# Mathematics

- Probability theory
The logic of science; coherence relations on quantitative degrees of belief.

- Well-calibrated probabilities
Even if you’re fairly ignorant, you can still strive to ensure that when you say “70% probability”, it’s true 70% of the time.

- Bayesian reasoning
A probability-theory-based view of the world; a coherent way of changing probabilistic beliefs based on evidence.

- Bayesian update
Bayesian updating: the ideal way to change probabilistic beliefs based on evidence.

- Extraordinary claims require extraordinary evidence
The people who adamantly claim they were abducted by aliens do provide some evidence for aliens. They just don’t provide quantitatively enough evidence.

- Extraordinary claims
What makes something an ‘extraordinary claim’ that requires extraordinary evidence?

- Extraordinary claims
- Bayesian view of scientific virtues
Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes’ rule, of course.

- Strength of Bayesian evidence
From a Bayesian standpoint, the strength of evidence can be identified with its likelihood ratio.

- Path: Insights from Bayesian updating
A learning-path placeholder page for insights derived from the Bayesian rule for updating beliefs.

- Ordinary claims require ordinary evidence
Extraordinary claims require extraordinary evidence, but ordinary claims

*don’t*.

- Extraordinary claims require extraordinary evidence
- Bayes' rule
Bayes’ rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.

- Bayes' rule examples
Interesting problems solvable by Bayes’ rule

- Realistic (Math 1)
Real-life examples of Bayesian reasoning

- Introductory Bayesian problems
Bayesian problems to try to solve yourself, before beginning to learn about Bayes’ rule.

- Diseasitis
20% of patients have Diseasitis. 90% of sick patients and 30% of healthy patients turn a tongue depressor black. You turn a tongue depressor black. What’s the chance you have Diseasitis?

- Blue oysters
A probability problem about blue oysters.

- Sparking widgets
- Sock-dresser search
There’s a

^{4}⁄_{5}chance your socks are in one of your dresser’s 8 drawers. You check 6 drawers at random. What’s the probability they’ll be in the next drawer you check?

- Diseasitis

- Realistic (Math 1)
- Waterfall diagram
Visualizing Bayes’ rule as the mixing of probability streams.

- Waterfall diagrams and relative odds
A way to visualize Bayes’ rule that yields an easier way to solve some problems

- Waterfall diagrams and relative odds
- Bayes' rule: Odds form
The simplest and most easily understandable form of Bayes’ rule uses relative odds.

- Introduction to Bayes' rule: Odds form
Bayes’ rule is simple, if you think in terms of relative odds.

- Introduction to Bayes' rule: Odds form
- Proof of Bayes' rule
Proofs of Bayes’ rule, with graphics

- Belief revision as probability elimination
Update your beliefs by throwing away large chunks of probability mass.

- Probability notation for Bayes' rule
The probability notation used in Bayesian reasoning

- Probability notation for Bayes' rule: Intro (Math 1)
How to read, and identify, the probabilities in Bayesian problems.

- Probability notation for Bayes' rule: Intro (Math 1)
- Bayes' rule: Vector form
For when you want to apply Bayes’ rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)

- Bayes' rule: Log-odds form
A simple transformation of Bayes’ rule reveals tools for measuring degree of belief, and strength of evidence.

- Bayes' rule: Functional form
Bayes’ rule for to continuous variables.

- Bayes' rule: Proportional form
The fastest way to say something both convincing and true about belief-updating.

- Bayes' rule: Guide
The Arbital guide to Bayes’ rule

- Comprehensive guide to Bayes' Rule
This is an arc that includes all Bayes content.

- Introduction to Bayes' Rule odds form
This is an arc that includes just enough content to teach about Bayes’s Rule odds form.

- Bayes' Rule and its different forms
This is an arc that includes different ways to look at Bayes’ Rule.

- Bayes' Rule and its implications
This is an arc that includes implications of the Bayes’s Rule.

- Wants to get straight to Bayes

- Comprehensive guide to Bayes' Rule
- Path: Multiple angles on Bayes's Rule
A learning-path placeholder page for learning multiple angles on Bayes’s Rule.

- Shift towards the hypothesis of least surprise
When you see new evidence, ask: which hypothesis is

*least surprised?* - Bayes' rule: Definition
- Bayes' rule: Probability form
The original formulation of Bayes’ rule.

- Frequency diagram
Visualizing Bayes’ rule by manipulating frequencies in large populations

- Frequency diagrams: A first look at Bayes
The most straightforward visualization of Bayes’ rule.

- Frequency diagrams: A first look at Bayes
- Bayes' rule: Beginner's guide
Beginner’s guide to learning about Bayes’ rule.

- High-speed intro to Bayes's rule
A high-speed introduction to Bayes’s Rule on one page, for the impatient and mathematically adept.

- Bayes' rule examples
- Prior probability
What we believed before seeing the evidence.

- Interest in mathematical foundations in Bayesianism
“Want” this requisite if you prefer to see extra information about the mathematical foundations in Bayesianism.

- Posterior probability
What we believe, after seeing the evidence and doing a Bayesian update.

- Humans doing Bayes
The human use of Bayesian reasoning in everyday life

- Explicit Bayes as a counter for 'worrying'
Explicitly walking through Bayes’s Rule can summarize your knowledge and thereby stop you from bouncing around pieces of it.

- Explicit Bayes as a counter for 'worrying'
- Ignorance prior
Key equations for quantitative Bayesian problems, describing exactly the right shape for what we believed before observation.

- Inductive prior
Some states of pre-observation belief can learn quickly; others never learn anything. An “inductive prior” is of the former type.

- Solomonoff induction
A simple way to superintelligently predict sequences of data, given unlimited computing power.

- Solomonoff induction: Intro Dialogue (Math 2)
An introduction to Solomonoff induction for the unfamiliar reader who isn’t bad at math

- Solomonoff induction: Intro Dialogue (Math 2)
- Laplace's Rule of Succession
Suppose you flip a coin with an unknown bias 30 times, and see 4 heads and 26 tails. The Rule of Succession says the next flip has a

^{5}⁄_{32}chance of showing heads. - Universal prior
A “universal prior” is a probability distribution containing

*all*the hypotheses, for some reasonable meaning of “all”. E.g., “every possible computer program that computes probabilities”.

- Solomonoff induction

- Inductive prior
- Strictly confused
A hypothesis is strictly confused by the raw data, if the hypothesis did much worse in predicting it than the hypothesis itself expected.

- Finishing your Bayesian path on Arbital
The page that comes at the end of reading the Arbital Guide to Bayes’ rule

- Prior
A state of prior knowledge, before seeing information on a new problem. Potentially complicated.

- Empirical probabilities are not exactly 0 or 1
“Cromwell’s Rule” says that probabilities of exactly 0 or 1 should never be applied to empirical propositions—there’s always some probability, however tiny, of being mistaken.

- Subjective probability
Probability is in the mind, not in the environment. If you don’t know whether a coin came up heads or tails, that’s a fact about you, not a fact about the coin.

- Probability distribution: Motivated definition
People keep writing things like P(sick)=0.3. What does this mean, on a technical level?

- Probability distribution: Motivated definition
- Likelihood
- Relative likelihood
How relatively likely an observation is, given two or more hypotheses, determines the strength and direction of evidence.

- Likelihood notation
- Likelihood function
- Likelihood ratio

- Relative likelihood

- Bayesian update
- Odds
Odds express a relative probability.

- Odds: Introduction
What’s the difference between probabilities and odds? Why is a 20% probability of success equivalent to 1 : 4 odds favoring success?

- Odds: Refresher
A quick review of the notations and mathematical behaviors for odds (e.g. odds of 1 : 2 for drawing a red ball vs. green ball from a barrel).

- Odds: Technical explanation
Formal definitions, alternate representations, and uses of odds and odds ratios (like a 1 : 2 chance of drawing a red ball vs. green ball from a barrel).

- Odds: Introduction
- Mutually exclusive and exhaustive
The condition needed for probabilities to sum to 1

- Probability
The degree to which someone believes something, measured on a scale from 0 to 1, allowing us to do math to it.

- Joint probability
The notation for writing the chance that both X and Y are true.

- Conditional probability
The notation for writing “The probability that someone has green eyes, if we know that they have red hair.”

- Conditional probability: Refresher
Is P(yellow | banana) the probability that a banana is yellow, or the probability that a yellow thing is a banana?

- Conditional probability: Refresher
- Interpretations of "probability"
What does it

*mean*to say that a fair coin has a 50% probability of coming up heads?- Subjective probability
Probability is in the mind, not in the environment. If you don’t know whether a coin came up heads or tails, that’s a fact about you, not a fact about the coin.

- Probability distribution: Motivated definition
People keep writing things like P(sick)=0.3. What does this mean, on a technical level?

- Probability distribution: Motivated definition
- Correspondence visualizations for different interpretations of "probability"
- Probability interpretations: Examples

- Subjective probability
- Report likelihoods, not p-values
- Likelihood functions, p-values, and the replication crisis
What’s the whole Bayesian-vs.-frequentist debate about?

- Report likelihoods not p-values: FAQ

- Likelihood functions, p-values, and the replication crisis

- Joint probability
- Normalization (probability)
That thingy we do to make sure our probabilities sum to 1, when they should sum to 1.

- Sample space
The set of possible things that could happen in a part of the world that you are uncertain about.

- Sample spaces are too large
Sample spaces are often large, so it is hard to do probabilistic computations using a raw distribution over the sample space.

- Uncountable sample spaces are way too large
We can’t define probability distributions over uncountable sample spaces by just assigning numbers to each point in the sample space.

- Sample spaces are too large
- Probability distribution (countable sample space)
A function assigning a probability to each point in the sample space.

- Square visualization of probabilities on two events
- Square visualization of probabilities on two events: (example) Diseasitis
But it

*seems*like the patient with the black tongue depressor has diseasitis…

- Square visualization of probabilities on two events: (example) Diseasitis
- Two independent events
What do pair of dice, pair of coins, and pair of people on opposite sides of the planet all have in common?

- Probability distribution: Motivated definition
People keep writing things like P(sick)=0.3. What does this mean, on a technical level?

- Well-calibrated probabilities
- Ability to read algebra
Do you have sufficient mathematical ability that you can read a sentence that uses some algebra or invokes a mathematical idea, without slowing down too much?

- Arithmetical hierarchy
The arithmetical hierarchy is a way of classifying logical statements by the number of clauses saying “for every object” and “there exists an object”.

- Ability to read logic
Can you read sentences symbolically stating “For all x: exists y: phi(x, y) or not theta(y)” without slowing down too much?

- Math 0
Are you not actively bad at math, nor traumatized about math?

- Math 1
Is math sometimes fun for you, and are you not anxious if you see a math puzzle you don’t know how to solve?

- Math 2
Do you work with math on a fairly routine basis? Do you have little trouble grasping abstract structures and ideas?

- Math 2 example statements
If you can read these formulas, you’re in Math 2!

- Math 2 example statements
- Math 3
Can you read the sort of things that professional mathematicians read, aka LaTeX formulas with a minimum of explanation?

- Math 3 example statements
If you can read these formulas, you’re in Math 3!

- Math 3 example statements
- Ability to read calculus
Can you take integral signs and differentiations in stride?

- Uncountability
Some infinities are bigger than others. Uncountable infinities are larger than countable infinities.

- Uncountability: Intro (Math 1)
Not all infinities are created equal. The infinity of real numbers is infinitely larger than the infinity of counting numbers.

- Uncountability: Intuitive Intro
Are all sizes of infinity the same? What does “the same” even mean here?

- Uncountability (Math 3)
Formal definition of uncountability, and foundational considerations.

- Real numbers are uncountable
The real numbers are uncountable.

- Uncountability: Intro (Math 1)
- Gödel encoding and self-reference
The formalism that mathematicians use to talk about arithmetic turns out to be able to talk about itself.

- Quine
A computer program that prints (or does other computations to) its own source code, using indirect self-reference.

- Quine
- Math playpen
Playpen page for Math domain

- Order theory
The study of binary relations that are reflexive, transitive, and antisymmetic.

- Partially ordered set
A set endowed with a relation that is reflexive, transitive, and antisymmetric.

- Poset: Examples
- Poset: Exercises
- Greatest lower bound in a poset
The greatest lower bound is an abstraction of the idea of the greatest common divisor to a general poset.

- Join and meet
- Lattice (Order Theory)
A poset that is closed under binary joins and meets.

- Totally ordered set
A set where all the elements can be compared as greater than or less than.

- Well-ordered set
An ordered set with an order that always has a “next element”.

- Well-ordered set
- Monotone function
An order-preserving map between posets.

- Complete lattice
A poset that is closed under arbitrary joins and meets.

- Partially ordered set
- Group theory
What kinds of symmetry can an object have?

- Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.

- Order of a group
- Abelian group
A group where the operation commutes. Named after Niels Henrik Abel.

- Group: Examples
Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today?

- Group: Exercises
Test your understanding of the definition of a group with these exercises.

- Group homomorphism
A group homomorphism is a “function between groups” that “respects the group structure”.

- Kernel of group homomorphism
- Image of the identity under a group homomorphism is the identity
All group homomorphisms preserve the identity.

- Under a group homomorphism, the image of the inverse is the inverse of the image
The operations of “taking inverses” and “applying a group homomorphism” commute: it does not matter in which order we do them.

- The image of a group under a homomorphism is a subgroup of the codomain
Group homomorphisms take groups to groups, but it is additionally guaranteed that the elements they hit form a group.

- The composition of two group homomorphisms is a homomorphism
The collection of group homomorphisms is closed under composition.

- Cyclic group
Cyclic groups form one of the most simple classes of groups.

- Cyclic Group Intro (Math 0)
A finite cyclic group is a little bit like a clock.

- Cyclic Group Intro (Math 0)
- Symmetric group
The symmetric groups form the fundamental link between group theory and the notion of symmetry.

- Cayley's Theorem on symmetric groups
The “fundamental theorem of symmetry”, forging the connection between symmetry and group theory.

- Cycle notation in symmetric groups
Cycle notation is a convenient way to represent the elements of a symmetric group.

- Disjoint cycle notation is unique
Disjoint cycle notation provides a canonical way to express elements of the symmetric group.

- Cycle type of a permutation
The cycle type is an invariant of a permutation in the symmetric group.

- Disjoint cycle notation is unique
- Disjoint cycles commute in symmetric groups
In cycle notation, if two cycles are disjoint, then they commute.

- Conjugacy class is cycle type in symmetric group
There is a neat characterisation of the conjugacy classes in the symmetric group on a finite set.

- Conjugacy classes of the symmetric group on five elements
The symmetric group on five elements is a group of just the right size to make a good example of a table of conjugacy classes.

- Transposition (as an element of a symmetric group)
A transposition is the simplest kind of permutation: it swaps two elements.

- Every member of a symmetric group on finitely many elements is a product of transpositions
This fact can often simplify arguments about permutations: if we can show that something holds for transpositions, and that it holds for products, then it holds for everything.

- The sign of a permutation is well-defined
This result is what allows the alternating group to exist.

- Sign homomorphism (from the symmetric group)
The sign homomorphism is how we extract the alternating group from the symmetric group.

- Cayley's Theorem on symmetric groups
- Group isomorphism
“Isomorphism” is the proper notion of “sameness” or “equality” among groups.

- Order of a group element
- Dihedral group
The dihedral groups are natural examples of groups, arising from the symmetries of regular polygons.

- Dihedral groups are non-abelian
The group of symmetries of the triangle and all larger regular polyhedra are not abelian.

- Dihedral groups are non-abelian
- Group conjugate
Conjugation lets us perform permutations “from the point of view of” another permutation.

- Normal subgroup
Normal subgroups are subgroups which are in some sense “the same from all points of view”.

- Subgroup is normal if and only if it is the kernel of a homomorphism
The “correct way” to think about normal subgroups is as kernels of homomorphisms.

- Quotient by subgroup is well defined if and only if subgroup is normal
- Subgroup is normal if and only if it is a union of conjugacy classes
A useful way to think about normal subgroups, which meshes with their “closed under conjugation” interpretation.

- Subgroup is normal if and only if it is the kernel of a homomorphism
- Alternating group
The alternating group is the only normal subgroup of the symmetric group (on five or more generators).

- The collection of even-signed permutations is a group
This proves the well-definedness of one particular definition of the alternating group.

- Alternating group is generated by its three-cycles
A useful result which lets us prove things about the alternating group more easily.

- The alternating groups on more than four letters are simple
The alternating groups are the most accessible examples of simple groups, and this fact also tells us that the symmetric groups are “complicated” in some sense.

- The alternating group on five elements is simple
The smallest (nontrivial) simple group is the alternating group on five elements.

- The alternating group on five elements is simple: Simpler proof
A proof which avoids some of the heavy machinery of the main proof.

- The alternating group on five elements is simple: Simpler proof
- Conjugacy classes of the alternating group on five elements
\(A_5\) has easily-characterised conjugacy classes, based on a rather surprising theorem about when conjugacy classes in the symmetric group split.

- Conjugacy classes of the alternating group on five elements: Simpler proof
A listing of the conjugacy classes of the alternating group on five letters, without using heavy theory.

- Conjugacy classes of the alternating group on five elements: Simpler proof
- Splitting conjugacy classes in alternating group
The conjugacy classes in the alternating group are usually the same as those in the symmetric group; there is a surprisingly simple condition for when this does not hold.

- The collection of even-signed permutations is a group
- Group coset
- Left cosets partition the parent group
In a group, every element has a unique coset in which it lies, allowing us to compress some of the information about the group.

- Left cosets are all in bijection
The left cosets of a subgroup in a parent group are all the same size.

- Left cosets partition the parent group
- Simple group
The simple groups form the “building blocks” of group theory, analogously to the prime numbers in number theory.

- Prime order groups are cyclic
This is the first step on the road to classifying the finite groups.

- Lagrange theorem on subgroup size
Lagrange’s Theorem is an important restriction on the sizes of subgroups of a finite group.

- Lagrange theorem on subgroup size: Intuitive version
Lagrange’s theorem strongly restricts the size a subgroup of a group can be.

- Lagrange theorem on subgroup size: Intuitive version
- Cauchy's theorem on subgroup existence
Cauchy’s theorem is a useful condition for the existence of cyclic subgroups of finite groups.

- Group orbit
- Cyclic Group Intro (Math 0)
A finite cyclic group is a little bit like a clock.

- Subgroup
A group that lives inside a bigger group.

- Group presentation
Presentations are a fairly compact way of expressing groups.

- Every group is a quotient of a free group
- Free group
The free group is “the purest way to make a group containing a given set”.

- Free groups are torsion-free
An easy way to determine that many groups are not free: free groups contain no non-identity elements of finite order.

- Formal definition of the free group
Van der Waerden’s trick lets us define the free groups in a slick but mostly incomprehensible way.

- Free group universal property

- Free groups are torsion-free

- Group theory: Examples
What does thinking in terms of group theory actually look like? And what does it buy you?

- Group action
“Groups, as men, will be known by their actions.”

- Group action induces homomorphism to the symmetric group
We can view group actions as “bundles of homomorphisms” which behave in a certain way.

- Orbit-stabiliser theorem
The Orbit-Stabiliser theorem tells us a lot about how a group acts on a given element.

- Orbit-Stabiliser theorem: External Resources
External resources on the Orbit-Stabiliser theorem.

- Orbit-Stabiliser theorem: External Resources
- Stabiliser is a subgroup
Given a group acting on a set, each element of the set induces a subgroup of the group.

- Group orbits partition
When a group acts on a set, the set falls naturally into distinct pieces, where the group action only permutes elements within any given piece, not between them.

- Stabiliser (of a group action)
If a group acts on a set, it is useful to consider which elements of the group don’t move a certain element of the set.

- Group action induces homomorphism to the symmetric group

- Group
- Closure
- Abstract algebra
The study of groups, fields, vector spaces, arithmetics, algebras, and more.

- Algebraic structure
- Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.

- Order of a group
- Abelian group
A group where the operation commutes. Named after Niels Henrik Abel.

- Group: Examples
Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today?

- Group: Exercises
Test your understanding of the definition of a group with these exercises.

- Group homomorphism
A group homomorphism is a “function between groups” that “respects the group structure”.

- Kernel of group homomorphism
- Image of the identity under a group homomorphism is the identity
All group homomorphisms preserve the identity.

- Under a group homomorphism, the image of the inverse is the inverse of the image
The operations of “taking inverses” and “applying a group homomorphism” commute: it does not matter in which order we do them.

- The image of a group under a homomorphism is a subgroup of the codomain
Group homomorphisms take groups to groups, but it is additionally guaranteed that the elements they hit form a group.

- The composition of two group homomorphisms is a homomorphism
The collection of group homomorphisms is closed under composition.

- Cyclic group
Cyclic groups form one of the most simple classes of groups.

- Cyclic Group Intro (Math 0)
A finite cyclic group is a little bit like a clock.

- Cyclic Group Intro (Math 0)
- Symmetric group
The symmetric groups form the fundamental link between group theory and the notion of symmetry.

- Cayley's Theorem on symmetric groups
The “fundamental theorem of symmetry”, forging the connection between symmetry and group theory.

- Cycle notation in symmetric groups
Cycle notation is a convenient way to represent the elements of a symmetric group.

- Disjoint cycle notation is unique
Disjoint cycle notation provides a canonical way to express elements of the symmetric group.

- Cycle type of a permutation
The cycle type is an invariant of a permutation in the symmetric group.

- Disjoint cycle notation is unique
- Disjoint cycles commute in symmetric groups
In cycle notation, if two cycles are disjoint, then they commute.

- Conjugacy class is cycle type in symmetric group
There is a neat characterisation of the conjugacy classes in the symmetric group on a finite set.

- Conjugacy classes of the symmetric group on five elements
The symmetric group on five elements is a group of just the right size to make a good example of a table of conjugacy classes.

- Transposition (as an element of a symmetric group)
A transposition is the simplest kind of permutation: it swaps two elements.

- Every member of a symmetric group on finitely many elements is a product of transpositions
This fact can often simplify arguments about permutations: if we can show that something holds for transpositions, and that it holds for products, then it holds for everything.

- The sign of a permutation is well-defined
This result is what allows the alternating group to exist.

- Sign homomorphism (from the symmetric group)
The sign homomorphism is how we extract the alternating group from the symmetric group.

- Cayley's Theorem on symmetric groups
- Group isomorphism
“Isomorphism” is the proper notion of “sameness” or “equality” among groups.

- Order of a group element
- Dihedral group
The dihedral groups are natural examples of groups, arising from the symmetries of regular polygons.

- Dihedral groups are non-abelian
The group of symmetries of the triangle and all larger regular polyhedra are not abelian.

- Dihedral groups are non-abelian
- Group conjugate
Conjugation lets us perform permutations “from the point of view of” another permutation.

- Normal subgroup
Normal subgroups are subgroups which are in some sense “the same from all points of view”.

- Subgroup is normal if and only if it is the kernel of a homomorphism
The “correct way” to think about normal subgroups is as kernels of homomorphisms.

- Quotient by subgroup is well defined if and only if subgroup is normal
- Subgroup is normal if and only if it is a union of conjugacy classes
A useful way to think about normal subgroups, which meshes with their “closed under conjugation” interpretation.

- Subgroup is normal if and only if it is the kernel of a homomorphism
- Alternating group
The alternating group is the only normal subgroup of the symmetric group (on five or more generators).

- The collection of even-signed permutations is a group
This proves the well-definedness of one particular definition of the alternating group.

- Alternating group is generated by its three-cycles
A useful result which lets us prove things about the alternating group more easily.

- The alternating groups on more than four letters are simple
The alternating groups are the most accessible examples of simple groups, and this fact also tells us that the symmetric groups are “complicated” in some sense.

- The alternating group on five elements is simple
The smallest (nontrivial) simple group is the alternating group on five elements.

- The alternating group on five elements is simple: Simpler proof
A proof which avoids some of the heavy machinery of the main proof.

- The alternating group on five elements is simple: Simpler proof
- Conjugacy classes of the alternating group on five elements
\(A_5\) has easily-characterised conjugacy classes, based on a rather surprising theorem about when conjugacy classes in the symmetric group split.

- Conjugacy classes of the alternating group on five elements: Simpler proof
A listing of the conjugacy classes of the alternating group on five letters, without using heavy theory.

- Conjugacy classes of the alternating group on five elements: Simpler proof
- Splitting conjugacy classes in alternating group
The conjugacy classes in the alternating group are usually the same as those in the symmetric group; there is a surprisingly simple condition for when this does not hold.

- The collection of even-signed permutations is a group
- Group coset
- Left cosets partition the parent group
In a group, every element has a unique coset in which it lies, allowing us to compress some of the information about the group.

- Left cosets are all in bijection
The left cosets of a subgroup in a parent group are all the same size.

- Left cosets partition the parent group
- Simple group
The simple groups form the “building blocks” of group theory, analogously to the prime numbers in number theory.

- Prime order groups are cyclic
This is the first step on the road to classifying the finite groups.

- Lagrange theorem on subgroup size
Lagrange’s Theorem is an important restriction on the sizes of subgroups of a finite group.

- Lagrange theorem on subgroup size: Intuitive version
Lagrange’s theorem strongly restricts the size a subgroup of a group can be.

- Lagrange theorem on subgroup size: Intuitive version
- Cauchy's theorem on subgroup existence
Cauchy’s theorem is a useful condition for the existence of cyclic subgroups of finite groups.

- Group orbit
- Cyclic Group Intro (Math 0)
A finite cyclic group is a little bit like a clock.

- Subgroup
A group that lives inside a bigger group.

- Group presentation
Presentations are a fairly compact way of expressing groups.

- Every group is a quotient of a free group
- Free group
The free group is “the purest way to make a group containing a given set”.

- Free groups are torsion-free
An easy way to determine that many groups are not free: free groups contain no non-identity elements of finite order.

- Formal definition of the free group
Van der Waerden’s trick lets us define the free groups in a slick but mostly incomprehensible way.

- Free group universal property

- Free groups are torsion-free

- Ring
- Ordered ring
A ring with a total ordering compatible with its ring structure.

- Irreducible element (ring theory)
This is the appropriate abstraction of the concept of “prime number” to general rings.

- Prime element of a ring
Despite the name, “prime” in ring theory refers not to elements which are “multiplicatively irreducible” but to those such that if they divide a product then they divide some term of the product.

- Integral domain
An integral domain is a ring where the only way to express zero as a product is by having zero as one of the terms.

- In a principal ideal domain, "prime" and "irreducible" are the same
Principal ideal domains have a very useful property that we don’t need to distinguish between the informal notion of “prime” (i.e. “irreducible”) and the formal notion.

- Unit (ring theory)
A unit in a ring is just an element with a multiplicative inverse.

- Principal ideal domain
A principal ideal domain is a kind of ring, in which all ideals have a certain nice form.

- Kernel of ring homomorphism
The kernel of a ring homomorphism is the collection of things which that homomorphism sends to 0.

- Ideals are the same thing as kernels of ring homomorphisms
- Euclidean domains are principal ideal domains
A Euclidean domain is one where we may somehow perform the division algorithm; this gives us access to some of the nicest properties of the integers.

- Unique factorisation domain
This is the correct way to abstract from the integers the fact that every integer can be written uniquely as a product of prime numbers.

- Ordered ring
- Abelian group
A group where the operation commutes. Named after Niels Henrik Abel.

- Monoid
- Algebraic structure tree
When is a monoid a semilattice? What’s the difference between a semigroup and a groupoid? Find out here!

- Group
- Underlying set

- Algebraic structure
- Associative operation
- Commutative operation
- String (of text)
- Function
- Operator
- Arity (of a function)
- Domain (of a function)
- concat (function)
- Binary function
- Codomain (of a function)
- Image (of a function)
- Range (of a function)
- Codomain vs image
- Function: Physical metaphor
- Ceiling
- Ackermann function
The slowest-growing fast-growing function.

- Bijective function
A bijective function is a function with an inverse.

- Bijective Function: Intro (Math 0)
Two boxes are bijective if they contain the same number of items.

- Bijective Function: Intro (Math 0)
- Injective function
- Surjective function
A surjective function is one which “hits everything in the codomain”.

- Inverse function
The inverse of a function returns an input of the original function when fed the original’s corresponding output.

- Exponential
Any function that constantly gets larger as a proportion of itself.

- Currying
Transforms a function of many arguments into a function into a function of a single argument

- Convex function
A function that only curves upward

- Partial function
A partial function is one which “might not be defined everywhere one might expect it to be”.

- Set
An unordered collection of distinct objects.

- Set builder notation
- Cardinality
The “size” of a set, or the “number of elements” that it has.

- Convex set
A set that contains all line segments between points in the set

- Operations in Set theory
- Cartesian product
- Absolute Complement
- Union
The union of two sets is the set of elements which are in one or the other, or both

- Intersection
The intersection of two sets is the set of elements they have in common

- Relative complement

- Cantor-Schröder-Bernstein theorem
This theorem tells us that comparing sizes of sets makes sense: if one set is smaller than another, and the other is smaller than the one, then they are the same size.

- Disjoint union of sets
One of the most basic ways we have of joining two sets together.

- Universal property of the disjoint union
Just as the empty set may be described by a universal property, so too may the disjoint union of sets.

- Universal property of the disjoint union
- Empty set
The empty set does what it says on the tin: it is the set which is empty.

- Set product
A fundamental way of combining sets is to take their product, making a set that contains all tuples of elements from the originals.

- Finite set
A finite set is one which is not infinite. Some of these are the least complicated sets.

- Category of finite sets
The category of finite sets is exactly what it claims to be. It’s a useful training ground for some of the ideas of category theory.

- Category of finite sets
- Extensionality Axiom
If two sets have exactly the same members, then they are equal

- Logarithm
- Logarithm: Examples
- Logarithm: Exercises
- Introductory guide to logarithms
- Logarithmic identities
- Logarithms invert exponentials
- Logarithm tutorial overview
- What is a logarithm?
- Log as generalized length
To estimate the log (base 10) of a number, count how many digits it has.

- Logarithm base 1
There is no log base 1.

- Exchange rates between digits
In terms of data storage, if a coin is worth \(1, a digit wheel is worth more than \)3.32, but less than $3.33. Why?

- Why is log like length?
- Fractional digits
- Log as the change in the cost of communicating
- The characteristic of the logarithm
- Properties of the logarithm
- Log base infinity
There is no log base infinity, but if there were, it would send everything to zero

- There is only one logarithm
All logarithm functions are the same, up to a multiplicative constant.

- The log lattice
- Life in logspace
- The End (of the basic log tutorial)
- Why is the decimal expansion of log2(3) infinite?
Because 2 and 3 are relatively prime.

- Relation
- Equivalence relation
A relation that allows you to partition a set into equivalence classes.

- Order relation
A way of determining which elements of a set come “before” or “after” other elements.

- Transitive relation
If a is related to b and b is related to c, then a is related to c.

- Reflexive relation
- Antisymmetric relation
A binary relation where no two distinct elements are related in both directions

- Equivalence relation
- Information theory
- Bit (of data)
- Fractional bits
- GalCom
- n-message
- Decit
Decimal digit

- How many bits to a trit?
- Trit
Trinary digit

- Intradependent encodings can be compressed
- Intradependent encoding
- Dependent messages can be encoded cheaply
- Information
- Data capacity
- Compressing multiple messages
- Communication: magician example
- Algorithmic complexity
When you compress the information, what you are left with determines the complexity.

- Most complex things are not very compressible
We can’t

*prove*it’s impossible, but it would be*extremely surprising*to discover a 500-state Turing machine that output the exact text of “Romeo and Juliet”.

- Most complex things are not very compressible

- Type theory
Modern foundations for formal mathematics.

- Programming in Dependent Type Theory
Working with simple types in Lean

- Programming in Dependent Type Theory
- Linear algebra
- Eigenvalues and eigenvectors
We applied this linear transformation to one of its eigenvectors; you won’t believe what happened next!

- Vector space
- Subspace

- Eigenvalues and eigenvectors
- Real analysis
The study of real numbers and real-valued functions.

- Colon-to notation
Find out what the notation “f : X → Y” means that everyone keeps using.

- Decit
Decimal digit

- Trit
Trinary digit

- Bit
- Bit (abstract)
- Digit wheel
- Natural number
The numbers we use to count: 0, 1, 2, 3, …

- Graham's number
A fairly large number, as numbers go.

- A googolplex
A moderately large number, as large numbers go.

- A googol
A pretty small large number.

- Prime number
The prime numbers are the “building blocks” of the counting numbers.

- Proof that there are infinitely many primes
Suppose there were finitely many primes. Then consider the product of all the primes plus 1…

- Proof that there are infinitely many primes

- Graham's number
- Uncomputability
The diagonal function and the halting problem

- Iff
If and only if…

- Derivative
How things change

- Integer
- Integers: Intro (Math 0)
The integers are the whole numbers extended into the negatives.

- Integers: Intro (Math 0)
- Pi
- Pi is irrational
The number pi is famously not rational, in spite of joking attempts at legislation to fix its value at 3 or

^{22}⁄_{7}.

- Pi is irrational
- Complexity theory
Study of the computational resources needed to compute something

- Complexity theory: Complexity zoo
Pass and see the exotic beasts coming from the lands of afar!

- P vs NP
Is creativity purely mechanical?

- P vs NP: Arguments against P=NP
Why we believe P and NP are different

- P vs NP: Arguments against P=NP
- Decision problem
Formalization of general problems

- P (Polynomial Time Complexity Class)
P is the class of problems which can be solved by algorithms whose run time is bounded by a polynomial.

- Complexity theory: Complexity zoo
- Expected value
Trying to assign value to an uncertain state? The weighted average of outcomes is probably the tool you need.

- Real number
- Real number (as Cauchy sequence)
There are several ways to construct real numbers; this is the most natural way to use them in computations.

- The reals (constructed as classes of Cauchy sequences of rationals) form a field
The reals are an archetypal example of a field, but if we are to construct them from simpler objects, we need to show that our construction does indeed have the right properties.

- The reals (constructed as classes of Cauchy sequences of rationals) form a field
- Real number (as Dedekind cut)
A way to construct the real numbers that follows the intuition of filling in the gaps.

- The reals (constructed as Dedekind cuts) form a field
The reals are an archetypal example of a field, but if we are to construct them from simpler objects, we need to show that our construction does indeed have the right properties.

- The reals (constructed as Dedekind cuts) form a field
- Real numbers are uncountable
The real numbers are uncountable.

- Real number (as Cauchy sequence)
- P vs NP
Is creativity purely mechanical?

- P vs NP: Arguments against P=NP
Why we believe P and NP are different

- P vs NP: Arguments against P=NP
- Conjugacy class
In a group, the elements can be partitioned naturally into certain classes.

- Church-Turing thesis
A thesis about computational models

- Strong Church Turing thesis
A strengthening of the Church Turing thesis

- Church-Turing thesis: Evidence for the Church-Turing thesis
Why do we believe in CT thesis?

- Strong Church Turing thesis
- Category theory
How mathematical objects are related to others in the same category.

- Category (mathematics)
A description of how a collection of mathematical objects are related to one another.

- Equaliser (category theory)
- Product (Category Theory)
How a product is characterized rather than how it’s constructed

- Universal property of the product
The product can be defined in a very general way, applicable to the natural numbers, to sets, to algebraic structures, and so on.

- Product is unique up to isomorphism
If something satisfies the universal property of the product, then it is uniquely specified by that property, up to isomorphism.

- Product is unique up to isomorphism

- Universal property of the product
- Object identity via interactions
If we think of objects as opaque “black boxes”, how can we tell whether two objects are different? By looking at how they interact with other objects!

- Universal property
A universal property is a way of defining an object based purely on how it interacts with other objects, rather than by any internal property of the object itself.

- Universal property of the empty set
The empty set can be characterised by how it interacts with other sets, rather than by any explicit property of the empty set itself.

- The empty set is the only set which satisfies the universal property of the empty set
This theorem tells us that the universal property provides a sensible way to define the empty set uniquely.

- The empty set is the only set which satisfies the universal property of the empty set
- Universal property of the product
The product can be defined in a very general way, applicable to the natural numbers, to sets, to algebraic structures, and so on.

- Product is unique up to isomorphism
If something satisfies the universal property of the product, then it is uniquely specified by that property, up to isomorphism.

- Product is unique up to isomorphism
- Universal property of the disjoint union
Just as the empty set may be described by a universal property, so too may the disjoint union of sets.

- Universal property of the empty set
- Category of finite sets
The category of finite sets is exactly what it claims to be. It’s a useful training ground for some of the ideas of category theory.

- Category (mathematics)
- Morphism
- Isomorphism
A morphism between two objects which describes how they are “essentially equivalent” for the purposes of the theory under consideration.

- Group isomorphism
“Isomorphism” is the proper notion of “sameness” or “equality” among groups.

- Isomorphism: Intro (Math 0)
Things which are basically the same, except for some stuff you don’t care about.

- Up to isomorphism
A phrase mathematicians use when saying “we only care about the structure of an object, not about specific implementation details of the object”.

- Group isomorphism
- Computer Programming Familiarity
Want to see programming analogies and applications in your math explanations? Mark this as known.

- n-digit
- Emulating digits
- Decimal notation
The winning architecture for numerals

- 0.999...=1
No, it’s not “infinitesimally far” from 1 or anything like that. 0.999… and 1 are literally the same number.

- 0.999...=1
- Quotient group
- Proportion
A representation of a value as a fraction or multiple of another value.

- Rational number
The rational numbers are “fractions”.

- The rationals form a field
- Rational numbers: Intro (Math 0)
- Arithmetic of rational numbers (Math 0)
How do we combine rational numbers together?

- Addition of rational numbers (Math 0)
The simplest operation on rational numbers is addition.

- Addition of rational numbers exercises
Test and cement your understanding of how we add rational numbers!

- Addition of rational numbers exercises
- Subtraction of rational numbers (Math 0)
In which we meet anti-apples.

- Division of rational numbers (Math 0)
“Division” is the idea of “dividing something up among some people so that we can give equal amounts to each person”.

- Rational arithmetic all works together
The various operations of arithmetic all play nicely together in a certain specific way.

- Order of rational operations (Math 0)
Our shorthand for all the operations on rationals is very useful, but full of brackets; this is how to get rid of some of the brackets.

- Addition of rational numbers (Math 0)
- Field structure of rational numbers
In which we describe the field structure on the rationals.

- Arithmetic of rational numbers (Math 0)
How do we combine rational numbers together?

- Addition of rational numbers (Math 0)
The simplest operation on rational numbers is addition.

- Addition of rational numbers exercises
Test and cement your understanding of how we add rational numbers!

- Addition of rational numbers exercises
- Subtraction of rational numbers (Math 0)
In which we meet anti-apples.

- Division of rational numbers (Math 0)
“Division” is the idea of “dividing something up among some people so that we can give equal amounts to each person”.

- Rational arithmetic all works together
The various operations of arithmetic all play nicely together in a certain specific way.

- Order of rational operations (Math 0)
Our shorthand for all the operations on rationals is very useful, but full of brackets; this is how to get rid of some of the brackets.

- Addition of rational numbers (Math 0)

- Arithmetic of rational numbers (Math 0)

- Currying
Transforms a function of many arguments into a function into a function of a single argument

- The set of rational numbers is countable
Although there are “lots and lots” of rational numbers, there are still only countably many of them.

- Elementary Algebra
- Modal logic
The logic of boxes and bots.

- Standard provability predicate
Encoding provability as a statement of arithmetic

- Normal system of provability logic
- Provability logic
Learn how to reason about provability!

- Fixed point theorem of provability logic
Deal with those pesky self-referential sentences!

- Solovay's theorems of arithmetical adequacy for GL
Using GL to reason about PA, and viceversa

- Fixed point theorem of provability logic
- Kripke model
The semantics of modal logic

- Modalized modal sentence
- Modal combat
Modal combat

- Standard provability predicate
- Intro to Number Sets
An introduction to number sets for people who have no idea what a number set is.

- Order of operations
Conventions used for disambiguating infix notation.

- Order of rational operations (Math 0)
Our shorthand for all the operations on rationals is very useful, but full of brackets; this is how to get rid of some of the brackets.

- Order of rational operations (Math 0)
- Irrational number
Real numbers that are not rational numbers

- Pi is irrational
The number pi is famously not rational, in spite of joking attempts at legislation to fix its value at 3 or

^{22}⁄_{7}. - The square root of 2 is irrational
The number whose square is 2 can’t be written is a quotient of natural numbers

- Pi is irrational
- Logistic function
A monotonic function from the real numbers to the open unit interval.

- Löb's theorem
Löb’s theorem

- Proof of Löb's theorem
Proving that I am Santa Claus

- Löb's theorem and computer programs
- Gödel II and Löb's theorem

- Proof of Löb's theorem
- The n-th root of m is either an integer or irrational
In other words, no power of a rational number that is not an integer is ever an integer.

- Binary notation
A way to write down numbers using powers of two.

- Boolean
A value in logic that evaluates to either “true” or “false”.

- Diagonal lemma
Constructing self-referential sentences

- Quine
A computer program that prints (or does other computations to) its own source code, using indirect self-reference.

- Quine
- Gödel's first incompleteness theorem
The theorem that destroyed Hilbert’s program

- Multiplication of rational numbers (Math 0)
“Multiplication” is the idea of “now do the same as you just did, but instead of doing it to one apple, do it to some other number”.

- Factorial
The number of ways you can order things. (Alternately subtitled: Is that exclamation point a factorial, or are you just excited to see me?)

- Provability predicate
- Freely reduced word
“Freely reduced” captures the idea of “no cancellation” in a free group.

- Exponential notation for function spaces
Why \(Y^X\) is good notation for the space of maps from \(X\) to \(Y\)

- Consistency
- Euclid's Lemma on prime numbers
A very basic but vitally important property of the prime numbers is that they “can’t be split between factors”: if a prime divides a product then it must divide one of the individual factors.

- Greatest common divisor
The greatest common divisor of two natural numbers is… the largest number which is a divisor of both. The clue is in the name, really.

- Bézout's theorem
Bézout’s theorem is an important link between highest common factors and the integer solutions of a certain equation.

- Bézout's theorem
- Metric
A metric is a function that defines a distance between elements in a set and follows some basic rules.

- Modular arithmetic
Addition as traveling around a circle, instead of along a line.

- Turing machine
A Turing Machine is a simple mathematical model of computation that is powerful enough to describe any computation a computer can do.

- Rice's Theorem
Rice’s Theorem tells us that if we want to determine pretty much anything about the behaviour of an arbitrary computer program, we can’t in general do better than just running it.

- Rice's theorem and the Halting problem
- Proof of Rice's theorem
A standalone proof of Rice’s theorem, including one surprising lemma.

- Rice's Theorem: Intro (Math 1)
You can’t write a program that looks at another programs source code, and tells you whether it computes the Fibonacci sequence.

- Turing machine: External resources

- Rice's Theorem
- Ordering of rational numbers (Math 0)
How do we know if one lot of apples is “more apples” than another lot?

- Fundamental Theorem of Arithmetic
The FTA tells us that natural numbers can be decomposed uniquely into prime factors; it is the basis of almost all number theory.

- Well-defined
A mathematical object is “well-defined” if we have given it a completely unambiguous definition.

- Ordered field
An ordered ring with division.

- LaTeX
- Mapsto notation
There’s an arrow called \mapsto in LaTeX. What does it mean?

- In notation
There’s a weird E-looking symbol called \in in LaTeX. What does it mean?

- Mapsto notation
- Mathematical object
- Bag
- List
- Number
An abstract object that expresses quantity or value of some sort.

- Complex number
- Whole number
A term that can refer to three different sets of “numbers that are not fractions”.

- Transcendental number
A transcendental number is one which is not the root of any integer-coefficient polynomial.

- Element
- Identity element
An element in a set with a binary operation that leaves every element unchanged when used as the other operand.

- Identity element
- Proof technique
- Proof by contradiction
Discover what ‘reductio ad absurdum’ means!

- Mathematical induction
Proving a statement about all positive integers by knocking them down like dominoes.

- Proof by contradiction
- Generalized element
A category-theoretic generalization of the notion of element of a set.

- Examination through isomorphism
- Least common multiple
- An introductory guide to modern logic
Logic, provability, Löb, Gödel and more!

- Axiom
- Axiom of Choice
The most controversial axiom of the 20th century.

- Axiom of Choice: Guide
Learn about the most controversial axiom of the 20th century.

- Axiom of Choice: Introduction
What is the axiom of choice and why do I care?

- Axiom of Choice: Definition (Formal)
Mathematically speaking, what does the Axiom of Choice say?

- Axiom of Choice Definition (Intuitive)
Definition of the Axiom of Choice, without using heavy mathematical notation.

- Axiom of Choice Definition (Intuitive)
- Axiom of Choice: History and Controversy
Really? The

*most*controversial axiom of the 20th century? Yes. - Axiom of Choice Definition (Intuitive)
Definition of the Axiom of Choice, without using heavy mathematical notation.

- Axiom of Choice: Guide

- Axiom of Choice
- Countability
Some infinities are bigger than others. Countable infinities are the smallest infinities.

- Featured math content
Some Arbital pages we think are great!

- Primer on Infinite Series
What does it mean to add things together forever?

- Lambda calculus
A minimal, inefficient, and hard-to-read, but still interesting and useful, programming language.

- Square root
What is the opposite of multiplying a number by itself?