What is the type of 2 + 2 = 4?
All elementary functions from a single operator
A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin\sin, cos\cos, \sqrt{\phantom{x}}, and log\log has always required multiple distinct operations. Here we show that a single binary operator,
Grade 3 » Operations & Algebraic Thinking » Represent and solve problems involving multiplication and division. » 1 | Common Core State Standards Initiative
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
Mathematics in type theory.
An explanation of how to set up mathematics using universes, types, and terms
Parametric Subtyping for Structural Parametric Polymorphism
Recursive types, generics (sometimes called parametric polymorphism), and subtyping are all essential features for modern programming languages across numerous paradigms. However, structural subtyp…
A theory of type polymorphism in programming
The aim of this work is largely a practical one. A widely employed style of programming, particularly in structure-processing languages which impose n…
ignore the code: Toggles Considered Harmful
I just read this post from Marcin Wichary's Unsung, where he points out a peculiar macOS dialog box that shows boolean toggles in four different shades of grey:
4-Clause BSD Explained in Plain English - TLDRLegal
4-Clause BSD summarized/explained in plain English.
Syntax and Semantics of Linear Dependent Types
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax...
Prologue: How to Program
Naturally, when DrRacket calculates for you, it uses the rules that you know and love from math. Like you, it can determine the result of an addition only when all the operands are plain numbers. If an operand is a parenthesized operator expression—something that starts with a “(” and an operation—it determines the result of that nested expression first. Unlike you, it never needs to ponder which expression to calculate first—because this first rule is the only rule there is.
With Category Theory, Mathematics Escapes From Equality | Quanta Magazine
Two monumental works have led many mathematicians to avoid the equal sign. The process has not always gone smoothly.
Variety (cybernetics)
In cybernetics, the term variety denotes the total number of distinguishable elements of a set, most often the set of states, inputs, or outputs of a finite-state machine or transformation, or the binary logarithm of the same quantity.[1] Variety is used in cybernetics as an information theory that is easily related to deterministic finite automata, and less formally as a conceptual tool for thinking about organization, regulation, and stability. It is an early theory of complexity in automata, complex systems,[1]: 6  and operations research.[2]
A modal analysis of staged computation | Journal of the ACM
We show that a type system based on the intuitionistic modal logic S4 provides an expressive framework for specifying and analyzing computation stages in the context of typed λ-calculi and functional languages. We directly demonstrate the sense in which ...
The Principal Type-Scheme of an Object in Combinatory Logic
R. Hindley, The Principal Type-Scheme of an Object in Combinatory Logic, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 29-60