doc/contexts.md

@@ -1,50 +1,57 @@
 # Context-oriented mathematics
 
 ## The problem
-A known problem for implementing mathematics in statically-typed languages (and not only in them) is that different 
+A known problem for implementing mathematics in statically-typed languages (but not only in them) is that different
-sets of mathematical operation could be defined on the same mathematical objects. Sometimes there is not single way to
+sets of mathematical operators can be defined on the same mathematical objects. Sometimes there is no single way to
-treat some operations like basic arithmetic operations on Java/Kotlin `Number`. Sometimes there are different ways to do
+treat some operations, including basic arithmetic operations, on a Java/Kotlin `Number`. Sometimes there are different ways to
-the same thing like Euclidean and elliptic geometry vector spaces defined over real vectors. Another problem arises when 
+define the same structure, such as Euclidean and elliptic geometry vector spaces over real vectors. Another problem arises when
-one wants to add some kind of behavior to existing entity. In dynamic languages those problems are usually solved
+one wants to add some kind of behavior to an existing entity. In dynamic languages those problems are usually solved
-by adding dynamic context-specific behaviors in runtime, but this solution has a lot of drawbacks.
+by adding dynamic context-specific behaviors at runtime, but this solution has a lot of drawbacks.
 
 ## Context-oriented approach
-One of possible solutions to those problems is to completely separate object numerical representations from behaviors.
+
-In terms of kotlin it means to have separate class to represent some entity without any operations, 
+One possible solution to these problems is to completely separate numerical representations from behaviors.
+One solution in Kotlin, is to define a separate class which represents some entity without any operations,
 for example a complex number:
 
 ```kotlin
 data class Complex(val re: Double, val im: Double)
 ```
-And a separate class or singleton, representing operation on those complex numbers:
+
+And then define a separate class or singleton, representing an operation on those complex numbers:
+
 ```kotlin
-object: ComplexOperations{
+object ComplexOperations {
     operator fun Complex.plus(other: Complex) = Complex(re + other.re, im + other.im)
     operator fun Complex.minus(other: Complex) = Complex(re - other.re, im - other.im)
 }
 ```
 
-In Java, application of such external operations could be very cumbersome, but Kotlin has a unique feature which allows
+In Java, applying such external operations could be very cumbersome, but Kotlin has a unique feature which allows
-to treat this situation: blocks with receivers. So in kotlin, operation on complex number could beimplemented as:
+to treat this situation: [extensions with receivers](https://kotlinlang.org/docs/reference/extensions.html#extension-functions).
+So in Kotlin, an operation on complex number could be implemented as:
+
 ```kotlin
-with(ComplexOperations){c1 + c2 - c3}
+with(ComplexOperations) { c1 + c2 - c3 }
 ```
+
 Kotlin also allows to create functions with receivers:
+
 ```kotlin
 fun ComplexOperations.doSomethingWithComplex(c1: Complex, c2: Complex, c3: Complex) = c1 + c2 - c3
 
 ComplexOperations.doComethingWithComplex(c1,c2,c3)
 ```
 
-In fact, whole parts of program could run in a mathematical context or even multiple nested contexts. 
+In fact, whole parts of a program may be run within a mathematical context or even multiple nested contexts.
 
-In `kmath` contexts are responsible not only for operations, but also for raw object creation and advanced features.
+In KMath, contexts are responsible not only for operations, but also for raw object creation and advanced features.
 
 ## Other possibilities
 
-An obvious candidate to get more or less the same functionality is type-class feature. It allows to bind a behavior to
+An obvious candidate to get more or less the same functionality is the type-class, which allows one to bind a behavior to
-a specific type without modifying the type itself. On a plus side, type-classes do not require explicit context 
+a specific type without modifying the type itself. On the plus side, type-classes do not require explicit context
 declaration, so the code looks cleaner. On the minus side, if there are different sets of behaviors for the same types,
-it is impossible to combine them in the single module. Also, unlike type-classes, context could have parameters or even
+it is impossible to combine them into one module. Also, unlike type-classes, context can have parameters or even
-state. For example in `kmath`, sizes and strides for `NDElement` or `Matrix` could be moved to context to optimize 
+state. For example in KMath, sizes and strides for `NDElement` or `Matrix` could be moved to context to optimize
-performance in case of large amount of structures.
+performance in case of a large amount of structures.