# One, Two, Infinity

Physicists and mathematicians count one, two, infinity.

We start with the simplest case, as a proof of principle. We take a stripped down toy model or simple calculation and show that our idea works. We count “one”, and we publish.

Next, we let things get a bit more complicated. In the next toy model, or the next calculation, new interactions can arise. We figure out how to deal with those new interactions, our count goes from “one” to “two”, and once again we publish.

By this point, hopefully, we understand the pattern. We know what happens in the simplest case, and we know what happens when the different pieces start to interact. If all goes well, that’s enough: we can extrapolate our knowledge to understand not just case “three”, but any case: any model, any calculation. We publish the general case, the general method. We’ve counted one, two, infinity.

Once we’ve counted “infinity”, we don’t have to do any more cases. And so “infinity” becomes the new “zero”, and the next type of calculation you don’t know how to do becomes “one”. It’s like going from addition to multiplication, from multiplication to exponentiation, from exponentials up into the wilds of up-arrow notation. Each time, once you understand the general rules you can jump ahead to an entirely new world with new capabilities…and repeat the same process again, on a new scale. You don’t need to count one, two, three, four, on and on and on.

Of course, research doesn’t always work out this way. My last few papers counted three, four, five, with six on the way. (One and two were already known.) Unlike the ideal cases that go one, two, infinity, here “two” doesn’t give all the pieces you need to keep going. You need to go a few numbers more to get novel insights. That said, we are thinking about “infinity” now, so look forward to a future post that says something about that.

A lot of frustration in physics comes from situations when “infinity” remains stubbornly out of reach. When people complain about all the models for supersymmetry, or inflation, in some sense they’re complaining about fields that haven’t taken that “infinity” step. One or two models of inflation are nice, but by the time the count reaches ten you start hoping that someone will describe all possible models of inflation in one paper, and see if they can make any predictions from that.

(In particle physics, there’s an extent to which people can actually do this. There are methods to describe all possible modifications of the Standard Model in terms of what sort of effects they can have on observations of known particles. There’s a group at NBI who work on this sort of thing.)

The gold standard, though, is one, two, infinity. Our ability to step back, stop working case-by-case, and move on to the next level is not just a cute trick: it’s a foundation for exponential progress. If we can count one, two, infinity, then there’s nowhere we can’t reach.

# The Multiverse Can Only Kill Physics by Becoming Physics

I’m not a fan of the multiverse. I think it’s over-hyped, way beyond its current scientific support.

But I don’t think it’s going to kill physics.

By “the multiverse” I’m referring to a group of related ideas. There’s the idea that we live in a vast, varied universe, with different physical laws in different regions. Relatedly, there’s the idea that the properties of our region aren’t typical of the universe as a whole, just typical of places where life can exist. It may be that in most of the universe the cosmological constant is enormous, but if life can only exist in places where it is tiny then a tiny cosmological constant is what we’ll see. That sort of logic is called anthropic reasoning. If it seems strange, think about a smaller scale: there are many planets in the universe, but only a small number of them can support life. Still, we shouldn’t be surprised that we live on a planet that can support life: if it couldn’t, we wouldn’t live here!

If we really do live in a multiverse, though, some of what we think of as laws of physics are just due to random chance. Maybe the quarks have the masses they do not for some important reason, but just because they happened to end up that way in our patch of the universe.

This seems to have depressing implications. If the laws of physics are random, or just consequences of where life can exist, then what’s left to discover? Why do experiments at all?

Well, why not ask the geoscientists?

These guys

We might live in one universe among many, but we definitely live on one planet among many. And somehow, this realization hasn’t killed geoscience.

That’s because knowing we live on a random planet doesn’t actually tell us very much.

Now, I’m not saying you can’t do anthropic reasoning about the Earth. For example, it looks like an active system of plate tectonics is a necessary ingredient for life. Even if plate tectonics is rare, we shouldn’t be surprised to live on a planet that has it.

Ok, so imagine it’s 1900, before Wegener proposed continental drift. Scientists believe there are many planets in the universe, that we live in a “multiplanet”. Could you predict plate tectonics?

Even knowing that we live on one of the few planets that can support life, you don’t know how it supports life. Even living in a “multiplanet”, geoscience isn’t dead. The specifics of our Earth are still going to teach you something important about how planets work.

Physical laws work the same way. I’ve said that the masses of the quarks could be random, but it’s not quite that simple. The underlying reasons why the masses of the quarks are what they are could be random: the specifics of how six extra dimensions happened to curl up in our region of the universe, for example. But there’s important physics in between: the physics of how those random curlings of space give rise to quark masses. There’s a mechanism there, and we can’t just pick one out of a hat or work backwards to it anthropically. We have to actually go out and discover the answer.

Similarly, we don’t know automatically which phenomena are “random”, which are “anthropic”, and which are required by some deep physical principle. Even in a multiverse, we can’t assume that everything comes down to chance, we only know that some things will, much as the geoscientists don’t know what’s unique to Earth and what’s true of every planet without actually going out and checking.

You can even find a notion of “naturalness” here, if you squint. In physics, we find phenomena like the mass of the Higgs “unnatural”, they’re “fine-tuned” in a way that cries out for an explanation. Normally, we think of this in terms of a hypothetical “theory of everything”: the more “fine-tuned” something appears, the harder it would be to explain it in a final theory. In a multiverse, it looks like we’d have to give up on this, because even the most unlikely-looking circumstance would happen somewhere, especially if it’s needed for life.

Once again, though, imagine you’re a geoscientist. Someone suggests a ridiculously fine-tuned explanation for something: perhaps volcanoes only work if they have exactly the right amount of moisture. Even though we live on one planet in a vast universe, you’re still going to look for simpler explanations before you move on to more complicated ones. It’s human nature, and by and large it’s the only way we’re capable of doing science. As physicists, we’ve papered this over with technical definitions of naturalness, but at the end of the day even in a multiverse we’ll still start with less fine-tuned-looking explanations and only accept the fine-tuned ones when the evidence forces us to. It’s just what people do.

The only way for anthropic reasoning to get around this, to really make physics pointless once and for all, is if it actually starts making predictions. If anthropic reasoning in physics can be made much stronger than anthropic reasoning in geoscience (which, as mentioned, didn’t predict tectonic plates until a century after their discovery) then maybe we can imagine getting to a point where it tells us what particles we should expect to discover, and what masses they should have.

At that point, though, anthropic reasoning won’t have made physics pointless: it will have become physics.

If anthropic reasoning is really good enough to make reliable, falsifiable predictions, then we should be ecstatic! I don’t think we’re anywhere near that point, though some people are earnestly trying to get there. But if it really works out, then we’d have a powerful new method to make predictions about the universe.

Ok, so with all of this said, there is one other worry.

Karl Popper criticized Marxism and Freudianism for being unfalsifiable. In both disciplines, there was a tendency to tell what were essentially “just-so stories”. They could “explain” any phenomenon by setting it in their framework and explaining how it came to be “just so”. These explanations didn’t make new predictions, and different people often ended up coming up with different explanations with no way to distinguish between them. They were stories, not scientific hypotheses. In more recent times, the same criticism has been made of evolutionary psychology. In each case the field is accused of being able to justify anything and everything in terms of its overly ambiguous principles, whether dialectical materialism, the unconscious mind, or the ancestral environment.

Or an elephant’s ‘satiable curtiosity

You’re probably worried that this could happen to physics. With anthropic reasoning and the multiverse, what’s to stop physicists from just proposing some “anthropic” just-so-story for any evidence we happen to find, no matter what it is? Surely anything could be “required for life” given a vague enough argument.

You’re also probably a bit annoyed that I saved this objection for last. I know that for many people, this is precisely what you mean when you say the multiverse will “kill physics”.

I’ve saved this for last for a reason though. It’s because I want to point out something important: this outcome, that our field degenerates into just-so-stories, isn’t required by the physics of the multiverse. Rather, it’s a matter of sociology.

If we hold anthropic reasoning to the same standards as the rest of physics, then there’s no problem: if an anthropic explanation doesn’t make falsifiable predictions then we ignore it. The problem comes if we start loosening our criteria, start letting people publish just-so-stories instead of real science.

This is a real risk! I don’t want to diminish that. It’s harder than it looks for a productive academic field to fall into bullshit, but just-so-stories are a proven way to get there.

What I want to emphasize is that we’re all together in this. We all want to make sure that physics remains scientific. We all need to be vigilant, to prevent a culture of just-so-stories from growing. Regardless of whether the multiverse is the right picture, and regardless of how many annoying TV specials they make about it in the meantime, that’s the key: keeping physics itself honest. If we can manage that, nothing we discover can kill our field.

# Boltzmann Brains, Evil Demons, and Why It’s Occasionally a Good Idea to Listen to Philosophers

There’s been a bit of a buzz recently about a paper Sean Carroll posted to the arXiv, “Why Boltzmann Brains Are Bad”. The argument in the paper isn’t new, it’s something Carroll has been arguing for a long time, and the arXiv post was just because he had been invited to contribute a piece to a book on Current Controversies in Philosophy of Science.

(By the way: in our field, invited papers and conference proceedings are almost always reviews of old work, not new results. If you see something on arXiv and want to know whether it’s actually new work, the “Comments:” section will almost always mention this.)

While the argument isn’t new, it is getting new attention. And since I don’t think I’ve said much about my objections to it, now seems like a good time to do so.

Carroll’s argument is based on theoretical beings called Boltzmann brains. The idea is that if you wait a very very long time in a sufficiently random (“high-entropy”) universe, the matter in that universe will arrange itself in pretty much every imaginable way, if only for a moment. In particular, it will eventually form a brain, or enough of a brain to have a conscious experience. Wait long enough, and you can find a momentary brain having any experience you want, with any (fake) memories you want. Long enough, and you can find a brain having the same experience you are having right now.

So, Carroll asks, how do you know you aren’t a Boltzmann brain? If the universe exists for long enough, most of the beings having your current experiences would be Boltzmann brains, not real humans. But if you really are a Boltzmann brain, then you can’t know anything about the universe at all: everything you think are your memories are just random fluctuations with no connection to the real world.

Carroll calls this sort of situation “cognitively unstable”. If you reason scientifically that the universe must be full of Boltzmann brains, then you can’t rule out that you could be a Boltzmann brain, and thus you shouldn’t accept your original reasoning.

The only way out, according to Carroll, is if we live in a universe that will never contain Boltzmann brains, for example one that won’t exist in its current form long enough to create them. So from a general concern about cognitive instability, Carroll argues for specific physics. And if that seems odd…well, it is.

For the purpose of this post, I’m going to take for granted the physics case: that a sufficiently old and random universe would indeed produce Boltzmann brains. That’s far from uncontroversial, and if you’re interested in that side of the argument (and have plenty of patience for tangents and Czech poop jokes) Lubos Motl posted about it recently.

Instead, I’d like to focus on the philosophical side of the argument.

Descartes wanted to start philosophy from scratch by questioning everything he thought he knew. In one of his arguments, he asks the reader to imagine an evil demon.

Probably Graz’zt. It’s usually Graz’zt.

Descartes imagines this evil demon exercising all its power to deceive. Perhaps it could confound your senses with illusions, or modify your memories. If such a demon existed, there would be no way to know if anything you believed or reasoned about the world was correct. So, Descartes asked, how do you know you’re not being deceived by an evil demon right now?

Amusingly, like Carroll, Descartes went on to use this uncertainty to argue for specific proposals in physics: in Descartes’ case, everything from the existence of a benevolent god to the idea that gravity was caused by a vortex of fluid around the sun.

Descartes wasn’t the last to propose this kind of uncertainty, and philosophers have asked more sophisticated questions over the years challenging the idea that it makes sense to reason from the past about the future at all.

Carroll is certainly aware of all of this. But I suspect he doesn’t quite appreciate the current opinion philosophers have on these sorts of puzzles.

The impression I’ve gotten from philosophers is that they don’t take this kind of “cognitive instability” very seriously anymore. There are specialists who still work on it, and it’s still of historical interest. But the majority of philosophers have moved on.

How did they move on? How have they dismissed these kinds of arguments?

That varies. Philosophers don’t tend to have the kind of consensus that physicists usually do.

Some reject them on pragmatic grounds: science works, even if we can’t “justify” it. Some use a similar argument to Carroll’s, but take it one step back, arguing that we shouldn’t worry that we could be deceived by an evil demon or be a Boltzmann brain because those worries by themselves are cognitively unstable. Some bite the bullet, that reasoning is impossible, then just ignore it and go on with their lives.

The common trait of all of these rejections, though? They don’t rely on physics.

Philosophers don’t argue “evil demons are impossible, therefore we can be sure we’re not deceived by evil demons”. They don’t argue “dreams are never completely realistic, so we can’t just be dreaming right now”.

And they certainly don’t try to argue the reverse: that consistency means there can never be evil demons, or never be realistic dreams.

I was on the debate team in high school. One popular tactic was called the “non-unique”. If your opponent argued that your plan had some negative consequences, you could argue that those consequences would happen regardless of whether you got to enact your plan or not: that the consequences were non-unique.

At this point, philosophers understand that cognitive instability and doubt are “non-unique”. No matter the physics, no matter how the world looks, it’s still possible to argue that reasoning isn’t justified, that even the logic we used to doubt the world in the first place could be flawed.

Carroll’s claim to me seems non-unique. Yes, in a universe that exists for a long time you could be a Boltzmann brain. But even if you don’t live in such a universe, you could still be a brain in a jar or a simulation. You could still be deceived by an “evil demon”.

And so regardless, you need the philosophers. Regardless, you need some argument that reasoning works, that you can ignore doubt. And once you’re happy with that argument, you don’t have to worry about Boltzmann brains.

# Science Is a Collection of Projects, Not a Collection of Beliefs

Read a textbook, and you’ll be confronted by a set of beliefs about the world.

(If it’s a half-decent textbook, it will give justifications for those beliefs, and they will be true, putting you well on the way to knowledge.)

The same is true of most science popularization. In either case, you’ll be instructed that a certain set of statements about the world (or about math, or anything else) are true.

If most of your experience with science comes from popularizations and textbooks, you might think that all of science is like this. In particular, you might think of scientific controversies as matters of contrasting beliefs. Some scientists “believe in” supersymmetry, some don’t. Some “believe in” string theory, some don’t. Some “believe in” a multiverse, some don’t.

In practice, though, only settled science takes the form of beliefs. The rest, science as it is actually practiced, is better understood as a collection of projects.

Scientists spend most of their time working on projects. (Well, or procrastinating in my case.) Those projects, not our beliefs about the world, are how we influence other scientists, because projects build off each other. Any time we successfully do a calculation or make a measurement, we’re opening up new calculations and measurements for others to do. We all need to keep working and publishing, so anything that gives people something concrete to do is going to be influential.

The beliefs that matter come later. They come once projects have been so successful, and so widespread, that their success itself is evidence for beliefs. They’re the beliefs that serve as foundational assumptions for future projects. If you’re going to worry that some scientists are behaving unscientifically, these are the sorts of beliefs you want to worry about. Even then, things are often constrained by viable projects: in many fields, you can’t have a textbook without problem sets.

Far too many people seem to miss this distinction. I’ve seen philosophers focus on scientists’ public statements instead of their projects when trying to understand the implications of their science. I’ve seen bloggers and journalists who mostly describe conflicts of beliefs, what scientists expect and hope to be true rather than what they actually work on.

Do scientists have beliefs about controversial topics? Absolutely. Do those beliefs influence what they work on? Sure. But only so far as there’s actually something there to work on.

That’s why you see quite a few high-profile physicists endorsing some form of multiverse, but barely any actual journal articles about it. The belief in a multiverse may or may not be true, but regardless, there just isn’t much that one can do with the idea right now, and it’s what scientists are doing, not what they believe, that constitutes the health of science.

Different fields seem to understand this to different extents. I’m reminded of a story I heard in grad school, of two dueling psychologists. One of them believed that conversation was inherently cooperative, and showed that, unless unusually stressed or busy, people would put in the effort to understand the other person’s perspective. The other believed that conversation was inherently egocentric, and showed that, the more you stressed or busy people are, the more they assume that everyone else has the same perspective they do.

Strip off the “beliefs”, and these two worked on the exact same thing, with the same results. With their beliefs included, though, they were bitter rivals who bristled if their grad students so much as mentioned the other scientist.

We need to avoid this kind of mistake. The skills we have, the kind of work we do, these are important, these are part of science. The way we talk about it to reporters, the ideas we champion when we debate, those are sidelines. They have some influence, dragging people one way or another. But they’re not what science is, because on the front lines, science is about projects, not beliefs.

# The Metaphysics of Card Games

I tend to be skeptical of attempts to apply metaphysics to physics. In particular, I get leery when someone tries to describe physics in terms of which fundamental things exist, and which things are made up of other things.

Now, I’m not the sort of physicist who thinks metaphysics is useless in general. I’ve seen some impressive uses of supervenience, for example.

But I think that, in physics, talk of “things” is almost always premature. As physicists, we describe the world mathematically. It’s the most precise way we have access to of describing the universe. The trouble is, slightly different mathematics can imply the existence of vastly different “things”.

To give a slightly unusual example, let’s talk about card games.

To defeat metaphysics, we must best it at a children’s card game!

Magic: The Gathering is a collectible card game in which players play powerful spellcasters who fight by casting spells and summoning creatures. Those spells and creatures are represented by cards.

If you wanted to find which “things” exist in Magic: The Gathering, you’d probably start with the cards. And indeed, cards are pretty good candidates for fundamental “things”. As a player, you have a hand of cards, a discard pile (“graveyard”) and a deck (“library”), and all of these are indeed filled with cards.

However, not every “thing” in the game is a card. That’s because the game is in some sense limited: it needs to represent a broad set of concepts while still using physical, purchasable cards.

Suppose you have a card that represents a general. Every turn, the general recruits a soldier. You could represent the soldiers with actual cards, but they’d have to come from somewhere, and over many turns you might quickly run out.

Instead, Magic represents these soldiers with “tokens”. A token is not a card: you can’t shuffle a token into your deck or return it to your hand, and if you try to it just ceases to exist. But otherwise, the tokens behave just like other creatures: they’re both the same type of “thing”, something Magic calls a “permanent”. Permanents live in an area between players called the “battlefield”.

And it gets even more complicated! Some creatures have special abilities. When those abilities are activated, they’re treated like spells in many ways: you can cast spells in response, and even counter them with the right cards. However, they’re not spells, because they’re not cards: like tokens, you can’t shuffle them into your deck. Instead, both they and spells that have just been cast live in another area, the “stack”.

So while Magic might look like it just has one type of “thing”, cards, in fact it has three: cards, permanents, and objects on the stack.

We can contrast this with another card game, Hearthstone.

Hearthstone is much like Magic. You are a spellcaster, you cast spells, you summon creatures, and those spells and creatures are represented by cards.

The difference is, Hearthstone is purely electronic. You can’t go out and buy the cards in a store, they’re simulated in the online game. And this means that Hearthstone’s metaphysics can be a whole lot simpler.

In Hearthstone, if you have a general who recruits a soldier every turn, the soldiers can be cards just like the general. You can return them to your hand, or shuffle them into your deck, just like a normal card. Your computer can keep track of them, and make sure they go away properly at the end of the game.

This means that Hearthstone doesn’t need a concept of “permanents”: everything on its “battlefield” is just a card, which can have some strange consequences. If you return a creature to your hand, and you have room, it will just go there. But if your hand is full, and the creature has nowhere to go, it will “die”, in exactly the same way it would have died in the game if another creature killed it. From the game’s perspective, the creature was always a card, and the card “died”, so the creature died.

These small differences in implementation, in the “mathematics” of the game, change the metaphysics completely. Magic has three types of “things”, Hearthstone has only one.

And card games are a special case, because in some sense they’re built to make metaphysics easy. Cards are intuitive, everyday objects, and both Magic and Hearthstone are built off of our intuitions about them, which is why I can talk about “things” in either game.

Physics doesn’t have to be built that way. Physics is meant to capture our observations, and help us make predictions. It doesn’t have to sort itself neatly into “things”. Even if it does, I hope I’ve convinced you that small changes in physics could lead to large changes in which “things” exist. Unless you’re convinced that you understand the physics of something completely, you might want to skip the metaphysics. A minor mathematical detail could sweep it all away.

# Thought Experiments, Minus the Thought

My second-favorite Newton fact is that, despite inventing calculus, he refused to use it for his most famous work of physics, the Principia. Instead, he used geometrical proofs, tweaked to smuggle in calculus without admitting it.

Essentially, these proofs were thought experiments. Newton would start with a standard geometry argument, one that would have been acceptable to mathematicians centuries earlier. Then, he’d imagine taking it further, pushing a line or angle to some infinite point. He’d argue that, if the proof worked for every finite choice, then it should work in the infinite limit as well.

These thought experiments let Newton argue on the basis of something that looked more rigorous than calculus. However, they also held science back. At the time, only a few people in the world could understand what Newton was doing. It was only later, when Newton’s laws were reformulated in calculus terms, that a wider group of researchers could start doing serious physics.

What changed? If Newton could describe his physics with geometrical thought experiments, why couldn’t everyone else?

The trouble with thought experiments is that they require careful setup, setup that has to be thought through for each new thought experiment. Calculus took Newton’s geometrical thought experiments, and took out the need for thought: the setup was automatically a part of calculus, and each new researcher could build on their predecessors without having to set everything up again.

This sort of thing happens a lot in science. An example from my field is the scattering matrix, or S-matrix.

The S-matrix, deep down, is a thought experiment. Take some particles, and put them infinitely far away from each other, off in the infinite past. Then, let them approach, close enough to collide. If they do, new particles can form, and these new particles will travel out again, infinite far away in the infinite future. The S-matrix then is a metaphorical matrix that tells you, for each possible set of incoming particles, what the probability is to get each possible set of outgoing particles.

In a real collider, the particles don’t come from infinitely far away, and they don’t travel infinitely far before they’re stopped. But the distances are long enough, compared to the sizes relevant for particle physics, that the S-matrix is the right idea for the job.

Like calculus, the S-matrix is a thought experiment minus the thought. When we want to calculate the probability of particles scattering, we don’t need to set up the whole thought experiment all over again. Instead, we can start by calculating, and over time we’ve gotten very good at it.

In general, sub-fields in physics can be divided into those that have found their S-matrices, their thought experiments minus thought, and those that have not. When a topic has to rely on thought experiments, progress is much slower: people argue over the details of each setup, and it’s difficult to build something that can last. It’s only when a field turns the corner, removing the thought from its thought experiments, that people can start making real collaborative progress.

# What Does It Mean to Know the Answer?

My sub-field isn’t big on philosophical debates. We don’t tend to get hung up on how to measure an infinite universe, or in arguing about how to interpret quantum mechanics. Instead, we develop new calculation techniques, which tends to nicely sidestep all of that.

If there’s anything we do get philosophical about, though, any question with a little bit of ambiguity, it’s this: What counts as an analytic result?

“Analytic” here is in contrast to “numerical”. If all we need is a number and we don’t care if it’s slightly off, we can use numerical methods. We have a computer use some estimation trick, repeating steps over and over again until we have approximately the right answer.

“Analytic”, then, refers to everything else. When you want an analytic result, you want something exact. Most of the time, you don’t just want a single number: you want a function, one that can give you numbers for whichever situation you’re interested in.

It might sound like there’s no ambiguity there. If it’s a function, with sines and cosines and the like, then it’s clearly analytic. If you can only get numbers out through some approximation, it’s numerical. But as the following example shows, things can get a bit more complicated.

Suppose you’re trying to calculate something, and you find the answer is some messy integral. Still, you’ve simplified the integral enough that you can do numerical integration and get some approximate numbers out. What’s more, you can express the integral as an infinite series, so that any finite number of terms will get close to the correct result. Maybe you even know a few special cases, situations where you plug specific numbers in and you do get an exact answer.

It might sound like you only know the answer numerically. As it turns out, though, this is roughly how your computer handles sines and cosines.

When your computer tries to calculate a sine or a cosine, it doesn’t have access to the exact solution all of the time. It does have some special cases, but the rest of the time it’s using an infinite series, or some other numerical trick. Type in a random sine into your calculator and it will be just as approximate as if you did a numerical integration.

So what’s the real difference?

Rather than how we get numbers out, think about what else we know. We know how to take derivatives of sines, and how to integrate them. We know how to take limits, and series expansions. And we know their relations to other functions, including how to express them in terms of other things.

If you can do that with your integral, then you’ve probably got an analytic result. If you can’t, then you don’t.

What if you have only some of the requirements, but not the others? What if you can take derivatives, but don’t know all of the identities between your functions? What if you can do series expansions, but only in some limits? What if you can do all the above, but can’t get numbers out without a supercomputer?

That’s where the ambiguity sets in.

In the end, whether or not we have the full analytic answer is a matter of degree. The closer we can get to functions that mathematicians have studied and understood, the better grasp we have of our answer and the more “analytic” it is. In practice, we end up with a very pragmatic approach to knowledge: whether we know the answer depends entirely on what we can do with it.