# Mathematics

Mathematics is the study of crisply specified formal objects — for example, numbers — and the ways of knowing their properties — such as proofs. We can see “logic” as the study of “which conclusions follow with certainty from which premises”. Using this definition of logic, we can also see mathematics as the study of logical objects in logical universes — entities whose properties follow from specifications about them, rather than from observation of the real world. The number 3 is a logical object because its behavior follows from axioms about addition and multiplication; Mount Everest is a physical object because we learn about it by physically measuring Mount Everest.

Children:

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

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

- 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.

- Gödel encoding and self-reference
The formalism that mathematicians use to talk about arithmetic turns out to be able to talk about itself.

- Math playpen
Playpen page for Math domain

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

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

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

- Associative operation
- Commutative operation
- String (of text)
- Function
- Set
An unordered collection of distinct objects.

- Logarithm
- Relation
- Information theory
- Type theory
Modern foundations for formal mathematics.

- Linear algebra
- 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, …

- Uncomputability
The diagonal function and the halting problem

- Iff
If and only if…

- Derivative
How things change

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

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

- Real number
- P vs NP
Is creativity purely mechanical?

- Conjugacy class
In a group, the elements can be partitioned naturally into certain classes.

- Church-Turing thesis
A thesis about computational models

- Category theory
How mathematical objects are related to others in the same category.

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

- 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

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

- Rational number
The rational numbers are “fractions”.

- 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.

- 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.

- Irrational number
Real numbers that are not rational numbers

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

- Löb's theorem
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

- 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.

- 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.

- 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
- Mathematical object
- Element
- Proof technique
- 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
- 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?