Primer on Infinite Series

What does it mean to add things together forever? While many get introduced to infinite series in Calculus class, it seems that everyone skips ahead to memorizing terminology without first motivating the subject, or considering that it's sorta weird. In reality, the long history of infinite series would suggest that the subject is subtle and strange. But by steaming ahead, memorizing the vocabulary, examples, and rules for convergence, learners miss a chance to really think about what these infinite sums really are or could be. They end up being able to more or less accurately parrot those rules, but if you speak with them a bit or overhear a conversation with others about the topic, you realize their understanding is a hollow shell. This article is a look at infinite series, adding things together forever, from more or less first principles.

Example One: Zeno Crosses the Room
----------------------------------

One of the origin stories for infinite series is attributed to the Greek Philosopher Zeno. It's a good place to start motivating why you end up dealing with infinite series and also a place from which you can begin to appreciate the need for a way of thinking about infinity that doesn't lead to pure nonsense. Although he had many paradoxes, they all boil down to certain weirdnesses that come up when you imagine space or time being infinitely divisible. Things that are obvious and simple from the point of view of everyday experience are contradicted by their analysis from the point of view of infinite divisibility. Consider the following scenario.

You want to cross the room.

That is you wish to cover one room's worth of distance. Easy enough right? You don't even need to stretch. But...(Zeno's voice) to cross the room, first you need to get halfway there. Before you cross the 2nd half of the room, you need to cross half of the 2nd half of the room. And so forth. At each stage you subdivide the remaining distance in half and cover that distance.

Zeno's problem was that this sequence of instants is infinite. If each turn took say 1 second, it would take infinity seconds to cross the room. From this perspective any motion—not just room crossing—appears impossible. Zeno's conclusion was that bringing infinity into the discussion, in this case dividing a given distance into infinitely many pieces, leads to nonsense and should be avoided.

%note:For this and other similar reasons, the official use of infinity in mathematical arguments was postponed until the innovation of Calculus. Essentially, due to the simultaneous creation of mechanics and all the practical uses to which it could be put, the problems with reasoning with infinite sums were swept under the rug for a while. Calculus uses infinity all the time and doesn't apologize for it. Newton, Leibniz, etc. didn't figure out how to solve Zeno's paradox, they just ignored it. It wasn't until a couple centuries later that mathematicians really confronted the problem.%

Zeno's room crossing can be written using numbers and symbols instead of sentences, where it might look a bit more familiar as an "infinite sum".

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots$$

And we can think of its sum either from the statement of the problem (where it is clear that the sum should be 1, a whole room) or by a direct calculation of the distance remaining to cover and noting where its limit is.

<colgroup>
<col style="text-align:left;"/>
<col style="text-align:left;"/>
<col style="text-align:left;"/>
</colgroup>

	Step
	Distance Covered
	Distance Remaining

	0
	0
	1

	1
	\(\frac{1}{2}\)
	\(\frac{1}{2}\)

	2
	\(\frac{1}{2}+\frac{1}{4}\)
	\(\frac{3}{4}\)

	3
	\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\)
	\(\frac{7}{8}\)

	4
	\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}\)
	\(\frac{15}{16}\)

	5
	\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}\)
	\(\frac{31}{32}\)

So, at this point, we have a good reason to want to be able to write

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots = 1$$

Metaphysical or existential doubts notwithstanding, it just looks like we should end up with one if this process of making up half the remaining distance were to continue. In this and many other situations, its clear that adding together infinitely many things should work out to give something finite. The way we would read these symbols in words is

>If you keep adding halves of halves ($\frac{1}{2}$ + \frac{1{4}$ and so on), you get closer and closer to 1. If somehow you could add forever, you'd have exactly one.

Exact and Approximate
---------------------

At this point we should clear up one thing about *exact* versus *approximate*.  You might be thinking that, regardless of the "infinite" part of the sum, this is a clear case of "good enough for all practical purposes". At some point you get close enough to touch the wall at the end of the room. So yes, we can think of the sum as approximating one. And these infinite sums were originally very useful because they gave ways of approximating difficult to calculate things with very easy to calculate things. But there is some subtlety here in the words we use. When we say,

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots = 1$$

We do not mean

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots \approx \text{ (is approximately equal to) }1$$

It *would* be correct to say

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8} \approx 1$$

or also

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32} \approx 1$$

but once we have infinitely many terms, once we are asking what would happen if the process were allowed to continue indefinitely, we are no longer taking about approximately, we are talking about exactly equal.

And this is where people, not just young students but the many mathematical thinkers through history, have had trouble. We are used to the things related by equal signs both being simple numbers. Finite things. Not infinite processes. So what does it mean for an infinite process to be *exactly equal* to a finite number? So far in this discussion we're not yet sure. But we have a clue. We have exactly one case where we have an infinite process that has an obvious limit. We'd like to equate them.

Next, let's look at a case where the infinite process, instead of telling us something we already know, helps us see something new.

Example 2: Archie and the Area of a Circle
------------------------------------------

You may have been asked to remember formulas for the circumference and area of a circle in school.
$$ \text{Circumference }(C) = \pi \cdot \text{Diameter }(D) = 2\pi \cdot \text{Radius }(r)$$

$$ \text{Area of a Circle}(A) = \pi \cdot r^2$$

But did you ever wonder why these formulas are true or why they both happen to have $\pi$ in them? Did you ever wonder how we figured out the area of a round shape like the circle in the first place?

Well, Archimedes did. And the way he figured it out was by creating an infinite sum.

The formula for circumference is easy. It's just a definition. Ancient Greeks and many others noticed that $\frac{C}{D}$ was the same ratio for any circle (*WHY?*). They didn't call this $\pi$ or even consider it a number, but we do. By definition, $\frac{C}{D} = \pi$, so of course $C=\pi D$.

The real mystery is with the area formula. And this is what Archimedes did. He imagined a regular polygon inscribed within a circle. Here's a picture.

![](some pic)

Now, in my picture, there are 8 sides. But there's nothing special about 8. It could be 6, or 12, or a million. But...the more sides there are, the more the polygon looks like a circle, the better approximation its area is for the area of the circle. And, as it turns out, the area of the polygon is easy to compute. Moreover, computing the area of the pylon, we can see how it changes with the number of sides and how it stays the same, *and* we can see what happens to this area if we somehow had infinitely many sides. Let's look:

First, we notice that a polygon can be split into a bunch of triangles, each of which is the same. How many? The same number as we have sides of the polygon. Just like slices of pie. Add up the area of all the slices to get the area of the original polygon.

$$\text{Area of polygon with }n \text{ sides} = \text{sum of } n \text{ triangles}$$

Of course, each triangle has the same area, and any triangle has area $= \frac{1}{2} \cdot \text{base}\cdot \text{height}$. Here, the base is the side of a polygon (let's call it $s$) and the height we can call $h$. Our formula for the area of a polygon becomes:

$$\text{Area of polygon with }n \text{ sides} = \underbrace{\frac{1}{2}sh+\frac{1}{2}sh+\ldots+\frac{1}{2}sh}_{n \text{ times}}$$

Now, factor the $\frac{1}{2}h$ out of the above expression to get something even nicer.

$$\text{Area of polygon with }n \text{ sides} = \frac{1}{2}h(\underbrace{s+s+\ldots+s}_{n \text{ times}})$$

And look, $n$ sides added together is nothing other than the perimeter ($P$) of the polygon.

$$\text{Area of polygon} = \frac{1}{2}hP$$

This formula is true regardless of how many sides we have. All we have to do now to recover the formula for the area of the circle is to imagine what happens to all of the parts when the number of sides, the extent of the approximation, the number of terms in the underlying sum, gets larger.

$$ \text{Area of polygon} \rightarrow \text{Area of a Circle}(A)$$

$$h \rightarrow r$$

$$P \rightarrow C$$

So our formula becomes

$$ A = \frac{1}{2}rC = \frac{1}{2}r(\pi D) = \frac{1}{2}r(\pi(2r))=\pi r^2$$

Amazing, no?

There are *tons* of other situations where you can answer a question by putting together infinitely many pieces, situations where we would like to be able to formally write down something of the sort:

$$\text{infinite sum} = \text{finite number}$$

but there are also obvious and not so obvious situations where doing so is nonsense. We would like to go so far as to extend our rules for algebra to be able to operate on these infinite sums (things like multiplying both sides of an equation by the same number or grouping terms), but the truth is we don't have enough information to know when that does and doesn't work out. Next we will take a look at this more general question.


Lambda calculus

What's the least we can put in a programming language and still have it be able to do anything other [computability languages can] %%note:i.e. [turing_complete]%%? It turns out we can get away with including only a single built in feature: [-3jy] definition, written using $\lambda$. The language that has only this one feature is called "lambda calculus" or "$\lambda$ calculus." We also need a way to apply our functions to inputs; we write $f\ x$ to indicate the value of the function $f$ on $x$, perhaps more familiarly written $f(x)$.

A $\lambda$ expression is something of the form $\lambda x.f(x)$, which means "take in an input $x$, and return $f(x)$. For example, $\lambda x.x+1$ takes in a number and returns one more than it. We only directly have functions with one input, but we can build functions with multiple inputs out of this. For example, applying $\lambda x.\lambda y.x+y$ to $3$, and applying the result to $4$ gives $3+4=7$ . We can think of this as a two-input function, and abbreviate it as $\lambda xy.x+y$; "$\lambda xy$" is simply a shorthand for "$\lambda x.\lambda y$." We don't lose any power by restricting ourselves to unary functions, because we can think of functions with multiple inputs as taking only one input at a time, and returning another function at each intermediate step.

We said the only feature we have is function definition, so what's this about "$+$" and "$1$"? These symbols don't exist in $\lambda$ calculus, so we'll have to define [-54y numbers] and operations on them using $\lambda$ %%note:The standard way of doing this is called the [6qb].%%. But $\lambda$ expressions are useful even when we have other features as well; programming languages such as Python allow function definition using $\lambda$. It's also useful when trying to understand what $\lambda$ means to see examples with functions you're more familiar with.

We're not going to put constraints on what inputs $\lambda$ expressions can take. If we allowed expressions like $\lambda x.x+1$, it would get confused when we try to give it something other than a number; this is why we can't have $+$ and $1$. So what sorts of objects will we give our functions? The only objects have we: other functions, of course!

[toc:]

#Formal definition

Let's formally define $\lambda$ calculus. First, the language of $\lambda$ calculus contains

- Parentheses ( and )
- Symbols $\lambda$ and .
- Variables $v_1,v_2,\dots$ (in practice we'll use other letters too).

What counts as a $\lambda$ expression? We define them recursively:

- For any variable $x$, $x$ is a $\lambda$ expression.
- For any variable $x$ and $\lambda$ expression $M$, $(\lambda x.M)$ is a $\lambda$ expression.
- If $M$ and $N$ are $\lambda$ expressions, so is $(M\ N)$.

We also want to define free and bound variables, and also do this recursively:

- The variable $x$ is free in the expression $x$.
- Every instance of $x$ is bound in the expression $(\lambda x.M)$. If an instance of $x$ is free in $M$, we say the initial $\lambda x$ in $(\lambda x.M)$ *binds* it. We say that $M$ is the *scope* of that $\lambda x$.
- An instance of $x$ in $(M\ N)$ is free if it is free in $M$ or $N$, and bound otherwise.

We can use the rules to build expressions like $((\lambda x.(\lambda y.(x\ y))))\ x)$, in which the first instance of $x$ and the only instance of $y$ are bound, and the second instance of $x$ is free. We have an oppressive amount of parentheses; we'll establish informal conventions to reduce them in the next section.

#Parenthesis conventions

If we want to be extremely careful, we should write the add-two-numbers function as $(\lambda x.(\lambda y.(x+y)))$, using parentheses as in the formal definition. But we often don't need them, and use the following rules:

- We don't need outermost parentheses. We can write $\lambda x.(\lambda y.(x+y))$ instead of $(\lambda x.(\lambda y.(x+y)))$.
- Application is, by default, left associative. $f\ x\ y$ means "the function $f$ with inputs $x$ and $y$," i.e. $(f\ x)\ y$, not $f\ (x\ y)$.
- The scope of any $\lambda$ is as large is possible. $\lambda x.\lambda y.x+y$ means $\lambda x.(\lambda y.(x+y))$, not $(\lambda x.\lambda y.x)+y$ or $\lambda x.(\lambda y.x)+y$ or anything like that.
- We can abbreviate sequences of $\lambda$s. As mentioned before, we can write $\lambda xy.x+y$ for $\lambda x.\lambda y.x+y$.

Of course, we don't have to omit parentheses whenever we can. These rules govern what it means when we don't have all the parentheses strictly necessary, but we can include them if we want to be extra clear.

#Reducing lambda expressions

So far, $\lambda$ expressions don't have any meaning; we've said what we're allowed to write down but not how the relate to each other. What we usually do with a $\lambda$ expression is reduce it, and there are three rules that we can use: [beta_reduction $eta$-reduction], [alpha_conversion $lpha$-conversion], and [eta_conversion $\eta$-conversion].

[beta_reduction $eta$-reduction] is the fancy name for "stick in the inputs to a function." For example, to evaluate $(\lambda x.\lambda y.x+y)\ 6\ 3$, we first notice that $6$ is the input to a $\lambda$ expression that starts $\lambda x$, and substitute $6$ for $x$ in the expression, giving us $(\lambda y.6+y)\ 3$. Now $3$ is substituted for $y$, giving us $6+3=9$.

Formally, for a variable $x$ and $\lambda$ expressions $M$ and $N$, $eta$-reduction converts $((\lambda x.M)\ N)$ to $M[N/x]$, i.e. $M$ with every free instance of $x$ replaced with $N$. The expressions $((\lambda x.M)\ N)$ and $M[N/x]$ are equivalent. %%todo: is there a more standard notation for this?%%

[alpha_conversion $lpha$-conversion] says we can rename variables; $\lambda x.f\ x$ is the same as $\lambda y.f\ y$. Formally, if $M$ is a $\lambda$ expression containing $\lambda x$, and no instance of $x$ bound by that $\lambda x$ is within the scope of a $\lambda y$, then we can replace every $x$ in the scope of the $\lambda x$ with a $y$ to get an equivalent expression. We need the second criterion because otherwise we could replace the $x$s in $\lambda x.\lambda y.x$ with $y$s to get $\lambda y.\lambda y.y$, and then replace the outer $y$ with $x$ to get $\lambda x.\lambda y.x$, which is not equivalent (the second substitution here is valid; the first one is the problem).

Finally, [eta_conversion $\eta$-conversion] says that if two expressions give the same result on any input, then they're equivalent. We should have $\lambda x.f\ x$ and $f$ be the same, since $eta$-reduction converts $(\lambda x.f\ x)\ x$ to $f\ x$, so they're the same on any input. Formally, if $x$ isn't free in $M$, then $(\lambda x.(M\ x))$ is equivalent to $M$.

#[50p]

Let's look again at the function that adds two numbers, $\lambda x.\lambda y.x+y$. There are two different ways we can think of this: first, as a function that takes in two numbers, and returns their sum. If we're talking about [45h natural numbers], this is a function $\mathbb N^2	o\mathbb N$.

We could also think of the function as taking one input at a time. It says "take in input $x$, return the expression $\lambda y.x+y$." For example, on input $6$ we have $(\lambda x.\lambda y.x+y)\ 6=\lambda y.6+y$. Now we have a function that takes in a number, and gives us a function that takes in a number and returns a number. We can write this as $\mathbb N	o(\mathbb N	o\mathbb N)$.

This equivalence between functions on two (or more) inputs, and functions on one input that return a new function, is known as [50p] and is important in $\lambda$ calculus, where the only objects are functions. We have to get used to the idea that a function can take other functions as arguments and give functions as return values.

#Defining variables

$\lambda$ calculus doesn't have a built in way to define a [-variable]. Suppose we have the function $d=\lambda f.\lambda x.f\ (f\ x)$ which applies its first input to its second input twice. Currying, $d$ takes in a function and returns that function composed with itself. Say we want to know what $d\ d$ is. We could just write $d$ twice, to get $(\lambda f.\lambda x.f\ (f\ x))\ (\lambda f.\lambda x.f\ (f\ x))$. $eta$-reduction eventually reduces this to $\lambda f.\lambda x.f\ (f\ (f\ (f\ x)))$, the function that applies its first input to its second input four times (as you might have expected, applying a function twice, twice, is the same as applying it four times).

We can also get $d\ d$ while only writing out $d$ once. Take the $\lambda$ expression $(\lambda x.x\ x)\ d$ or, writing out $d$, $(\lambda x.x\ x)\ (\lambda f.\lambda x.f\ (f\ x))$. A single step of $eta$-reduction turns this into the previous version, where we wrote $d$ out twice. The expression we started with was shorter, and expressed better the idea that we wanted to use the same value of $d$ twice. In general, if we want to say something like $x = a; f\ x$ (i.e. define the variable $x$, and then use it in some expression), we can write $(\lambda x.f\ x)\ a$, first building the expression we want to use the value of $x$ in, and then substituting in the value using function application. This will be very useful in the next section when we make [recursion recursive] functions.

#Loops

It might seem like $\lambda$ expressions are very finite, and $eta$-reduction will always finish after a reasonable amount of time. But for $\lambda$ calculus to be as powerful as normal programming languages, there must be $\lambda$ expressions that don't reduce in a [46h finite amount of time]. The way to resolve this is to figure out how to build while loops in $\lambda$ calculus; if you can make a while loop, you can write a program that runs forever pretty easily.

Since all we have is functions, we'll use [-recursion] to make loopy programs. For example, consider this factorial function in Python:

    factorial = lambda n: 1 if n==0 else n*factorial(n-1)

How can we write this in lambda calculus? Let's assume we have [54y numbers], [-arithmetic], [57f booleans], %%todo: possibly change these links to church encoding pages once they exist%%
and a function $Z$ that returns True on $0$ and False on anything else. Let's also a assume we have an "if" function $I$ that takes in three inputs: $I = (\lambda p.\lambda x.\lambda y.$if $p$, then $x$, else $y)$. So $I\ True\ x\ y=x$ and $I\ False\ x\ y=y$.

Starting to translate from Python to $\lambda$ calculus, we have $F = \lambda n.I\ (Z\ n)\ 1\ (n	imes(F\ (n-1)))$, which says "if $n$ is $0$, then $1$, else $n	imes F(n-1)$. But we can't define $F$ like this; we have to use the method described in the previous section. In order for each recursive call of the factorial function to know how to call further recursive calls, we need to pass in $F$ to itself. Consider the expression

$$F=(\lambda x.x\ x)\ (\lambda r.\lambda n.I\ (Z\ n)\ 1\ (n	imes(r\ r\ (n-1))))$$

Let $g=\lambda r.\lambda n.I\ (Z\ n)\ 1\ (n	imes(r\ r\ (n-1)))$. Then $F=(\lambda x.x\ x)\ g=g\ g$ is $g$ applied to itself. Plugging in $g$ for $r$, $g\ g=\lambda n.I\ (Z\ n)\ 1\ (n	imes(g\ g\ (n-1)))$. But since $g\ g=F$, this is just $\lambda n.I\ (Z\ n)\ 1\ (n	imes(F\ (n-1)))$, exactly what we wanted $F$ to be.

Let's generalize this. Say we want to define a recursive formula $f=\lambda x.h\ f\ x$, where $h$ is any expression (in the factorial example $h=\lambda f.\lambda n.I\ (Z\ n)\ 1\ (n	imes(f\ (n-1)))$). We define

egin{align*}
    g&=\lambda r.h\ (r\ r)
ewline
    &=\lambda r.\lambda x.h\ (r\ r)\ x
\end{align*}

and use this to build

egin{align*}
    f&=(\lambda x.x\ x)\ g
ewline
    &=g\ g
ewline
    &=(\lambda r.\lambda x.h\ (r\ r)\ x)\ g
ewline
    &=\lambda x.h\ (g\ g)\ x
ewline
    &=\lambda x.h\ f\ x.
\end{align*}

We can express this process taking any $h$ to the recursive function $f$ as a $\lambda$ expression:

egin{align*}
    Y&=\lambda h.(\lambda x.x\ x)\ (\lambda r.h\ (r\ r))
ewline
    &=\lambda h.(\lambda r.h\ (r\ r))\ (\lambda r.h\ (r\ r)).
\end{align*}

For any $h$, which takes as arguments $f$ and $x$ and returns some expression using them, $Y\ h$ recursively calls $h$. Specifically, letting $f=Y\ h$, we have $f=\lambda x.h\ f\ x$.

For fun, let's make a $\lambda$ expression that enters an infinite loop by reducing to itself. If we make $h$ the function that applies $f$ to $x$, recursively calling it should accomplish this, since it's analogous to the Python function `f = lambda x: f(x)`, which just loops forever. So let $h=\lambda f.\lambda x.f x=\lambda f.f$. Now egin{align*}
    Y\ h&=(\lambda x.x\ x)\ (\lambda r.(\lambda f.f)\ (r\ r))
ewline
    &=(\lambda x.x\ x)\ (\lambda r.r\ r)
ewline
    &=(\lambda r.r\ r)\ (\lambda r.r\ r).
\end{align*}

Continued $eta$-reduction doesn't get us anywhere. This expression is also interesting because it's a simple [-322], a program that outputs its own code, and this is a sort of template you can use to make quines in other languages.

This strategy of building recursive functions results in some very long intermediate computations. If $F$ is the factorial function, we compute $F(100)$ as $100	imes F(99)=100	imes 99	imes F(98)$, etc. By the time the recursive calls finally stop, we have the very long expression $100	imes99	imes\dots	imes2	imes1$, which takes a lot of space to write down or store in a computer. Many programming languages will give you an error if you try to do this, complaining that you exceeded the max recursion depth.

The fix is something called [-tail_recursion], which roughly says "return only a function evaluation, so you don't need to remember to do something with it when it finishes." The general way do accomplish this is to pass along the intermediate computation to the recursive calls, instead of applying it to the result of the recursive calls. We can make a tail recursive version of our Python factorial like so %%note:Python will still give you a "maximum recursion depth exceeded" error with this, but the strategy works in lambda calculus and lazy languages like Lisp and Haskell, and Python is easier to demonstrate with.%%:

    f = lambda n, k: k if n==0 else f(n-1, k*n)

We pass along $k$ as an intermediate value. It has to start with $k=1$; we can do this by defining a new function `g = lambda n: f(n,1)` or using Python's default value feature to have $k$ be $1$ be default. Now if we want $100$ factorial, we get $f(100,1)=f(99,1*100)=f(99,100)=f(98,9900)=\dots$. We compute the products as we go instead of all at the end, and only ever have one call of $f$ active at a time.

Writing a tail recursive version of factorial in $\lambda$ calculus is left as an exercise for the reader.

#[Y_combinator]

You might be wondering why we called the recursive-function-maker $Y$, so let's look at that again. We said earlier that $Y\ h=\lambda x.h\ (Y\ h)\ x$. By $\eta$-conversion, $Y\ h=h\ (Y\ h)$. That means $Y\ h$ is a fixed point of $h$! For any function $h$, $Y$ finds a value $f$ such that $h\ f=f$. This kind of object is called a [-fixed_point_combinator]; this particular one is called the [Y_combinator], conventionally denoted $Y$.

Why is a fixed point finder the right way to make recursive functions? Consider again the example of factorial. If a function $f$ says "do factorial for $k$ recursive steps," then $h\ f$ says "do factorial for $k+1$ recursive steps;" $h$ was designed to be the function which, calling recursively, gives factorial. Naming the [identity_map identity function] $i=\lambda n.n$, $i$ does factorial for $0$ steps, so $h\ i$ does it for $1$ step, $h\ (h\ i)$ for $2$, and so on. To completely compute factorial, we need a function $F$ that says "do factorial for infinity recursive steps." But doing one more than infinity is the same as just doing infinity, so it must be true that $h\ F=F$; i.e. that $F$ is a fixed point of $h$. There may be other fixed points of $h$, but the Y combinator finds the fixed point that corresponds to recursively calling it.

## With $Y$ and without $Y$

What do we get, and what do we lose, if we allow the use of $Y$?

$Y$ takes a lambda-term which may have corresponded to a [-primitive_recursive] function, and outputs a lambda term which need not correspond to a primitive recursive function.
That is because it allows us to perform an *unbounded* search: it expresses fully-general looping strategies.
(As discussed earlier, without a fixed-point finder of some kind, we can only create lambda-terms corresponding to programs which provably halt.)

On the other side of the coin, $Y$ may produce non-[well_typed well-typed] terms, while terms without a fixed-point finder are guaranteed to be well-typed.
That means it's hard or even impossible to perform type-checking (one of the most basic forms of static analysis) on terms which invoke $Y$.

Strongly relatedly, in a rather specific sense through the [curry_howard Curry-Howard correspondence], the use of $Y$ corresponds to accepting non-[constructive_proof constructiveness] in one's proofs.
Without a fixed-point finder, we can always build a constructive proof corresponding to a given lambda-term; but with $Y$ involved, we can no longer keep things constructive.


Monotone function: exercises

Try these exercises and become a *deity* of monotonicity.

Monotone composition
-----

Let $P, Q$, and $R$ be posets and let $f : P 	o Q$ and $g : Q 	o R$ be monotone
functions. Prove that their composition $g \circ f$ is a monotone function from $P$ to $R$.

Evil twin
-------

Let $P$ and $Q$ be posets. A function $f : P 	o Q$ is called **antitone** if it *reverses* order: that is, $f$ is antitone whenever $p_1 \leq_P p_2$ implies  $f(p_1) \geq_Q f(p_2)$. Prove that the composition of two antitone functions is monotone.

Partial monotonicity
----------

A two argument function $f : P 	imes A 	o Q$ is called  *partially monotone in the 1st argument* whenever $P$ and $Q$ are posets and for all $a \in A$, $p_1 \leq_P p_2$ implies $f(a, p_1) \leq_Q f(a, p_2)$. Likewise a 2-argument function $f : A 	imes P 	o Q$ is called *partially monotone in the second argument* whenever $P$ and $Q$ are posets and for all $a \in A$, $p_1 \leq_P p_2$ implies $f(p_1, a) \leq_Q f(p_2, a)$.

Let $P, Q, R$, and $S$ be posets, and let $f : P 	imes Q 	o R$ be a function that is partially monotone in both of its arguments. Furthermore, let $g_1 : S 	o P$ and $g_2 : S 	o Q$ be monotone functions.

Prove that the function $h : S 	o R$ defined as $h(s) \doteq f(g_1(s), g_2(s))$ is monotone.

Brain storm
------

List all of the commonly used two argument functions you can think of that are partially monotone in both arguments. Also, list all of the commonly used two argument functions you can think of that are partially monotone in one argument and partially antitone in the other.


Complete lattice

[summary:
A **complete lattice** is a [3rb poset] that is closed under arbitrary [3rc joins and meets]. A complete lattice, being closed under arbitrary joins and meets, is closed in particular under binary joins and meets. A complete lattice is thus a specific type of [46c lattice], and hence satisfies [3h4 associativity], [3jb commutativity], idempotence, and absorption of joins and meets. Complete lattices can be equivalently formulated as posets which are closed under arbitrary joins; it then follows that complete lattices are closed under arbitrary meets as well.

Because complete lattices are closed under all joins, a complete lattice $L$ must contain both $igvee \emptyset$ and $igvee L$ as elements. Since $igvee \emptyset$ is a lower bound of $L$ and $igvee L$ is an upper bound of $L$,  complete lattices are bounded.
]

A **complete lattice** is a [3rb poset] that is closed under arbitrary [3rc joins and meets]. A complete lattice, being closed under arbitrary joins and meets, is closed in particular under binary joins and meets; a complete lattice is thus a specific type of [46c lattice], and hence satisfies [3h4 associativity], [3jb commutativity], idempotence, and absorption of joins and meets.

Complete lattices can be equivalently formulated as posets which are closed under arbitrary joins; it then follows that complete lattices are closed under arbitrary meets as well.

%%hidden(Proof):
Suppose that $P$ is a poset which is closed under arbitrary joins. Let $A \subseteq P$. Let $A^L$ be the set of lower bounds of $A$, i.e. the set $\{ p \in P \mid orall a \in A. p \leq a \}$. Since $P$ is closed under joins, we have the existence of $igvee A^L$ in $P$.  We will now show that $igvee A^L$ is the meet of $A$.

First, we show that $igvee A^L$ is a lower bound of $A$. Let $a \in A$. By the definition of $A^L$, $a$ is an upper bound of $A^L$. Because $igvee A^L$ is less than or equal to any upper bound of $A^L$, we have $igvee A^L \leq a$. $igvee A^L$ is therefore a lower bound of $A$.

Now we will show that $igvee A^L$ is greater than or equal to any lower bound of $A$. Let $p \in P$ be a lower bound of $A$. Then $p \in A^L$.  Because $igvee A^L$ is an upper bound of $A^L$, we have $p \leq igvee A^L$.
%%

Complete lattices are bounded
========================

As a consequence of closure under arbitrary joins, a complete attice $L$ contains both $igvee \emptyset$ and $igvee L$. The former is the least element of $L$ satisfying a vacuous set of constraints; every element of $L$ satisfies a vacuous set of constraints, so this is really the minimum element of $L$. The latter is an upper bound of all elements of $L$, and so it is a maximum. A lattice with both minimum and maximum elements is called bounded, and as this discussion has shown, all complete lattices are bounded.

Basic examples
=============

Finite Lattices
------------------------

The collection of all subsets of a finite lattice coincides with its collection of finite subsets. A finite lattice, being a finite poset that is closed under finite joins, is then necessarily closed under arbitrary joins. All finite lattices are therefore complete lattices.

Powersets
------------------

For any set $X$, consider the poset $\langle \mathcal P(X), \subseteq
angle$ of $X$'s powerset ordered by inclusion.
This poset is a complete lattice in which for all $Y \subset \mathcal P(X)$, $igvee Y = igcup Y$.

To see that $igvee Y = igcup Y$, first note that because union contains all of its constituent sets, for all $A \in Y$, $A \subseteq igcup Y$. This makes $igcup Y$ an upper bound of $Y$. Now suppose that $B \in \mathcal P(X)$ is an upper bound of $Y$; i.e., for all $A \in Y$, $A \subseteq B$. Let $x \in igcup Y$. Then $x \in A$ for some $A \in Y$. Since $A \subseteq B$, $x \in B$. Hence, $igcup Y \subseteq B$, and so $igcup Y$ is the least upper bound of $Y$.

The Knaster-Tarski fixpoint theorem
=============================

Suppose that we have a poset $X$ and a [5jg monotone function] $F : X 	o X$. An element $x \in X$ is called $F$**-consistent** if $x \leq F(x)$ and is called $F$**-closed** if $F(x) \leq x$. A fixpoint of $F$ is then an element of $X$ which is both $F$-consistent and $F$-closed.

Let $A \subseteq X$ be the set of all fixpoints of $F$. We are often interested in the maximum and minimum elements of $A$, if indeed it has such elements. Most often it is the minimum element of $A$, denoted $\mu F$ and called the **least fixpoint** of $F$, that holds our interest. In the deduction system example from [5lf], the least fixpoint of the deduction system $F$ is equal to the set of all judgments which can be proven without assumptions. Knowing $\mu F$ may be first step toward testing a judgment's membership in $\mu F$, thus determining whether or not it is provable. In less pedestrian scenarios, we may be interested in the set of all judgments which can be proven without assumption using *possibly infinite proof trees*; in these cases, it is the **greatest fixpoint** of $F$, denoted $
u F$, that we are interested in.

Now that we've established the notions of the least and greatest fixpoints, let's try an exercise. Namely, I'd like you to think of a lattice $L$ and a monotone function $F : L 	o L$ such that neither $\mu F$ nor $
u F$ exists.

%%hidden(Show solution):
Let $L = \langle \mathbb R, \leq
angle$ and let $F$ be the identity function $F(x) = x$. $x \leq y \implies F(x) = x \leq y = F(y)$, and so $F$ is monotone. The fixpoints of $F$ are all elements of $\mathbb R$. Because $\mathbb R$ does not have a maximum or minimum element, neither $\mu F$ nor $
u F$ exist.
%%

If that was too easy, here is a harder exercise: think of a complete lattice $L$ and monotone function $F : L 	o L$ for which neither $\mu F$ nor $
u F$ exist.

%%hidden(Show solution):
There are none. :p
%%

In fact, every monotone function on a complete lattice has both least and greatest fixpoints. This is a consequence of the **Knaster-Tarski fixpoint theorem**.

**Theorem (The Knaster-Tarski fixpoint theorem)**: Let $L$ be a complete lattice and $F : L 	o L$ a monotone function on $L$. Then $\mu F$ exists and is equal to $igwedge \{x \in L \mid F(x) \leq x\}$. Dually, $
u F$ exists and is equal to $igvee \{x \in L \mid x \leq F(x) \}$.

%%hidden(Proof):
We know that both $igwedge \{x \in L \mid F(x) \leq x\}$ and $igvee \{x \in L \mid F(x) \leq x \}$ exist due to the closure of complete lattices under meets and joins. We therefore only need to prove that $igwedge \{x \in L \mid F(x) \leq x\}$ is a fixpoint of $F$ that is less or equal to all other fixpoints of $F$. The rest follows from duality.

Let $U = \{x \in L \mid F(x) \leq x\}$ and $y = igwedge U$. We seek to show that $F(y) = y$. Let $V$ be the set of fixpoints of $F$. Clearly, $V \subseteq U$. Because $y \leq u$ for all $u \in U$, $y \leq v$ for all $v \in V$. In other words, $y$ is less than or equal to all fixpoints of $F$.

For $u \in U$, $y \leq u$, and so $F(y) \leq F(u) \leq u$. Since $F(y)$ is a lower bound of $U$, the definition of $y$ gives $F(y) \leq y$. Hence, $y \in U$. Using the monotonicity of $F$ on the inequality $F(y) \leq y$ gives $F(F(y)) \leq F(y)$, and so $F(y) \in U$. By the definition of $y$, we then have $y \leq F(y)$. Since we have established  $y \leq F(y)$ and $F(y) \leq y$, we can conclude that $F(y) = y$.
%%

TODO: Prove the knaster tarski theorem and explain these images

add !'s in front of the following two lines
[A Knaster-Tarski-style view of complete latticess](http://i.imgur.com/wKq74gC.png)
[More Knaster-Tarski-style view of complete latticess](http://i.imgur.com/AYKyxlF.png)


Square root

##Introduction
When you try to get the area of a square  %%note: A shape where all sides are equal%%  and know the length of a **side** %%note: Otherwise known as its **root**%% you multiply the length of one side by itself or [4ts square] it.

![area of square](http://2.bp.blogspot.com/-hkA5woVs-NQ/VRw85xkXlDI/AAAAAAAAAFc/Xc7utStLO7E/s1600/C%2BProgram%2BArea%2BOf%2BSquare.jpg)

What if you don't know the length of a side and instead only know the area of a square and want to get the length of a side?
You would have to do something to the area that transforms it in the opposite direction that squaring the length of a side does. This is just like how placing 2 apples to a pile transforms a pile of apples by increasing the amount of apples and eating 2 apples transforms the pile back to the way it was. So in other words adding is to subtracting as squaring is to.... what?

Well this kind of transformation is called a **Square Root**.


Rationality

The subject domain for [ epistemic] and [ instrumental] rationality.


Central examples

The "central examples" for a subject are examples that are referred to over and over again in the course of making several different points.  They should usually come with their own tutorials.


The empiricist-theorist false dichotomy

No discussion here yet:  See https://www.facebook.com/groups/674486385982694/permalink/784664101631588/


Intension vs. extension

To give an "intensional definition" is to define a word or phrase in terms of other words, as a dictionary does.  To give an "extensional definition" is to point to examples, as adults do when teaching children.  The preceding sentence gives an intensional definition of "extensional definition", which makes it an extensional example of "intensional definition".  See http://lesswrong.com/lw/nh/extensions_and_intensions/

In the context of AI, an "intensional concept" is the code or statistical pattern that executes to determine whether something is a member of the concept, while the "extension" is the set of things that are thus determined to belong to the concept.  The intensional concept "test: does 2 evenly divide x?" recognizes the even numbers 0, 2, 4, 6... as its extension.

Given the modern level of visual recognition technology, a neural network that tries to classify cat photos vs. noncat photos would have some cat photos in its extension, but almost certainly also many things we think are 'cat photos' that would fail to be in its extension and many non-cat-photos that did end up in the extension of that particular neural network.  The intensional concept would be the classifier network itself - its weights, propagation rules, and so on.


Strained argument

A phenomenological feeling associated with a step of reasoning going from X to Y where it feels like somebody is stretching a hand across a cliff and not quite making it.

Example:

Alice:  "Hey, do you think that guy there is stealing that bike?"
Bob:  "They could be the bike's owner."
Alice:  "They're cutting the lock off with a hacksaw."
Bob:  "Maybe they lost the key.  They'd have to get the bike back somehow, after that."
Alice:  "So what do you think will happen if I go over there and ask to see their driver's license and maybe take a picture of them using my cellphone, just in case?"
Bob:  "Maybe they'll have a phobia of being photographed and so they'll react with anger or maybe even run away."

At the point where Bob suggested that maybe the bicycle thief(?) would have a phobia of being photographed, you might have detected a sense of *strained argument* - even by comparison to 'maybe they lost the key', which should sound improbable or like an excuse, but not have the same sense of 'somebody trying to reach across a cliff that is actually too far to cross', or overcomplicatedness, or 'at this point you're trying too hard', as "Maybe they have a phobia of being photographed".


Perfect rolling sphere

A "perfect rolling sphere" is metaphorically an object that is too simple or idealized to exist in real life.  It derives from an old joke about a physicist who tries to bet on horse races, starting from the assumption that each horse is a perfect rolling sphere.  Nonetheless, in a lot of cases, a perfect rolling sphere is a pretty good approximation to a roughly spherical wooden ball moving down an iron trough inside an atmosphere.  More importantly, you're probably not going to be able to analyze the 'realistic' case until you understand the perfect rolling sphere.

[summary(Brief): A "perfect rolling sphere" is metaphorically an object that is too simple or idealized to exist in real life.  (But which might be illuminating to consider.)]


Aaron Gertler

[summary: Nothing here yet.]

Automatically generated page for "Aaron Gertler" user.
If you are the owner, click [here to edit](https://arbital.com/edit/2d8).


Introduction to Effective Altruism

Effective altruism (EA) means using evidence and reason to take actions that help others as much as possible.

The world has a lot of terrible problems, but we can't work on them all at once. And we don't agree on what the worst problems are.

Despite this, most people want to "make the world a better place". What should those people actually be doing?

Some of the questions EA sets out to answer:

 - Is it better to help people in your own country, or abroad?
 - Is it better to take a high-paying job and give money to charity, or to work for a charity directly?
 - Is it better to support an intervention with a lot of strong evidence, or to support research on a promising new intervention?

The answers to these kinds of questions will be different for each person. What you should do depends on who you are: What are your skills? What are your values? How much time do you have?

But no matter who you are, effective altruism can help you figure out how to improve the world. To learn more, follow the links below.

##More on EA##

Are you a high school student? Are you a college student?

Do you work for a nonprofit or a business with a social mission?

Are you religious?

Which of the following issues most concerns you? (Politics, poverty, health, education, animals, environment, far future)

Are you interested in donating? In direct work?

Are you financially comfortable?

##Common Questions##

 - What beliefs do all EAs have in common?
 - What are the different causes EAs support?
 - What are some actions EAs take to improve the world?
 - What makes EA different from other ways of helping people?
 - What are some common criticisms of EA, and how do EAs respond?
 - What is the philosophical justification for EA?

Here are some really good articles from outside Arbital:

 - [What is Effective Altruism?](https://whatiseffectivealtruism.com) (The best general introduction to the topic.)

**This page is a work in progress. If you have questions, or you'd like to see a topic added, please contact the author: aaronlgertler (at) gmail.com.**


Strictly factual question

A *strict question of fact* has an answer determined solely by the state of the material universe, and sufficiently straightforward math, in a clearly understandable way - none of our uncertainty about this question is about definitions, values, or viewpoints; we are just wondering which quarks go where.

Questions of strict fact:

- What happens if I press the giant red button?(\*)
- Did Sally claim more charitable donations as an income tax deduction than Bob?
- Are there unicorns anywhere on Earth?
- Will Peano arithmetic prove a contradiction if I search all the proofs less than a billion steps long?

Questions not yet of strict fact:

- Should I press the giant red button?
- Is Sally a more charitable person than Bob?
- Do unicorns exist?
- Will Zermelo-Frankel set theory prove a contradiction if I search for one forever?

Why the second group aren't yet questions of strict fact:

- Because the notion of *should* has not yet been fully specified or determined.
- Because we haven't said exactly what it means to be charitable.
- Because "exist" is a much more ambiguous notion than "exist inside our galaxy".  For example, there could be unicorns $10^{1,000,000}$ lightyears away, or inside some other mathematical universe that has as much "existence" as this one does.  The notion of "X exists" is less firmly nailed down than "X is inside our closet".
- Because the notion of 'forever' is more mathematically fraught than 'for ten to the billionth power years'.  For example, nailing down exactly which infinity we're talking about can't be done in a system of first-order logic.

(\*)  At least, this is a straightforward question so long as we don't poke too hard at the nature of [ what-if counterfactuals].  In most real-life situations, the question "What happens if I turn on this blender with a fork inside?" is something that has a sufficiently straightforward material answer for us to say that it's just a question of material facts.


Proving too much

An argument "proves too much" when you can just as easily or naturally extend it to prove false things as true things.  For example, the argument "Nobody should engage in genetic engineering because Nazis" can just as easily be extended to say "Nobody should eat vegetables because Hitler ate vegetables", so the general rule *proves too much.*

See also:

- http://en.wikipedia.org/wiki/Proving_too_much
- http://slatestarcodex.com/2013/04/13/proving-too-much/


Bulverism

[Bulverism](https://en.wikipedia.org/wiki/Bulverism) is when you [43h explain what goes so horribly wrong in people's minds] when they believe X, before you've actually explained why X is wrong.  It can be seen as a subspecies of the fallacy of "[poisoning the well](https://en.wikipedia.org/wiki/Poisoning_the_well)".  **Bulverism is forbidden on Arbital.**  Every time you talk about what goes wrong in people's minds that makes them believe X, make sure you first discuss why X is, in fact, wrong, or route your readers to a page that does discuss it.  Also see the entry on [43h].


Emphemeral premises

A purported [43h psychological] fallacy wherein people confronted with an incongruent proposition X will say "Oh, not-X because Y", and perhaps a week later, say, "Oh, not-X because Z."  [Ephemeral Premises](https://www.facebook.com/yudkowsky/posts/10154268184344228) is when you make up assumptions to deflect X, don't write them down or list them anywhere, and since there's no central list, you might be making up different assumptions when talking to someone else.  It's a disposable premise being used as a parry in short-term combat, not part of a long-term edifice being made with consistent, well-defended materials.  Needs to be paired with understanding of the [multiple_stage_fallacy Multiple-Stage Fallacy] so that you don't fall for the reverse fallacy of making people list out lots of Ys and Zs to make arbitrary propositions sound very improbable.


Multiple stage fallacy

[summary:  The purported "Multiple-Stage Fallacy" is when you list multiple 'stages' that need to happen on the way to some final outcome, assign probabilities to each 'stage', multiply the probabilities together, and end up with a small final answer.  The alleged problem is that you can do this to almost *any* proposition by staring at it hard enough, including propositions that turn out to be true.

An alleged example would be that in August 2015, renowned forecaster Nate Silver wrote "[Trump's Six Stages of Doom](http://fivethirtyeight.com/features/donald-trumps-six-stages-of-doom/)" in which he gave Donald Trump a 2% chance of getting the Republican *nomination* (not the presidency), based on six stages each with at most a 50% probability of being passed.  In late March, Trump had passed the first four stages Nate listed, and prediction markets gave Trump a 75% chance of clinching the Republican nomination.  By Nate Silver's logic, Trump's probability of passing the remaining stages should have been $0.50 \cdot 0.50 = 0.25.$

The three main purported problems with the Multiple-Stage Fallacy are that:

- To get the final estimate, you logically need to multiply [1rj conditional probabilities], not unconditional probabilities.  Getting your mind to give good conditional probabilities isn't easy; Nate Silver might have failed to imagine how much he'd *actually* update after *actually* seeing Trump pass the first four stages.
- Often, people fail to consider disjunctive alternatives, or ways that not all the stages may need passing.
- People tend to assign middle-tending probabilities.   So if you list enough stages, you can drive the apparent probability of anything down to zero, *even if you seem to be soliciting probabilities from the reader.*]

In August 2015, renowned statistician and predictor Nate Silver wrote "[Trump's Six Stages of Doom](http://fivethirtyeight.com/features/donald-trumps-six-stages-of-doom/)" in which he gave Donald Trump a 2% chance of getting the Republican *nomination* (not the presidency).  Silver reasoned that Trump would need to pass through six stages to get the nomination, "Free-for-all", "Heightened scrutiny", "Iowa and New Hampshire", "Winnowing", "Delegate accumulation", and "Endgame."  Nate Silver argued that Trump had at best a 50% chance of passing each stage, implying a final nomination probability of at most 2%.

In late March, Trump had passed the first four stages, while prediction markets gave him a 75% chance of clinching the Republican nomination.  By Nate Silver's logic, Trump's probability of passing the remaining stages should have been $0.50 \cdot 0.50 = 0.25$ *conditional* on Trump passing the first four stages.

Nate Silver might not have been wrong to assign Trump a low advance probability of being nominated.  Many people were surprised by Trump's nomination.  But it seems more likely that Silver committed an error when he said, specifically, that if we'd observed Trump to pass the first four stages, this would probably be taking place in a world where Trump had a 50% probability of passing the two remaining stages.

The purported "Multiple-Stage Fallacy" is when you list multiple 'stages' that need to happen on the way to some final outcome, assign probabilities to each 'stage', multiply the probabilities together, and end up with a small final answer.  The alleged problem is that you can do this to almost any kind of proposition by staring at it hard enough, including things that actually happen.

On a probability-theoretic level, the three problems at work in the usual Multiple-Stage Fallacy are as follows:

- You need to multiply *conditional* probabilities rather than the absolute probabilities. When you're considering a later stage, you need to assume that the world was such that every prior stage went through. Nate Silver was [43h probably] trying to simulate his prior model of Trump accumulating enough delegates in March through June, not imagining his *updated* beliefs about Trump and the world after seeing Trump be victorious up to March.
  - Even if you're aware in principle that you need to use conditional probabilities - Nate Silver certainly knew about them - it's hard to update far *enough* when you imagine the pure hypothetical possibility that Trump wins stages 1-4 for some reason - compared to how much you actually update when you actually see Trump winning! (Some sort of reverse hindsight bias or something? We don't realize how much we'd need to update our current model if we were already that surprised?)
- Often, people neglect to consider disjunctive alternatives - there may be more than one way to reach a stage, so that not *all* the listed things need to happen. Trump accumulated enough delegates in Nate's fifth stage that there was no "Endgame" convention fight in the supposed sixth stage.
- People have tendencies to assign middle-tending probabilities. So if you list enough stages, you can drive the apparent probability of anything down to zero, *even if you seem to be soliciting probabilities from the reader.*
 - If you're a [motivated skeptic](https://wiki.lesswrong.com/wiki/Motivated_skepticism), you will be tempted to list more 'stages'.

The Multiple-Stage Fallacy is particularly dangerous for people who've read studies on the dangers of probabilistic overconfidence.  In late March, the 75% prediction-market probabilities must have corresponded to, e.g., something like an 80% chance of getting enough delegates and a 90% chance of passing the convention conditional on getting enough delegates.  Imagine how overconfident this might have sounded without the prediction market to establish a definite probability - "Oh, don't you know that what people assign 90% confidence doesn't usually happen 90% of the time?"

Instances of the Multiple-Stage Fallacy may also sound more persuasive to readers who've read about the [Conjunction Fallacy](http://lesswrong.com/lw/ji/conjunction_fallacy/).

Yudkowsky [argues](https://www.facebook.com/yudkowsky/posts/10154036150109228):

> If you're not willing to make "overconfident" probability assignments like those, then you can drive the apparent probability of anything down to zero by breaking it down into enough 'stages'. In fact, even if someone hasn't heard about overconfidence, people's probability assignments often trend toward the middle, so you can drive down their "personally assigned" probability of anything just by breaking it down into more stages...

> From beginning to end, I've never used this style of reasoning and I don't recommend that you do so either. \[Since\] even Nate Silver couldn't get away with it, I think we just shouldn't try. It's a doomed methodology.


Mind projection fallacy

[summary:  One commits the mind projection fallacy when they postulate that features of how their model of the world works are actually features of the world.

Suppose you flip a coin, slap it against your wrist, and don't look at the result.  Does it make sense to say that the probability of the coin being heads is 50%?  How can this be true, when the coin has already landed, and is either definitely heads or definitely tails? One who says "the coin is fundamentally uncertain; it is a feature of the coin that it is always 50% likely to be heads" commits the mind projection fallacy. Uncertainty is in the mind, not in reality. It makes sense that brains have an internal measure of how uncertain they are about the world, but that uncertainty is not a fact about the coin, it's a fact about the uncertain person. The coin itself is not sure or unsure.]

The "mind projection fallacy" occurs when somebody expects an overly direct resemblance between the intuitive language of the mind, and the language of physical reality.

Consider the [map_territory map and territory] metaphor, in which the world is a like a territory and your mental model of the world is like a map of that territory. In this metaphor, the  mind projection fallacy is analogous to thinking that the territory can be folded up and put into your pocket.

As an archetypal example:  Suppose you flip a coin, slap it against your wrist, and don't yet look at it.  Does it make sense to say that the probability of the coin being heads is 50%?  How can this be true, when the coin itself is already either definitely heads or definitely tails?

One who says "the coin is fundamentally uncertain; it is a feature of the coin that it is always 50% likely to be heads" commits the mind projection fallacy. Uncertainty is in the mind, not in reality.  If you're ignorant about a coin, that's not a fact about the coin, it's a fact about you. It makes sense that your brain, the map, has an internal measure of how it's more or less sure of something.  But that doesn't mean the coin itself has to contain a corresponding quantity of increased or decreased sureness; it is just heads or tails.

The [-ontology] of a system is the elementary or basic components of that system. The ontology of your model of the world may include intuitive measures of uncertainty that it can use to represent the state of the coin, used as primitives like [float floating-point numbers] are primitive in computers. The mind projection fallacy occurs whenever someone reasons as if the territory, the physical universe and its laws, must have the same sort of ontology as the map, our models of reality.

See also:

- [4vr]
- The LessWrong sequence on [Reductionism](https://wiki.lesswrong.com/wiki/Reductionism_%28sequence%29), especially:
 - [How an algorithm feels from the inside](http://lesswrong.com/lw/no/how_an_algorithm_feels_from_inside/)
 - [The Mind Projection Fallacy](http://lesswrong.com/lw/oi/mind_projection_fallacy/)
 - [Probability is in the mind](http://lesswrong.com/lw/oj/probability_is_in_the_mind/)


Invisible background fallacies

[summary:  Considering *all* the implications often gives us a very different picture from considering *some* of the implications, and the latter is a common species of fallacy.

Suppose that you went outdoors and jumped up into the air, and then, as you were jumping, gravity suddenly switched off.  Would your feet hit the ground again?

You might initially imagine that gravity no longer pulls you down toward the Earth, and that therefore your jump would keep you continuing up into the air.  If you were thinking more broadly, you might imagine that the air on Earth's surface, previously kept pressurized by gravity, might start to rush off into space.

Actually, the ground under your feet is also currently being pulled downward by gravity.  It follows that other pressures on the ground immediately below your feet--from the air above it, and from the rock below--must net out to an upward force that would, if *not* for gravity, push the ground upward at $9.8m/s^2.$  So if you jumped up into the air, and then gravity switched off, the ground and everything previously motionless around you would accelerate upward at $9.8m/s^2,$ causing your feet to hit the ground at the expected time.  That is, if gravity switched off everywhere in the universe, you wouldn't notice anything for a couple of minutes.]

Suppose that you went outdoors and jumped up into the air, and then, as you were jumping, the law of gravity suddenly switched off--for whatever weird reason, masses everywhere in the Universe stopped attracting one another.  Would your feet hit the ground again?

As an initial thought about this problem, you might imagine that gravity no longer pulls you down toward the Earth, and that therefore your jump would keep you continuing up into the air.  If you were thinking more broadly, you might realize that the concept of 'air pressure' means that the air on Earth's surface is kept pressurized by gravity, and in the absence of gravity holding down all that air, the atmosphere might start to rush off into space as it expanded to relieve the pressure.  Although (you then reason further) the speed of sound is finite, so the atmosphere might start rushing out at the edges, and would need to disappear there before the air started to expand away from the Earth's surface.  And then that rushing air would only carry you further away from the ground.

This answer is incorrect.  Why?  Because the ground under your feet is also currently being pulled downward by gravity.

Indeed, the ground under your feet is currently in a state of [ equilibrium] despite being pulled downward by gravity at what would otherwise be an acceleration of $9.8m/s^2.$  It follows that other pressures on the ground immediately below your feet--from the air above it, and from the rock below--must net out to an upward force that would, if not for gravity, push the ground upward at $9.8m/s^2.$  This logic applies to the dirt below your feet, the air around you, and the rocks immediately underneath the dirt under your feet.  Everything around you that seemed motionless was previously in equilibrium with gravity, so as soon as gravity vanishes, it all accelerates upward in near unison.

So your feet hit the ground again, as the Earth expands beneath you.  And then the Earth continues to accelerate upward at $9.8m/s^2$, pressuring your feet and accelerating you as well.  So in fact, if gravity had switched off everywhere in the universe 10 seconds earlier as you read this, you probably would not have noticed anything different, yet.  (Though you *would* notice shortly afterward as things began to expand far enough to change the balance of forces, and as the Sun finished exploding.)

In many situations and scenarios, there are rules or laws or considerations that properly apply to *everything*--including background objects and processes and concepts we don't usually think about--and then considering the *complete* effect gives us a very different picture from thinking only about the effect on *visible* things, with the latter being a common species of fallacy.

An essay now famous in the field of [ economics] is Frederic Bastiat's "[That Which Is Seen, And That Which Is Not Seen](http://bastiat.org/en/twisatwins.html)" which illustrated this idea and introduced the now-famous 'broken window fallacy'.  Suppose you heave a rock through somebody's window; have you done a good deed or a bad deed?  You can imagine somebody arguing that they've done a good deed, despite the initial inconvenience, because the homeowner needs to pay a glazier to fix the window, and then the glazier spends money at the baker, and so the whole economy is helped.  This is a fallacy because, if the window hadn't been broken, the homeowner would have spent the money somewhere else.  But the broken window and the payment to the glazier are highly visible; while the disappearing of whatever economic activity would otherwise have occurred, is not seen.  If we consider the effect on the whole picture, including things that are ordinarily hard to see or fading into the background or invisible for some other reason, we arrive at a very different picture of the net effect of breaking somebody's window.


Ideological Turing test

The [Ideological Turing Test](https://en.wikipedia.org/wiki/Ideological_Turing_Test) is when, to try to make sure you really understand an opposing position, you explain it such that *an advocate of the position cannot tell whether the explanation was written by you or a real advocate.*  This is often run online as a formal test where an audience tries to distinguish which pro-X and anti-X essays have been written by real pro-X and anti-X advocates, and which essays were actually written by advocates for the opposing side.  If you're pro-X, and you write an intelligent essay advocating anti-X, and other people can't tell that you were really pro-X, you win the Ideological Turing Test by having demonstrated that you really and honestly understand the arguments for anti-X.


Gotcha button

A "gotcha button" is a conversational point which, when pressed, causes some substantial fraction of listeners to leap up and shout "Gotcha!  That's obviously wrong and shows you don't know what you're talking about!", after which they have [motivated_skepticism license to dismiss] the conversation.

This behavior does not *necessarily* mean the listener is wrong.  If you're describing your brilliant invention to a physics-literate person, and you mention that it can accelerate forward without pushing anything else in the opposite direction, the listener will shout "Gotcha!  You can't violate Conservation of Momentum!"  They'll be right, and their implied inference about your level of physics literacy will probably also be right.

Nonetheless, there are a *lot* of ideas that function as Gotcha Buttons for some significant fraction of various audiences, which do *not* correspond to a claimed known falsity.

- Some true/plausible claims will be seen as 'obviously false' by many listeners.
- Some phrases or ideas will be predictably misinterpreted to a claim that actually is false.

Example of the first type:  If you were explaining evolutionary biology to a novice audience, you might be wise to start with some example other than how hominids in particular evolved from primate ancestors.  This isn't wrong, but a fraction of the audience thinks it's obviously wrong and diagnostic of you as someone they shouldn't listen to.

Example of the second type:  If you were discussing [42g Artificial General Intelligence], or AI with [7mt par-human performance] in some domain, you might be wise to avoid the term "human-level AI"; because that particular phrase will cause some fraction of the audience to jump up and shout "Gotcha!  AIs won't have a humanlike balance of abilities!  There's no such thing as the human level!"  (They can then dismiss the rest of your argument, without trying to carefully consider whether your further arguments are actually refuted by accepting "AIs won't have a humanlike balance of abilities", because you've made such an obvious and severe mistake that you just lose.)


Harmless supernova fallacy

[summary:  "Harmless supernova" fallacies are arguments that [3tc can equally be used to conclude] than any real phenomenon, including a supernova, is harmless.

- **Bounded, therefore harmless:**  "A supernova isn't *infinitely* energetic--that would violate the laws of physics! Just wear a flame-retardant jumpsuit and you'll be fine."
- **Continuous, therefore harmless:**  "Temperature is continuous; there's no qualitative threshold where something becomes 'super' hot. We just need better versions of our existing heat-resistant materials."
- **Varying, therefore harmless:**  "A supernova wouldn't heat up all areas of the solar system to exactly the same temperature!  It'll be hotter closer to the center, and cooler toward the outside. We just need to stay on a part of Earth that's further way from the Sun."
- **Mundane, therefore harmless** aka **straw superpower**:  "Contrary to what many non-astronomers seem to believe, a supernova can't burn hot enough to sear through time itself.  So we'll be fine."
- **Precedented, therefore harmless:** "Really, we've already had supernovas around for a while; there already exist devices that produce tons of energy by fusion, and they're called thermonuclear weapons.  Society has proven itself able to handle those."
- **Undefinable, therefore harmless:**  "What exactly is heat?  Different people could call different temperatures 'too warm'.  When exactly does a sun shining become so 'super' that it's a supernova?"]

Harmless supernova fallacies are a class of arguments, usually a subspecies of false dichotomy or continuum fallacy, which [3tc can equally be used to argue] that almost any physically real phenomenon--including a supernova--is harmless / manageable / safe / unimportant.

- **Bounded, therefore harmless:**  "A supernova isn't infinitely energetic--that would violate the laws of physics! Just wear a flame-retardant jumpsuit and you'll be fine."  (All physical phenomena are finitely energetic; some are nonetheless energetic enough that a flame-retardant jumpsuit won't stop them.  "Infinite or harmless" is a false dichotomy.)
- **Continuous, therefore harmless:**  "Temperature is continuous; there's no qualitative threshold where something becomes 'super' hot. We just need better versions of our existing heat-resistant materials, which is a well-understood engineering problem."  (Direct instance of the standard continuum fallacy: the existence of a continuum of states between two points does not mean they are not distinct.  Some temperatures, though not qualitatively distinct from lower temperatures, exceed what we can handle using methods that work for lower temperatures.  "Quantity has a quality all of its own.")
- **Varying, therefore harmless:**  "A supernova wouldn't heat up all areas of the solar system to exactly the same temperature!  It'll be hotter closer to the center, and cooler toward the outside. We just need to stay on a part of Earth that's further way from the Sun."  (The temperatures near a supernova vary, but they are all quantitatively high enough to be far above the greatest manageable level; there is no nearby temperature low enough to form a survivable valley.)
- **Mundane, therefore harmless** or **straw superpower**: "Contrary to what many non-astronomers seem to believe, a supernova can't burn hot enough to sear through time itself, so we'll be fine."  (False dichotomy: the divine ability is not required for the phenomenon to be dangerous / non-survivable.)
- **Precedented, therefore harmless:**  "Really, we've already had supernovas around for a while: there are already devices that produce 'super' amounts of heat by fusing elements low in the periodic table, and they're called thermonuclear weapons.  Society has proven well able to regulate existing thermonuclear weapons and prevent them from being acquired by terrorists; there's no reason the same shouldn't be true of supernovas."  (Reference class tennis / noncentral fallacy / continuum fallacy: putting supernovas on a continuum with hydrogen bombs doesn't make them able to be handled by similar strategies, nor does finding a class such that it contains both supernovas and hydrogen bombs.)
- **Undefinable, therefore harmless:**  "What is 'heat', really?  Somebody from Greenland might think 295 Kelvin was 'warm', somebody from the equator might consider the same weather 'cool'.  And when exactly does a 'sun shining' become a 'supernova'?  This whole idea seems ill-defined."  (Someone finding it difficult to make exacting definitions about a physical process doesn't license the conclusion that the physical process is harmless. %%note: It also happens that the distinction between the runaway process in a supernova and a sun shining is empirically sharp; but this is not why the argument is invalid--a super-hot ordinary fire can also be harmful even if we personally are having trouble defining "super" and "hot". %%)


Epistemology

"Epistemology" is the subject matter that deals with truth on a meta level: e.g., questions such as "What is truth?" and "What methods of reasoning are most likely to lead to true beliefs?"  Epistemology can be considered as a child subject of [9l].  Many [9n Arbital Discussion Practices] have their roots in a thesis about epistemology - about what forms of reasoning are most likely to lead to true beliefs - and you may have ended up looking at this subject after trying to track down how a Discussion Practice is justified.


Conceivability

A hypothetical scenario is 'conceivable' or 'imaginable' when it is not *immediately* incoherent, although it may still contain some subtle flaw that makes it [ logically impossible].  If you haven't yet checked for factors, it's conceivable to you that 91 is either a [ prime number] or a [ composite number], even though only one of these scenarios is [ logically possible].

Whether 91 is prime or compositive is a logical consequence of the definitions of 'prime' and 'composite', but it's not literally and explicitly written into our brain's representations of the definitions of 'prime' and 'composite' that 91 is a prime/composite number. If you can conceive of 91 being prime, this illustrates that, in your own mind, the property of 91-ness is not the same writing on your map as the property of composite-ness, but it doesn't demonstrate that it's logically or physically possible for 91 to be composite.  If you can *conceive* of a scenario in which X is true and Y is false, this demonstrates mainly that X and Y have different [10b intensional definitions] inside your own mind; you have separate concepts in your mind for X and Y.  It doesn't demonstrate that a world where 'X but not Y' is logically possible, let alone physically possible, or that X and Y refer to two different things, etcetera.

Labeling a scenario 'conceivable' or saying 'we can conceive that' introduces a scenario for discussion or refutation, *without* assuming in the introduction that the scenario is logically possible (let alone physically possible, or at all plausible in our world).  Since many true facts are also plausible, physically possible, logically possible, and conceivable, introducing a scenario as 'conceivable' also does not imply that it is false.  It means that someone found themselves able to imagine something, perhaps a true thing, perhaps without visualizing it in the complete detail that would reveal its impossibility; but if all we know at the start is that someone felt like they could imagine something, we'll call it 'conceivable' or 'imaginable'.


Mathematical object


Proof technique


Comprehensive guide to Bayes' Rule

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Introduction to Bayes' Rule odds form

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Bayes' Rule and its implications

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Axiom of Choice Definition (Intuitive)

#Getting the Heavy Maths out the Way: Definitions#
Intuitively, the [-axiom_mathematics axiom] of choice states that, given a collection of *[-5zc non-empty]* [-3jz sets], there is a [-3jy function] which selects a single element from each of the sets.

More formally, given a set $X$ whose [-5xy elements] are only non-empty sets, there is a function
$$
f: X
ightarrow igcup_{Y \in X} Y
$$
from $X$ to the [-5s8 union] of all the elements of $X$ such that, for each $Y \in X$, the [-3lh image] of $Y$ under $f$ is an element of $Y$, i.e., $f(Y) \in Y$.

In [-logical_notation logical notation],
$$
orall_X
\left(
\left[orall_{Y \in X} Y
ot=  \emptyset
ight]
\Rightarrow
\left[\exists
\left( f: X
ightarrow igcup_{Y \in X} Y
ight)
\left(orall_{Y \in X}
\exists_{y \in Y} f(Y) = y
ight)
ight]

ight)
$$

#Axiom Unnecessary for Finite Collections of Sets#
For a [-5zy finite set]  $X$ containing only [-5zy finite] non-empty sets, the axiom is actually provable (from the [-zermelo_fraenkel_axioms Zermelo-Fraenkel axioms] of set theory ZF), and hence does not need to be given as an [-axiom_mathematics axiom]. In fact, even for a finite collection of possibly infinite non-empty sets, the axiom of choice is provable (from ZF), using the [-axiom_of_induction axiom of induction]. In this case, the function can be explicitly described. For example, if the set $X$ contains only three, potentially infinite, non-empty sets $Y_1, Y_2, Y_3$, then the fact that they are non-empty means they each contain at least one element, say $y_1 \in Y_1, y_2 \in Y_2, y_3 \in Y_3$. Then define $f$ by $f(Y_1) = y_1$, $f(Y_2) = y_2$ and $f(Y_3) = y_3$. This construction is permitted by the axioms ZF.

The problem comes in if $X$ contains an infinite number of non-empty sets. Let's assume $X$ contains a [-2w0 countable] number of sets $Y_1, Y_2, Y_3, \ldots$. Then, again intuitively speaking, we can explicitly describe how $f$ might act on finitely many of the $Y$s (say the first $n$ for any natural number $n$), but we cannot describe it on all of them at once.

To understand this properly, one must understand what it means to be able to 'describe' or 'construct' a function $f$. This is described in more detail in the sections which follow. But first, a bit of background on why the axiom of choice is interesting to mathematicians.