Tag Archives: physics

Math Is the Art of Stating Things Clearly

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Why do we use math?

In physics we describe everything, from the smallest of particles to the largest of galaxies, with the language of mathematics. Why should that one field be able to describe so much? And why don’t we use something else?

The truth is, this is a trick question. Mathematics isn’t a language like English or French, where we can choose whichever translation we want. We use mathematics because it is, almost by definition, the best choice. That is because mathematics is the art of stating things clearly.

An infinite number of mathematicians walk into a bar. The first orders a beer. The second orders half a beer. The third orders a quarter. The bartender stops them, pours two beers, and says “You guys should know your limits.”

That was an (old) joke about infinite series of numbers. You probably learned in high school that if you add up one plus a half plus a quarter…you eventually get two. To be a bit more precise:

\sum_{i=0}^\infty \frac{1}{2^i} = 1+\frac{1}{2}+\frac{1}{4}+\ldots=2

We say that this infinite sum limits to two.

But what does it actually mean for an infinite sum to limit to a number? What does it mean to sum infinitely many numbers, let alone infinitely many beers ordered by infinitely many mathematicians?

You’re asking these questions because I haven’t yet stated the problem clearly. Those of you who’ve learned a bit more mathematics (maybe in high school, maybe in college) will know another way of stating it.

You know how to sum a finite set of beers. You start with one beer, then one and a half, then one and three-quarters. Sum N beers, and you get

\sum_{i=0}^N \frac{1}{2^i}

What does it mean for the sum to limit to two?

Let’s say you just wanted to get close to two. You want to get \epsilon close, where epsilon is the Greek letter we use for really small numbers.

For every \epsilon>0 you choose, no matter how small, I can pick a (finite!) N and get at least that close. That means that, with higher and higher N, I can get as close to two as a I want.

As it turns out, that’s what it means for a sum to limit to two. It’s saying the same thing, but more clearly, without sneaking in confusing claims about infinity.

These sort of proofs, with \epsilon (and usually another variable, \delta) form what mathematicians view as the foundations of calculus. They’re immortalized in story and song.

And they’re not even the clearest way of stating things! Go down that road, and you find more mathematics: definitions of numbers, foundations of logic, rabbit holes upon rabbit holes, all from the effort to state things clearly.

That’s why I’m not surprised that physicists use mathematics. We have to. We need clarity, if we want to understand the world. And mathematicians, they’re the people who spend their lives trying to state things clearly.

When to Trust the Contrarians

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One of my colleagues at the NBI had an unusual experience: one of his papers took a full year to get through peer review. This happens often in math, where reviewers will diligently check proofs for errors, but it’s quite rare in physics: usually the path from writing to publication is much shorter. Then again, the delays shouldn’t have been too surprising for him, given what he was arguing.

My colleague Mohamed Rameez, along with Jacques Colin, Roya Mohayaee, and Subir Sarkar, wants to argue against one of the most famous astronomical discoveries of the last few decades: that the expansion of our universe is accelerating, and thus that an unknown “dark energy” fills the universe. They argue that one of the key pieces of evidence used to prove acceleration is mistaken: that a large region of the universe around us is in fact “flowing” in one direction, and that tricked astronomers into thinking its expansion was accelerating. You might remember a paper making a related argument back in 2016. I didn’t like the media reaction to that paper, and my post triggered a response by the authors, one of whom (Sarkar) is on this paper as well.

I’m not an astronomer or an astrophysicist. I’m not qualified to comment on their argument, and I won’t. I’d still like to know whether they’re right, though. And that means figuring out which experts to trust.

Pick anything we know in physics, and you’ll find at least one person who disagrees. I don’t mean a crackpot, though they exist too. I mean an actual expert who is convinced the rest of the field is wrong. A contrarian, if you will.

I used to be very unsympathetic to these people. I was convinced that the big results of a field are rarely wrong, because of how much is built off of them. I thought that even if a field was using dodgy methods or sloppy reasoning, the big results are used in so many different situations that if they were wrong they would have to be noticed. I’d argue that if you want to overturn one of these big claims you have to disprove not just the result itself, but every other success the field has ever made.

I still believe that, somewhat. But there are a lot of contrarians here at the Niels Bohr Institute. And I’ve started to appreciate what drives them.

The thing is, no scientific result is ever as clean as it ought to be. Everything we do is jury-rigged. We’re almost never experts in everything we’re trying to do, so we often don’t know the best method. Instead, we approximate and guess, we find rough shortcuts and don’t check if they make sense. This can take us far sometimes, sure…but it can also backfire spectacularly.

The contrarians I’ve known got their inspiration from one of those backfires. They saw a result, a respected mainstream result, and they found a glaring screw-up. Maybe it was an approximation that didn’t make any sense, or a statistical measure that was totally inappropriate. Whatever it was, it got them to dig deeper, and suddenly they saw screw-ups all over the place. When they pointed out these problems, at best the people they accused didn’t understand. At worst they got offended. Instead of cooperation, the contrarians are told they can’t possibly know what they’re talking about, and ignored. Eventually, they conclude the entire sub-field is broken.

Are they right?

Not always. They can’t be, for every claim you can find a contrarian, believing them all would be a contradiction.

But sometimes?

Often, they’re right about the screw-ups. They’re right that there’s a cleaner, more proper way to do that calculation, a statistical measure more suited to the problem. And often, doing things right raises subtleties, means that the big important result everyone believed looks a bit less impressive.

Still, that’s not the same as ruling out the result entirely. And despite all the screw-ups, the main result is still often correct. Often, it’s justified not by the original, screwed-up argument, but by newer evidence from a different direction. Often, the sub-field has grown to a point that the original screwed-up argument doesn’t really matter anymore.

Often, but again, not always.

I still don’t know whether to trust the contrarians. I still lean towards expecting fields to sort themselves out, to thinking that error alone can’t sustain long-term research. But I’m keeping a more open mind now. I’m waiting to see how far the contrarians go.

In Defense of the Streetlight

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If you read physics blogs, you’ve probably heard this joke before:

A policeman sees a drunk man searching for something under a streetlight and asks what the drunk has lost. He says he lost his keys and they both look under the streetlight together. After a few minutes the policeman asks if he is sure he lost them here, and the drunk replies, no, and that he lost them in the park. The policeman asks why he is searching here, and the drunk replies, “this is where the light is”.

The drunk’s line of thinking has a name, the streetlight effect, and while it may seem ridiculous it’s a common error, even among experts. When it gets too tough to research something, scientists will often be tempted by an easier problem even if it has little to do with the original question. After all, it’s “where the light is”.

Physicists get accused of this all the time. Dark matter could be completely undetectable on Earth, but physicists still build experiments to search for it. Our universe appears to be curved one way, but string theory makes it much easier to study universes curved the other way, so physicists write a lot of nice proofs about a universe we don’t actually inhabit. In my own field, we spend most of our time studying a very nice theory that we know can’t describe the real world.

I’m not going to defend this behavior in general. There are real cases where scientists trick themselves into thinking they can solve an easy problem when they need to solve a hard one. But there is a crucial difference between scientists and drunkards looking for their keys, one that makes this behavior a lot more reasonable: scientists build technology.

As scientists, we can’t just grope around in the dark for our keys. The spaces we’re searching, from the space of all theories of gravity to actual outer space, are much too vast to search randomly. We need new ideas, new mathematics or new equipment, to do the search properly. If we were the drunkard of the story, we’d need to invent night-vision goggles.

Is the light better here, or is it just me?

Suppose you wanted to design new night-vision goggles, to search for your keys in the park. You could try to build them in the dark, but you wouldn’t be able to see what you were doing: you’d lose pieces, miss screws, and break lenses. Much better to build the goggles under that convenient streetlight.

Of course, if you build and test your prototype goggles under the streetlight, you risk that they aren’t good enough for the dark. You’ll have calibrated them in an unrealistic case. In all likelihood, you’ll have to go back and fix your goggles, tweaking them as you go, and you’ll run into the same problem: you can’t see what you’re doing in the dark.

At that point, though, you have an advantage: you now know how to build night-vision goggles. You’ve practiced building goggles in the light, and now even if the goggles aren’t good enough, you remember how you put them together. You can tweak the process, modify your goggles, and make something good enough to find your keys. You’re good enough at making goggles that you can modify them now, even in the dark.

Sometimes scientists really are like the drunk, searching under the most convenient streetlight. Sometimes, though, scientists are working where the light is for a reason. Instead of wasting their time lost in the dark, they’re building new technology and practicing new methods, getting better and better at searching until, when they’re ready, they can go back and find their keys. Sometimes, the streetlight is worth it.

The Real E=mc^2

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It’s the most famous equation in all of physics, written on thousands of chalkboard stock photos. Part of its charm is its simplicity: E for energy, m for mass, c for the speed of light, just a few simple symbols in a one-line equation. Despite its simplicity, E=mc^2 is deep and important enough that there are books dedicated to explaining it.

What does E=mc^2 mean?

Some will tell you it means mass can be converted to energy, enabling nuclear power and the atomic bomb. This is a useful picture for chemists, who like to think about balancing ingredients: this much mass on one side, this much energy on the other. It’s not the best picture for physicists, though. It makes it sound like energy is some form of “stuff” you can pour into your chemistry set flask, and energy really isn’t like that.

There’s another story you might have heard, in older books. In that story, E=mc^2 tells you that in relativity mass, like distance and time, is relative. The more energy you have, the more mass you have. Those books will tell you that this is why you can’t go faster than light: the faster you go, the greater your mass, and the harder it is to speed up.

Modern physicists don’t talk about it that way. In fact, we don’t even write E=mc^2 that way. We’re more likely to write:

E=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}

“v” here stands for the velocity, how fast the mass is moving. The faster the mass moves, the more energy it has. Take v to zero, and you get back the familiar E=mc^2.

The older books weren’t lying to you, but they were thinking about a different notion of mass: “relativistic mass” m_r instead of “rest mass” $m_0$, related like this:

m_r=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}

which explains the difference in how we write E=mc^2.

Why the change? In part, it’s because of particle physics. In particle physics, we care about the rest mass of particles. Different particles have different rest mass: each electron has one rest mass, each top quark has another, regardless of how fast they’re going. They still get more energy, and harder to speed up, the faster they go, but we don’t describe it as a change in mass. Our equations match the old books, we just talk about them differently.

Of course, you can dig deeper, and things get stranger. You might hear that mass does change with energy, but in a very different way. You might hear that mass is energy, that they’re just two perspectives on the same thing. But those are stories for another day.

I titled this post “The Real E=mc^2”, but to clarify, none of these explanations are more “real” than the others. They’re words, useful in different situations and for different people. “The Real E=mc^2” isn’t the E=mc^2 of nuclear chemists, or old books, or modern physicists. It’s the theory itself, the mathematical rules and principles that all the rest are just trying to describe.

A Field That Doesn’t Read Its Journals

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Last week, the University of California system ended negotiations with Elsevier, one of the top academic journal publishers. UC had been trying to get Elsevier to switch to a new type of contract, one in which instead of paying for access to journals they pay for their faculty to publish, then make all the results openly accessible to the public. In the end they couldn’t reach an agreement and thus didn’t renew their contract, cutting Elsevier off from millions of dollars and their faculty from reading certain (mostly recent) Elsevier journal articles. There’s a nice interview here with one of the librarians who was sent to negotiate the deal.

I’m optimistic about what UC was trying to do. Their proposal sounds like it addresses some of the concerns raised here with open-access systems. Currently, journals that offer open access often charge fees directly to the scientists publishing in them, fees that have to be scrounged up from somebody’s grant at the last minute. By setting up a deal for all their faculty together, UC would have avoided that. While the deal fell through, having an organization as big as the whole University of California system advocating open access (and putting the squeeze on Elsevier’s profits) seems like it can only lead to progress.

The whole situation feels a little surreal, though, when I compare it to my own field.

At the risk of jinxing it, my field’s relationship with journals is even weirder than xkcd says.

arXiv.org is a website that hosts what are called “preprints”, which originally meant papers that haven’t been published yet. They’re online, freely accessible to anyone who wants to read them, and will be for as long as arXiv exists to host them. Essentially everything anyone publishes in my field ends up on arXiv.

Journals don’t mind, in part, because many of them are open-access anyway. There’s an organization, SCOAP3, that runs what is in some sense a large-scale version of what UC was trying to set up: instead of paying for subscriptions, university libraries pay SCOAP3 and it covers the journals’ publication costs.

This means that there are two coexisting open-access systems, the journals themselves and arXiv. But in practice, arXiv is the one we actually use.

If I want to show a student a paper, I don’t send them to the library or the journal website, I tell them how to find it on arXiv. If I’m giving a talk, there usually isn’t room for a journal reference, so I’ll give the arXiv number instead. In a paper, we do give references to journals…but they’re most useful when they have arXiv links as well. I think the only times I’ve actually read an article in a journal were for articles so old that arXiv didn’t exist when they were published.

We still submit our papers to journals, though. Peer review still matters, we still want to determine whether our results are cool enough for the fancy journals or only good enough for the ordinary ones. We still put journal citations on our CVs so employers and grant agencies know not only what we’ve done, but which reviewers liked it.

But the actual copy-editing and formatting and publishing, that the journals still employ people to do? Mostly, it never gets read.

In my experience, that editing isn’t too impressive. Often, it’s about changing things to fit the journal’s preferences: its layout, its conventions, its inconvenient proprietary document formats. I haven’t seen them try to fix grammar, or improve phrasing. Maybe my papers have unusually good grammar, maybe they do more for other papers. And maybe they used to do more, when journals had a more central role. But now, they don’t change much.

Sometimes the journal version ends up on arXiv, if the authors put it there. Sometimes it doesn’t. And sometimes the result is in between. For my last paper about Calabi-Yau manifolds in Feynman diagrams, we got several helpful comments from the reviewers, but the journal also weighed in to get us to remove our more whimsical language, down to the word “bestiary”. For the final arXiv version, we updated for the reviewer comments, but kept the whimsical words. In practice, that version is the one people in our field will read.

This has some awkward effects. It means that sometimes important corrections don’t end up on arXiv, and people don’t see them. It means that technically, if someone wanted to insist on keeping an incorrect paper online, they could, even if a corrected version was published in a journal. And of course, it means that a large amount of effort is dedicated to publishing journal articles that very few people read.

I don’t know whether other fields could get away with this kind of system. Physics is small. It’s small enough that it’s not so hard to get corrections from authors when one needs to, small enough that social pressure can get wrong results corrected. It’s small enough that arXiv and SCOAP3 can exist, funded by universities and private foundations. A bigger field might not be able to do any of that.

For physicists, we should keep in mind that our system can and should still be improved. For other fields, it’s worth considering whether you can move in this direction, and what it would cost to do so. Academic publishing is in a pretty bizarre place right now, but hopefully we can get it to a better one.

What Science Would You Do If You Had the Time?

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I know a lot of people who worry about the state of academia. They worry that the competition for grants and jobs has twisted scientists’ priorities, that the sort of dedicated research of the past, sitting down and thinking about a topic until you really understand it, just isn’t possible anymore. The timeline varies: there are people who think the last really important development was the Standard Model, or the top quark, or AdS/CFT. Even more optimistic people, who think physics is still just as great as it ever was, often complain that they don’t have enough time.

Sometimes I wonder what physics would be like if we did have the time. If we didn’t have to worry about careers and funding, what would we do? I can speculate, comparing to different communities, but here I’m interested in something more concrete: what, specifically, could we accomplish? I often hear people complain that the incentives of academia discourage deep work, but I don’t often hear examples of the kind of deep work that’s being discouraged.

So I’m going to try an experiment here. I know I have a decent number of readers who are scientists of one field or another. Imagine you didn’t have to worry about funding any more. You’ve got a permanent position, and what’s more, your favorite collaborators do too. You don’t have to care about whether your work is popular, whether it appeals to the university or the funding agencies or any of that. What would you work on? What projects would you personally do, that you don’t have the time for in the current system? What worthwhile ideas has modern academia left out?

Congratulations to Arthur Ashkin, Gérard Mourou, and Donna Strickland!

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The 2018 Physics Nobel Prize was announced this week, awarded to Arthur Ashkin, Gérard Mourou, and Donna Strickland for their work in laser physics.

nobel2018Some Nobel prizes recognize discoveries of the fundamental nature of reality. Others recognize the tools that make those discoveries possible.

Ashkin developed techniques that use lasers to hold small objects in place, culminating in “optical tweezers” that can pick up and move individual bacteria. Mourou and Strickland developed chirped pulse amplification, the current state of the art in extremely high-power lasers. Strickland is only the third woman to win the Nobel prize in physics, Ashkin at 96 is the oldest person to ever win the prize.

(As an aside, the phrase “optical tweezers” probably has you imagining two beams of laser light pinching a bacterium between them, like microscopic lightsabers. In fact, optical tweezers use a single beam, focused and bent so that if an object falls out of place it will gently roll back to the middle of the beam. Instead of tweezers, it’s really more like a tiny laser spoon.)

The Nobel announcement emphasizes practical applications, like eye surgery. It’s important to remember that these are research tools as well. I wouldn’t have recognized the names of Ashkin, Mourou, and Strickland, but I recognized atom trapping, optical tweezers, and ultrashort pulses. Hang around atomic physicists, or quantum computing experiments, and these words pop up again and again. These are essential tools that have given rise to whole subfields. LIGO won a Nobel based on the expectation that it would kick-start a vast new area of research. Ashkin, Mourou, and Strickland’s work already has.