m not holding my breath.

The problem today is that it’s impossible to keep up. Pilot studies, kids maker studios, programming toys and apps, … These are all beautiful, but the growth of STEM education has now outpaced even the technology. The curriculum cannot keep up, and so the onus of STEM education, however you want to define it, is largely on the parents.

Again, the efforts of STEM education researchers are impressive, but a parent cannot assume that their child will happen to be in the school that benefits from these one-off pilot studies or incursions. The education system in most developed countries has been too long taken for granted and is now depleted from underfunding. No doubt there are many great principals and great teachers out there. Two days from now, I’m going to go speak with a dozen principals and teachers about STEM education. But there are almost 1500 primary schools in Sydney alone (over 3000 in New South Wales). There is much that needs to be done at the larger scale—but even if I said that was being done, it is little comfort for parents today.

So—in the end—this is what I both want and expect from the book: the elimination of doubt and fear. I want quantum physics, indeed all physics and math and science, to be normal for a child to take interest in. When your child asks about going to Canada for a summer school on quantum cryptography, that should be seen as normal request. When she asks to help her set up an account for a quantum cloud computing service, you should be like, no worries I already have one.

Today, when 1 in 3 Americans would rather clean a toilet then do a math problem, when a search for “quantum physics” brings up Deepak Chopra instead of Stephen Hawking, and when the facts pointing to climate change are seen as equally compelling as a celebrity’s argument for a flat earth, we need all the help we can get. And we need to start that conversation as early as possible.

Quantum Physics for Babies was just the beginning…IMG_20170606_175007

What does it mean to excel at math?

“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,balls_to_sphere 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!

sphere_to_ballWhy 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.