Tag Archives: philosophy of science

Which Things Exist in Quantum Field Theory

If you ever think metaphysics is easy, learn a little quantum field theory.

Someone asked me recently about virtual particles. When talking to the public, physicists sometimes explain the behavior of quantum fields with what they call “virtual particles”. They’ll describe forces coming from virtual particles going back and forth, or a bubbling sea of virtual particles and anti-particles popping out of empty space.

The thing is, this is a metaphor. What’s more, it’s a metaphor for an approximation. As physicists, when we draw diagrams with more and more virtual particles, we’re trying to use something we know how to calculate with (particles) to understand something tougher to handle (interacting quantum fields). Virtual particles, at least as you’re probably picturing them, don’t really exist.

I don’t really blame physicists for talking like that, though. Virtual particles are a metaphor, sure, a way to talk about a particular calculation. But so is basically anything we can say about quantum field theory. In quantum field theory, it’s pretty tough to say which things “really exist”.

I’ll start with an example, neutrino oscillation.

You might have heard that there are three types of neutrinos, corresponding to the three “generations” of the Standard Model: electron-neutrinos, muon-neutrinos, and tau-neutrinos. Each is produced in particular kinds of reactions: electron-neutrinos, for example, get produced by beta-plus decay, when a proton turns into a neutron, an anti-electron, and an electron-neutrino.

Leave these neutrinos alone though, and something strange happens. Detect what you expect to be an electron-neutrino, and it might have changed into a muon-neutrino or a tau-neutrino. The neutrino oscillated.

Why does this happen?

One way to explain it is to say that electron-neutrinos, muon-neutrinos, and tau-neutrinos don’t “really exist”. Instead, what really exists are neutrinos with specific masses. These don’t have catchy names, so let’s just call them neutrino-one, neutrino-two, and neutrino-three. What we think of as electron-neutrinos, muon-neutrinos, and tau-neutrinos are each some mix (a quantum superposition) of these “really existing” neutrinos, specifically the mixes that interact nicely with electrons, muons, and tau leptons respectively. When you let them travel, it’s these neutrinos that do the traveling, and due to quantum effects that I’m not explaining here you end up with a different mix than you started with.

This probably seems like a perfectly reasonable explanation. But it shouldn’t. Because if you take one of these mass-neutrinos, and interact with an electron, or a muon, or a tau, then suddenly it behaves like a mix of the old electron-neutrinos, muon-neutrinos, and tau-neutrinos.

That’s because both explanations are trying to chop the world up in a way that can’t be done consistently. There aren’t electron-neutrinos, muon-neutrinos, and tau-neutrinos, and there aren’t neutrino-ones, neutrino-twos, and neutrino-threes. There’s a mathematical object (a vector space) that can look like either.

Whether you’re comfortable with that depends on whether you think of mathematical objects as “things that exist”. If you aren’t, you’re going to have trouble thinking about the quantum world. Maybe you want to take a step back, and say that at least “fields” should exist. But that still won’t do: we can redefine fields, add them together or even use more complicated functions, and still get the same physics. The kinds of things that exist can’t be like this. Instead you end up invoking another kind of mathematical object, equivalence classes.

If you want to be totally rigorous, you have to go a step further. You end up thinking of physics in a very bare-bones way, as the set of all observations you could perform. Instead of describing the world in terms of “these things” or “those things”, the world is a black box, and all you’re doing is finding patterns in that black box.

Is there a way around this? Maybe. But it requires thought, and serious philosophy. It’s not intuitive, it’s not easy, and it doesn’t lend itself well to 3d animations in documentaries. So in practice, whenever anyone tells you about something in physics, you can be pretty sure it’s a metaphor. Nice describable, non-mathematical things typically don’t exist.

Science as Hermeneutics: Closer Than You’d Think

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This post is once again inspired by a Ted Chiang short story. This time, it’s “The Evolution of Human Science”, which imagines a world in which super-intelligent “metahumans” have become incomprehensible to the ordinary humans they’ve left behind. Human scientists in that world practice “hermeneutics“: instead of original research, they try to interpret what the metahumans are doing, reverse-engineering their devices and observing their experiments.

Much like a blogger who, out of ideas, cribs them from books.

It’s a thought-provoking view of what science in the distant future could become. But it’s also oddly familiar.

You might think I’m talking about machine learning here. It’s true that in recent years people have started using machine learning in science, with occasionally mysterious results. There are even a few cases of physicists using machine-learning to suggest some property, say of Calabi-Yau manifolds, and then figuring out how to prove it. It’s not hard to imagine a day when scientists are reduced to just interpreting whatever the AIs throw at them…but I don’t think we’re quite there yet.

Instead, I’m thinking about my own work. I’m a particular type of theoretical physicist. I calculate scattering amplitudes, formulas that tell us the probabilities that subatomic particles collide in different ways. We have a way to calculate these, Feynman’s famous diagrams, but they’re inefficient, so researchers like me look for shortcuts.

How do we find those shortcuts? Often, it’s by doing calculations the old, inefficient way. We use older methods, look at the formulas we get, and try to find patterns. Each pattern is a hint at some new principle that can make our calculations easier. Sometimes we can understand the pattern fully, and prove it should hold. Other times, we observe it again and again and tentatively assume it will keep going, and see what happens if it does.

Either way, this isn’t so different from the hermeneutics scientists practice in the story. Feynman diagrams already “know” every pattern we find, like the metahumans in the story who already know every result the human scientists can discover. But that “knowledge” isn’t in a form we can understand or use. We have to learn to interpret it, to read between the lines and find underlying patterns, to end up with something we can hold in our own heads and put into action with our own hands. The truth may be “out there”, but scientists can’t be content with that. We need to get the truth “in here”. We need to interpret it for ourselves.

Unification That Does Something

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I’ve got unification on the brain.

Recently, a commenter asked me what physicists mean when they say two forces unify. While typing up a response, I came across this passage, in a science fiction short story by Ted Chiang.

Physics admits of a lovely unification, not just at the level of fundamental forces, but when considering its extent and implications. Classifications like ‘optics’ or ‘thermodynamics’ are just straitjackets, preventing physicists from seeing countless intersections.

This passage sounds nice enough, but I feel like there’s a misunderstanding behind it. When physicists seek after unification, we’re talking about something quite specific. It’s not merely a matter of two topics intersecting, or describing them with the same math. We already plumb intersections between fields, including optics and thermodynamics. When we hope to find a unified theory, we do so because it does something. A real unified theory doesn’t just aid our calculations, it gives us new ways to alter the world.

To show you what I mean, let me start with something physicists already know: electroweak unification.

There’s a nice series of posts on the old Quantum Diaries blog that explains electroweak unification in detail. I’ll be a bit vaguer here.

You might have heard of four fundamental forces: gravity, electromagnetism, the strong nuclear force, and the weak nuclear force. You might have also heard that two of these forces are unified: the electromagnetic force and the weak nuclear force form something called the electroweak force.

What does it mean that these forces are unified? How does it work?

Zoom in far enough, and you don’t see the electromagnetic force and the weak force anymore. Instead you see two different forces, I’ll call them “W” and “B”. You’ll also see the Higgs field. And crucially, you’ll see the “W” and “B” forces interact with the Higgs.

The Higgs field is special because it has what’s called a “vacuum” value. Even in otherwise empty space, there’s some amount of “Higgsness” in the background, like the color of a piece of construction paper. This background Higgs-ness is in some sense an accident, just one stable way the universe happens to sit. In particular, it picks out an arbitrary kind of direction: parts of the “W” and “B” forces happen to interact with it, and parts don’t.

Now let’s zoom back out. We could, if we wanted, keep our eyes on the “W” and “B” forces. But that gets increasingly silly. As we zoom out we won’t be able to see the Higgs field anymore. Instead, we’ll just see different parts of the “W” and “B” behaving in drastically different ways, depending on whether or not they interact with the Higgs. It will make more sense to talk about mixes of the “W” and “B” fields, to distinguish the parts that are “lined up” with the background Higgs and the parts that aren’t. It’s like using “aft” and “starboard” on a boat. You could use “north” and “south”, but that would get confusing pretty fast.

My cabin is on the west side of the ship…unless we’re sailing east….

What are those “mixes” of the “W” and “B” forces? Why, they’re the weak nuclear force and the electromagnetic force!

This, broadly speaking, is the kind of unification physicists look for. It doesn’t have to be a “mix” of two different forces: most of the models physicists imagine start with a single force. But the basic ideas are the same: that if you “zoom in” enough you see a simpler model, but that model is interacting with something that “by accident” picks a particular direction, so that as we zoom out different parts of the model behave in different ways. In that way, you could get from a single force to all the different forces we observe.

That “by accident” is important here, because that accident can be changed. That’s why I said earlier that real unification lets us alter the world.

To be clear, we can’t change the background Higgs field with current technology. The biggest collider we have can just make a tiny, temporary fluctuation (that’s what the Higgs boson is). But one implication of electroweak unification is that, with enough technology, we could. Because those two forces are unified, and because that unification is physical, with a physical cause, it’s possible to alter that cause, to change the mix and change the balance. This is why this kind of unification is such a big deal, why it’s not the sort of thing you can just chalk up to “interpretation” and ignore: when two forces are unified in this way, it lets us do new things.

Mathematical unification is valuable. It’s great when we can look at different things and describe them in the same language, or use ideas from one to understand the other. But it’s not the same thing as physical unification. When two forces really unify, it’s an undeniable physical fact about the world. When two forces unify, it does something.

The Parameter Was Inside You All Along

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Sabine Hossenfelder had an explainer video recently on how to tell science from pseudoscience. This is a famously difficult problem, so naturally we have different opinions. I actually think the picture she draws is reasonably sound. But while it is a good criterion to tell whether you yourself are doing pseudoscience, it’s surprisingly tricky to apply it to other people.

Hossenfelder argues that science, at its core, is about explaining observations. To tell whether something is science or pseudoscience you need to ask, first, if it agrees with observations, and second, if it is simpler than those observations. In particular, a scientist should prefer models with fewer parameters. If your model has so many parameters that you can fit any observation, you’re not being scientific.

This is a great rule of thumb, one that as Hossenfelder points out forms the basis of a whole raft of statistical techniques. It does rely on one tricky judgement, though: how many parameters does your model actually have?

Suppose I’m one of those wacky theorists who propose a whole new particle to explain some astronomical mystery. Hossenfelder, being more conservative in these things, proposes a model with no new particles. Neither of our models fit the data perfectly. Perhaps my model fits a little better, but after all it has one extra parameter, from the new particle. If we want to compare our models, we should take that into account, and penalize mine.

Here’s the question, though: how do I know that Hossenfelder didn’t start out with more particles, and got rid of them to get a better fit? If she did, she had more parameters than I did. She just fit them away.

The problem here is closely related to one called the look-elsewhere effect. Scientists don’t publish everything they try. An unscrupulous scientist can do a bunch of different tests until one of them randomly works, and just publish that one, making the result look meaningful when really it was just random chance. Even if no individual scientist is unscrupulous, a community can do the same thing: many scientists testing many different models, until one accidentally appears to work.

As a scientist, you mostly know if your motivations are genuine. You know if you actually tried a bunch of different models or had good reasons from the start to pick the one you did. As someone judging other scientists, you often don’t have that luxury. Sometimes you can look at prior publications and see all the other attempts someone made. Sometimes they’ll even tell you explicitly what parameters they used and how they fit them. But sometimes, someone will swear up and down that their model is just the most natural, principled choice they could have made, and they never considered anything else. When that happens, how do we guard against the look-elsewhere effect?

The normal way to deal with the look-elsewhere effect is to consider, not just whatever tests the scientist claims to have done, but all tests they could reasonably have done. You need to count all the parameters, not just the ones they say they varied.

This works in some fields. If you have an idea of what’s reasonable and what’s not, you have a relatively manageable list of things to look at. You can come up with clear rules for which theories are simpler than others, and people will agree on them.

Physics doesn’t have it so easy. We don’t have any pre-set rules for what kind of model is “reasonable”. If we want to parametrize every “reasonable” model, the best we can do are what are called Effective Field Theories, theories which try to describe every possible type of new physics in terms of its effect on the particles we already know. Even there, though, we need assumptions. The most popular effective field theory, called SMEFT, assumes the forces of the Standard Model keep their known symmetries. You get a different model if you relax that assumption, and even that model isn’t the most general: for example, it still keeps relativity intact. Try to make the most general model possible, and you end up waist-deep in parameter soup.

Subjectivity is a dirty word in science…but as far as I can tell it’s the only way out of this. We can try to count parameters when we can, and use statistical tools…but at the end of the day, we still need to make choices. We need to judge what counts as an extra parameter and what doesn’t, which possible models to compare to and which to ignore. That’s going to be dependent on our scientific culture, on fashion and aesthetics, there just isn’t a way around that. The best we can do is own up to our assumptions, and be ready to change them when we need to.

Bottomless Science

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There’s an attitude I keep seeing among physics crackpots. It goes a little something like this:

“Once upon a time, physics had rules. You couldn’t just wave your hands and write down math, you had to explain the world with real physical things.”

What those “real physical things” were varies. Some miss the days when we explained things mechanically, particles like little round spheres clacking against each other. Some want to bring back absolutes: an absolute space, an absolute time, an absolute determinism. Some don’t like the proliferation of new particles, and yearn for the days when everything was just electrons, protons, and neutrons.

In each case, there’s a sense that physicists “cheated”. That, faced with something they couldn’t actually explain, they made up new types of things (fields, relativity, quantum mechanics, antimatter…) instead. That way they could pretend to understand the world, while giving up on their real job, explaining it “the right way”.

I get where this attitude comes from. It does make a certain amount of sense…for other fields.

An an economist, you can propose whatever mathematical models you want, but at the end of the day they have to boil down to actions taken by people. An economist who proposed some sort of “dark money” that snuck into the economy without any human intervention would get laughed at. Similarly, as a biologist or a chemist, you ultimately need a description that makes sense in terms of atoms and molecules. Your description doesn’t actually need to be in terms of atoms and molecules, and often it can’t be: you’re concerned with a different level of explanation. But it should be possible in terms of atoms and molecules, and that puts some constraints on what you can propose.

Why shouldn’t physics have similar constraints?

Suppose you had a mandatory bottom level like this. Maybe everything boils down to ball bearings, for example. What happens when you study the ball bearings?

Your ball bearings have to have some properties: their shape, their size, their weight. Where do those properties come from? What explains them? Who studies them?

Any properties your ball bearings have can be studied, or explained, by physics. That’s physics’s job: to study the fundamental properties of matter. Any “bottom level” is just as fit a subject for physics as anything else, and you can’t explain it using itself. You end up needing another level of explanation.

Maybe you’re objecting here that your favorite ball bearings aren’t up for debate: they’re self-evident, demanded by the laws of mathematics or philosophy.

Here for lack of space, I’ll only say that mathematics and philosophy don’t work that way. Mathematics can tell you whether you’ve described the world consistently, whether the conclusions you draw from your assumptions actually follow. Philosophy can see if you’re asking the right questions, if you really know what you think you know. Both have lessons for modern physics, and you can draw valid criticisms from either. But neither one gives you a single clear way the world must be. Not since the days of Descartes and Kant have people been that naive.

Because of this, physics is doing something a bit different from economics and biology. Each field wants to make models, wants to describe its observations. But in physics, ultimately, those models are all we have. We don’t have a “bottom level”, a backstop where everything has to make sense. That doesn’t mean we can just make stuff up, and whenever possible we understand the world in terms of physics we’ve already discovered. But when we can’t, all bets are off.

The Point of a Model

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I’ve been reading more lately, partially for the obvious reasons. Mostly, I’ve been catching up on books everyone else already read.

One such book is Daniel Kahneman’s “Thinking, Fast and Slow”. With all the talk lately about cognitive biases, Kahneman’s account of his research on decision-making was quite familiar ground. The book turned out to more interesting as window into the culture of psychology research. While I had a working picture from psychologist friends in grad school, “Thinking, Fast and Slow” covered the other side, the perspective of a successful professor promoting his field.

Most of this wasn’t too surprising, but one passage struck me:

Several economists and psychologists have proposed models of decision making that are based on the emotions of regret and disappointment. It is fair to say that these models have had less influence than prospect theory, and the reason is instructive. The emotions of regret and disappointment are real, and decision makers surely anticipate these emotions when making their choices. The problem is that regret theories make few striking predictions that would distinguish them from prospect theory, which has the advantage of being simpler. The complexity of prospect theory was more acceptable in the competition with expected utility theory because it did predict observations that expected utility theory could not explain.
Richer and more realistic assumptions do not suffice to make a theory successful. Scientists use theories as a bag of working tools, and they will not take on the burden of a heavier bag unless the new tools are very useful. Prospect theory was accepted by many scholars not because it is “true” but because the concepts that it added to utility theory, notably the reference point and loss aversion, were worth the trouble; they yielded new predictions that turned out to be true. We were lucky.
Thinking Fast and Slow, page 288

Kahneman is contrasting three theories of decision making here: the old proposal that people try to maximize their expected utility (roughly, the benefit they get in future), his more complicated “prospect theory” that takes into account not only what benefits people get but their attachment to what they already have, and other more complicated models based on regret. His theory ended up more popular, both than the older theory and than the newer regret-based models.

Why did his theory win out? Apparently, not because it was the true one: as he says, people almost certainly do feel regret, and make decisions based on it. No, his theory won because it was more useful. It made new, surprising predictions, while being simpler and easier to use than the regret-based models.

This, a theory defeating another without being “more true”, might bug you. By itself, it doesn’t bug me. That’s because, as a physicist, I’m used to the idea that models should not just be true, but useful. If we want to test our theories against reality, we have a large number of “levels” of description to choose from. We can “zoom in” to quarks and gluons, or “zoom out” to look at atoms, or molecules, or polymers. We have to decide how much detail to include, and we have real pragmatic reasons for doing so: some details are just too small to measure!

It’s not clear Kahneman’s community was doing this, though. That is, it doesn’t seem like he’s saying that regret and disappointment are just “too small to be measured”. Instead, he’s saying that they don’t seem to predict much differently from prospect theory, and prospect theory is simpler to use.

Ok, we do that in physics too. We like working with simpler theories, when we have a good excuse. We’re just careful about it. When we can, we derive our simpler theories from more complicated ones, carving out complexity and estimating how much of a difference it would have made. Do this carefully, and we can treat black holes as if they were subatomic particles. When we can’t, we have what we call “phenomenological” models, models built up from observation and not from an underlying theory. We never take such models as the last word, though: a phenomenological model is always viewed as temporary, something to bridge a gap while we try to derive it from more basic physics.

Kahneman doesn’t seem to view prospect theory as temporary. It doesn’t sound like anyone is trying to derive it from regret theory, or to make regret theory easier to use, or to prove it always agrees with regret theory. Maybe they are, and Kahneman simply doesn’t think much of their efforts. Either way, it doesn’t sound like a major goal of the field.

That’s the part that bothered me. In physics, we can’t always hope to derive things from a more fundamental theory, some theories are as fundamental as we know. Psychology isn’t like that: any behavior people display has to be caused by what’s going on in their heads. What Kahneman seems to be saying here is that regret theory may well be closer to what’s going on in people’s heads, but he doesn’t care: it isn’t as useful.

And at that point, I have to ask: useful for what?

As a psychologist, isn’t your goal ultimately to answer that question? To find out “what’s going on in people’s heads”? Isn’t every model you build, every theory you propose, dedicated to that question?

And if not, what exactly is it “useful” for?

For technology? It’s true, “Thinking Fast and Slow” describes several groups Kahneman advised, most memorably the IDF. Is the advantage of prospect theory, then, its “usefulness”, that it leads to better advice for the IDF?

I don’t think that’s what Kahneman means, though. When he says “useful”, he doesn’t mean “useful for advice”. He means it’s good for giving researchers ideas, good for getting people talking. He means “useful for designing experiments”. He means “useful for writing papers”.

And this is when things start to sound worryingly familiar. Because if I’m accusing Kahneman’s community of giving up on finding the fundamental truth, just doing whatever they can to write more papers…well, that’s not an uncommon accusation in physics as well. If the people who spend their lives describing cognitive biases are really getting distracted like that, what chance does, say, string theory have?

I don’t know how seriously to take any of this. But it’s lurking there, in the back of my mind, that nasty, vicious, essential question: what are all of our models for?

Bonus quote, for the commenters to have fun with:

I have yet to meet a successful scientist who lacks the ability to exaggerate the importance of what he or she is doing, and I believe that someone who lacks a delusional sense of significance will wilt in the face of repeated experiences of multiple small failures and rare successes, the fate of most researchers.
Thinking Fast and Slow, page 264

Understanding Is Translation

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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.

A Scale of “Sure-Thing-Ness” for Experiments

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No experiment is a sure thing. No matter what you do, what you test, what you observe, there’s no guarantee that you find something new. Even if you do your experiment correctly and measure what you planned to measure, nature might not tell you anything interesting.

Still, some experiments are more sure than others. Sometimes you’re almost guaranteed to learn something, even if it wasn’t what you hoped, while other times you just end up back where you started.

The first, and surest, type of experiment, is a voyage into the unknown. When nothing is known about your target, no expectations, and no predictions, then as long as you successfully measure anything you’ll have discovered something new. This can happen if the thing you’re measuring was only recently discovered. If you’re the first person who manages to measure the reaction rates of an element, or the habits of an insect, or the atmosphere of a planet, then you’re guaranteed to find out something you didn’t know before.

If you don’t have a total unknown to measure, then you want to test a clear hypothesis. The best of these are the theory killers, experiments which can decisively falsify an idea. History’s most famous experiments take this form, like the measurement of the perihelion of Mercury to test General Relativity or Pasteur’s tests of spontaneous generation. When you have a specific prediction and not much wiggle room, an experiment can teach you quite a lot.

“Not much wiggle room” is key, because these tests can all to easily become theory modifiers instead. If you can tweak your theory enough, then your experiment might not be able to falsify it. Something similar applies when you have a number of closely related theories. Even if you falsify one, you can just switch to another similar idea. In those cases, testing your theory won’t always teach you as much: you have to get lucky and see something that pins your theory down more precisely.

Finally, you can of course be just looking. Some experiments are just keeping an eye out, in the depths of space or the precision of quantum labs, watching for something unexpected. That kind of experiment might never see anything, and never rule anything out, but they can still sometimes be worthwhile.

There’s some fuzziness to these categories, of course. Often when scientists argue about whether an experiment is worth doing they’re arguing about which category to place it in. Would a new collider be a “voyage into the unknown” (new energy scales we’ve never measured before), a theory killer/modifier (supersymmetry! but which one…) or just “just looking”? Is your theory of cosmology specific enough to be “killed”, or merely “modified”? Is your wacky modification of quantum mechanics something that can be tested, or merely “just looked” for?

For any given experiment, it’s worth keeping in mind what you expect, and what would happen if you’re wrong. In science, we can’t do every experiment we want. We have to focus our resources and try to get results. Even if it’s never a sure thing.

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.

The Teaching Heuristic for Non-Empirical Science

4 gravitons

Stories about physics from someone who's been there

Tag Archives: philosophy of science

Which Things Exist in Quantum Field Theory

Science as Hermeneutics: Closer Than You’d Think

Unification That Does Something

The Parameter Was Inside You All Along

Bottomless Science

The Point of a Model

Understanding Is Translation

A Scale of “Sure-Thing-Ness” for Experiments

Math Is the Art of Stating Things Clearly

The Teaching Heuristic for Non-Empirical Science