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more adjustments to contexts doc

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# Context-oriented mathematics
## The problem
A known problem for implementing mathematics in statically-typed languages (but not only in them) is that different
sets of mathematical operators can be defined on the same mathematical objects. Sometimes there is no single way to
treat some operations, including basic arithmetic operations, on a Java/Kotlin `Number`. Sometimes there are different ways to
@ -10,15 +11,15 @@ by adding dynamic context-specific behaviors at runtime, but this solution has a
## Context-oriented approach
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:
One possible solution to these problems is to divorce numerical representations from behaviors.
For example in Kotlin one can define a separate class which represents some entity without any operations,
ex. a complex number:
```kotlin
data class Complex(val re: Double, val im: Double)
```
And then define a separate class or singleton, representing an operation on those complex numbers:
And then to define a separate class or singleton, representing an operation on those complex numbers:
```kotlin
object ComplexOperations {
@ -27,15 +28,15 @@ object ComplexOperations {
}
```
In Java, applying such external operations could be very cumbersome, but Kotlin has a unique feature which allows
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:
In Java, applying such external operations could be very cumbersome, but Kotlin has a unique feature which allows us
implement this naturally: [extensions with receivers](https://kotlinlang.org/docs/reference/extensions.html#extension-functions).
In Kotlin, an operation on complex number could be implemented as:
```kotlin
with(ComplexOperations) { c1 + c2 - c3 }
```
Kotlin also allows to create functions with receivers:
Kotlin also allows the creation of functions with receivers:
```kotlin
fun ComplexOperations.doSomethingWithComplex(c1: Complex, c2: Complex, c3: Complex) = c1 + c2 - c3
@ -45,13 +46,13 @@ ComplexOperations.doComethingWithComplex(c1,c2,c3)
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 not only responsible 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 the type-class, which allows one to bind a behavior to
a specific type without modifying the type itself. On the plus side, type-classes do not require explicit context
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 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 into one module. Also, unlike type-classes, context can 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
performance in case of a large amount of structures.