# Newtonmas in Uncertain Times

Three hundred and eighty-two years ago today (depending on which calendars you use), Isaac Newton was born. For a scientist, that’s a pretty good reason to celebrate.

Last month, our local nest of science historians at the Niels Bohr Archive hosted a Zoom talk by Jed Z. Buchwald, a Newton scholar at Caltech. Buchwald had a story to tell about experimental uncertainty, one where Newton had an important role.

If you’ve ever had a lab course in school, you know experiments never quite go like they’re supposed to. Set a room of twenty students to find Newton’s constant, and you’ll get forty different answers. Whether you’re reading a ruler or clicking a stopwatch, you can never measure anything with perfect accuracy. Each time you measure, you introduce a little random error.

Textbooks worth of statistical know-how has cropped up over the centuries to compensate for this error and get closer to the truth. The simplest trick though, is just to average over multiple experiments. It’s so obvious a choice, taking a thousand little errors and smoothing them out, that you might think people have been averaging in this way through history.

They haven’t though. As far as Buchwald had found, the first person to average experiments in this way was Isaac Newton.

What did people do before Newton?

Well, what might you do, if you didn’t have a concept of random error? You can still see that each time you measure you get a different result. But you would blame yourself: if you were more careful with the ruler, quicker with the stopwatch, you’d get it right. So you practice, you do the experiment many times, just as you would if you were averaging. But instead of averaging, you just take one result, the one you feel you did carefully enough to count.

Before Newton, this was almost always what scientists did. If you were an astronomer mapping the stars, the positions you published would be the last of a long line of measurements, not an average of the rest. Some other tricks existed. Tycho Brahe for example folded numbers together pair by pair, averaging the first two and then averaging that average with the next one, getting a final result weighted to the later measurements. But, according to Buchwald, Newton was the first to just add everything together.

Even Newton didn’t yet know why this worked. It would take later research, theorems of statistics, to establish the full justification. It seems Newton and his later contemporaries had a vague physics analogy in mind, finding a sort of “center of mass” of different experiments. This doesn’t make much sense – but it worked, well enough for physics as we know it to begin.

So this Newtonmas, let’s thank the scientists of the past. Working piece by piece, concept by concept, they gave use the tools to navigate our uncertain times.

# Halloween Post: Superstimuli for Physicists

For Halloween, this blog has a tradition of covering “the spooky side” of physics. This year, I’m bringing in a concept from biology to ask a spooky physics “what if?”

In the 1950’s, biologists discovered that birds were susceptible to a worryingly effective trick. By giving them artificial eggs larger and brighter than their actual babies, they found that the birds focused on the new eggs to the exclusion of their own. They couldn’t help trying to hatch the fake eggs, even if they were so large that they would fall off when they tried to sit on them. The effect, since observed in other species, became known as a supernormal stimulus, or superstimulus.

Can this happen to humans? Some think so. They worry about junk food we crave more than actual nutrients, or social media that eclipses our real relationships. Naturally, this idea inspires horror writers, who write about haunting music you can’t stop listening to, or holes in a wall that “fit” so well you’re compelled to climb in.

(And yes, it shows up in porn as well.)

But this is a physics blog, not a biology blog. What kind of superstimulus would work on physicists?

Well for one, this sounds a lot like some criticisms of string theory. Instead of a theory that just unifies some forces, why not unify all the forces? Instead of just learning some advanced mathematics, why not learn more, and more? And if you can’t be falsified by any experiment, well, all that would do is spoil the fun, right?

But it’s not just string theory you could apply this logic to. Astrophysicists study not just one world but many. Cosmologists study the birth and death of the entire universe. Particle physicists study the fundamental pieces that make up the fundamental pieces. We all partake in the euphoria of problem-solving, a perpetual rush where each solution leads to yet another question.

Do I actually think that string theory is a superstimulus, that astrophysics or particle physics is a superstimulus? In a word, no. Much as it might look that way from the news coverage, most physicists don’t work on these big, flashy questions. Far from being lured in by irresistible super-scale problems, most physicists work with tabletop experiments and useful materials. For those of us who do look up at the sky or down at the roots of the world, we do it not just because it’s compelling but because it has a good track record: physics wouldn’t exist if Newton hadn’t cared about the orbits of the planets. We study extremes because they advance our understanding of everything else, because they give us steam engines and transistors and change everyone’s lives for the better.

Then again, if I had fallen victim to a superstimulus, I’d say that anyway, right?

*cue spooky music*

# Formal Theory and Simulated Experiment

There are two kinds of theoretical physicists. Some, called phenomenologists, make predictions about the real world. Others, the so-called “formal theorists”, don’t. They work with the same kinds of theories as the phenomenologists, quantum field theories of the sort that have been so successful in understanding the subatomic world. But the specific theories they use are different: usually, toy models that aren’t intended to describe reality.

Most people get this is valuable. It’s useful to study toy models, because they help us tackle the real world. But they stumble on another point. Sure, they say, you can study toy models…but then you should call yourself a mathematician, not a physicist.

I’m a “formal theorist”. And I’m very much not a mathematician, I’m definitely a physicist. Let me explain why, with an analogy.

As an undergrad, I spent some time working in a particle physics lab. The lab had developed a new particle detector chip, designed for a future experiment: the International Linear Collider. It was my job to test this chip.

Naturally, I couldn’t test the chip by flinging particles at it. For one, the collider it was designed for hadn’t been built yet! Instead, I had to use simulated input: send in electrical signals that mimicked the expected particles, and see what happens. In effect, I was using a kind of toy model, as a way to understand better how the chip worked.

I hope you agree that this kind of work counts as physics. It isn’t “just engineering” to feed simulated input into a chip. Not when the whole point of that chip is to go into a physics experiment. This kind of work is a large chunk of what an experimental physicist does.

As a formal theorist, my work with toy models is an important part of what a theoretical physicist does. I test out the “devices” of theoretical physics, the quantum-field-theoretic machinery that we use to investigate the world. Without that kind of careful testing on toy models, we’d have fewer tools to work with when we want to understand reality.

Ok, but you might object: an experimental physicist does eventually build the real experiment. They don’t just spend their career on simulated input. If someone only works on formal theory, shouldn’t that at least make them a mathematician, not a physicist?

Here’s the thing, though: after those summers in that lab, I didn’t end up as an experimental physicist. After working on that chip, I didn’t go on to perfect it for the International Linear Collider. But it would be rather bizarre if that, retroactively, made my work in that time “engineering” and not “physics”.

Oh, I should also mention that the International Linear Collider might not ever be built. So, there’s that.

Formal theory is part of physics because it cares directly about the goals of physics: understanding the real world. It is just one step towards that goal, it doesn’t address the real world alone. But neither do the people testing out chips for future colliders. Formal theory isn’t always useful, similarly, planned experiments don’t always get built. That doesn’t mean it’s not physics.

# Understanding Is Translation

Kernighan’s Law states, “Debugging is twice as hard as writing the code in the first place. Therefore, if you write the code as cleverly as possible, you are, by definition, not smart enough to debug it.” People sometimes make a similar argument about philosophy of mind: “The attempt of the mind to analyze itself [is] an effort analogous to one who would lift himself by his own bootstraps.”

Both points operate on a shared kind of logic. They picture understanding something as modeling it in your mind, with every detail clear. If you’ve already used all your mind’s power to design code, you won’t be able to model when it goes wrong. And modeling your own mind is clearly nonsense, you would need an even larger mind to hold the model.

The trouble is, this isn’t really how understanding works. To understand something, you don’t need to hold a perfect model of it in your head. Instead, you translate it into something you can more easily work with. Like explanations, these translations can be different for different people.

To understand something, I need to know the algorithm behind it. I want to know how to calculate it, the pieces that go in and where they come from. I want to code it up, to test it out on odd cases and see how it behaves, to get a feel for what it can do.

Others need a more physical picture. They need to know where the particles are going, or how energy and momentum are conserved. They want entropy to be increased, action to be minimized, scales to make sense dimensionally.

Others in turn are more mathematical. They want to start with definitions and axioms. To understand something, they want to see it as an example of a broader class of thing, groups or algebras or categories, to fit it into a bigger picture.

Each of these are a kind of translation, turning something into code-speak or physics-speak or math-speak. They don’t require modeling every detail, but when done well they can still explain every detail.

So while yes, it is good practice not to write code that is too “smart”, and too hard to debug…it’s not impossible to debug your smartest code. And while you can’t hold an entire mind inside of yours, you don’t actually need to do that to understand the brain. In both cases, all you need is a translation.

# Math Is the Art of Stating Things Clearly

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

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

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.

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

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

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?

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?