Bayesian view of scientific virtues

A number of scientific virtues are explained intuitively by Bayes' rule, including:

- __Falsifiability:__ A good scientist should say what they do *not* expect to see if a theory is true.
- __Boldness:__ A good theory makes bold experimental predictions (that we wouldn't otherwise expect)
- __Precision:__ A good theory makes precise experimental predictions (that turn out correct)
- __Falsificationism:__ Acceptance of a scientific theory is always provisional; rejection of a scientific theory is pretty permanent.
- __Experimentation:__ You find better theories by making observations, and then updating your beliefs.

# Falsifiability

"Falsifiability" means saying which events and observations should definitely *not* happen if your theory is true.

This was first popularized as a scientific virtue by Karl Popper, who wrote, in a famous critique of Freudian psychoanalysis:

> Neither Freud nor Adler excludes any particular person’s acting in any particular way, whatever the outward circumstances. Whether a man sacrificed his life to rescue a drowning child (a case of sublimation) or whether he murdered the child by drowning him (a case of repression) could not possibly be predicted or excluded by Freud’s theory; *the theory was compatible with everything that could happen.*

In a Bayesian sense, we can see a hypothesis's falsifiability as a requirement for obtaining strong [likelihood ratios](https://arbital.com/p/1rq) in favor of the hypothesis, compared to, e.g., the alternative hypothesis "I don't know."

Suppose you're a very early researcher on gravitation, named Grek.  Your friend Thag is holding a rock in one hand, about to let it go.  You need to predict whether the rock will move downward to the ground, fly upward into the sky, or do something else.  That is, you must say how your theory $Grek$ assigns its probabilities over $up, down,$ and $other.$

As it happens, your friend Thag has his own theory $Thag$ which says "Rocks do what they want to do."  If Thag sees the rock go down, he'll explain this by saying the rock wanted to go down.  If Thag sees the rock go up, he'll say the rock wanted to go up.  Thag thinks that the Thag Theory of Gravitation is a very good one because it can explain any possible thing the rock is observed to do.  This makes it superior compared to a theory that could *only* explain, say, the rock falling down.

As a Bayesian, however, you realize that since $up, down,$ and $other$ are [mutually exclusive and exhaustive](https://arbital.com/p/1rd) possibilities, and *something* must happen when Thag lets go of the rock, the [conditional probabilities](https://arbital.com/p/1rj) $\mathbb P(\cdot\mid Thag)$ must sum to $\mathbb P(up\mid Thag) + \mathbb P(down\mid Thag) + \mathbb P(other\mid Thag) = 1.$

If Thag is "equally good at explaining" all three outcomes - if Thag's theory is equally compatible with all three events and produces equally clever explanations for each of them - then we might as well call this $1/3$ probability for each of $\mathbb P(up\mid Thag), \mathbb P(down\mid Thag),$ and $\mathbb P(other\mid Thag)$.  Note that Thag theory's is isomorphic, in a probabilistic sense, to saying "I don't know."

But now suppose Grek make falsifiable prediction!  Grek say, "Most things fall down!"

Then Grek not have all probability mass distributed equally!  Grek put 95% of probability mass in $\mathbb P(down\mid Grek)!$  Only leave 5% probability divided equally over $\mathbb P(up\mid Grek)$ and $\mathbb P(other\mid Grek)$ in case rock behave like bird.

Thag say this bad idea.  If rock go up, Grek Theory of Gravitation disconfirmed by false prediction!  Compared to Thag Theory that predicts 1/3 chance of $up,$ will be likelihood ratio of 2.5% : 33% ~ 1 : 13 against Grek Theory!  Grek embarrassed!

Grek say, she is confident rock *does* go down.  Things like bird are rare.  So Grek willing to stick out neck and face potential embarrassment.  Besides, is more important to learn about if Grek Theory is true than to save face.

Thag let go of rock.  Rock fall down.

This evidence with likelihood ratio of 0.95 : 0.33 ~ 3 : 1 favoring Grek Theory over Thag Theory.

"How you get such big likelihood ratio?" Thag demand.  "Thag never get big likelihood ratio!"

Grek explain is possible to obtain big likelihood ratio because Grek Theory stick out neck and take probability mass *away* from outcomes $up$ and $other,$ risking disconfirmation if that happen.  This free up lots of probability mass that Grek can put in outcome $down$ to make big likelihood ratio if $down$ happen.

Grek Theory win because falsifiable and make correct prediction!  If falsifiable and make wrong prediction, Grek Theory lose, but this okay because Grek Theory not Grek.

# Advance prediction

On the next experiment, Thag lets go of the rock, watches it fall down, and then says, "Thag Theory assign 100% probability to $\mathbb P(down\mid Thag)$."

Grek replies, "Grek think if Thag see rock fly up instead, Thag would've said $\mathbb P(up\mid Thag) = 1.$  Thag engage in [hindsight bias](https://en.wikipedia.org/wiki/Hindsight_bias)."

"Grek can't prove Thag biased," says Thag.  "Grek make ad hominem argument."

"New rule," says Grek.  "Everyone say probability assignment *before* thing happens.  That way, no need to argue afterwards."

Thag thinks.  "Thag say $\mathbb P(up\mid Thag) = 1$ and $\mathbb P(down\mid Thag) = 1$."

"Thag violate [probability axioms](https://en.wikipedia.org/wiki/Probability_axioms)," says Grek.  "Probability of all [mutually exclusive](https://arbital.com/p/1rd) outcomes must sum to $1$ or less.  But good thing Thag say in advance so Grek can see problem."

"That not fair!" objects Thag.  "Should be allowed to say afterwards so nobody can tell!"

## Formality

The rule of advance prediction is much more pragmatically important for informal theories than formal ones; and for these purposes, a theory is 'formal' when the theory's predictions are produced in a sufficiently mechanical and determined way that anyone can plug the theory into a computer and get the same answer for what probability the theory assigns.

When Newton's Theory of Gravitation was proposed, it was considered not-yet-fully-proven because retrodictions such as the tides, elliptical planetary orbits, and Kepler's Laws, had all been observed *before* Newton proposed the theory.  Even so, a pragmatic Bayesian would have given Newton's theory a lot of credit for these retrodictions, because unlike, say, a psychological theory of human behavior, it was possible for *anyone* - not just Newton - to sit down with a pencil and derive exactly the same predictions from Newton's Laws.  This wouldn't completely eliminate the possibility that Newton's Theory had in some sense been overfitted to Kepler's Laws and the tides, and would then be incapable of further correct new predictions.  But it did mean that, as a formal theory, there could be less pragmatic worry that Thagton was just saying, "Oh, well, of course my theory of 'Planets go where they want' would predict elliptical orbits; elliptical orbits look nice."

## Asking a theory's adherents what the theory says.

Thag picks up another rock.  "I say in advance that Grek Theory assign 0% probability to rock going down."  Thag drops the rock.  "Thag disprove Grek Theory!"

Grek shakes her head.  "Should ask advocates of Grek Theory what Grek Theory predicts."  Grek picks up another rock.  "I say Grek Theory assign $\mathbb P(down\mid Grek) = 0.95$."

"I say Grek Theory assign $\mathbb P(down\mid Grek) = 0$," counters Thag.

"That not how science work," replies Grek.  "Thag should say what Thag's Theory says."

Thag thinks for a moment.  "Thag Theory says rock has 95% probability of going down."

"What?" says Grek.  "Thag just copying Grek Theory!  Also, Thag not say that *before* seeing rocks go down!"

Thag smiles smugly.  "Only Thag get to say what Thag Theory predict, right?"

Again for pragmatic reasons, we should *first* ask the adherents of an informal theory to say what the theory predicts (a formal theory can simply be operated by anyone, and if this is not true, we will not call the theory 'formal').

Furthermore, since you can find a fool following any cause, you should ask the smartest or most technically adept advocates of the theory.  If there's any dispute about who those are, ask separate representatives from the leading groups.  Fame is definitely not the key qualifier; you should ask Murray Gell-Mann and not Deepak Chopra about quantum mechanics, even if more people have heard of Deepak Chopra's beliefs about quantum mechanics than have heard about Murray Gell-Mann.  If you really can't tell the difference, ask them both, don't ask only Chopra and then claim that Chopra gets to be *the* representative because he is most famous.

These types of courtesy rules would not be necessary if we were dealing with a sufficiently advanced Artificial Intelligence or ideal rational agent, but it makes sense for human science where people may be motivated to falsely construe another theory's probability assignments.

This informal rule has its limits, and there may be cases where it seems really obvious that a hypothesis's predictions ought not to be what the hypothesis's adherents claim, or that the theory's adherents are just stealing the predictions of a more successful theory.  But there ought to be a *large* (if defeasible) bias in favor of letting a theory's adherents say what that theory predicts.

# Boldness

A few minutes later, Grek is picking up another rock.  "$\mathbb P(down\mid Grek) = 0.95$," says Grek.

"$\mathbb P(down\mid Thag) = 0.95$," says Thag.  "See, Thag assign high probability to outcomes observed.  Thag win yet?"

"No," says Grek.  "[Likelihood ratios](https://arbital.com/p/1rq) 1 : 1 all time now, even if we believe Thag.  Thag's theory not pick up advantage.  Thag need to make *bold* prediction other theories not make."

Thag frowns.  "Thag say... rock will turn blue when you let go this time?  $\mathbb P(blue\mid Thag) = 0.90$."

"That very bold," Grek says.  "Grek Theory not say that (nor any other obvious 'common sense' or 'business as usual' theories).  Grek think that $\mathbb P(blue\mid \neg Thag) < 0.01$ so Thag prediction definitely has virtue of boldness.  Will be big deal if Thag prediction come true."

$\dfrac{\mathbb P(Thag\mid blue)}{\mathbb P(\neg Thag\mid blue)} > 90 \cdot \dfrac{\mathbb P(Thag)}{\mathbb P(\neg Thag)}$

"Thag win now?" Thag says.

Grek lets go of the rock.  It falls down to the ground.  It does not turn blue.

"Bold prediction not correct," Grek says.  "Thag's prediction virtuous, but not win.  Now Thag lose by 1 : 10 likelihood ratio instead.  Very science, much falsification."

"Grek lure Thag into trap!" yells Thag.

"Look," says Grek, "whole point is to set up science rules so *correct* theories can win.  If wrong theorists lose quickly by trying to be scientifically virtuous, is feature rather than bug.  But if Thag try to be good and loses, we shake hands and everyone still think well of Thag.  Is normative social ideal, anyway."

# Precision

At a somewhat later stage in the development of gravitational theory, the Aristotelian synthesis of Grek and Thag's theories, "Most things have a final destination of being at the center of the Earth, and try to approach that final destination" comes up against Galileo Galilei's "Most unsupported objects accelerate downwards, and each second of passing time the object gains another 9.8 meters per second of downward speed; don't ask me why, I'm just observing it."

"You're not just predicting that rocks are observed to move downward when dropped, are you?" says Aristotle.  "Because I'm already predicting that."

"What we're going to do next," says Galileo, "is predict *how long* it will take a bowling ball to fall from the Leaning Tower of Pisa."  Galileo takes out a pocket stopwatch.  "When my friend lets go of the ball, you hit the 'start' button, and as soon as the ball hits the ground, you hit the 'stop' button.  We're going to observe exactly what number appears on the watch."

After some further calibration to determine that Aristotle has a pretty consistent reaction time for pressing the stopwatch button if Galileo snaps his fingers, Aristotle looks up at the bowling ball being held from the Leaning Tower of Pisa.

"I think it'll take anywhere between 0 and 5 seconds inclusive," Aristotle says.  "Not sure beyond that."

"Okay," says Galileo.  "I measured this tower to be 45 meters tall.  Now, if air resistance is 0, after 3 seconds the ball should be moving downward at a speed of 9.8 * 3 = 29.4 meters per second.  That speed increases continuously over the 3 seconds, so the ball's *average* speed will have been 29.4 / 2 = 14.7 meters per second.  And if the ball moves at an average speed of 14.7 meters per second, for 3 seconds, it will travel downward 44.1 meters.  So the ball should take just a little more than 3 seconds to fall 45 meters.  Like, an additional 1/29th of a second or so."

![tower cartoon](https://i.imgur.com/kUeMG6w.png?0)

"Hm," says Aristotle.  "This pocketwatch only measures whole seconds, so your theory puts all its probability mass on 3, right?"

"Not literally *all* its probability mass," Galileo says.  "It takes you some time to press the stopwatch button once you see the ball start to fall, but it also takes you some time to press the button after you see the ball hit the ground.  Those two sources of measurement error should *mostly* cancel out, but maybe they'll happen not to on this particular occasion.  We don't have all *that* precise or well-tested an experimental setup here.  Like, if the stopwatch breaks and we observe a 0, then that will be a *defeat* for Galilean gravity, but it wouldn't imply a *final* refutation - we could get another watch and make better predictions and make up for the defeat."

"Okay, so what probabilities do you assign?" says Aristotle.  "I think my own theory is about equally good at explaining any falling time between 0 and 5 seconds."

![aristotle probability](https://i.imgur.com/LQNDTki.png?0)

Galileo ponders.  "Since we haven't tested this setup yet... I think I'll put something like 90% of my probability mass on a falling time between 3 and 4 seconds, which corresponds to an observable result of the watch showing '3'.  Maybe I'll put another 5% probability on air resistance having a bigger effect than I think it should over this distance, so between 4 and 5 seconds or an observable of '4'.  Another 4% probability on this watch being slower than I thought, so 4% for a *measured* time between 2 and 3 and an observation of '2'.  0.99% probability on the stopwatch picking this time to break and show '1' (or '0', but that we both agree shouldn't happen), and 0.01% probability on an observation of '2' which basically shouldn't happen for any reason I can think of."

![galileo probability](https://i.imgur.com/4CoLkuW.png?0)

"Well," says Aristotle, "your theory certainly has the scientific virtue of *precision,* in that, by concentrating most of its probability density on a narrow region of possible precise observations, it will gain a great likelihood advantage over *vaguer* theories like mine, which roughly say that 'things fall down' and have made 'successful predictions' each time things fall down, but which don't predict exactly how long they should take to fall.  If your prediction of '3' comes true, that'll be a 0.9 : 0.2 or 4.5 : 1 likelihood ratio favoring Galilean over Aristotelian gravity."

"Yes," says Galileo.  "Of course, it's not enough for the prediction to be precise, it also has to be correct.  If the watch shows '4' instead, that'll be a likelihood ratio of 0.05 : 0.20 or 1 : 4 against my theory.  It's better to be vague and right than to be precise and wrong."

Aristotle nods.  "Well, let's test it, then."

The bowling ball is dropped.

The stopwatch shows 3 seconds.

"So do you believe my theory yet?" says Galileo.

"Well, I believe it somewhere in the range of four and a half times as much as I did previously," says Aristotle.  "But that part where you're plugging in numbers like 9.8 and calculations like the *square* of the time strike me as kinda complicated.  Like, if I'm allowed to plug in numbers that precise, and do things like square them, there must be hundreds of different theories I could make which would be that complicated.  By the [quantitative form of Occam's Razor](https://arbital.com/p/11w), we need to [penalize the prior probability](https://arbital.com/p/) of your theory for its [algorithmic complexity](https://arbital.com/p/5v).  One observation with a likelihood ratio of 4.5 : 1 isn't enough to support all that complexity.  I'm not going to believe something that complicated because I see a stopwatch showing '3' just that one time!  I need to see more objects dropped from various different heights and verify that the times are what you say they should be.  If I say the prior complexity of your theory is, say, [20 bits](https://arbital.com/p/1zh), then 9 more observations like this would do it.  Of course, I expect you've already made more observations than that in private, but it only becomes part of the public knowledge of humankind after someone replicates it."

"But of course," says Galileo.  "I'd like to check your experimental setup and especially your calculations the first few times you try it, to make sure you're not measuring in feet instead of meters, or forgetting to halve the final speed to get the average speed, and so on.  It's a formal theory, but in practice I want to check to make sure you're not making a mistake in calculating it."

"Naturally," says Aristotle.  "Wow, it sure is a good thing that we're both Bayesians and we both know the governing laws of probability theory and how they motivate the informal social procedures we're following, huh?""

"Yes indeed," says Galileo.  "Otherwise we might have gotten into a heated argument that could have lasted for hours."

# Falsificationism

One of the reasons why Karl Popper was so enamored of "falsification" was the observation that falsification, in science, is more definite and final than confirmation.  A classic parable along these lines is Newtonian gravitation versus General Relativity (Einsteinian gravitation) - despite the tons and tons of experimental evidence for Newton's theory that had accumulated up to the 19th century, there was no sense in which Newtonian gravity had been *finally* verified, and in the end it was finally discarded in favor of Einsteinian gravity.  Now that Newton's gravity has been tossed on the trash-heap, though, there's no realistic probability of it *ever* coming back; the discard, unlike the adoption, is final.

Working in the days before Bayes became widely known, Popper put a *logical* interpretation on this setup.  Suppose $H \rightarrow E,$ hypothesis H logically implies that evidence E will be observed.  If instead we observe $\neg E$ we can conclude $\neg H$ by the law of the contrapositive.  On the other hand, if we observe $E,$ we can't logically conclude $H.$  So we can logically falsify a theory, but not logically verify it.

Pragmatically, this often isn't how science works.

In the nineteenth century, observed anomalies were accumulating in the observation of Uranus's orbit.  After taking into account all known influences from all other planets, Uranus still was not *exactly* where Newton's theory said it should be.  On the logical-falsification view, since Newtonian gravitation said that Uranus ought to be in a certain precise place and Uranus was not there, we ought to have become [infinitely certain](https://arbital.com/p/) that Newton's theory was false.  Several theorists did suggest that Newton's theory might have a small error term, and so be false in its original form.

The actual outcome was that Urbain Le Verrier and John Couch Adams independently suspected that the anomaly in Uranus's orbit could be accounted for by a previously unobserved eighth planet.  And, rather than *vaguely* say that this was their hypothesis, in a way that would just spread around the probability mass for Uranus's location and cause Newtonian mechanics to be not *too* falsified, Verrier and Adams independently went on to calculate where the eighth planet ought to be.  In 1846, Johann Galle observed Neptune, based on Le Verrier's observations - a tremendous triumph for Newtonian mechanics.

In 1859, Urbain Le Verrier recognized another problem: Mercury was not exactly where it should be.  While Newtonian gravity did predict that Mercury's orbit should precess (the point of closest approach to the Sun should itself slowly rotate around the Sun), Mercury was precessing by 38 arc-seconds per century more than it ought to be (later revised to 43).  This anomaly was harder to explain; Le Verrier thought there was [a tiny planetoid orbiting the Sun inside the orbit of Mercury](https://en.wikipedia.org/wiki/Vulcan_) (hypothetical_planet)).

A bit more than half a century later, Einstein, working on the equations for General Relativity, realized that Mercury's anomalous precession was exactly explained by the equations in their simplest and most elegant form.

And that was the end of Newtonian gravitation, permanently.

If we try to take Popper's logical view of things, there's no obvious difference between the anomaly with Uranus and the anomaly with Mercury.  In both cases, the straightforward Newtonian prediction seemed to be falsified.  If Newtonian gravitation could bounce back from one logical disconfirmation, why not the other?

From a Bayesian standpoint, we can see the difference as follows:

In the case of Uranus, there was no attractive alternative to Newtonian mechanics that was making better predictions.  The current theory seemed to be [strictly confused](https://arbital.com/p/227) about Uranus, in the sense that the current Newtonian model was making confident predictions about Uranus that were much wronger than the theory expected to be on average.  This meant that there ought to be *some* better alternative.  It didn't say that the alternative had to be a non-Newtonian one.  The low $\mathbb P(UranusLocation\mid currentNewton)$ created a potential for some modification of the current model to make a better prediction with higher $\mathbb P(UranusLocation\mid newModel)$, but it didn't say what had to change in the new model.

Even after Neptune was observed, though, this wasn't a *final* confirmation of Newtonian mechanics.  While the new model assigned very high $\mathbb P(UranusLocation\mid Neptune \wedge Newton),$ there could, for all anyone knew, be some unknown Other theory that would assign equally high $\mathbb P(UranusLocation\mid Neptune \wedge Other).$  In this case, Newton's theory would have no likelihood advantage versus this unknown Other, so we could not say that Newton's theory of gravity had been confirmed over *every other possible* theory.

In the case of Mercury, when Einstein's formal theory came along and assigned much higher $\mathbb P(MercuryLocation\mid Einstein)$ compared to $\mathbb P(MercuryLocation\mid Newton),$ this created a huge likelihood ratio for Einstein over Newton and drove the probability of Newton's theory very low.  Even if someday some other theory turns out to be better than Einstein, to do equally well at $\mathbb P(MercuryLocation\mid Other)$ and also get even better $\mathbb P(newObservation\mid Other),$ the fact that Einstein's theory did do much better than Newton on Mercury tells us that it's *possible* for simple theories to do much better on Mercury, in a simple way, that's definitely not Newtonian.  So whatever Other theory comes along will also do better on Mercury than $\mathbb P(MercuryLocation\mid Newton)$ in a non-Newtonian fashion, and Newton will just be at a new, huge likelihood disadvantage against this Other theory.

So - from a Bayesian standpoint - after explaining Mercury's orbital precession, we can't be sure Einstein's gravitation is correct, but we *can* be sure that Newton's gravitation is wrong.

But this doesn't reflect a logical difference between falsification and verification - everything takes place inside a world of probabilities.

## Possibility of permanent confirmation

It's worth noting that although Newton's theory of gravitation was false, something *very much like it* was true.  So while the belief "Planets move *exactly* like Newton says" could only be provisionally accepted and was eventually overturned, the belief, "All the kind of planets we've seen so far, in the kind of situations we've seen so far, move *pretty much* like Newtonian gravity says" was much more strongly confirmed.

This implies that, contra Popper's rejection of the very notion of confirmation, *some* theories can be finally confirmed, beyond all reasonable doubt.  E.g., the DNA theory of biological reproduction.  No matter what we wonder about quarks, there's no plausible way we could be wrong about the existence of molecules, or about there being a double helix molecule that encodes genetic information.  It's reasonable to say that the theory of DNA has been forever confirmed beyond a reasonable doubt, and will never go on the trash-heap of science no matter what new observations may come.

This is possible because DNA is a non-fundamental theory, given in terms like "molecules" and "atoms" rather than quarks.  Even if quarks aren't exactly what we think, there will be something enough *like* quarks to underlie the objects we call protons and neutrons and the existence of atoms and molecules above that, which means the objects we call DNA will still be there in the new theory.  In other words, the biological theory of DNA has a "something *sort of like this* must be true" theory *underneath* it.  The hypothesis that what Joseph Black called 'fixed air' and we call 'carbon dioxide', is in fact made up of one carbon atom and two oxygen atoms, has been permanently confirmed in a way that Newtonian gravity was not permanently confirmed.

There's some amount of observation which would convince us that all science was a lie and there were fairies in the garden, but short of that, carbon dioxide is here to stay.

Nonetheless, in ordinary science when we're trying to figure out controversies, working to Bayes' rule implies that a virtuous scientist should think like Karl Popper suggested:

- Treat disconfirmation as stronger than confirmation;
- Only provisionally accept hypotheses that have a lot of favorable-seeming evidence;
- Have some amount of disconfirming evidence and prediction-failures that makes you permanently put a hypothesis on the trash-heap and give up hope on its resurrection;
- Require a qualitatively *more* powerful kind of evidence than that, with direct observation of the phenomenon's parts and processes in detail, before you start thinking of a theory as 'confirmed'.

# Experimentation

Whatever the likelihood for $\mathbb P(observation\mid hypothesis)$, it doesn't change your beliefs unless you actually execute the experiment, learn whether $observation$ or $\neg observation$ is true, and [condition your beliefs in order to update your probabilities](https://arbital.com/p/1y6).

In this sense, Bayes' rule can also be said to motivate the experimental method.  Though you don't necessarily need a lot of math to realize that drawing an accurate map of a city requires looking at the city.  Still, since the experimental method wasn't quite obvious for a lot of human history, it could maybe use all the support it can get - including the central Bayesian idea of "Make observations to update your beliefs."

Eliezer Yudkowsky
Alexei Andreev
Eric Rogstad
Eric Bruylant
Nate Soares
Dewi Morgan
Patrick LaVictoire
Otto Mossberg
Adom Hartell

Summary: The classic scientific virtues of falsifiability (saying what we should *not* expect to see if a hypothesis is true), making bold experimental predictions (that aren't predicted by other theories), and precision (making narrow predictions about quantitative measurements) can be seen in the light of [Bayes' rule](https://arbital.com/p/1lz) as properties which allow theories to gain stronger evidence (greater [likelihood ratios](https://arbital.com/p/1rq)) in their favor when their predictions are correct.  We can also interpret traditional ideas like falsificationism - acceptance being provisional, and rejection being more forceful - as features that arise from the probability theory of Bayes' rule.


Strictly confused

A hypothesis is "strictly confused" by the data if the hypothesis does much worse at predicting the data than it expected to do.  If, on average, you expect to assign around 1% likelihood to the exact observation you see, and you actually see something to which you assigned 0.000001% likelihood, you are strictly confused.

%%knows-requisite([https://arbital.com/p/1r6](https://arbital.com/p/1r6)):
I.e., letting $H$ be a hypothesis and $e_0$ be the data observed from some set $E$ of possible observations, we say that $H$ is "strictly confused" when

$$ \log \mathbb P(e_0 \mid H) \ll \sum_{e \in E} \mathbb P(e \mid H) \cdot \log \mathbb P(e \mid H)$$

%%

# Motivation and examples

In Bayesian reasoning, the main reason to reject a hypothesis is when we find a better hypothesis.  Suppose we think a coin is fair, and we flip it 100 times, and we see that the coin comes up "HHHHHHH..." or all heads.  After doing this 100 times, the hypothesis "This is a double-headed coin" has a likelihood ratio of $2^{100} : 1$ favoring it over the "fair coin" hypothesis, and the "double-headed coin" hypothesis isn't *more* improbable than $2^{-100}$ a priori.

But this relies on the insight that there's a simple / a priori plausible *alternative* hypothesis that does better.  What if the coin is producing TTHHTTHHTTHH and we just never happen to think of 'alternating pairs of tails and heads' as a hypothesis?  It's possible to do better by thinking of a better hypothesis, but so far as the 'fair coin' hypothesis sees the world, TTHHTTHH... is no more or less likely than any other possible sequence it could encounter; the first eight coinflips have a probability of $2^{-8}$ and this would have been true no matter which eight coinflips were observed.  After observing 100 coinflips, the fair coin hypothesis will assign them a collective probability of $2^{-100},$ and in this sense, no sequence of 100 coinflips is any more 'surprising' or 'confusing' than any other from *within* the perspective of the fair coin hypothesis.

We can't say that we're 'confused' or 'surprised' on seeing a long sequence of coinflips to which we assigned some very low probability on the order of $2^{-100} \approx 10^{-30},$ because we expected to assign a probability that low.

On the other hand, suppose we think that a coin is biased to produce 90% heads and 10% tails, and we flip it 100 times and get some fair-looking sequence like  "THHTTTHTTTTHTHTHHH..." (courtesy of random.org).  Then we *expected* to assign the observed sequence a probability in the range of $0.9^{90} \cdot 0.1^{10} \approx 7\cdot 10^{-15},$ but we *actually* saw a sequence we assigned probability around $0.9^{50} \cdot 0.1^{50} \approx 5 \cdot 10^{-53}.$  We don't need to consider any other hypotheses to realize that we are very confused.  We don't need to have *invented* the concept of a 'fair coin', or know that the 'fair coin' hypothesis would have assigned a much higher likelihood in the region of $7 \cdot 10^{-31},$ to realize that there's something wrong with the current hypothesis.

In the case of the supposed fair coin that produces HHHHHHH, we only do poorly relative to a better hypothesis 'all heads' that makes a superior prediction.  In the case of the supposed 90%-heads coin that produces a random-looking sequence, we do poorly than we expected to do from inside the 90%-heads hypothesis, so we are doing poorly in an absolute, non-relative sense.

Being strictly confused is a sign that tells us to look for *some* alternative hypothesis in advance of our having any idea whatsoever what that alternative hypothesis might be.

# Distinction from frequentist p-values

The classical frequentist test for rejecting the null hypothesis involves considering the probability assigned to particular 'obvious'-seeming partitions of the data, and asking if we ended up inside a low-probability partition.

Suppose you think some coin is fair, and you flip the coin 100 times and see a random-looking sequence "THHTTTHTT..."

Someone comes along and says, "You know, this result is very surprising, given your 'fair coin' theory.  You really didn't expect that to happen."

"How so?" you reply.

They say, "Well, among all sequences of 1000 coins, only 1 in 16 such sequences start with a string like THHT TTHTT, a palindromic quartet followed by a palindromic quintet.  You confidently predicted that had a 15/16 chance of *not* happening, and then you were surprised."

"Okay, look," you reply, "if you'd written down that *particular* prediction in advance and not a lot of others, I might be interested.  Like, if I'd already thought that way of partitioning the data — namely, 'palindrome quartet followed by palindrome quintet' vs. '*not* palindrome quartet followed by palindrome quintet' — was a specially interesting and distinguished one, I might notice that I'd assigned the second partition 15/16 probability and then it failed to actually happen.  As it is, it seems like you're really reaching."

We can think of the frequentist tests for rejecting the fair-coin hypothesis as a *small* set of 'interesting partitions' that were written down in advance, which are supposed to have low probability given the fair coin.  For example, if a coin produces HHHHH HTHHH HHTHH, a frequentist says, "*Partitioning by number of heads*, the fair coin hypothesis says that on 15 flips we should get between 12 and 3 heads, inclusive, with a probability of 98.6%.  You are therefore surprised because this event you assigned 98.6% probability failed to happen.  And yes, we're just checking the number of heads and a few other obvious things, not for palindromic quartets followed by palindromic quintets."

Part of the point of being a Bayesian, however, is that we try to only reason on the data we actually observed, and not put that data into particular partitions and reason about those partitions.  The partitioning process introduces potential subjectivity, especially in an academic setting fraught with powerful incentives to produce 'statistically significant' data - the equivalent of somebody insisting that palindromic quartets and quintets are special, or that counting heads isn't special.

E.g., if we flip a coin six times and get HHHHHT, this is "statistically significant p < 0.05" if the researcher decided to flip coins until they got at least one T and then stop, in which case a fair coin has only a 1/32 probability of requiring six or more steps to produce a T.  If on the other hand the researcher decided to flip the coin six times and then count the number of tails, the probability of getting 1 or fewer T in six flips is 7/64 which is not 'statistically significant'.

The Bayesian says, "If I use [the Rule of Succession](https://arbital.com/p/21c) to denote the hypothesis that the coin has an unknown bias between 0 and 1, then the sequence HHHHHT is assigned 1/30 probability by the Rule of Succession  and 1/64 probability by 'fair coin', so this is evidence with a likelihood ratio of ~ 2 : 1 favoring the hypothesis that the coin is biased - not enough to [overcome](https://arbital.com/p/) any significant [prior improbability](https://arbital.com/p/)."

The Bayesian arrives at this judgment by only considering the particular, exact data that was observed, and not any larger partitions of data.  To compute the probability flow between two hypotheses $H_1$ and $H_2$ we only need to know the likelihoods of our *exact* observation given those two hypotheses, not the likelihoods the hypotheses assign to any partitions into which that observation can be put, etcetera.

Similarly, the Bayesian looks at the sequence HHHHH HTHHH HHTHH and says: this specific, exact data that we observed gives us a likelihood ratio of (1/1680 : 1/32768) ~ (19.5 : 1) favoring "[The coin has an unknown bias](https://arbital.com/p/21c) between 0 and 1" over "The coin is fair".  With that already said, the Bayesian doesn't see any need to talk about the total probability of the fair coin hypothesis producing data inside a partition of similar results that could have been observed but weren't.

But even though Bayesians usually try avoid thinking in terms of rejecting a null hypothesis using partitions, saying "I'm strictly confused!" gives a Bayesian a way of saying "Well, I know *something's* wrong..." that doesn't require already having the insight to propose a better alternative, or even the insight to realize that some particular partitioning of the data is worth special attention.

Roland G
Eliezer Yudkowsky
Alexei Andreev
Eric Bruylant
Nate Soares


Diseasitis

A nurse is screening a student population for a certain illness, Diseasitis (lit. "inflammation of the disease").

- Based on prior epidemiological studies, you expect that around 20% of the students in a screening population like this one will actually have Diseasitis.

You are testing for the presence of the disease using a color-changing tongue depressor with a sensitive chemical strip.

- Among students with Diseasitis, 90% turn the tongue depressor black.
- 30% of the students without Diseasitis will also turn the tongue depressor black.

One of your students comes into the office, takes your test, and turns the tongue depressor black.

Given only that information, what is the probability that they have Diseasitis?

This problem is used as a central example in several introductions to Bayes's Rule, including all paths in the [Arbital Guide to Bayes' Rule](https://arbital.com/p/1zq) and the [High-Speed Intro to Bayes' Rule](https://arbital.com/p/693).  A simple, unnecessarily difficult calculation of the answer can be found in [https://arbital.com/p/55z](https://arbital.com/p/55z).

Eric Bruylant
Eliezer Yudkowsky
Nate Soares
Alexei Andreev

Summary:
A nurse is screening a set of patients for an illness, Diseasitis, using a tongue depressor that tends to turn black in the presence of the disease.

- On average, around 20% of the patients in a population like this one will actually have Diseasitis.  (20% prior prevalence.)
- Among the patients with Diseasitis, 90% turn the tongue depressor black.  (90% true positives.)
- 30% of patients without Diseasitis also turn the tongue depressor black.  (30% false positives.)

The probability that a patient with a blackened tongue depressor has Diseasitis can be found using [Bayes' rule](https://arbital.com/p/693).


Just a requisite

A tag which indicates that a page is mainly meant to serve as an anonymous requisite inside Arbital's requisite system.


Introductory Bayesian problems

Introductory problems:

1. [https://arbital.com/p/22s](https://arbital.com/p/22s)
2. [https://arbital.com/p/55b](https://arbital.com/p/55b)
3. [https://arbital.com/p/559](https://arbital.com/p/559)
4. [https://arbital.com/p/54v](https://arbital.com/p/54v)

(Hover or click to read them.)

Eliezer Yudkowsky
Alexei Andreev
Eric Bruylant
Nate Soares
Vojta Kovarik


Strength of Bayesian evidence

From a Bayesian standpoint, the strength of evidence can be identified with its [relative likelihood](https://arbital.com/p/1rq).  See the [log odds form of Bayes' rule](https://arbital.com/p/1zh) for a representation which makes this unusually clear.


Path: Insights from Bayesian updating

You've finished reading our current collection of Bayesian insights!  There aren't many here yet, but there might be more later.  Please continue on to the next part of this path!

Eric Bruylant
Nate Soares
Eliezer Yudkowsky

Summary: A small page that collects the currently explained list of Bayesian insights as requisites, to help Arbital auto-generate a learning path that includes all those requisites.


Finishing your Bayesian path on Arbital

You've now reached the end of whatever you asked the [Bayes' rule guide](https://arbital.com/p/1zq) to explain to you!

If you'd like to join in and help create more math content like this, we'd [love to have you](https://arbital.com/p/4d6)!

%%!knows-requisite([https://arbital.com/p/25y](https://arbital.com/p/25y)): If you'd like to keep reading about Bayesian reasoning here on Arbital, there's a couple of less polished pages on real-world insights derived from Bayes' rule — [click here to start on a path to learn them](https://arbital.com/p/https://arbital.com/learn/?path=bayes_update_details,bayes_guide_end).%%

You might also be interested in reading [this explanation of Arbital's goals.](https://arbital.com/p/58p)

Thank you!

Eric Bruylant
Nate Soares
Alexei Andreev
Eliezer Yudkowsky


Extraordinary claims

# Summary

What makes something count as an 'extraordinary claim' for the purposes of determining whether it [requires extraordinary evidence](https://arbital.com/p/21v)?

Broadly speaking:

- Not-yet-supported complexity or precise details; by [Occam's Razor](https://arbital.com/p/occams_razor), this requires added support from the evidence.
- Violation of (deep, causal) generalizations; behaving in a way that's out of character for (the lowest level of) the universe.

Some properties that do *not* make a claim inherently 'extraordinary' in the above sense:

- Having future consequences that are important, or extreme-sounding, or that would imply we need to take costly policies.
- Whether few or many people already believe it.
- Whether there might be bad reasons to believe it.

The ultimate grounding for the notion of 'extraordinary claim' would come from [Solomonoff induction](https://arbital.com/p/11w), or some generalization of Solomonoff induction to handle more [https://arbital.com/p/-naturalistic](https://arbital.com/p/-naturalistic) reasoning about the world.  Since this page is a [Work In Progress](https://arbital.com/p/4v), it currently only lists out the derived heuristics, rather than trying to explain in any great detail how those heuristics might follow from Solomonoff-style reasoning.

# Example 1: Anthropogenic global warming

Consider the idea of [anthropogenic global warming](https://en.wikipedia.org/wiki/Global_warming) as it might have been analyzed in *advance* of observing the actual temperature record.  Would the claim, "Adding lots of carbon dioxide to the atmosphere will result in increased global temperatures and climate change", be an 'extraordinary' claim requiring extraordinary evidence to verify, or an ordinary claim not requiring particularly strong evidence?  We assume in this case you can do all the physics reasoning or ecological reasoning you want, but you can't actually look at the temperature record yet.

The core argument (in *advance* of looking at the temperature record) will be:  "Carbon dioxide is a greenhouse gas, so adding sufficient amounts of it to the atmosphere ought to trap more heat, which ought to raise the equilibrium temperature of the Earth."

To evaluate the ordinariness or extraordinariness of this claim:

We *don't* ask whether the future consequences of the claim are extreme or important.  Suppose that adding carbon dioxide actually did trap more heat; would the Standard Model of physics think to itself, "Uh oh, that has some extreme consequences" and decide to let the heat radiate away anyway?  Obviously not; the laws of physics have no privileged tendency to avoid consequences that are, on a human scale, very important in a positive or negative direction.

We *don't* ask whether the policies required to address the claim are very costly - this isn't something that would prevent the causal mechanisms behind the claim from operating, and more generally, reality doesn't try to avoid inconveniencing us, so it doesn't affect the prior probability we assign to a claim in advance of seeing any evidence.

We *don't* ask whether someone has a motive to lie to us about the claim, or if they might be inclined to believe it for crazy reasons.  If someone has a motive to lie to us about the evidence, this affects the strength of evidence, rather than lowering the [prior probability](https://arbital.com/p/1rm).  Suppose somebody said, "Hey, I own an apartment in New York, and I'll rent it to you for $2000/month."  They might be lying and trying to trick you out of the money, but this doesn't mean "I own an apartment in New York" is an extraordinary claim.  Lots of people own apartments in New York.  It happens all the time, even.  The monetary stake means that the person might have a motive to lie to you, but this affects the likelihood ratio, not the prior odds.  If we're just considering their unsupported word, the probability that they'll say "I own an apartment in New York", given that they *don't* own an apartment in New York, might be unusually high because they could be trying to run a rent scam.  But this doesn't mean we have to call in physicists to check out whether the apartment is really there - we just need stronger, but ordinary, evidence.  Similarly, even if there was someone tempted to lie about global warming, we'd consider this as a potential weakness of the evidence they offer us, but not a weakness in the prior probability of the proposition "Adding carbon dioxide to the atmosphere heats it up."

(Similarly, wanting strong evidence about a subject doesn't always coincide with the underlying claim being improbable.  Maybe you're considering *buying* a house in San Francisco, and *millions* of dollars are at stake.  This implies a high [value of information](https://arbital.com/p/value_of_information) and you might want to invest in extra-strong evidence like having a third party check the title to the house.  But this isn't because it's a Lovecraftian anomaly for anyone to own a house in San Francisco.  The money at stake just means that we're willing to pay more to eliminate small residues of improbability from this very ordinary claim.)

We *do* ask whether "adding carbon dioxide warms the atmosphere" or "carbon dioxide doesn't warm the atmosphere" seems more consonant with the previously observed behavior of carbon dioxide.

After we finish figuring out how carbon dioxide molecules and infrared photons usually behave, we *don't* give priority to generalizations like, "For as long as we've observed it, the average summer temperature in Freedonia has never gone over 30C."  It's true that the predicted consequences of carbon dioxide behaving like it usually does, are violating another generalization about how Freedonia usually behaves.  But we generally give priority to *deeper* generalizations continuing, i.e., generalizations that are *lower-level* or *closer to the start of causal chains.*

- The behavior of carbon dioxide is lower-level - it's generalizing over a class of molecules that we can observe in very great detail and make very precise generalizations about.  The weight of a carbon dioxide molecule (with the standard isotopes in both cases), or the amount of infrared light the corresponding gas allows to pass, is something that varies much less than the summer temperature in Freedonia - it's a very precise, very strong generalization.
- The behavior of carbon dioxide is closer to the start of the chain of cause and effect.  The summer temperature in Freedonia is something that's caused by, or happens as a result of, a particular level of carbon dioxide in Freedonia's atmosphere.  We'd expect changes that happened toward the start of the causal chain to produce changes in the effects at the end of the causal chain.  Conversely, it would be very surprising if the Freedonia-surface-temperature generalization can reach back and force carbon dioxide to have a different permeability to infrared.

We *don't* consider whether lots of prestigious scientists believe in global warming.  If you expect that lots of prestigious scientists usually won't believe in a proposition like global warming in worlds where global warming is false, then observing an apparent scientific consensus might be moderately strong *evidence* favoring the claim.  But that isn't part of the *prior probability* before seeing any evidence.  For that, we want to ask about how complicated the claim is, and whether it violates or obeys generalizations we already know about.

Another way of looking at a test of extraordinariness is to ask whether the claim's truth or falsity would imply learning more about the universe that we didn't already know.  If you'd never observed the temperature record, and had only guessed *a priori* that adding carbon dioxide would warm the atmosphere, you wouldn't be too surprised to go look at the temperature record and find that nothing seemed to be happening.  In this case, rather than imagining that you were wrong about the behavior of infrared light, you might suspect, for example, that plants were growing more and absorbing the carbon dioxide, keeping the total atmospheric level in equilibrium.  But in this case you would have learned *a new fact not already known to you (or science)* which explained why global temperatures were not rising.  So to expect that outcome in *advance* would be a more extraordinary claim than to not expect it.  If we can imagine some not-too-implausible ways that a claim could be wrong, but they'd all require us to postulate new facts we don't solidly know, then this doesn't make the original claim 'extraordinary'.  It's still a very ordinary claim that we'd start believing in after seeing an ordinary amount of evidence.

Eric Bruylant
Eliezer Yudkowsky
Alexei Andreev

Summary:
To determine whether something is an 'extraordinary claim', for purposes of deciding whether it [requires extraordinary evidence](https://arbital.com/p/21v), we consider / don't consider these factors:

- Do consider:  Whether the claim requires lots of complexity that isn't already implied by other evidence.
- Don't consider:  Whether the claim has future consequences that are extreme or important-sounding.
- Don't consider:  Whether the claim's truth would imply that we ought to do costly or painful things.
- Do consider:  Whether the claim violates low-level or causal generalizations.
- Don't consider:  Whether the claim, if locally true, would imply downstream effects that violate high-level or surface generalizations.

We don't automatically *believe* claims that pass such tests; we just consider them as ordinary claims which we'd believe on obtaining ordinary evidence.


Prior

Our (potentially rich or complex) state of knowledge and *propensity to learn,* before seeing the evidence, expressed as a [probability function](https://arbital.com/p/1zj).  This is a deeper and more general concept than '[https://arbital.com/p/-1rm](https://arbital.com/p/-1rm)'.  A prior probability is like guessing the chance that it will be cloudy outside, in advance of looking out a window.  The more general notion of a Bayesian prior would include probability distributions that answered the question, "*Suppose* I saw the Sun rising on 999 successive days; would I afterwards think the probability of the Sun rising on the next day was more like 1000/1001, 1/2, or 1 - 10^-6?"  In a sense, a baby can be said to have a 'prior' before it opens its eyes, and then to develop a model of the world by updating on the evidence it sees after that point.  The baby's 'prior' expresses not just its current ignorance, but the different kinds of worlds the baby would end up believing in, depending on what sensory evidence they saw over the rest of their lives.  Key subconcepts include [ignorance priors](https://arbital.com/p/219) and [inductive priors](https://arbital.com/p/21b), and key examples are [Laplace's Rule of Succession](https://arbital.com/p/21c) and [Solomonoff induction](https://arbital.com/p/11w).


Kevin Clancy

summary: Nothing here yet.

Automatically generated page for "Kevin Clancy" user.
If you are the owner, click [here to edit](https://arbital.com/edit/299).


Travis Rivera

summary: Machine Learning and statistics expert.


Uncountability

Sizes of infinity fall into two broad classes: *countable* infinities, and *uncountable* infinities.

A set $S$ is *uncountable* if it is not [countable](https://arbital.com/p/6f8).

Jaime Sevilla Molina
Jason Gross
Alexei Andreev
Eric Bruylant
Malcolm McCrimmon
Patrick Stevens


Uncountability: Intro (Math 1)

[Collections of things](https://arbital.com/p/3jz) which are the same [size](https://arbital.com/p/4w5) as or smaller than the collection of all [natural numbers](https://arbital.com/p/-45h) are called *countable*, while larger collections (like the set of all [real numbers](https://arbital.com/p/-4bc)) are called *uncountable*.

All uncountable collections (and some countable collections) are [infinite](https://arbital.com/p/infinity). There is a meaningful and [well-defined](https://arbital.com/p/5ss) way to compare the sizes of different infinite collections of things %%note: Specifically, mathematical systems which use the [https://arbital.com/p/69b](https://arbital.com/p/69b), see the [technical](https://arbital.com/p/4zp) page for details.%%. To demonstrate this, we'll use a 2d grid.

[https://arbital.com/p/toc:](https://arbital.com/p/toc:)

## Real and Rational numbers

[Real numbers](https://arbital.com/p/4bc) are numbers with a [decimal expansion](https://arbital.com/p/4sl), for example 1, 2, 3.5, $\pi$ = 3.14159265...  Every real number has an infinite decimal expansion, for example 1 = 1.0000000000..., 2 = 2.0000000000..., 3.5 = 3.5000000000...  Recall that the rational numbers are [fractions](https://arbital.com/p/fraction) of [integers](https://arbital.com/p/48l), for example $1 = \frac11$, $\frac32$, $\frac{100}{101}$, $\frac{22}{7}$.  The positive integers are the integers greater than zero (i.e. 1, 2, 3, 4, ..).

There is a [https://arbital.com/p/-theorem](https://arbital.com/p/-theorem) in math that states that the rational numbers are *countable* %%note: You can see the theorem [here](https://arbital.com/p/511).%%, that is, that the set of rational numbers is the same size as the set of positive integers, and another theorem which states that the real numbers are *uncountable*, that is, that the set of real numbers is strictly bigger.  By "same size" and "strictly bigger", we mean that it is possible to match every rational number with some positive integer in a way so that there are no rational numbers, nor positive integers, left unmatched, but that any matching you make between real numbers and positive integers leaves some real numbers not matched with anything.

## Rational grid

If you imagine laying the rational numbers out on a two-dimensional grid, so that the number $p / q$ falls at $(p, q)$, then we may match the positive integers with the rational numbers by walking in a spiral pattern out from zero, skipping over numbers that we have already counted (or that are undefined, such as zero divided by any number).  The beginning of this sequence is $\frac01$, $\frac11$, $\frac12$, $\frac{-1}{2}$, $\frac{-1}{1}$, ...  Graphically, this is:

![A counting of the rational numbers](//i.imgur.com/OS5hr4U.png?1)

This shows that the rational numbers are countable.

## Reals are uncountable

The real numbers, however, cannot be matched with the positive integers.  I show this by [contradiction](https://arbital.com/p/46z). %%note:That is to say, I show that if there is such a matching, then we can conclude nonsensical statements (and if making a new assumption allows us to conclude nonsense, then the assumption itself must be nonsense.%%

Suppose we have such a matching.  We can construct a new real number that differs in its $n^\text{th}$ decimal digit from the real number matched with $n$.

For example, if we were given a matching that matched 1 with 1.8, 2 with 1.26, 3 with 5.758, 4 with 1, and 5 with $\pi$, then our new number could be 0.11111, which differs from 1.8 in the first decimal place (the 0.1 place), 1.26 in the second decimal place (the 0.01 place), and so on.  It is clear that this number cannot be matched with any number under the matching we are given, because, if it were matched with $n$, then it would differ from itself in the $n^\text{th}$ decimal digit, which is nonsense.  Thus, there is no matching between the real numbers and the positive integers.

## See also

If you enjoyed this explanation, consider exploring some of [Arbital's](https://arbital.com/p/3d) other [featured content](https://arbital.com/p/6gg)!

Arbital is made by people like you, if you think you can explain a mathematical concept then consider [https://arbital.com/p/-4d6](https://arbital.com/p/-4d6)!

Eric Bruylant
Jason Gross
Mark Chimes
Patrick Stevens

Summary:
[Collections of things](https://arbital.com/p/3jz) which are the same [size](https://arbital.com/p/4w5) as or smaller than the collection of all [natural numbers](https://arbital.com/p/-45h) are called *countable*, while larger collections (like the set of all [real numbers](https://arbital.com/p/-4bc)) are called *uncountable*.

All uncountable collections (and some countable collections) are [infinite](https://arbital.com/p/infinity). There is a meaningful and [well-defined](https://arbital.com/p/5ss) way to compare the sizes of different infinite collections of things, and some infinite collections are larger than others.


Uncountability: Intuitive Intro

[Collections ](https://arbital.com/p/3jz) which have less than or the same [number of items](https://arbital.com/p/4w5) than the collection of all [natural numbers](https://arbital.com/p/-45h) are called *countable*, while larger collections (like the set of all [real numbers](https://arbital.com/p/-4bc)) are called *uncountable*.

All uncountable collections (and some countable collections) are [infinite](https://arbital.com/p/infinity) and some infinite collections are larger than others %%note: At least, within mathematical systems which include the [https://arbital.com/p/69b](https://arbital.com/p/69b), see the [technical](https://arbital.com/p/4zp) page for details.%%. To demonstrate this, we'll explore a graphical demonstration with tiles and paths.

[https://arbital.com/p/toc:](https://arbital.com/p/toc:)

## Tiles and paths

![A colored sidewalk, the top row red, the bottom row blue](//i.imgur.com/8HDHt19.png)

Consider, as shown above, a sidewalk that goes on forever in one direction, which is made up of equal-sized square tiles.  The sidewalk is two squares across.  Consider a person who walks forever on it, obeying the following rule: Each step the person takes must be to one of the two tiles immediately in front of that person; no going backwards, no skipping tiles, no going sideways, no standing in place forever. The following is the beginning of one possible path:
![A zig-zagging path that begins on blue.](//i.imgur.com/1oO6YeD.png)

Now let's ask two questions:

 1. How many tiles are there?
 2. How many possible paths are there?

In both cases, you could just say that there are infinitely many, and that would be correct. But now let's consider a third question:

 3. Is the number of tiles the same as the number of possible paths?

It turns out that there is a meaningful and [well-defined](https://arbital.com/p/5ss) way to compare the sizes of different infinite [collections of things](https://arbital.com/p/3jz), and some infinite collections are larger than others. In particular, some infinite collections are *countable* (like the [set](https://arbital.com/p/3jz) of all [https://arbital.com/p/-45h](https://arbital.com/p/-45h)s), while others are *uncountable* (like the set of all [https://arbital.com/p/-4bc](https://arbital.com/p/-4bc)s). As we will see, it can be shown that the number of tiles on our infinite sidewalk is countable, but that the number of possible paths one could take, following the rules above, is uncountable. So there are in fact *more* possible paths than there are tiles.

Let's dig into exactly what this means and why it's true.

## Pairing off

We say that two collections of things are the "same size" if you can match the items together completely: you can pair each of the things in the first collection with exactly one of the things in the second collection, in such a way that there is nothing left unpaired. For example, given two sets of three things each, we may pair them.  Here is an example of such a pairing:

![Example pairing showing that 3 = 3: Three things on each side, all matched up: a cow (http://www.faqs.org/photo-dict/phrase/348/cow.html) matched to a racecar (http://www.zcars.com.au/wrc/), an airplane (http://www.penziononyx.cz/) matched to a watermelon (http://www.free-extras.com/tags/1/watermelon.htm),the earth (http://www.treehugger.com/2010/04/18-week/) matched to a computer (http://www.sb.fsu.edu/~xray/Xrf/anaconda.html)](//i.imgur.com/JphY27q.png)

You might think it obvious, then, that the number of paths our person can walk is bigger than the number of tiles.  We can match each tile with the path that starts on a tile the same color as it, and changes to the other color after it hits this tile.  For example, we would match the third red tile with the path
![An example of a path](//i.imgur.com/s7yjGBX.png)

It is important to note, however, that it is not sufficient that we find some matching that leaves things left over.  We must show that *every* matching leaves things left over.  For example, an infinite sidewalk that is one tile across has just as many tiles as an infinite sidewalk that is two tiles across, as we can see from the picture below by matching the 1R on top with the 1R on bottom, the 1B on top with the 1B on bottom, the 2R on top with the 2R on bottom, and so on.

![A two-tile-wide sidewalk, with each column having a B tile and an R tile](//i.imgur.com/4YecTkO.png)
![A one-tile-wide sidewalk, alternating B and R tiles](//i.imgur.com/DDGBX7G.png)

In fact, if we were only to require that *some* matching leave extra tiles, then the number of tiles in a sidewalk that is one tile wide would not be equal to itself, for we could match the first tile with 1B (in the bottom picture above), the second tile with 2B, and so on, and we would leave over half the tiles!

In fact, even if we had a *field* of tiles that is infinite in every direction, we would still have no more tiles than if we had only a sidewalk that is one tile across.  The following matching shows this:

![A one-tile-wide sidewalk, with each tile numbered starting at 1](//i.imgur.com/P4b0CPc.png)
![A field of tiles](//i.imgur.com/glPTsSG.png)

## An unpairable path

You might wonder, given that there are so many different ways to match up infinitely many things, how we can know that there is no matching that catches everything.  I will now prove that, no matter how you try to match paths (ways of walking) and tiles, you will miss some paths.  Since we have already seen that the number of tiles in a sidewalk two tiles wide is the same as the number of tiles in a sidewalk one tile wide, I will show that any matching between paths and tiles in a sidewalk one tile wide misses some paths.  I will do this by creating a path that does not match the path we have chosen for any tile %%note: This type of proof is known as a [https://arbital.com/p/-46z](https://arbital.com/p/-46z).%%.

Suppose we are given a matching between tiles and paths.  Since we have numbered the tiles in a sidewalk one tile wide ($\fbox{1}\fbox{2}\fbox{3}\fbox{4}\fbox{5}\fbox{6}\fbox{7}\fbox{8}\overline{\underline{\vphantom{1234567890}\cdots}}$), we also have a numbering of the paths in our matching.  Consider a new path that differs from the [$n^\text{th}$](https://arbital.com/p/nth) path in our matching on the $n^\text{th}$ tile, that is, the $n^\text{th}$ step that you take.  For example, if our first eight paths are

![8 paths: 1: BRBRBRBR, 2: BBBBBBBB, 3: RRRRRRRR, 4: BRRRRRBB, 5: RBBRBRRB, 6: RBRRBBRR, 7: BRRRRRRB, 8: BRRBBBBB](//i.imgur.com/0Snexa1.png)

then our new path is

![The path RRBBRRBR](//i.imgur.com/s7yjGBX.png)

Clearly, this path is not any of the ones in the matching, because it differs from every single path at some point (in particular, it differs from the $n^\text{th}$ path on the $n^\text{th}$ tile, the $n^\text{th}$ step you take, which is highlighted in yellow).

Because we can repeat this exercise no matter what matching we're given, that means *any* possible matching will always leave out at least one path.  Thus, the number of paths a person can take must be strictly larger than the number of tiles in the sidewalk.

## See also

If you enjoyed this explanation, consider exploring some of [Arbital's](https://arbital.com/p/3d) other [featured content](https://arbital.com/p/6gg)!

Arbital is made by people like you, if you think you can explain a mathematical concept then consider [https://arbital.com/p/-4d6](https://arbital.com/p/-4d6)!

Jason Gross
Eric Rogstad
Eric Bruylant
Malcolm McCrimmon
Joe Zeng


Gödel encoding and self-reference

A Gödel encoding is a way of using the formal machinery of proving theorems in arithmetic to discuss that very machinery. [Kurt Gödel](https://en.wikipedia.org/wiki/Kurt_G%C3%B6del) rocked the world of mathematics in 1931 by showing that this was possible, and that it had surprising consequences for the sort of things that could ever be formally proven in a given system.

The original formulation of this self-referential framework was really messy (involving prime factorizations of huge numbers), but we can think about it more easily now by thinking about computer programs. In order to get mathematical statements that refer to themselves, we can first show that we can talk interchangeably about mathematical proofs and computer programs, and then show how to write computer programs that refer to themselves.

To be specific about the machinery of proving theorems, we'll use the formal system of [Peano Arithmetic](https://arbital.com/p/3ft).

What kind of self-reference do we want for our computer program? We would like to be able to write a program that takes its own source code and performs computations on it. But it doesn't count for the program to access its own location in memory; we'd like something we could write directly into an interpreter, with no other framework, and get the same result.

This kind of program is known as a [quine](https://arbital.com/p/322).

Gödel encoding is equivalent to an encoding of a computer program as a binary string.

If we think about any conjecture in mathematics that can be stated in terms of arithmetic, you can write a program that loops over all possible strings, checks whether any of them is a valid proof of the conjecture, and halts if and only if it finds one. In the other direction, we can imagine proving a theorem about whether a particular computer program ever halts.


Nate Soares

Automatically generated page for "Nate Soares" user. Click [here to edit](https://arbital.com/edit/32).


Quine

A quine is a computer program that 'knows' its own entire source code via indirect self-reference, rather than by getting it as input. The classic exercise in this domain is writing a (non-empty) program that takes no input and which outputs its own entire source code. But the same trick can be used to do more than that; for any program that takes a string as input and performs some operations on it, we can write a quining program that takes no input and performs the same operations on the program's own source code.

The trick to write a quining program is to take a recipe for substituting a string into various places in a program, and then to apply this recipe to itself.  If you've not encountered this idea before, but you know a programming language, it is a nice exercise to try and write a quine which prints its own source code.

An example of a quine in Python (due to [https://arbital.com/p/32](https://arbital.com/p/32)) which prints its own source code:

	template = 'template = {hole}\nprint(template.format(hole=repr(template)))'
	print(template.format(hole=repr(template)))

Wikipedia has a [list of examples of quining programs](https://en.wikipedia.org/wiki/Quine_) (computing)#Examples) in various languages.

Named after the logician [W.V.O. Quine](https://en.wikipedia.org/wiki/Willard_Van_Orman_Quine), as coined by [Douglas Hofstadter](https://en.wikipedia.org/wiki/Douglas_Hofstadter) in [Gödel, Escher, Bach](https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach), in the informal context of English phrases like

> "Yields falsehood when preceded by its quotation" yields falsehood
> when preceded by its quotation.

Quining has been used for practical purposes and even seen in the wild, most famously in [Ken Thompson's illustration of a C compiler Trojan backdoor](http://en.wikipedia.org/wiki/Backdoor_) (computing)#Compiler_backdoors).

Nate Soares
Patrick LaVictoire
Jaime Sevilla Molina


Math playpen

$$\mathcal{P}e^{i \oint_C A_\mu dx^\mu} \to g(x) \mathcal{P}e^{i \oint_C A_\mu dx^\mu} g^{-1}(x)\,$$

Numbers, woo!

Can I bork the page? Trying to bork the page… 3

asdf

Alexei Andreev
Kevin Clancy
Eric Rogstad
Patrick Stevens


Arbital

Arbital is a platform for finding, reading, and creating crowdsourced, intuitive explanations for <s>everything</s> math. Several existing websites ([Wikipedia](https://www.wikipedia.org/), [MathOverflow](http://mathoverflow.net/), and [Wolfram MathWorld](http://mathworld.wolfram.com/) among others) already work well as references, but it's hard to learn from them unless you are already familiar with the subject. Other websites, like [Khan Academy](https://www.khanacademy.org/math) and [Better Explained](http://betterexplained.com/), have some great explanations (at least for K-12 material) but they are limited by the fact that they are closed platforms. Arbital is fundamentally a collaborative platform, which allows everyone to contribute explanations, add examples, and share their expertise.

## Ideal explanation platform

What would the ideal platform for (math) explanations look like? For example, if you wanted to learn Bayes' theorem, what would you want to see behind [this link](https://arbital.com/p/1zq)?

The ideal platform would use each reader's existing knowledge, expertise, and learning preferences to generate a tailored explanation. If they were comfortable with mathematical notation, the ideal platform would show them equations instead of wordy explanations. If they were missing certain prerequisites, the ideal platform would make it easy to catch up on those. If they already knew half the explanation, the ideal platform would skip past it. If one explanation didn't work for them, the ideal platform would offer another.

The ideal platform would be interactive, as if you were studying with a personal tutor. Asking for a reminder about what a [https://arbital.com/p/-3rb](https://arbital.com/p/-3rb) is would be as easy as hovering your mouse over a link. It would be easy to ask questions and instantly get detailed answers. It would be easy to speed up an explanation or slow it down.

The ideal platform would have intuitive, memorable, mind-blowing explanations that reliably produce that magical "click" feeling. The one you get when something finally makes perfect sense.
%%note:
A few examples:
[What if? - Relativistic baseball](http://what-if.xkcd.com/1/)
[Wait Buy Why - Fermi paradox](http://waitbutwhy.com/2014/05/fermi-paradox.html)
[Better Explained - Intuitive trigonometry](http://betterexplained.com/articles/intuitive-trigonometry/)
[Scott Aarson - Who can name the bigger number?](http://www.scottaaronson.com/writings/bignumbers.html)
And videos too:
[Vi Hart - Spirals and Fibonacci numbers](https://www.youtube.com/watch?v=ahXIMUkSXX0)
[US auto industry - How differential steering works](https://www.youtube.com/watch?v=yYAw79386WI)
%%
It would be easy to give feedback to authors to help them incrementally improve each page to make that happen.

That ideal platform doesn't exist yet, but we're building it, and it's called [Arbital](https://arbital.com/p/58p). What you see before you is a beta version, and we're going to continue working hard to improve it and make our vision a reality.

## Enthusiastic community and effective tools

Our top priority is to grow our community of like-minded people who love helping others learn. We are avid learners ourselves, and some of the most interesting things we've learned are the ones we've taught ourselves outside the context of classrooms and courses. We like explaining things and sharing our knowledge with others. If you look at the success of Wikipedia and MathOverflow, it's clear that communities with a shared mission can produce amazing, useful, and lasting resources that benefit the whole world.

We aim to amplify each author's effort by building a new set of tools tailored specifically for reading and collaboratively creating online explanations. We've already implemented a [number of useful features](https://arbital.com/p/14q). We also aim to build an ecosystem of modular pages that will make it easy for authors to reuse existing content and create explanations that can build on each other. We want to empower our authors to write in a way that works best for them, while preserving overall quality.

## Future of Arbital

Currently, we are focused on math explanations. Eventually, we plan to move beyond math to computer science, physics, statistics, economics, health, e-sports, and everything else. One day, you'll be able to use Arbital to get to the very frontier of human knowledge. Then, we'll extend the platform into one that can foster discussions and help all humanity push that frontier forward.

If you want to see that day come sooner, [you can help](https://arbital.com/p/4d6)! You can contribute individual pages, incrementally improve existing pages by proposing edits, and craft entire explanation [paths](https://arbital.com/p/1rt). You can also read, learn, and provide feedback. Each improvement you make helps the entire platform.

Let's work together to build [the best explanation platform this world has ever seen!](https://arbital.com/p/58p)

Anna Salamon
Eliezer Yudkowsky
Alexei Andreev
Eric Rogstad
Alex Montel
Eric Bruylant
Nate Soares
mrkun
Tom Brown

Summary: Arbital is a platform for finding, reading, and creating crowdsourced, intuitive explanations for <s>everything</s> math. It is fundamentally collaborative, which allows everyone to contribute and share their expertise. To see what the platform can do, check out [Arbital's guide to Bayes' rule](https://arbital.com/p/1zq). To help out with the content, take a look at [https://arbital.com/p/4d6](https://arbital.com/p/4d6).


Order theory

Introduction
==========

Order pervades mathematics and the sciences. Often, a reader's intuitive notion of order is all that is necessary for understanding a particular invocation of the notion of order. However, a deeper examination of order reveals a rich taxonomy over the types of orderings that can arise and leads to powerful theorems with applications in mathematics, computer science, the natural sciences, and the social sciences.

The following are examples of orders that one might encounter in life:

- In a work place, employees may have a *rank* that determines which other employees they have authority over. IT workers report to the IT manager, marketers report to the marketing manager, managers report to the CEO, etc.

- In a grocery store, someone planning to buy corn flake cereal may consider the ordering on the *prices* of various brands of shredded wheat to decide which to purchase.

- At lunchtime, someone with a craving for burritos may order the nearby burrito restaurants by their *distance* from her workplace. She may also order these burrito joints by *quality*, and consider both orderings in her decision.

Now that we've seen some concrete examples of order, we can begin working toward a rigorous mathematical definition. In each of the above examples, we have some *underlying set* of objects that we are comparing: employees, corn flake brands, and burrito restaurants, respectively. We also have an *ordering*, which is a [binary relation](https://arbital.com/p/3nt) determining whether or not one object is "less than" another. This suggests that order esentially a pair of a set and some binary relation defined on it. Is this all we need to capture the notion of order?

Here are a few examples of binary relations which may or may not be orderings. Do they agree with your intuitive notion of ordering?

* 1.) The relation which includes all pairs of people $(a,b)$ such that $a$ and $b$ are friends.
* 2.) The relation of pairs of people $(a,b)$ such that $a$ is a genetic descendant of $b$.
* 3.) The relation on days of the week which contains the exactly pairs $(a,b)$ such that a directly precedes b: $\{ (Monday, Tuesday), (Tuesday, Wednesday), ... \}$

It turns out that only item 2 agrees with the mathematical definition of ordering. Intuitively, item 1 is not an ordering because of its symmetry: a friendship between two people does not distinguish one friend as being "less than" the other (not a healthy friendship, at least). Friendship is actually an instance of another important class of binary relations called the [equivalence relations](https://arbital.com/p/equivalence_relation_mathematics). Item 3 is not an ordering because it lacks transitivity: Monday directly precedes Tuesday and Tuesday directly precedes Wednesday, but Monday does not directly precede Wednesday.

Posets
=====

We're now ready to provide a formal, mathematical definition, for a class of objects called *posets* (a contraction of partially ordered set), which captures the idea of ordering.

--

__Definition: Poset__

A *poset* is a pair $\langle P, \leq \rangle$ of a set $P$ and a binary relation $\leq$ on $P$ such that for all $p,q,r \in P$, the following properties are satisfied:

- Reflexivity: $p \leq p$
- Transitivity: $p \leq q$ and  $q \leq r$ implies $p \leq r$
- Anti-Symmetry: $p \leq q$ and $q \leq p$ implies $p = q$

$P$ is referred to as the poset's *underlying set* and $\leq$
 is referred to as its *order relation*.

--

The above definition may strike some readers as more general than expected. Indeed, in both mathematics and conversational English, when someone states that a set of objects is ordered, they often mean that it is totally ordered. A total order is a poset for which any two elements of $a$ and $b$ of the underlying set are *comparable*; that is, $a \leq b$ or $b \leq a$. But our definition of a poset allows the possibility that two elements are *incomparable*. Recall our example of employees in a work place. In our *reports-to* relation, both an IT manager and a marketing manager report to the CEO; however, it would not make sense for an IT manager to report to a marketing manager or vice versa. The marketing and IT managers are thus incomparable. We write $a \parallel b$ to state that two poset elements $a$ and $b$ are incomparable.

Another important distinction must be made. Partial orders as we have described them are not *strict* orders. From any poset $\langle P, \leq \rangle$, we can derive an strict order relation $<$, which includes exactly those pairs of $\leq$ relating two elements of $P$ that are distinct. It should be noted that, while strict orders are transitive and vacuously anti-symmetric, they are *not* partial orders because they are not reflexive. In everyday situations, strict orders seem to be more useful, but in mathematics non-strict orderings are of more use, and so non-strictness is built into the definition of poset.

Summary: The study of binary relations that are reflexive, transitive, and anti-symmetric.


Group theory

Group theory is the study of the [algebraic structures](https://arbital.com/p/3gx) known as "[groups](https://arbital.com/p/3gd)". A group $G$ is a collection of elements $X$ paired with an operator $\bullet$ that combines elements of $X$ while obeying certain laws. Roughly speaking, $\bullet$ treats elements of $X$ as composable, invertible actions.

Group theory has many applications. Historically, groups first appeared in mathematics as groups of "substitutions" of mathematical functions; for example, the group of [integers](https://arbital.com/p/48l) $\mathbb{Z}$ [acts](https://arbital.com/p/3t9) on the set of functions $f : \mathbb{R} \to \mathbb{R}$ via the substitution $n : f(x) \mapsto f(x - n)$, which corresponds to translating the graph of $f$ $n$ units to the right. The functions which are invariant under this group action are precisely the functions which are periodic with period $1$, and group theory can be used to explain how this observation leads to the expansion of such a function as a [Fourier series](https://arbital.com/p/Fourier_series) $f(x) = \sum \left( a_n \cos 2 \pi n x + b_n \sin 2 \pi n x \right)$.

Groups are used as a building block in the formalization of many other mathematical structures, including [fields](https://arbital.com/p/481), [vector spaces](https://arbital.com/p/3w0), and [integers](https://arbital.com/p/48l). Group theory has various [applications to physics](https://arbital.com/p/group_theory_and_physics). For a list of example groups, see the [examples page](https://arbital.com/p/algebraic_group_examples). For a list of the key theorems in group theory, see the [main theorems page](https://arbital.com/p/algebraic_group_theorems).

# Interpretations, visualizations, and uses

Group theory abstracts away from the elements in the [underlying set](https://arbital.com/p/3gz) $X$: the group axioms do not care about what sort of thing is in $X$; they care only about the way that $\bullet$ relates them. $X$ tells us how many elements $G$ has; all other information resides in the choices that $\bullet$ makes to combine two group elements into a third group element. Thus, group theory can be seen as the study of possible relationships between objects that obey the group laws; regardless of what the objects themselves are. To visualize a group, we need only visualize the way that the elements relate to each other via $\bullet$. This is the approach taken by [group multiplication tables](https://arbital.com/p/group_multiplication_table) and [group diagrams](https://arbital.com/p/cayley_diagram).

Group theory is interesting in part because the constraints on $\bullet$ are at a "sweet spot" between "too lax" and "too restrictive." Group structure crops up in many areas of physics and mathematics, but the group axioms are still restrictive enough to make groups fairly easy to work with and reason about. For example, if the [order](https://arbital.com/p/3gg) of $G$ is prime then there is only one possible group that $G$ can be (up to isomorphism). There are only 2, 2, 5, 2, and 2 groups of order 4, 6, 8, 9, and 10 (respectively). There are only 16 groups of order 100. If a group structure can be found in an object, this makes the behavior of the object fairly easy to analyze (especially if the order of the group is small). Group structure is relatively common in math and physics; for example, the solutions to a polynomial equation are acted on by a group called the [Galois group](https://arbital.com/p/Galois_group) (a fact from which the [unsolvability of quintic polynomials by radicals](https://arbital.com/p/unsolvability_of_the_quintic) was proven). Group theory is thus a useful tool for figuring out how various mathematical and physical objects behave. For more on this idea, see the page on [group actions](https://arbital.com/p/3t9).

Roughly speaking, the constraints on $\bullet$ can be interpreted as saying that $\bullet$ has to treat the elements like they are a "complete set of transformations" of a physical object. The axiom of identity then says "one of the elements has to be interpreted as a transformation that does nothing"; the axiom of inversion says "each transformation must be invertible;" and so on. In this light, group theory can be seen as the study of possible complete sets of transformations. Because the set of possible groups is relatively limited and well-behaved, group theory is a useful tool for studying the relationship between transformations of physical objects. For more discussion of this interpretation of group theory, see [https://arbital.com/p/groups_and_transformations](https://arbital.com/p/groups_and_transformations).

Groups can also be seen as a tool for studying symmetry. Roughly speaking, given a set $X$ and some redundant structure atop $X$, a symmetry of $X$ is a function from $X$ to itself that preserves the given structure. For example, a physicist might build a model of two planets interacting in space, where an arbitrary point picked out as the origin. The behavior of the planets in relation to each other shouldn't depend on which point they label $(0, 0, 0)$, but many questions (such as "where is the planet now?") depend on the choice of origin. To study facts that are true regardless of where the origin is (rather than being artifacts of the representation), they might let $X$ be the possible states of the model, with then ask what properties of $X$ are preserved when the origin is shifted — i.e., they ask "what facts about this model would still be true if I had picked a different origin?" Translation of the origin is a "symmetry" of those properties: It takes one state of $X$ to another state of $X$, and preserves all properties of the physical system that are independent of the choice of origin. When a model has multiple symmetries (such as rotation invariance as well as translation invariance), those symmetries form a group under  composition. Group theory tells us about how those symmetries must relate to one another. For more on this topic, see [https://arbital.com/p/groups_and_symmetries](https://arbital.com/p/groups_and_symmetries).

In fact, laws of physics can be interpreted as "properties of the universe that are true at all places and all times, regardless of how you label things." Thus, the laws of physics can be studied by considering the properties of the universe that are preserved under certain transformations (such as changing labels). Group theory puts constraints on how these symmetries in the laws of physics relate to one another, and thus helps constrain the search for possible laws of physics. For more on this topic, see [https://arbital.com/p/groups_and_physics](https://arbital.com/p/groups_and_physics).

Finally, in a different vein, groups can be seen as a building block for other mathematical concepts. As an example, consider numbers. "Number" is a fairly fuzzy concept; the term includes the integers [$\mathbb Z$](https://arbital.com/p/48l), the rational numbers [$\mathbb Q$](https://arbital.com/p/4zq) (which include fractions), the real numbers [$\mathbb R$](https://arbital.com/p/4bc) (which include numbers such as $\sqrt{2}$), and so on. The arrow from $\mathbb Z \to \mathbb Q \to \mathbb R$ points in the direction of _specialization:_ Each step adds constraints (such as "now you must include fractions") and thus narrows the scope of study (there are fewer things in the world that act like real numbers than there are that act like integers). Group theory is the result of following the arrow in the opposite direction: groups are like integers but _more permissive._ Groups have an identity element that corresponds to $0$ and an operation that corresponds to $+$, but they don't need to be infinitely large and they don't need to have anything corresponding to multiplication. From this point of view, "group" is a generalization of "integer", and from that generalization, we can build up many different mathematical objects, including [rings](https://arbital.com/p/3gq), [fields](https://arbital.com/p/481), and [vector spaces](https://arbital.com/p/3w0). Empirically, many interesting and useful mathematical objects use groups as one of their building blocks, and thus, some familiarity with group theory is helpful when learning about other mathematical structures. For more on this topic, see the [tree of algebraic structures](https://arbital.com/p/5dz).

Tsvi BT
Alexei Andreev
Eric Rogstad
Stephanie Zolayvar
Qiaochu Yuan
Nate Soares
Patrick Stevens

Summary: Group theory is the study of the [algebraic structures](https://arbital.com/p/3gx) known as "[groups](https://arbital.com/p/3gd)". A group $G$ is a collection of elements $X$ paired with an operator $\bullet$ that combines elements of $X$ while obeying certain laws. Roughly speaking, $\bullet$ treats elements of $X$ as composable, invertible actions. Group theory has many applications in pure and applied mathematics, as well as in the hard sciences like physics and chemistry. Groups are used as a building block in the formalization of many other mathematical structures, including [vector spaces](https://arbital.com/p/3w0) and [integers](https://arbital.com/p/48l).


Group

summary(technical): A group $G$ is a pair $(X, \bullet)$ where $X$ is a [set](https://arbital.com/p/3jz) and $\bullet$ is a [operation](https://arbital.com/p/3h7) obeying the following laws:

 1. **[Closure](https://arbital.com/p/3gy):** The operation is a function. For all $x, y$ in* $X$, $x \bullet y$ is defined and in $X$. We abbreviate $x \bullet y$ as $xy$.
 2. **[Associativity](https://arbital.com/p/3h4):** $x(yz) = (xy)z$ for all $x, y, z \in X$.
 3. **[Identity](https://arbital.com/p/54p):** There is an element $e$ such that $xe=ex=x$ for all $x \in X$.
 4. **[Inverses](https://arbital.com/p/-inverse_element):** For each $x$ in $X$, there is an element $x^{-1} \in X$ such that $xx^{-1}=x^{-1}x=e$.

The operation need not be [commutative](https://arbital.com/p/-3jb), but if it is then the group is called [abelian](https://arbital.com/p/-3h2).

A group is an abstraction of a collection of symmetries of an object. The collection of symmetries of a triangle (rotating by $120^\circ$ or $240^\circ$ degrees and flipping), rearrangements of a collection of objects (permutations), or rotations of a sphere, are all examples of groups. A group abstracts from these examples by forgetting what the symmetries are symmetries of, and only considers how symmetries behave.

A group $G$ is a pair $(X, \bullet)$ where:

 - $X$ is a [set](https://arbital.com/p/3jz), called the "underlying set." By abuse of notation, $X$ is usually denoted by the same symbol as the group $G$, which we will do for the rest of the article.
 - $\bullet : G \times G \to G$ is a binary [operation](https://arbital.com/p/3h7). That is, a function that takes two elements of a set and returns a third. We will abbreviate $x \bullet y$ by $xy$ when not ambiguous.  This operation is subject to the following axioms:
- **[Closure](https://arbital.com/p/3gy):** $\bullet$ is a function. For all $x, y$ in $X$, $x \bullet y$ is defined and in $X$. We abbreviate $x \bullet y$ as $xy$.
- **[Identity](https://arbital.com/p/-54p):** There is an element $e$ such that $xe=ex=x$ for all $x \in X$.
- **[Inverses](https://arbital.com/p/-inverse_element):** For each $x$ in $X$, there is an element $x^{-1} \in X$ such that $xx^{-1}=x^{-1}x=e$.
- **[Associativity](https://arbital.com/p/3h4):** $x(yz) = (xy)z$ for all $x, y, z \in X$.

1) The set X is the collection of abstract symmetries that this group represents. "Abstract," because these elements aren't necessarily symmetries *of* something, but almost all examples will be.

2) The operation $\bullet$ is the abstract composition operation.

3) The axiom of closure is redundant, since $\bullet$ is defined as a function $G \times G \to G$, but it is useful to emphasize this, as sometimes one can forget to check that a given subsets of symmetries of an object is closed under composition.

4) The axiom of identity says that there is an element $e$ in $G$ that is a do-nothing symmetry: If you apply $\bullet$ to $e$ and $x$, then $\bullet$ simply returns $x$. The identity is unique: Given two elements $e$ and $z$ that satisfy axiom 2, we have $ze = ez = z.$ Thus, we can speak of "the identity" $e$ of $G$. This justifies the use of $e$ in the axiom of inversion: axioms 1 through 3 ensure that $e$ exists and is unique, so we can reference it in axiom 4.

$e$ is often written $1$ or $1_G$, because $\bullet$ is often treated as an analog of multiplication on the set $X$, and $1$ is the multiplicative [identity](https://arbital.com/p/54p). (Sometimes, e.g. in the case of [rings](https://arbital.com/p/3gq), $\bullet$ is treated as an analog of addition, in which case the identity is often written $0$ or $0_G$.)

5) The axiom of inverses says that for every element $x$ in $X$, there is some other element $y$ that $\bullet$ treats like the opposite of $x$, in the sense that $xy = e$ and vice versa. The inverse of $x$ is usually  written $x^{-1}$, or sometimes $(-x)$ in cases where $\bullet$ is analogous to addition.

6) The axiom of associativity says that \bullet behaves like composition of functions. When composing a bunch of functions, it doesn't matter what order the individual compositions are computed in. When composing $f$, $g$, and $h$, we can compute $g \circ f$, and then compute $h \circ (g \circ f)$, or we can compute $h \circ g$ and then compute $(h \circ g) \circ f$, and we will get the same result.

%%%knows-requisite([https://arbital.com/p/3h3](https://arbital.com/p/3h3)):
Equivalently, a group is a [monoid](https://arbital.com/p/3h3) which satisfies "every element has an inverse".
%%%

%%%knows-requisite([https://arbital.com/p/4c7](https://arbital.com/p/4c7)):
Equivalently, a group is a category with exactly one object, which satisfies "every arrow has an inverse"; the arrows are viewed as elements of the group. This justifies the intuition that groups are collections of symmetries. The object of this category can be thought of an abstract object that the isomorphisms are symmetries of. A functor from this category into the category of sets associates this object with a set, and each of the morphisms a permutation of that set.
%%%

# Examples

The most familiar example of a group is perhaps $(\mathbb{Z}, +)$, the integers under addition. To see that it satisfies the group axioms, note that:

1. (a) $\mathbb{Z}$ is a set, and (b) $+$ is a function of type $\mathbb Z \times \mathbb Z \to \mathbb Z$
2. $(x+y)+z=x+(y+z)$
3. $0+x = x = x + 0$
4. Every element $x$ has an inverse $-x$, because $x + (-x) = 0$.

For more examples, see the [examples page](https://arbital.com/p/3t1).

# Notation

Given a group $G = (X, \bullet)$, we say "$X$ forms a group under $\bullet$." $X$ is called the [underlying set](https://arbital.com/p/3gz) of $G$, and $\bullet$ is called the _group operation_.

$x \bullet y$ is usually abbreviated $xy$.

$G$ is generally allowed to substitute for $X$ when discussing the group. For example, we say that the elements $x, y \in X$ are "in $G$," and sometimes write "$x, y \in G$" or talk about the "elements of $G$."

The [order of a group](https://arbital.com/p/3gg), written $|G|$, is the size $|X|$ of its underlying set: If $X$ has nine elements, then $|G|=9$ and we say that $G$ has order nine.

# Resources

Groups are a ubiquitous and useful algebraic structure. Whenever it makes sense to talk about symmetries of a mathematical object, or physical system, groups pop up. For a discussion of group theory and its various applications, refer to the [group theory](https://arbital.com/p/3g8) page.

A group is a [monoid](https://arbital.com/p/3h3) with inverses, and an associative [loop](https://arbital.com/p/algebraic_loop). For more on how groups relate to other [algebraic structures](https://arbital.com/p/3gx), refer to the [tree of algebraic structures](https://arbital.com/p/5dz).

Daniel Satanove
Louis Paquin
Mark Chimes
Qiaochu Yuan
Eric Bruylant
Nate Soares
Eric Rogstad
Dylan Hendrickson
m g
Patrick Stevens
Joe Zeng

Summary:
A group is an abstraction of a collection of symmetries of an object. The collection of symmetries of a triangle (rotating by $120^\circ$ or $240^\circ$ degrees and flipping), rearrangements of a collection of objects (permutations), or rotations of a sphere, are all examples of groups.

A group abstracts from these examples by forgetting what the symmetries are symmetries *of*, and only considers how symmetries behave.

- Any two symmetries can be composed. For the symmetries of flipping and rotating a triangle, the whole action of rotating, and then flipping a triangle at once is a symmetry. So there is an **[operation](https://arbital.com/p/3h7)** on a group which represents composition of symmetries.
- There is always a do-nothing symmetry. When composed with another symmetry, the do-nothing symmetry doesn't change that symmetry. So the operation has an **[identity](https://arbital.com/p/-54p)**.
- Any symmetry can be reversed. That is, for any symmetry, there is another symmetry which when composed with the first, gives the do-nothing symmetry. Flipping a triangle, and then flipping it again, is the same as doing nothing. So the operation has an **[inverse](https://arbital.com/p/-inverse_mathematics)** operation.
- Like any collection of functions, when composing a bunch of functions, it doesn't matter what order the individual compositions are computed in, as long as the *overall* composition ends up in the right order. When composing $f$, $g$, and $h$, we can compute $g \circ f$, and then compute $h \circ (g \circ f)$, or we can compute $h \circ g$ and then compute $(h \circ g) \circ f$, and we will get the same result. So the operation is **[associative](https://arbital.com/p/-3h4)**.

Note that it is not necessarily the case that the operation is [commutative](https://arbital.com/p/-3jb). Flipping and then rotating a triangle will give different symmetry than rotating and then flipping. If it is commutative, then the group is called [abelian](https://arbital.com/p/-3h2).


Order of a group

The order $|G|$ of a [group](https://arbital.com/p/3gd) $G$ is the size of its [underlying set](https://arbital.com/p/3gz). For example, if $G=(X,\bullet)$ and $X$ has nine elements, we say that $G$ has order $9$. If $X$ is infinite, we say $G$ is infinite; if $X$ is finite, we say $G$ is finite.

The [order of an element](https://arbital.com/p/4cq) $g \in G$ of a group is the smallest nonnegative integer $n$ such that $g^n = e$, or $\infty$ if there is no such integer. The relationship between this usage of order and the above usage of order is that the order of $g \in G$ in this sense is the order of the [https://arbital.com/p/subgroup](https://arbital.com/p/subgroup) $\langle g \rangle = \{ 1, g, g^2, \dots \}$ of $G$ [generated by](https://arbital.com/p/generating_set) $g$ in the above sense.

Alexei Andreev
Qiaochu Yuan
Eric Bruylant
Nate Soares
Patrick Stevens